Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1183 Algebra, Algebraic Topology and thetr Interactions Proceedings of a Conference held in Stockholm, Aug. 3-13, 1983, and later developments

Edited by J.-E. Roos I

Springer-Verlag Berlin Heidelberg New York Tokyo

Editor

Jan-Erik Roos Department of Mathematics, University of Stockholm Box 6701, 113 85 Stockholm, Sweden

Mathematics Subject Classification (1980): 13-06, 13D03, 13E05, 13H99, 13J10, 14-06, 14F35, 16A24, 17B70, 18G15, 18G20, 20F05, 20F10, 55-06, 55P35, 55Q15, 55S30, 5?-xx ISBN 3-540-16453-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16453-? Springer-Verlag New York Heidelberg Berlin Tokyo

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A MATHEMATICAL

INTRODUCTION

These notes contain the outcome and later developments arising from a Nordic Summer School and Research Symposium held in Stockholm, August 3-13 th, 1983 on "ALGEBRA, ALGEBRAIC TOPOLOGY AND THEIR INTERACTIONS". Let me first give a brief indication of the main ideas behind this symposium. During the last decade several striking analogies between algebraic topology (at least rational homotopy theory) and algebra (at least local algebra) had been observed. Let me just give two examples.

(More examples and details can be found in the paper

Z~irough the looking glass: A dictionary between rational homotopy theory and local algebra by L. AVRAMOV and S. HALPERIN in these proceedings.) First some preliminaries. Let X be a finite, simply-connected CW-complex, ~X the space of loops on X and H.(~X,~) the rational homology algebra of ~X. (q~is algebra is even a Hopf algebra.) At the same time, let (R,m) be a local commutative noetherian ring R with maximal ideal m and residue field k = R/m, and let Ext~(k,k) be the graded vector space

@ Ext~(k,k) equipped with the algebra structure coming from the Yoneda n>O

composition Ext~(k,k) @ Ext~(k,k)

> Ext~+J(k,k). This Yoneda Ext-algebra

ExtR(k,k) is also a Hopf algebra and it is even the enveloping algebra of a certain graded Lie algebra w*(R) over k. On the other hand, it is also known that H.(~X,~) is the enveloping algebra of the rational homotopy Lie algebra ~.(~X)@z~.

(Note that the

Samelson product on this Lie algebra corresponds under the isomorphism Wn_~(gLX)m~n(X) to the Whitehead product on the ~ (X).) We are now ready for the examples: n Example I.- Let F "'"

+ Zn+1(B)

> E

> B be a Serre fibration and

~ > Zn(F) ----> Zn(E)

> Zn(B) ----> ...

(I)

the corresponding homotopy exact sequence. In [7] Halperin proved (under some minor extra conditions) that, if H*(F,~) is finite dimensional, then (I) breaks u p into exact sequences of 6 terms if we tensor it with ~. (More precisely, ~(Zodd(B)) is torsion.) On the other hand, if A

> B is a homomorphism of local commutative

noetherian rings such that B is A-flat and if B = B @A k is the "fibre" ring (assuming for simplicity that the local rings have the same residue field k) then, using earlier partial results of Oulliksen, Avramov proved in [3] that there is an exact sequence ...___>

n(~)

> ~(H)

> n(A )

6 > n+~(~)

____>

...

(2)

where 6 has properties similar to those of ~. (This time we do not have to tensor (2) with anything.) It should be remarked that neither Avramov nor Halperin knew about the other's work at the time. By now there are much more complete results and a common

IV

explanation in terms of differential graded algebras. Example 2.- It was asked by Serre whether the series dim$(Hn(~,~))-Z n ,

(3)

n>O and

by Kaplansky and Serre whether [ dimk(Ext~(k,k)).zn n>0

(4)

behaved in a nice way, e.g. whether they were rational functions of Z. (X and R are as in the preliminaries above.)

I proved in []3] that~ for spaces X with dim X ~ 4 and

for local rings (R,m) with _m3 = O, the two questions were equivalent (even more precise results were proved...). Thus, when Anick found a counterexample to the rationality of (3), it was immediately obvious how to produce a eoLunterexample to the rationality of (4). In [13] the algebra structures of H . ( ~ X ~ )

and ExtR(k,k) were also related to

each other. By now there are much more general results, at least about how the series (3) and (4) are related. It has turned out that even for X arbitrary (finite, simplyconnected) and (R,m) arbitrary (local noetherian), the series (3) and (4) are all "rationally related" I) to corresponding series (4) of local rings (S,[) with n 3 = 0 and to corresponding series (3) of finite Y:s with dim Y < 4 and thereby also "rationally related" to series

~ dimk(Fn).z n , where ~ is a finitely presented graded n>O

(l,2)-Hopf algebra, i.e. the quotient of a free associative algebra k< XI,...,X m > on generators XI,...,X n of degree l, by the two-sided ideal generated by some "quadratic Hopf relations"

:

j ~ c~j[Xi,X ] _

, where cij C k

and where [Xi,X j] : ~

1<_J -

XiXj + XjXi ' if i < j

LX~

For more details, cf. [2].[In what follows, if --

@ V is a graded k-vector space, with n n>0

dir~(Vn)'Z n

dimk(V n) < ~ for all n _> O, we will call n>0

~ if i = j.

the Hilbert series of V = ~ V n. n>0

With these and other examples in mind, it was clear that, if algebraists and (algebraical) algebraical topologists could meet for a longer period of time, then a fruitful interaction between their ideas might take place. Here are just a few examples of results obtained at or after the Stockholm conference that are published here for the first time: In B~gvad-Ha!perinPs paper an algebraist and an algebraical topologist cooperate to prove that, if H,(~X,$) ( recall that X is a finite, simply-connected CW-complex) is noetherian (left or right noetherian does not matter, since we are dealing with a Hopf

I )For the meaning of "rationally related", cf. "LOOKING AHEAD" below.

V

algebra)~

then there are only a finite number of non-zero rational homotopy groups of

X. (The converse

is evident.)

local commutative

noetherian

Ext~(k,k)

is noetherian

On the other hand they also prove that, if (R,m) is a ring with residue field k (no restrictions

(if and) only if R is a local complete

case only the two lowest 7i(R) can be different from algebraic topological

topology.

on k!), then

intersection.

(In this

from 0.) The idea of the proof comes

The Lusternik-Schnirelmann

(L.-S.) category

(an old

concept from the 1930:s), which had been introduced quite recently

rational homotopy theory

(and thus in the theory of differential

also used here for Avramov~s these proceedings

minimal models in a nice way.

contains an up-to-date

goes beyond the beautiful

in

graded algebras)

[~e

paper by Lemaire

is in

survey of L.-S. category, that completes

earlier survey of I.M. James

and

[9] from 1978.]

In order to present the next new result, I first have to recall an old result of Levin [10], which combined with later results of Avramov and L~fwall can be formulated as saying that, for any local commutative

(cf. these proceedings)

noetherian

Lie algebra 7*(R) is closely related to the Lie algebra 7*(R/@n) R/m n , provided n is bi 6 enough

ring (R,m) , the

of the artinian ring

(precisely how big n should be depends on the Artin-

Rees lemma, which Levin uses in [10] in a very clever way). More precisely,

if n £

some n(R), then the natural Lie algebra map * ( R / m_ n )

7"

-->

~*(R)

is onto, and the kernel of 7" is a free graded Lie algebra. precise results.)

[In technical terms one says that R

(~ere

are even more

~> R/m n is a Golod map.

A very ~eneral theor L of Golod maps is presented for the first time in the paper by Avramov in these proceedings.]

Here is one algebraical

topological

version of all this

(it is proved in the joint paper by Halperin and Levin in these proceedings): a simply-connected

CW-complex

(not necessarily

in each dimension and such that H*(X,~) noetherian homotopy

Let X be

finite) with a finite number of cells

is a finitely generated algebra

(i.e. a

ring). Then there exists an no(X) such that, for all n ~ no(X), the rational

fibre of the inclusion of the n-skeleton

Note the analogy:

the rational homology

is a free associative

algebra.

X = BU(m) and, more generally~

Xn

> X is a wedge of spheres.

ring of the loop space of a wedge of spheres

Results of this type had previously been known only for for X = certain products of Eilenberg-MacLane

The earlier ideas of Levin are essential

for the general proof.

There are many more examples of interaction The analogy is often not perfect,

spaces.

between algebra and algebraic

topology.

and this inevitably leads to more work if one wants

to go from one side to another. Let me say a few words about some other papers in these proceedings.

LSfwall's paper

is a corrected version of about one half of his 1976 thesis, and this half was never published, rationality

presumably because L6fwall first wanted to prove by his methods the of the series

(4) in general.

Now, as we have said above, we know better

as (4) is not always rational, but it was a genuine surprise when it turned out in

Vl

1984 [2] that the special cases studied by LSfwall, presented graded

(1,2)-Hopf algebras

related" to the general

series

and in particular

and their Hilbert

finitely

series, were "rationally

(4) for general local rings

(R,m). Thus with hindsight

one might say that in a sense L6fwallls thesis did treat the most general case. L6fwall~s thesis has been used by many workers

in the field and, in particular,

by

LSfwall himself []]] in his construction of counterexamples to a conjecture by v v Kostrikin and Safarevic. The papers by Anick-L6fwall and FrSberg-Gulliksen-L6fwall these proceedings their Hilbert

are recent studies of how finitely presented graded algebras

series can behave.

In particular

there exists a finite simply-connected orders

(buick and Avramov,

teristics.

0, whereas,

CW-complex

proceedings,

comments

theory

side, we often work over a field of

in local algebra, we can have residue fields of all characfor this (cf__~ ~. however with "LOOKING AHEAD" below, where

are given).

Indeed,

we find the first theorems

Morgan-Sullivan

X, whose H,(~X,~) has torsion of all

to appear).

There are reasons

more optimistic

and

the last paper can be used to prove that

The reader may have noticed that, on the homotopy characteristic

in

in Torsten Ekedahlts paper in these

showing that the beautiful Deligne-Griffiths-

[6], that a K~hlerian

compact manifold

its real homotopy type is a formal consequence

is "formal" over ~, i.e.

of its real cohomology in these proceedings

ring, is false

in characteristic

p. For the remaining papers

(some in algebra,

some in algebraic

topology and some being a mixture of both), we refer the reader to

the table of contents. LOOKING AHEA]) Here are some further directions

of research that seem to be fruitful:

I) Two formal power series P(Z) = ~ pn Z n (Po = I, Pi integers) n>0 (qo = 12 qi integers) are said to be "rationally matrix

(Aik(Z)) whose entries are polynomials

that det(Aik(Z))

n

if there exists a 2 x 2

in Z with integral coefficients

such

@ 0 and such that

A11(Z)Q(Z) + A12(Z) P(Z) = A21(Z)Q(Z)

+ A22(Z)

All (Z) + A12(Z) ( thus

These matrices modulo the diagonal ones

multiplication

related"

and Q(Z) = !0qnZn

A21(Z) + A22(Z)

{A(Z) 0 ) ~ 0 A(Z)

= I if Z = 0 ) .

form a group under matrix

and it would be interesting to try to classify the orbits of this group

acting on, say, the set of power series that are rationally related Hilbert finitely presented graded (1,2)-Hopf algebras.

series of

The old question of Kaplansky-Serre

mentioned above is equivalent to asking whether there is just one orbit. Now we know that there are many orbits. analog of the Serret theorem

Could we find nice representatives (cf. e.~.

for them? Is there an

[12], p. 55)? Here we have only been talking

Vll

about rational relationship between Hilbert series of graded algebras. Is there an underlying theory of "rational relationship" between the algebras themselves? If so, it might be easier to get more precise results about the H,(~X,~) than in the papers by Halperin et al. in these proceedings. 2) Torsten Ekedahl has recently developed the analog of rational homotopy theory for spaces "over Z", using cosimplicial algebras. This theory seems very promising, but nothing has yet been published about it. 3) Growth series and growth al~ebras of ~roups. Let G be a finitely generated group, with a fixed set of generators S, where we suppose that S is closed under the operation of taking inverses in G. Let k be a field and introduce a filtration on the group ring k[G] by means of F-I(k[G]) = 0 , Fn(k[G]) = the sub vector space of k[G], spanned by products of ~ n (n ~ O) elements from S. Then ¢ Fn(k[G])/ Fn-I(k[G]) dsf grs(k[G]) n>0 becomes a finitely generated graded algebra (the growth algebra of (G,S)) [h]. ffT~e Hilbert series of this graded algebra is the growth series of (G,S). Under some conditions (c_~f. e.g. [I] for the commutative noetherian case) there is a spectral sequence of algebras (k~k) E *I = Ext* grs(k[O])

=>

gr EXtk[G](k,k).

(5)

Could (5) be useful in some cases to relate the growth series of G to the cohomology of G? Another problem[!: It is known that, if G is fimitely presented, then grs(k[G]) is not necessarily so. Indeed, if grs(k[G]) is finitely presented, then its Hilbert series is primitive recursive [8] and then [5] G must have a solvable word problem. But there are finitely presented groups whose word problem is unsolvable. Thus we are led to the following PROBLEM: Is it true that the Hilbert series of finitely presented graded algebras are always rationally related to growth series of finitely presented groups with a solvable word problem (and conversely)

?

Stockholm, autumn 1985 JAN-ERIK ROOS

REFEEENCE

S:

[I]

R. ACHILLES - L. AVRAMOV, Relations between ~_~2perties of a ring and its associated graded ring, Seminar Eisenbud, Singh, Vogel, vol. 23 Teubner-Texte der Mathematik, vol. 48, 1982, 5-29, Teubner, Leipzig.

[2]

D. ANICK - T. GULLIKSEN~ Rational dependence among Hilbert and Poincar@ series, Journ. of Pure and Appl. Algebra, 38, 1985, 135-157.

VIII

[3]

L. AVRAMOV, Homolq6y of local flat extensions and complete intersection Math. Ann., 228, 1977, 27-37.

defects,

[4]

N. BILLINGTON, Growth of ~roups and ~raded a l ~ b r a s , Commun. 1984, 2579-2588.(Correction later in the same journal.)

[5]

J.W. CANNON, The ~rowth of the closed surface groups and the compact hyperbolic Coxeter groups (preprint, cf. Theorem 9.1).

[6]

P. DELIGNE - Pin. GRIFFITHS - J. MORGAN - D. SULLIVAN, Real homotopy theory of K~hler manifolds, Invent. Math.~ 29, 1975, 245-274.

[7]

S. HALPERIN, Rational fibrations~ minimal models and fibrin~s of homogeneous spaces, Trans. Amer. Math. Sot., 244, ]978, 199-224.

[8]

C. JACOBSSON - V. STOLTENBERG-HANSEN, Poincar@-Betti series are primitive recursive, Journ. London Math. Soc., ser. 2, 31, 1985, I-9.

[9]

I.M. JAMES, On__cate~ory, in the sense of Lusternik-Schnirelmann, 17, 1978, 331-348.

in Algebra,

12,

Topology,

[10]

G. LEVIN, Local rings and Golod homomorphisms, 266-289.

Journ. of Algebra, 37, 1975,

[11]

C. LOFWALL, Une a l ~ b r e nilpotente dont la s~rie de Poincar@-Betti est non rationnelle, Comptes rendus Acad. Sc. Paris, 288, s$rie A, 1979, 327-330.

[12]

O. PERRON, Die Lehre yon den Kettenbr~chen, Stuttgart.

[13]

J.-E. ROOS, Relations between the Poincar$-Betti series of loop spaces and of local rings, Lecture Notes in Mathematics~ 740~ 1979, 285-322, Springer-Verlag, Berlin, Heidelberg, New York.

Band I, Dritte Aufl., 1954, Teubner,

Jan-Erik Roos Department of Mathematics University of Stockholm Box 6701

S-113 85 STOCKHOLM (SWEDEN)

ACKNOWLEDGEMENTS

AND GENERAL

INFORMATION

The Nordic Summer School and Research Symposium on "ALGEBRA, ALGEBRAIC TOPOLOGY AND THEIR INTERACTIONS" received support from two sources: I) The Swedish Natural Science Research Council (NFR) and 2) The Nordic Governments, through "Nordiska Forskarkurser", which supports The Nordic Summer School of Mathematics, an organization with one director from each of the Nordic Countries and which works "with a minimum of bureaucracy"

(these are the words

of the founder of the school) and selects subjects and sites for Summer Schools. The founder and main animator is Professor Lars G~rding (Department of Mathematics, University of Lund, LUND, Sweden). Since the start in ]966, 13 summer schools, covering the following other subjects have been arranged: Harmonic Analysis (twice), Several Complex Variables, Algebraic Topology, Pseudodifferential operators and applications to index problems, Algebraic Geometry (twice), Discrete Groups and Quasiconformal maps, Operator Algebras and their Applications to Quantum Mechanics and Group representations, Singularities, Value Distribution of Holomorphic maps into Complex Projective Space (the Cartan-Ahlfors-Weyl theory) and Differential Geometry. I wish to thank both NFR and "Nordiska Forskarkurser" for their generous support. I also wish to thank Lars G~rding for his original (]966) initiative, which has turned out to be so extremely useful and valuable. The Summer School and Research Symposium took place at the University of Stockholm in Frescati, August 3 - August 13th~ ]983. The morning sessions consisted mainly of survey lectures, intended to bring the audience to the level of the research symposium (in the afternoons), which successively grew more and more advanced. The following survey lecture series were given: David ANICK, Basic algebraic topology. Luchezar AVRAMOV, Local algebra and algebraic topology. David EISENBUD, Commutative algebra thr~gugh e x a m p l e s i n a l ~ e b r a i c g e o m e t r y . Tor H. GULLIKSEN, Local algebra and differential gra@ed algebra. Stephen HALPERIN, Rational homotopy the or ~. Melvin HOCHSTER, The homological con~cjtures for ipsal rings. Christer LECH, Relations between a local rin~ and its completion. Jean-Michel LEMAIRE, Lusternik-Schnirelmann cate~0r~ and related topics. Rodney Y. SHARP, Basic commutative al~ebra. Richard STANLEY, Commutative al~ebra and combinatorics. In the afternoons there were both problem sessions (exercises) for some of the morning lectures as well as lectures in the research symposium. The following research

symposium lectures did not lead to a publication in these proceedings: R. FROBERG, Koszul algebras a~d Veronese embeddings. M. FIORENTINI, Alg~bres gradu@es associ@es aux suites r@guliSres. T. OGOMA, A note on unmixed domains, usjin6_Poincar@ series. D. EISENBUD, Linear series on reducible curves and applications. A.R. KUSTIN, Deformation and linkage of Gorenst_3in algebras. A. HOLME, Chern numbers of smooth codimension 2 subvarieties of pN (N ~ 6) M. HOCHSTER, Modules of finite homological dimension with negative intersection mult iplic ities. R.Y. SHARP, Generalized fractions and the monomial conjecture. A.R. PRINCE, Local rings and finite projectij~e planes. M. BRODMANN, Remarks on the connectedness of al~ebraic varieties. A. SLETSJ~E, Toroidal embeddings and Poincar@ series. K. BEHNKE, Infinitesimal deformations of cusp singularities. R. STANLEY, Symmetric functions and representations of SL(n,~). A. BJ~RNER, On the Stanley-Reisner.ring of a Tits building. N. SUZUKI, d-sequences.

I wish to thank all the participants

(in total about 100 people) and in particular all

the lecturers for their interest in this meeting. I also wish to g~ve special thanks to the following people who helped with practical details before, during and after the conference: Maje ARONSSON, J6rgen BACKELIN, Rickard B~GVAD, Tnrsten EKEDAHL,

Ir$ne

FLOD~N, Ralf FROBERG, Inez HJELM, Clas LO~WALL, June YAMAZAKI and Calle JACOBSSON (main organizer of an excursion by boat in the Stockholm archipelago). I also thank Hubert SHUTRICK for linguistic help. Finally I wish to thank Springer-Verlag for their cooperation. I hope their patience will be rewarded.

Stockholm autumn 1985 JA~-ER~ ROOS

TABLE

A mathematical

introduction

Acknowledgements

OF

CONTENTS

(by J.-E. ROOS) ......................................

and general information .........................................

TABLE OF CONTENTS ................................................................ L. AVR~MOV - S. HALPERIN, Through the looking glass: A dictionary between rational homotopy theory and local algebra .................................... D.J. ANICK, A rational homotopy analog of Whitehea§~s D.J. ANICK - C. L~FWALL,

Hilbert

Y. AOYAM~, On endomorphism L. A V ~ M O V ,

B~GVAD

-

28 32

rings of canonical modules

On the rates of growth of the homologies

S. HALPERiN,

I

problem ....................

(joint work with S. GOTO)..

of Verohese

subrings .......

J. BACKELIN - J.-E. ROOS, When is the double Yoneda Ext-algebra of a local noetherian ring again noetherian ? ............................................ R.

IX XI

series of finitely presented a!gebras ...........

Golod homomorphisms ..................................................

J. BACKELIN,

III

On a conjecture

of Roos .................................

56 59 79 101 120

T. EKEDAHL, Two examples of smooth projective varieties with non-zero Massey products ...............................................................

128

Y. FELIX, S. HALPERIN,

133

D. TANR~ and J.-C. THOMAS, The radical of ~,(~S)@~ ........

Y. FELIX - J.-C. THOMAS, Sur l'op$ration

rationnelle .................

~36

R. FR~BERG, T. GULLIKSEN and C. L~FWALL, Flat families of local, artinian algebras with an infinite number of Poincar@ series ...........................

170

T. GULLIKSEN,

192

A note on intersection

d~holonomie

multiplicities ..............................

T. GULLIKSEN, Reducing the Poincar@ series of loca ! rings to the case of quadratic relations ...........................................................

195

S. HALPERIN,

~99

S.

HALPERIN-

The radical of ~ , ( ~ S ) ~ ,

II .........................................

G. LEVIN, High skeleta of CW-complexes

O.A. LAUDAL, Matric Massey products C. LECH, A method for constructing

.............................

and formal moduli I .......................... bad noetherian

local rings ....................

211 218 241

C. LECH, Yet another pro pf of a result of Ogoma ..................................

248

D. L E H I g h ,

250

ModUle minimal reiatif des feuilletages ..............................

J.-M. LEMAIRE, Lusternik-Schnirelmann

category:

an introduction ..................

259

J. LESCOT, S@ries de Bass des modules de syzygie .................................

277

C. LOFWALL~ On the suhalgebra generated by one-dimensional elements in the Yoneda Ext-algebra ............................................................

291

D. REES, The general extension of a local ring and mixed multiplicities ..........

B39

D. TANR~, Cohomologie

361

de Harrison et type d~homotopie

rationnelle ................

M. VIGU~-POIRRIER, Cohomologie de l~esj~ace des sections d'un fibr@ et cohomologie de Gelfand-Fuchs d~une varigt@ ....................................

371

A RATIONAL HOMOTOPY ANALOG OF WHITEHEAD'S PROBLEM

by David J. Anick

This note will state and prove a theorem in rational homotopy which is an analog of the famous unsolved problem due to J.H.C. Whitehead as to whether or not subcomplexes of aspherical two-dimensional CW complexes are aspherical [9]. We first rephrase Whitehead's question so that it has a natural generalization to higher homotopy. Let

Y

be an aspherical two-dimensional CW complex and let

be a subcomplex. It is well-known that we need only consider the case where Y

share the same

l-skeleton

obtained by attaching to

W.

(resp. (iwy) #)

W

W

2-cells to

and base point

w 0 , so we may assume that

X , which in turn is gotten by attaching

and Y

is equivalent to the surjectivity of the homomorphism

induced on

~,(

)

by the inclusion

iWX: W ~ X

(resp.

(iwx) #

is

2-cells

has the homotopy type of a wedge of circles, so the asphericity of

Y)

X

X

X

(resp.

iwy) .

Whitehead's question becomes the following. Question I • Let to

W

and

Y

jective, is

W

be a wedge of

by attaching

S I 's and let

2-cells to

X . If

(iwx)#: v,(W,w 0) ~ ~,(X,w0)

X

be obtained by attaching

2-cells

(iwy)#: ~,(W,w 0) ~ ~,(Y,w O)

is sur-

necessarily surjective ?

In rational homotopy, we generally consider simply connected spaces only and tensor all homotopy groups with Q . In place of a wedge of circles we get a wedge da of spheres V S , d ~ 2 , and each attached cell may have any dimension three aEI or greater. The natural analog to Question I is Qustion 2. Let Y

V S d~ where I is any indexing set and d_ > 2 . Suppose ~EI ' (b6) ~ -is obtained by attaching cells to W , Y = W Uf(B~Jen ) for some indexing set

J

and dimensions

If

W =

b 6 ~ 3 , and suppose

(iwy)#: ~,(W,w 0) e Q ~ ~,(Y,w0) ~ Q ~,(X,w 0) @ Q

X

is a subcomplex of

is surjective, is

containing

W .

(iwx)#: ~,(W,w 0) @ Q

necessarily surjective ?

We may answer Question 2 affirmatively for locally finite argument. Let

Y

(Aw, dW)

denote the Adams-Hilton model [2] over

H,(Aw, dW) ~ H,(~W;Q) , and likewise for

(Ax,d X)

and

Y

by the following

Q

for

W

so that

(Ay,dy) . Using the equi-

valence between the rational homology of the loop space and the universal enveloping algebra of the rational homotopy Lie algebra of a space [see e.g. 5], we see that (iwy) #

(resp.

(iwx)#)

surjects if and only if

(iwy),: H,(Aw, dW) ~ H,(Ay,dy)

29

(resp.

(iwx) ,)

surjects.

The latter condition discussed

is a familiar one to rational homotopy

and only if the images

It is

(~f~),(z B)

(i~)* is onto if wI b~-I z~ E HbB_2(~S ~ ;Q) of

of the generators

spherical

loop space homology under the attaching maps

"strongly

free set" in

the cells of

Y - W

this property. When

Y

the collection

and the subset

This is equivalent

is easily accomplished

{(~fB),(zB)IB

K c J

the argument

of

J

indexes

X - W , also has

(iwx) , .

still works, but the proofs

in [3],

series, must be replaced by more general ones. Happily this and we will only outline how. For

in the (possibly locally infinite)

to-one for some (equivalently,

B

a set of homogeneous

connected graded

he classified as "strongly free" if and only if

(H/HBH) ~ H

constitute a

C K} , where

indexes the cells of

to the surjectivity

is not locally finite,

which relied on Hilbert

f$: e• (b~) ~ W

H,(~W;Q) . Because by [3, Lemma 2.7] a subset of a strongly

free set is strongly free,

elements

theorists.

in [7] and by [7] and [3, Theorem 2.9] we know that

(I~):

k-algebra

H , B

k < B >i~ (H/HBH) ~ H

should is one-

every) choice of graded vector space homomorphism

which is a right inverse to the projection

p:

p: H ~ H/HBH . [3, Ler~na

2.7] is easily reproved and the proof of [3, Theorem 2.9] remains valid. We have shown Theorem

I. The answer to Question

In the rational homotopy ment that

W

2 is "yes".

case we can take this further by relaxing the require-

be a wedge of spheres.

an equivalent

question

if

W

Qustion

is permitted

and the answer to the original Whitehead

I is trivially

seen to be replaced by

to be any twodimensional

question becomes "no" if

have dimension three [I]. For Question 2, however,

CW W

complex,

is allowed to

the effect of these substitutions

is less clear. We therefore formulate Question 3. Same as Question 2, except that

W

may be any simply connected

CW

complex. To answer this, we will use the following lemma. Lemma. Let field

(L,6)

be any associative connected differential

k , and let

B c ker (6)

= (L H k < C> , ~) , where deg (T(E)) = deg (x) - I and

and

be any subset of homogeneous

T : C ~ B ~

is a one-to-one

extends

~ = H,(~,~) , there is a natural map

i: L ~ ~

of chain algebras.

= {~ly C B} ~ G

Then

i,

is strongly free in

6

via

graded algebra over a elements.

correspondence

with

~(x) = T(x) . Writing

i,: G ~ ~

Let

G = H,(L,6)

induced by the inclusion

is onto if and only if the set of cycles

B =

G .

Note. The "if" direction of the lemma is essentially

proved in [8]. In his talk at

the 1983 Nordic Summer School, J.-M. Lemaire proved this lemma under the non-essential

30

restriction

that

B

be countable.

We offer here a simple, Proof of Lemma. Let

~(0) = L

{uxvlu C L,x C C,v E L } ~

His proof appears

completely

~

and

in [6, Section 2].

general proof based on [3].

~(n) = (LCL) n , where

and likewise

for

LCL

denotes

(LCL) n. Writing

Span

~p,q = I~(p) N Lq+p ,

we have =

@ ~ p,q~0 P'q

and

~(~

We obtain from this bigraded

P'q

) c • -- Lp-I'q

complex a spectral

E I = GHk

has bidegree

r

If

B

with

E,• ,

which converges

to

E0 = ~ with N0(C) = 0 , p,q P,q = 0 , ~1(x) = T--~) E G for x G C .

~I(G)

are

(-r,r - I ) .

is strongly

and the spectral

< C>

m

sequence

= H,(~,~) . Its first terms and differentials ~01L = ~ ' and

p,q-1

free in

G , then by

sequence degenerates,

[3, T h e o r e m 2.9]

E2 = 0 p,*

for

p > 0

yielding

= E °° = E 2 = E 2 ~ = im(i,) 0,, = E0, * as desired.

If instead

free in G by [3, 2.9] E2 # 0 . The ' I,* E2 # 0 also shows, when s is the minimal degree I,* having E2 # 0 that E 2 = 0 for p > 2 and q < s . Thus El2 s persists to 1,s ' p,q -El, s , and im(i,) = E0, * is not the whole of ~ . same reasoning

Theorem

which

B

is not strongly

shows that

2. The answer to Question

Proof of Theorem 2. As before, H,(~X;Q) precisely and

it suffices

surjects when we k n o w that the map

i,

to show that the map

(~iwy) ,

of the Lemma if we set

B = {(~fB),(zB)}BC J • Since

= H,(f~W;Q) . Again we deduce that (~iwx) ,

3 is "yes".

(~iwy) ,

surjects.

Using

(L,~) = (Aw, dw),

surjects,

{(~fB),(zB)}BE K

B

(~iwx),: H,(~W;Q) [2],

is strongly

is strongly

(~iWy),

is

(~,~) = ( A y , ~ ) free in

free in

,

G =

G , so

is onto as well.

As a final remark we notice allow the cells of

Y - X

y = S 2 x S 2 x S 2 , X = y4,

that the answer

to attach to

X

to Question

instead of to

2 becomes

"no" if we

W . An example

is

W = y2 .

REFERENCES [I]

J.F. Adams, A n e w proof of a theorem of W.H. Cockcroft. (1955), 482-488.

J. London Math.

[2]

J.F. Adams and P.J. Hilton, Helv. 30 (1955), 305-330.

On the chain algebra of a loop space. Comm. Math.

[3]

D.J. Anick, Non-commutative 78 (1982), 120-140.

graded algebras

[4]

H. Cartan and S. Eilenberg, N.J., 1956.

H omological

and their Hilbert

Algebra.

Soc. 3 0

series. J. Algebra

Princeton Univ. Press,

Princeton

31

[5]

S. Halperin, Lectures on Minimal Models, Publications de I'U.E.R. Math~matiques Pures et Appliqu~es, Universit~ des Sciences et Techniques de Lille, Vol. 3 (1981). (Also published as Mem. de la Soc. Math. de France 9/10 (1983).

[6]

S. Halperin and J.-M. Lemaire, Suites inertes dans les alg~bres de Lie gradu~es. Publications de l'Univ, de Nice, 1984, No. 22; also scheduled to appear in Math. Scand.

[7]

J.-M. Lemaire, Alg~bres connexes et homologie des espaces de lacets, Lecture Notes in Math. No. 422, Springer-Verlag, Berlin-Heidelberg-New York, 1974.

[8]

J.-M. Lemaire, Autopsie d'~meurtre dans l'homologie d'un~ Ann. Sci. Ecole Norm. Sup (4), 11 (1978), 93-100.

[9]

J.H.C. Whitehead, On adding relations to homotopy groups. Ann. Math. 42 (1941), 409-428.

D.J. Anick Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A.

alg~bre de chaines.

THROUGH THE LOOKING GLASS:

A DICTIONARY BETWEEN

RATIONAL HOMOTOPY THEORY AND LOCAL ALGEBRA

by Luchezar Avramov and Stephen Halperin

§ O. INTRODUCTION "Now, if you'll only attend, Kitty,... I'll tell you all my ideas about Looking-glass House. First, there's the room you can see through the glass - that's just the same as our drawing room, only the things go the other way." Alice [C]

Homological methods,

originally

invented as tools for algebraic

topologists,

have

almost from their inception played an important role in the study of rings. This has led to any number of analogies between the two subjects and to a certain overlap of terminology. More recently it has developed topy theory (within topology) particularly

that if one restricts

and to commutative

coherent analogy of unusual

to use intuition and techniques

rings

attention to rational homo-

(within algebra)

one gets a

scope and power. This has made it possible

from topology to prove theorems

in algebra, and con-

versely. In

[Av I]

for example homotopy-theoretic

ideas are applied to the study of local

rings in an essential way. That article also contains B~gvad's

article in these proceedings

Lusternik-Schnirelmann

is a complete intersection.

zero and defers the positive characteristic

An example in the other direction ([Le-Av])

on factoring

formal manifolds

by Avramov

theorem in topology: X

If the Lie algebra

~(~M)

® ~

~

Ease to later.)

ring. Translated

[AVl] , then for general manifolds

[Ha-Le] ) it becomes the following

generated.

(That article restricts

is the result of Levin and Avramov

the socle in a local Gorenstein

closed 1-connected manifold with

generated by

there the idea of

category in topology is used in algebra to prove that a local

ring with noetherian Ext-algebra to characteristic

a brief sketch of the analogy.

is another instance;

an

n-I

n.(~X) 0 ~

in 1978

(first for

by Halperin and Lemaire

Suppose

complex. Assume

i)

M = X U en H*(M;~)

is filtered by the "powers"

is a

is not singly of the ideal

then the associated graded Lie algebra is the free product of

and the free Lie algebra

L(~)

on

~ .

Our aim here is to develop the main ingredients

of this analogy with some care,

so that the reader who wishes may be in a position to continme translating

on his own.

I) Note by the editor: The general case (arbitrary characteristic) is now solved, and B~gvad's paper has been replaced by a joint paper by B~gvad and Halperin in these proceedings.

From time to time we surmnarize our conclusions

in the form of a dictionary.

In particular we have tried to avoid overlap with [AVl], for which this article might well be considered mathematical

as preparation.

substance presented

We should also make clear that much of the

here is not new and is deliberately

form. This is because of our mutual experience gist's)

standard triviality confuses a topologist

are hoping for comprehension

homotopy,

(resp. algebraist)

in elementary (resp. topoloand because we

from both.

We refer the reader to [Su],

mutative

that an algebraist's

[B-G] and [Ha] for the missing details

and to [Av 2] for its algebraic analogue.

in rational

The basic homotopy theory and com-

algebra we use can be found in [W] and in [A-M] and [Ma].

To describe

the central

ideas behind the analogy we need first to recall the

following essential observation Quillen's principle:

of Quillen

Differential

a tool for the calculation

([QI ]) which we state as

graded algebras

of (co) homology.

ought not to be regarded merely as

In fact a reasonable DGA category will

also carry a "homotopy theory" and with it a number of other invariants. Here this principle

is implemented

twice, once for topology and once for algebra.

In each case we shall arrive at a category of DGA's (commutative with a homotopy theory. The strong similarities

in the graded sense)

between the two categories will then

form the basis for our analogy. (This homotopy theory before, by Andr~

(on the algebra side) does not seem to have been described

although the ideas are clearly present [A 2] and Quillen

[Q2 ] in a simplicial

in [QI ]. Homotopy theory has been used context for the study of commutative

rings. This yields a theory which in positive characteristic

is distinct from the one

presented here.) On the topology side we use the Thom-Sullivan Functor

ApL

logical spaces and continuous maps to DGA's and DGA morphisms mutative

in the graded sense, graded non-negatively

to pass from topo-

(defined over

upstairs).

~, com-

This faithfully con-

verts rational homotopy theory to Quillen's algebraic homotopy theory. On the algebra side we may simply regard commutative in degree zero. When we come to homomorphisms sidue field

k), however,

~:R ~ S

rings as DGA's concentrated

(say of local rings with re-

we are faced with the graded algebra

TorR(s,k).

It turns

out that this is the common homology algebra of a certain canonical class of DGA's F

(again commutative The DGA's

F

in the graded sense, but now non-negatively

lost when we pass to homology. to "measure the difference" appears

graded downstairs).

then turn out to contain other information about

if we look at

In particular we can form

between

TorH(F)(~,k)

TorR(~,k) instead.

and

TorF(~,~)

TorS(~,k),

(Recall

~ , which is which turns out

a measure which dis-

H(F) = TorR(s,k).)

Thus on the algebra side we implement Quillen's principle by expanding the category of commutative construction

of such

rings to a certain category of DGA's, F's . As in the topological

large enough to permit the

case we get a homotopy theory (al-

though it is not clear that all the Quillen axioms hold).

Before going further we emphasize (i)

three points:

In order to get a proper analogy we need to consider

one hand and a certain category of DGA's (ii)

Our analogy is contravariant

and reverses arrows~) (iii)

because

(containing

topological

commutative

spaces on the

rings) on the other.

(for instance it converts pullbacks

ApL

is a contravariant

The two sides (topology and algebra)

to pushouts

functor.

are not simple mirror images. A theorem

on one side may translate to a false statement or a triviality on the other and even when it does translate addition,

to a good theorem its proof may not translate

to a proof.

In

the normal imprecision on any lexicon is present here. A good translation

may not always be possible or may depend on the context,

and the taste of the trans-

lator. With this in mind we look at the analogy at work. Homomorphisms continuous maps and the class of DGA's determined "homotopy fibre" of a continous map. Passing ponding

to the cohomology

~:R ~ S

correspond

corresponds

to homology we get

to

to the

TorR(s,k)

corres-

of the fibre.

At a more complicated is the universal

by

level the Ext algebra of a local ring (or suitable DGA)

enveloping algebra of a graded Lie algebra corresponding

to the homo-

topy Lie algebra of a space. Finally,

two examples we did not include in the body of the paper:

cover of a topological topological

space corresponds

to the Koszul complex of a local DGA and a

space with rational Poinear~ duality corresponds

In fact the caveats above notwithstanding, goes back at least to Quillen's homotopical any number of mathematicians Lemaire,

ring.

the basic idea works pretty well. It

have worked at its elaboration

Since then

(including Anick, Felix,

LSfwall and Thomas to name a few).

beginning with his fundamental of this conference.

article

of this idea is surely due to Roos,

[R] and continuing up through his organizing

Aside from this the two of us have, both individually

tively, and personally as well as professionally We take this opportunity

In the literature a

DGA

is a

which is a degree-i

Its homology algebra,

reason to be grateful

and collec-

to Jan-Erik.

to say thank-you.

§ i. DIFFERENTIAL GRADED ALGEBRAS

(DGA's)

Z-graded ring, R, together with a differential, (d: R n ~ R n _ l )

derivation

H(R), is the graded ring

R n = R_n; when degrees are written upstairs In any given context, category,

to a Gorenstein

algebra in the late sixties.

But the real credit for the exploitation

(d 2= 0)

the universal

however,

then

Kerd/Imd. d

(d(ab)=(da)b+(-l)degaa(db)). We adopt the convention

has degree

one is usually interested

and it is normal under those circumstances

d,

+i.

in some particular

to reserve

DGA

sub-

for objects in

that subcategory.

Here we shall be simultaneously

gories,

on whether we are in the algebra,

depending

For reasons expression

interested

in one of several cate-

or topology context.

of space and simplicity we adopt the device of permitting

to have two meanings:

one for each of our two contexts.

a single

Thus we establish

the

i.I

Convention.

(a)

In the alsebra write

• = 2 .

(ii)

With rare

R

=

@

R

n>O (b)

Graded

objects are graded over the non-negative

context:

(i)

(and clearly

(ii)

indicated)

(i)

With rare (and clearly @

is no restriction

exceptions

But

as to coefficient,

the grading

and we

is downstairs:

n

In the topology context:

R=

There

integers.

Rn

The coefficients

indicated)

are always

exceptions

~

the grading

and we write

is upstairs:

.

n>O In either case we write

@

1.2

DGA

d

Definition

(i).

A

and

Hom

for

@A

and

Hom A .

is a graded A - a l g e b r a

R, together with a differential

such that

(i)

ab = (-i) dega degbba

(ii)

a2=O

(iii)

d

(a,b E R),

(a E R, dega odd), and

is a derivation

of degree

-i

(resp.

+I)

in the algebra

(resp.

topology)

context.

(2)

The DGA's

in the algebra

(resp.

topology)

context are called

DGA~'s

(resp.

DGA ~' s) . (3)

In any category with differentials

isomorphism

(~)

will be called a homology

m o r p h i s m which is a homology (4)

Sometimes

case the underlying

~

isomorphism

R

(instead of

A-algebra

(R,d))

will be denoted

H ° = ~ . On the other hand we are primarily

which

satisfy two additional

these

(existence

essential

H(~)

and denoted by ~

is an A DGA-

conditions

of divided powers)

for us to have a homotopy

to denote a

DGA;

successfully

are those sa-

interested

- which are automatic

is standard; theory.

in this

R

Now it turns out that the DGA~'s we can handle most tisfying

for which

isomorphism will be called a quism.

we shall use graded

a morphism

in those D G A ' s

for q-algebras.

One of

the other is less well known but

Explicitly,

we m a k e the

1.3

Definition:

{yP}

(i)

of divided (ii)

A

DGP,

powers

in

R

is a

DGA,,

for which DGF, 's

By a quism between

(R;d),

d(yPa)

together

= (da).

we shall mean a

with an assigned

yP-l(a) DGA *

system

.

quism preserving

divi-

ded powers. (iii)

some

An

DGF,

h-DGF,

morphism

(iv)

An

h-DGA*

(v)

An

h-DGA

is a R' ~ R is a

(F-)

algebra

ab = (-l)dega'degbba

(ii)

For

p >_ O, n _> i

such that

y°(a)

in

and

or an

h-DGF

if

dega

there are assigned = a

see [G-L]

on a symbol

powers

x

If

degx

is odd then

fA<x> = ~ 0 fA • x

(iii)

If

degx

is even and

>O

if

~<X>

{x } on

is any

P-algebra,

[Av 2] if

R -, S,T

S ® R T . Finally,

X

n

algebra

A<x> =

R

is a

q-vector

is given by ~[x]

.

is the exterior • A'~P(x); p>_O

algebra

on

x .

yP(x)'~q(x)=(P+q~'P+q(x). \ H 7

&-module.

is a basis

for a free graded

is the tensor product:

then are

When

p'~fP(a)=a p,

structure.

of degree

(ii)

then

yP: R2n -~ R2n p , (or

for instance.)

is the polynomial

is a free graded

A divided

is odd (a,b C R) .

set maps

degx = 0 , &<x>

~<x>

powers.

and, among other properties,

is the unique divided ~<x>

review divided

for which

a 2= 0

= i, yl(a)

is in the image of

H° = @ .

If

P-algebra R

algebra

a E R+

= O .

(i)

More generally,

If

h-DGA*

definition

yP(a) = ~ . "a p

In any case

rally

H+(R')

for which

is a graded

The free F-algebra

free

DGA*

such that each

of the reader we briefly

p > 2 . (For a complete space

for which

(i)

R 2n -~ R 2rip)

(R,d),

is either an

For the convenience powers

DGP,,

R<X> = R @ ~ < X >

inherits

F-algebra morphisms

then a

in the topology

context

A-module

X

then the

O 1A<x~>; we also write

(~=Q)

a

F-structure;

F-structure

it is common

~<{x~}>.

more gene-

is determined

to write

~<X>= AX

and R<X> = R O AX = R O (exterior The essential For the moment x C R2n+l, structure

1.4

Homotopy

role of condition

we simply observe

n ~ 1 from

algebra

then

yP(dx)

(ii) in definition

that it implies is a boundary

algebra

1.3 will become

that if

R

is an

(p ~ I) . Thus

H(R)

X even) clear

h-DGP,

. in § 3.

and

inherits

a

F-

R .

type.

Two topological

if there are maps

~: M ~ N, ~: N ~ M

identity;

~

~

X °dd) O (symmetric

and

A weaker notion

are called

spaces

M

and

such that

~nverse

homotopy

is that of weak homotopy

~

N

have the same homotopv and

~

are homotopic

type

to the

equivalences.

equivalence;

these are the continuous

maps which yield isomorphisms

~i(M)--*~i(N )

(i ~ 0, each base point). By a theorem

of Whitehead, these two notions coincide for CW complexes. But for general spaces weak equivalences may not have "inverses" and so two spaces

M,N

are said to have

the same weak homotopy type if they are connected by a chain M = M

o

÷ ..... ÷ M

= N

n

of weak equivalences. A second theorem of Whitehead asserts that a map between simply connected spaces is a weak homotopy equivalence if and only if it induces an isomorphism of integral homology. By analogy, it is called a rational homotopy equivalence if it induces an isomorphism of rational homology. Two simply connected spaces

M,N

have the same

rational homotopy type if they are connected by a chain of rational equivalences. If one thinks of DGA's (resp. morphisms) as spaces (resp. continuous maps) then quisms will correspond to rational equivalences. Thus we say that two DGA's

R,S

have

the same homotopy type if they are connected by a chain of quisms R = R(O)

.......

R(n)

= S

this definition applies equally to: R(i)

are then

;

DGA*'s, DGA.'s, DGF.'s and h-DGF.'s, but the

required to belong to the appropriate subcategory.

In particular, we can begin the dictionary with ToDolo~¥

Algebra

Topological spaces, continuous maps

; h-DGF.'s, DGF,-morphisms

Rational homotopy equivalence

Quism

Rational homotopy type

Homotopy type

1.5

DGA's in topqlogy,.

In topology

DGA*'s were first envisaged as a means to the

calculation of the cohomology ring of a space (in our case, with rational coefficients).

The standard "differential algebra" used is the algebra of singular cochains,

which fails to be commutative in our sense. This failure is in some sense necessary in positive characteristic, because of the Steenrod operations, and it is for this reason we restrict to rational coefficients in topology. Once this restriction is made, however, we get the very good (contravariant) functor

ApL

from topological spaces to

DGA*'s. Originally described by Thom and

rediscovered over a decade later by Sullivan [Su], chain constructions:

The algebra

H(ApL())

ApL

has the property of all co-

is naturally isomorphic with rational

(singular) cohomology. Briefly, if

M

is a space then an element

singular simpliees of q-form on the standard fy two conditions:

M . If

o

is an

~ 6 A~L(M)

n-simplex, then

n-simplex, A n . The collection

is a function on the

~(O) {~(O)}

is a differential is required to satis-

(i)

~(O) 6 subalgebra generated over

their exterior derivatives, (ii)

{~(o)}

(~+~)(o)

and differentiation

: ~(O)

+ ~(o);

Now the fact which makes theory to algebraic homotopy to quisms

rational (1.6)

Theorem

ApL

(~AP)(O)

: ~(o)

valuable

is that it converts

theory.

(clearly)

In fact,

([Su],

dim H P ( ; ~ )

[B-G]).

1.8

type of

ApL

H°

M

is path connected

finite

phism

Im g

dim Hp

finite

.

invariants of a space

M

(eg.

DGA

in the

if and only if ideal

(H ° M

It turns out that

~'-~ M , to an augmentation

ApL(Pt)=~

ApL(M) ~ ~.

H ° = ~, which in turn is equivalent

is quasi-local).

is path connected

By analogy a

if and only if

DGA ~ R ApL(M)

is has

DGA ~ . h-DGF,'s

~ . Given one such we set

the elements of

R ~ R

[each

s

and

...) can be extracted from any

On the algebra side we consider augmented

by inverting

DGA

H° = ~, H1 = 0

spaces and local rings.

R ° = ~; then

the field

sends rational homotopy

.

the homotopy of a connected

generates

rational homotopy

Jwith

level, all the rational

has a unique maximal if

: d(~(O)

[homotopy t y p e s o f

converts a base point inclusion,

called connected

ApL

(d~)(O)

The map 1.6 is a bijection.

Base point s~ path connected

Moreover to:

ApL(M)

A ~(O);

since

--~

~ (M) @ ~, rational Whitehead products,

and so

xi, and

it induces a map

homotopy t y p e s o f

At a more computational

homotopy

functions,

are defined simpiexwise:

simply connected spaces with each

1.7

by the barycentric

is compatible with the face and degeneracy operators.

Addition, multiplication

equivalences

Q

dx.. 1

g: R ~ k

m = (ker ~)o

R ° - m ° There results a new

which is a quism if and only if

Ho(R)

such that

and localize h-DGF

R

Im at

, R m, and a m o r -

is quasi local.

(In this case

is sur~ective.) We now extend our dictionary: Pointed

spaces

Path connected Connected Finally,

DGA~'s a space

For such spaces R 1 = O; such

spaces

M

ApL(M)

DGA~'s

*--+

Augmented

=

~

h-DGF's

with

h-DGF's

:

~

h-DGF's

R

H°

with

is simply connected

quasi-local R

o

quasi-local.

if it is path connected and

has the homotopy type of a

DGA ~ R

satisfying

~I(M) = 0. R° = Q

and

are called simply connected.

Since path connected

spaces correspond

to

h-DGF~'s R

with

Ho(R)

quasi-local,

it seems Ho(R)

appropriate

a field.

a field,

ef.

to make simply connected spaces correspond

Such

h-DGF~'s

(2.18). Thus our dictionary

Simply connected

spaces

Simply connected

1.9

Finiteness

is noetherian if

R°

have the same homotopy

conditions:

and each

Ri

is quasi-local

and

h-DGF~'s

with

h-DGF~ R

with

R

if

R°

R

with

H (R)

+-~ h-DGF~'s R

with

R°

a field

a field.

A graded ring, R, is called piecewise noetherian is a noetherian R

Ro-module.

is piecewise noetherian.

finite type if each homogeneous

o

continues as follows:

~-~ h - D G r ' s

DGA~'s

to

type as an

component

A graded ring is called local A graded vector space has

has finite dimension.

has finite @-type if its rational cohomology

A topological

space

has finite type. Our dictionary conti-

nues with: Path connected DGA~'s R

spaces

M

of finite q-type

of finite type with

A space (resp. a

DGA ~, a

its rational cohomology all higher degrees.

R° = Q

+-~ h - D G F ' s

R

with

~-~ h - D G F ' s

R

which are local

DGA~) will be said to have formal dimension

(resp. homology)

is non-zero in degree

spaces of finite

h-DGF's

formal dimension

Differential

Tor.

Tot

to modules

over a

DGA ~

such that

H°(R) = ~ .

(I)

An

DGA

([M],

R-module,

rential, d, of degree (-I = dr.m

n

and vanishing

R

if in

of finite formal

We end with a brief review of the Eilenberg-Moore

of

Definition

n

dimension.

a

i. Ii

local

Thus

Path connected

i.iO

H(R)

[G-M]).

Suppose that

M, is a graded

in algebra,

+I

R

R#-module

in topology)

extension

is either a

DGA

or

together with a diffe-

such that

d(r-m) =

+(-l)degrr .dm .

(2)

The tensor product

M ~R N

is the

R#-module,

M @R# N, together with

d(m®n) = dm @ n + (-l)degmm @ dn . (3) {e }

An

R-module

of cycles (4)

An

is called free if it is free as an

R-module,

O = F 1 c F ° c ...

by

C, is called semi-free R-submodules

(In this case the filtration

(5)

on a basis

A semi-free

if it admits a filtration

C = U ~. and each Fi/Fi 1 is free. i I is called semi-free as well.) Note that a semi-free

R-module will always be free as an

if

R#-module

(de S = O).

filtration

such that

R#-module but may not be a free Fi

of

C

R-module.

is called an Eilenber$-Moore

filtration

H(Fo) -~H(C) (6)

and

ker{H(F i) -~ H(C)} = ker{H(F i) -* H(Fi+ I)} , i > 0 .

A semi-free resolution of an R-module

modules:

p: C -~ M

we call

0

with

C

semi-free.

an E_ilenberg-Moore

If

C

M

is a homology

isomorphism of

admits an Eilenberg-Moore

R-

filtration

resolution.

Exactly as in the classical

case we have the easy exercises:

Lifting property:

In a diagram of R-modules B ....

~ M

C ....... ,~ N with

i

an inclusion,

there is a morphism Existence ~ropert~: P/M

C/B

semi-free and

C -~ M

extending

Every morphism

admitting an Eilenberg--Moore

morphism.

(When

R

is a

DGA*

}

z

M -~ N

of

1.12

Lemma.

Suppose

this requires

~: C~-~C '

particular,

for any

R-moduke

factors as

P -~ N

M~

P-7-~N with

a surjective homolozy

iso-

H°(R) = ~[).

case now establishes

is a homology

isomorphism

} .

R-modules

resolution and

The same argument as in the classical

There is then a homology

a surjective homology isomorphism,

and lifting

C' ~ C

isomorphism

of semi-free

which is "homotopy

N , ~ @ id: C @R N ~ C' ~R N

R-modules.

inverse"

to

~ . In

is also a homology iso-

morphism.

1.13

Definition

(cf [G-M]~, [M]).

If

are

M,N

R-modules

then the differential

torsion functor is defined by TorR(M,N) where

C ~ M

1.14

= H(C ~R N)

is a semi-free resolution.

Remark____. It follows as in the classical

dent of the choice of resolution, M

and

functorial

and symmetric

in

N .

Finally,

suppose

sulting spectral

{Fi}

sequence,

is a semi-free filtration of together with

dI dI .... H(Fi+I/F i) ----~H(Fi/Fi_I) Since

case from (1.12) that this is indepenin all three variables

H(Fi/Fi_ I)

is a free

solution is precisely the classical

sense).

H(R)-module,

H(Fo) -+ H(C)

C. The

El-term of the re-

has the form

dI d1 . . . . . H(F o) -~ H(C) -~ O . the condition

that this sequence be a free

H(R)

that

{F i}

be an

resolution of

E-M

H(C)

re(in

10

Now if resolution

U

and

"'.~X i

TorK(u,v) = H(X,@KV) write

V are graded modules over a graded ring K then di ~Xi_ I ~ ' ' ' by free graded K modules such that

p

is the homological degree and

i, internal degree

In particul~r, suppose The filtration

{F i ®RN}

C ~ M

and

is the internal degree.

of

j-i, and total degree

is a resolution with

C @R N

(N

EM

Hj(Fi/Fi_ I)

has

j. filtration

{F i}

of

C.

a second R-module) then yields a spectral

sequence independent of the choice of resolution or M

q

is called the total degree. In the resolution above

homological degree

admits a

acquires a second gradation, called the internal degree. We

Tor~,q(U,V);~ here

The sum, p+q

U

deg d'=0"1 Thus

EM

filtration, and symmetric in

N . It is called the Eilenberg-Moore spectral sequence (E.M.s.s.). Moreover,

for DGA~'s_: The E.M.s.s. is a first quadrant homology spectral sequence with E2 = [TorH(R)(H(M), H(N))] ~P,q P,q

naively convergent to

for DGA~'s:

to identify the E.M.s.s. as a second quadrant (in fact

Write

F -i = F i

third octant) cohomology spectral sequence with in

E? p'q

E2 p'q = [TorH(R)(H(M), H(N))]. Naive p,-q i > p. Set E~P,q = lim E -p'q, then

convergence may fail, but

d.=O

E

module associated with the induced f i l t r a t ~

l

is the bigraded

H(R)

for

TorR(M,N).

i

."

of

TorR(M,N).

In this sense the E.M.s.s. is convergent. Finally, either directly or via the E.M.s.s. we get

1.15

Theorem.

Suppose

Let

R ~ R'

M ~ M' , N ~ N'

be a quism, either of

DGA's

are homology isomorphisms of

TorR(M,N) ~ Tor m

!

or of

R'-modules

DGA*'s

with

H°=~.

Then

(M',N')

is an isomorphism.

§ 2. FREE EXTENSIONS AND HOMOTOPY

Recall the definition of or a

DGF,

R<X>

for a graded

with underlying ring

2.1

Definition:

(i)

S# = R~<X>

(ii)

X

F-algebra, R. Now suppose

A free extension of

R

is a morphism

R ~ S

admits a well ordered homogeneous basis, x , such that

We shall abuse notation and write

R

is a DGA

R~ .

R<X>

for

g

and for

in which

dx

6 R# <{x~}~< > .

S# .

Condition (ii) may be unfamiliar to algebraists, because it is automatic for DGP 's:

simply order the

is essential for DGA*'s:

x~

so

~ < B

~<Xl,X2,X3>

if

with

degx

< degx B . On the other hand it

degxi=l, dXl=X2X3, dx2=x3xl, dx3=xlx 2

11

is not a free extension of Free extensions

rings. Their systematic Free extensions of resolutions, Lifting

Suppose

Existence ween

in a commutative

~

, property:

~

and lifting Suppose

satisfying

The m o r p h i s m

Next suppose morphism

R -* T

R -~ R<X>

2.2

Lemma:

to those

or

DGF,'s,

quism. Then there is a m o r p h i s m

= .

is a m o r p h i s m ¢

this factorization

is a free extension of category)

DGF, ' s

either between

factors as the composite

can be chosen so

DGA*'s

or bet-

R - ~I R<X> m--~ S m

~ T<X>, arrows

(i)

R<X>

(ii)

Assume

are

T<X>

is sur-

@ . or of

we get a pushout

free

is a semi-free R<X> ~ - - R

= T @RR<X>

DGF,'s.

For any

square

diagram of

DGA*

or

~

extensions.

R-module. ....,-T ..

R'<X'>~-- R'

~'

DGF,

T' morphisms,

are quisms and the left arrows are free extensions.

in which the vertical

arrows

Then the induced m o r p h i s m

T<X> ~ T' <X'> is a quism. Proof:

of

1

vertical

is a commutative

DGA*'s

"> T

t both

quite analogous

is called a free model of

(in the respective

R<X> which

through

~: R -~ S

H°= 9. Then

m: R<X>~-~S

R

in

square of

is a surjective ~

a free extension and a quism; moreover jective.

properties

~ T

is a free extension and extending

DGA*'s

study of local

-------* S

R<X>

i

[T] for the homological

have lifting and existence

R

that

by Tate

use in rational homotopy theory is due to Sullivan.

and almost as easily established.

property:

R<X> ~ S

9-

were introduced

(i) follows by induction on the well-ordered

This implies

that the m o r p h i s m

TorR(R<X>,T)

~ TorR'(R'<X'>,T').

H(T<X>) ~ H(T'<X'>)

basis

{x } of

X , cf.

(2.7).

can be identified with the m a p

But this is an isomorphism

by Theorem

1.15. O

12

2.3

Elementary

properties

of

DGF.'s.

Thus

side; i.e., with quism if

degx = 1

or

a quism

for

A < C > ~ A;

A n element morphism

x

R' ~ R

in a with

(infinite)

(i)

(ii)

R

(iii)

There

contains

= O. If extends

h-DGF R2n

to

A

Proposition:

in a

if and only

free extensions

DGF~, R, form a sub DGF

, R, which is

R.

free extension

~

in

R.

and a m o r p h i s m

DGF

, R, is an

and a m o r p h i s m

Suppose

~: R ~ S

h-DGF,

~ ~ R

whose image

if and only if there is an acyc-

A ~ R

is a

whose

DGF~

morphism.

If

R

is an

h-DGF~

and

~+

is surjective

(ii)

If

S

is an

h-DGF

and

~

is a quism then

(iii)

In a commutative T

~

extends

is a quism, to

(i) is trivial.

to

R

is an

is an

h-DGF..

h-DGF~.

~ R

S i

is a free extension and

For

(ii) choose an acyclic

~: R ~ S. Now

and it follows

defines an inverse quism N o w for

S

S

is an

h-DGF~

we have:

~': T<X> -~ R.

this gives a surjection }

then

DGF~ diagram

f%

T<X> - 7 in which

R+ @d(R1).

image contains

R.

cover for

(i)

extends

is admissible

R+ @ d R 1 .

lic free extension

{l,~,dw

ad-

we deduce

in

Such a m o r p h i s m will be called an acyclic

Proof:

x

~ R. Since acyclic

(n ~ 0), RI, and the cycles

is an acyclic

~

if it is in the image of a

then

~

elements

contained

The lemma shows that a

2.5

x, dx ~ O; this is a

are called acyclico

degx > 0

tensor products

The admissible

contains

by

is an example of a free extension

DGF~, R, is ca]led admissible

are closed under

Lemma.

~ . This

H+(R')

A<x,dx> ~ R

in fact the maximal

we work only on the algebra

~<x,dx>-~@ $

such free extensions

if the m o r p h i s m

2.4

For the moment

A = ~. Define

degx = 2n, n > i. In any case we m a y choose a free model

~ = A < x , d x > < X > ~-~A mitting

h-DGF~'s.

that

R J-~R

cover

has an

is a quism.

~ < C > ~ S. Tensored with A-basis

¢

of the form

The quism

A ~ A

then

p: R ~ R.

x E R n (n ~ I) we get B: A < C ( x ) > ~ S. Since

y: A < C ( x ) > ~ R

~

extending

j od.

o: A < x , d x > ~ R; because ~= ~j

is a quism,

so is

Hence

po: ~ < C ( x ) > ~ R

S

is an

~. Thus extends

h-DGF., B

O

¢~

lifts to a and shows

x

is admissible. To prove

(iii) construct

a surjective

quism

R ~-~S

extending

~

as in (ii).

13

Lift

B

to

%: T<X> ~ R

and set

~' = py . D

Note that proposition is always

2.~

acyclic,

Corollary.

S @R T

(i)

is also an

(ii)

If

h-DGF~

2.7

every

R<X>-~ S

Example. m

where

if

R

DGF

because,

~<x, dx>

since

cover.

morphisms

of a

DGF.

and

morphism

(concentrated

S,T

are

R ~ S

{dx i C m}

represent h-DGF~;

are symbols

h-DGF's

and if

S

then

is an

R

a basis

of degree

is any

KR

is the

for

zero) with maximal DGA~ R<Xl,...,Xn>

m/m 2 . It satisfies

h-DGF,

zero then the is an with

DGF

,

Ho(KR)=~.

it turns out that so is its Koszul

K R = R @~<{yi}>A<{Yi,Xj}> if

in degree

~. The Koszul complex

is itself an

More generally, an

are

be a local ring field

and R

Thus

R ~ S,T

is a free model

Yl ..... Yn

is acyclic.

has an acyclic

DGA~'s

R<X>.

Let

degx i =i

Indeed

If

and residue

Of course

DGA ~

for all

h-DGF~.

then so is

ideal

2.5 (iii) holds

complex.

A<{Yi,Xj}>

(dxi=Y i)

h-DGF, Ho(R)

a local ring we can construct

h-DGF~ K R = R<Xl,...,Xn > ,

such that h-DGF~,

2.8

H ~ R ) < x I ..... Xn > = ~ R )

R. It satisfies

Homotopy.

model

for

Fix an

%o,%1:

we write

contravariance

~o ~ ~I

object

IWR

implies

that

homotopic.

for the

~: W ~ R

and let

m: W-~ R

be a free

(W ~wW

= Q)

IwR = W--~ R . denote

the left and right

and call

space rather of

Definition.

write

morphism

(IwR,%o,%I)

IR; it corresponds

in a topological

2.9

complex

W @W W ~ R

W ~ IwR

We follow Quillen W = A

h-DGA

it to a free model mi:

Let

. This will be called a Koszul

Ho(K R) = ~.

~ . Consider m.m:

and extend

degx i = ]

inclusions.

a relative

(as we shall

cylinder

see in (3.3)

than to the cylinder

object

for

~. When

to the space of paths

i x M; this is again due to the

ApL .

Two

h-DGA

morphisms

~o,~i:

(rel W) - if for some free model

there is a morphism ~i o ~ = ~ o ° ~[)

~: I w R ~ S

When

W = A

R ~ S of

are homotopic ~

such that we write

(rel W) - we

and some relative ~o%i ~o ~ ~I

= ~i ° m. snd call

cylinder

(Note that this }o

and

~I

14

If that

W

is a quism restricting

W ~wW ~ W @wW

straightforward

consequence

to

id W

then Lemma 2.2 (ii) implies

is also a quism. Thus the next proposition

of the existence and lifting properties

sions, and of (2.5 (iii)). The arguments are very similar to those of Quillen

2.10

Proposition.

(i)

Homotopy

(rel W) is an equivalence

is a

for free exten-

relation,

[QI ].

independent

of

the choices of free model and cylinder object. (ii) ~o

~ ~i

If

~: S ~ T

is an

h-DGA morphism

then

(rel W). The reverse implication holds if

(iii)

If

~

factors as

W ~--~T ~--~R then

(rel W). The reverse implication holds if (iv)

If

~o ~ q~l (rel W)

then

~

~o % ~I ~

(rel W) implies

is a quism.

~o ~ ~I (rel W)

implies

#o~ ~ ¢i ~

is a quism.

~o ~ ~i " O

As an immediate corollary we deduce an

h-DGF

2.11

(recall an

h-DGA

is either an

h-DGA ~

or

.)

Theorem.

Let W

>R

W<¥>

~ S

B be a commutative quism. Then

~

diagram of extends to

second such extension then

2.12

Homotopympushouts.

free model

m: R<X> ~ S .

h-DGA's

in which

~': W ~ R

i

is a free extension and

such that

~'

~ ~ (rel W). if

~

is a

~"

is a

~' ~ ~" (rel W).

Suppose If

R ~ S, R ~ T

are

h-DGA

morphisms

and choose a

m': R<X'>~-~S

is s second free model then by Theorem w 2.11 we obtain a unique homotopy class (rel R) ef quisms R<X>--~R<X'> such that

m'~ ~ m (rel R). These homotopy classes are closed under composition to homotopy T @R

(re] R)) identification

and so provide a unique

of the homotopy types of the

(up

R<X>. Performing

yields unique homotopy classes (rel T) of quisms

(2.13)

2.14

T<X>~-~T<X'>

Definition.

out of

.

The common homotopy

type of the

Of course we would equally well have considered

duced unique homotopy classes quisms

T<X>

is called the homotopy push-

S ~ R ~ T.

(rel S) of quisms

free models

S ~-~S.

R-~ T

and de-

By Lemma 2.2 we have

15

(2.15)

T<X> ~-R<X,Y>~-~S

which are compatible with the identifications

(up to homotopy)

ticular we can canonically use any

R<X,Y>

S

or

on either side. In par-

to represent the homotopy push-

out. Finally, we remark that (by Lemma 2.2 (i)) the homology of the homotopy pushout is given by (2.16)

H(T<X>) = TorR(S,T)

and that the identification

H(T<X>) = TorR(S,T) = H(T<X'>)

is precisely the isomor-

phism induced from the homotopy class of quisms (2.13). Similarly the identification TorR(S,T) = TorR(T,S) It follows,

is induced by the quisms (2.15).

in particular,

that

TorR(S,T)

has naturally the structure of a

algebra, and that the symmetry identification is an isomorphism of

2.17

Coproducts:

the coproduct,

If

R

and

S

are

h-DGF.'s

F-algebras.

their tensor product

or pushout, with respect to the initial object

the homotopy type canonically represented by R

and

~<X> O A

R 0 S

is just

~ .

On the other hand, we have just defined the homotopy pushout of

ly free models of

F-

A ~ R,S; it is

where these are respective-

S. It is important to note that the canonical morphism

A<X> 0 A -~ R @ S need -not be a quism. Indeed when R 0 S = •

. On the other hand

R = S = Zp R

(and

S)

concentrated in degree zero then also has a free model

~<x>

with

degx = 1

P and

dx = p; it follows that H(~<x> 0 Z<x>) ~ Z <x> . P

2.18

Fields.

some

h-DGF. R

We are frequently faced with the problem that the homology contains a field

might be a Koszu] complex, another

2.19

h-DGF,

Lemma.

Proof:

cf.

(2.7),)

R

itself does not.

Every field

~

Ho(R)

(For example,

Free extensions permit, us to replace

of the same homotopy type, which does contain

has a free model of the form

R

of R by

k. We need the

~[Xo]<XI> .

The lemma follows via a direct limit argument from the following three obser-

vations:

(i)

I<x>, dx =p

is a free model of

model for an algebraic extension b E R

~, although

then

R, dx =ab -I

Now suppose p: ~<Xo,XI> Z-~k

R

is an

and lift

k(a);

~p; if

to

and

(ii) R =R

is a free model for

h-DGF, p

(iii)

is a free

o R(i/b).

Ho(R) ~ a field

~: I[X o] -* R °

k, dx =f(a)

is an integral domain and

so that

~. Choose a free model [~x] = px. Then for

y E X

16

[4dy] and

so

= pay

~ d y C dR 1 . B e c a u s e

= 0 X1

is

7-free

we c a n

4

extend

to a m o r p h i s m

4: Z<Xo,XI > ~ R . Finally,

extend

~

to a free model

of

R . Since

P

is a quism we can apply

Lemma 2.2 (ii) to get k~--Z<Xo,XI>~-*R in which

as

the

left

hand morphism

is

also

a quism.

Thus

R

has the same homotopy

type

k.

§ 3. FIBRATIONS

Fibrations DGA's.

AND FREE EXTENSIONS

play the same role in the topological

We recall that b~ definition a fibration

Lifting

In a commutative

property: A

~

category

that free extensions ~: E ~ B

is a continuous map

do for with the

square of continuous maps

E

X--TB in which map

A

X ~ E

is

a closed

strong

which lifts

property:

composite

Suppose

runs as follows:

compact open topology. where

f~y If

denotes

of

extends

X,

4: B 1 ~ B

let

Then

BI

is a continuous map. Then

equivalence

be the space of continuous

path at

E---,~B is a fibration and

y

Wy

is a fibration.

,,

Y~B

,

I ~ B

w(f,y) =f(1);

is a continuous map,

E

B

paths

factors as the w . (The conwith the

j(y)=(f~y,y)

~y.)

square y × B E ,,

4

j, and a fibration

E c B I × B 1 = {(f,y) If(O) =@y};

the constant

wY1

to a continuous

we have the

B1J-~*E ~---~B of a homotopy

struction

retract

4 •

As with free extensions Existence

deformation

Y×B E = {(y,e) l~y = we}

If (cf. Lemma 2.2 (ii))

,

then in the pullback

17

E

>B

-Y

<

E '----+B ' +---Y ' is a commutative

square of continuous maps

topy equivalences

in which the vertical

and the left arrows are fibrations

arrows are weak homo-

then

YXBE +--Y' XB,E' is a weak homotopy

equivalence.

Suppose now (as in (2.12))

that

tions of the same continuous map and a fibration. equivalences

X -~ E ~--~B and

X ~ B

X -~ E ' ~-~' B

as the composite

There is then a uniquely determined

~: E-U-~E '

(such that

~'~ = ~)

are two faetoriza-

of a homotopy

equivalence

homotopy class of weak homotopy

and hence

(for any continuous

Y ~ B)

a unique homotopy class of weak equivalences (3. i)

YXBE - ~ Y X B E ' The common weak h o m o t o p y

X ~ B ÷ Y. Had we factored have obtained

canonical

(3.2)

. type of these spaces

Y -~ B

as

represents

(3.1) and (3.2) are the precise analogues to homotopy pushout

3.3

Ex~p!e:

that the space M.

for

is just the ordinary

spaces corresponds

no__t to the coproduct

the homotopy pullback.

X ÷ pt + Y

product of

are fibrations

X x y. Thus the product

h-DGF,'s

If

of all continuous maps

I ~ M

~: M I ~ M x M; f~-~ (f(O),

f(1))

M

and that their of topological

but to their homotopy pushout.

is a topological

space recall

is called the free path space on

The p r o j e c t i o n

is a fibration and can be identified with the fibration map

Of course

h-DGF~' s.

Free path space and homotopy. MI

then we would

of (2.13) and (2.15), with h o m o t o p y pullback

we note that the projections

homotopy pullback

a fibration)

~--EIXBX

so that any of these spaces naturally

Finally,

(~I

weak equivalences

YXBE - ~ E I X B E

corresponding

is called the homotopy pullback of

Y Z-~EI --~IB

obtained from the diagonal

M ~ M x M . By definition,

there is a m a p

two continuous maps

~: N ~ M I

~

=

such that

(~o,~i)

•

~o,~i: N ~ M

are homotopic

if and only if

18

N o w consider m: A < X > ~-~R

the case of

h-DGF,'s.

then we translate

(see just above) while the diagonal R ÷ A<X>

Converting of

®A<X>:

M ~ M x M

If

R

the product

is one such with free model

M x M

M ~ M x M m'm

to the homotopy p u s h o u t A < X >

translates

@A<X>

to

.

to a flbration m e a n s

that we should construct

a free model

m.m , R+m(A<X>

The projections

® A<X>) = IR .

M x M ~ M

correspond

to the inclusions

I.: i

®A<X>

,

which we m a y think of as the inclusions If now we were to translate two m o r p h i s m s such that

~i: R ~ S

%.: ~ < X > ~ IR . I the topological d e f i n i t i o n

~<X> ~A<X>

were homotopic

of homotopy

we would

precisely when there was a m o r p h i s m

say

~: IR ~ S

~ o li = ~i o m . This is of course exactly the definition we gave in (2.8).

Thus in our dictionary Two continuous maps are homotopic ~

3.4

The model for a fibration.

extensions

(in algebra)

the passage h-DGF,'s, This,

ApL:

We have seen that fibrations

have analogous

topology ~,~ DGA*'s

one might hope that however,

fails.

Two h-DGF, morphisms

ApL

properties.

are homotopic.

(in topology)

Since our dictionary

together with the comparison mapped

fibrations

Instead we consider

and free

is based on

DGA*'s vs.

to free extensions.

fibrations

E ~---~B of path connected

spaces and form the free model

m: /ApL(B)<X> -/> JApL(E ) of

ApL(~)

. Then if

together with

Y ~ B

is any continuous map the m o r p h i s m

m, determines

a morphism

~: A p L ( Y ) < X > ~ ApL(YXBE)

•

In [Ha; § 20] is proved the

3.5

Theorem.

Suppose that

(i)

The fibre, F, of

(ii)

Either

(iii)

~I(B)

F

~

is path connected. Y

acts nilpotently

Then the m o r p h i s m

3.6

or both

Corollary.

~

and

B

on each

have finite HP(F;~)

q-type.

.

is a quism.

H*(Y×BE;~ ) ~ TorApL(B)(ApL(y),

ApL(E)).

ApL(B) ~ ApL(Y)

,

19

One certainly has the right to expect that theorems on motopy pushouts will correspond

to theorems on

h-DGA~'s

h-DGF~'s

and their ho-

and their homotopy pushouts.

On the other hand, the effect of Theorem 3.5 is to assure that the homotopy pushout of

h-DGA~'s

is a model for the homotopy pullback of topological

this reason that it is possible to translate topological

remark:

Corollary 3.6 is a variant of the original

of Eilenberg and Moore which "began"

Fibre and homotopy fibre.

tion

E ~ B

at

not depend on

b E B

the subject of differential

Fix a path connected

is the pull-back

if

~: Y ~ B

is the homotopy pull-back of the homotopy class of space is constructed

~

(i)

Convert

~

Convert

{b} ~ B

space, B. The fibre of a fibra-

{b} x B E = ?r-l(b) . Its hemotopy type does

is any continuous map then the homotopy fibre of ~

and (any) inclusion

and is an extremely

to a fibration

The analogue for to an augmentation

E-~B

h-DGF,

h-DGF~ s

R ~--~k

morphism

homotopy fibre of

{b} ~ B. It depends only on

important

invariant. A representative

and take a fibre, ~-l(b)

into a fibration

P ~ B

h-DGF~

(cf. 1.8) such that

R

at

c

(2.12) of

~ . As in topology it is an extremely

less attention,

presumably because even when

A representative

and

for this homotopy fibre can be constructed R -~ S

of

#; then

~

Choose a free model

R<X>~

of

E; then

S<X> represents

S<X>

invariant.

It has

in two ways:

Choose a free model

and

~: R ~ S

is called the

are classical commutative

(ii)

k

~

k .

h-DGF

(i)

in proving

corresponds

generates

k . If

~

important

R,S

P x B Y.

{b} ~ B Im E

is the pushout

then the homotopy pushout

rings the homotopy fibre is usually a genuine

.

and take the pull-back

should be clear. The inclusion

of an

Thus the fibre of a free extension

The fact that

algebra.

in one of two ways:

(ii)

received

theorem

homological

b .

More generally,

is any

It is for rings and

spaces.

One other historical

3.7

spaces.

theorems between commutative

have the same homotopy

represents

the homotopy fibre. the homotopy fibre.

type plays an important

role

some theorems.

3.8

The Serre spectral

and

B

sequence.

Let

E ~---~B be a fibration with

E

path connected

simply connected. Form a free model ApL(B)<X>~-~ApL(E). Filtering the left ~p def ~p F p = ApL(B)<X> = ApL(B) @ Ap~(B)ApL(B)<X> we get a first quadrant co-

hand side by homology

spectral

sequence converging

to

H (E;~). Its

E2-term is just

EP,q = HP(B) @ Hq(~<X>) 2 If the hypotheses

of Theorem 3.4

(Y =pt.)

are satisfied

then

H(~<X>) = H~(F;~)

and this is the 8erre spectral sequence. Analogously

suppose

R ~---+S is an

h-DGF~

morphism with

R

augmented

to

20

and

Ho(R) = ~ , cf. (1.8). Let

H-DGF., R<X>, by

R<X> ~ S

be a free model for

Fp = R @ ~A<X>]

R

~

and filter the

is not stable under

This produces a first quadrant homology spectral sequence in which

d~)

E1 = P,q

= Hq(R) @ [A<X>]p = Hq(R) ®kk<X>p , and E 2p,q = Hq(R) @kHp(k<X>) . This spectral sequence converges to

H(S); by analogy we also call it a Serre spectral

sequence. Observe that if the requirement that

B

$S simply connected is removed then the

E2-term of the topological spectral sequence becomes "cohomology with twisted coefficients". The identical phenomenon occurs if the requirement that placed by

Ho(R ) D ~ . This is another reason for the parallel:

Ho(R) = ~

in (1.8).

Ho(R) = k

is re-

simple connectivity ~-~

The analogy between the two sides is reinforced by the well known

3.9

Proposition.

Let

~: E ~ B

connected, and either the fibre h-DGF,

morphism with

R

be a fibration with F, or

B

of finite

augmented to

~

and

B

simply connected, E

@-type. Let

R ~ S

path

be an

Ho(R) = ~ . The following assertions

are then equivalent: (i)

The Serre spectral sequence collapses at

(ii)

The morphism

(iii)

H*(E;~)

Proof:

H*(E;~) ~ H*(F;~)

is a free

H~(B;~)

(resp.

module

In the topological case denote

H*(E) ~ H~(F)

is identified with

E2 . H(S) ~ TorR(s;~))

(resp.

H(S)

ApL(B) ~ ApL(E)

is surjective.

is a free

also by

H(R)-module).

R ~ S. Then

H(S) ~ TorR(s;k). In either case, this is one edge

homomorphism for the spectral sequence; the other is

H(R) ~ H(S). Now clearly (i)

(ii) and (iii). To show (ii) ~ (i) we need to reduce to the case topology side we may simply replace side we apply 2.18. Let

~

R

stand for

Now choose cycles

~i E R @ ~ < X >

sequences from

E2

(E2)

(H(S),~)

R

o

~ ~). On the

on the topology side. Then on either side R ~<X>

.

which project to a

R @~

~ R @~<X>

K-basis of

H(k<X>); let

gives an isomorphism of spectral

on.

To see that (iii) ~ (ii) we note

H(R)

Tor**

(or

by an appropriate free model; on the algebra ~

the spectral sequence arises from a model

be their span and observe that

R° = ~

that if (iii) holds the bigraded algebra

is concentrated in bidegrees

tral sequence (cf. (i. IO)) collapses and

(0,*). Thus the Eilenberg-Moore spec-

H(S) ~ TorR(s;~)

H(S) ~ TorH(R)(H(S);~), which in this case is surjective.

is identified with

21

A fibration

satisfying

the conclusions

(TNCZ = totally non cohomologous sions precisely when

3.10

H(S)

The dictionary

is

of (3.9) is said to have TNCZ fibre

to zero). A morphism H(R)-flat.

continued.

R ~ S

satisfies the conclu-

Thus these conditions

are analogous.

We summarize § 3 by the following

table in our dic-

tionary: Fibrations

<

>

Free extensions

Homotopy pullback Products

M × N

Homotopy pushout of

M

and

N

Homotopy pushouts The

Cohomology algebra of the homotopy pullback

F-algebra

Homotopy fibre

Homotopy fibre

Cohomology of the homotopy fibre

TorR(s;k)

Homotopy fibre has finite formal dimension

S

Serre spectral

sequence

Homotopy fibre is

of A ~R,S

TorR(S,T)

has finite flat dimension over

Serre spectral

TNCZ

A<X> @A

H(S)

is

R

sequence

H(R)-flat.

§ 4. LOOP SPACES

4.1

Topology.

Let

topy equivalent

(M,*)

be a pointed space. The inclusion of

to the path space fibration PM = {f: I ~ Mlf(O ) = *} ;

Its fibre, ~M, space on

is the space of pointed maps

M. By definition, ~ M

*

in

M

is homo-

z: PM ~ M: zf = f(1).

(sl,*) ~ (M,*)

and is called the loop

is the homotopy pullback of

Evidently any continuous map

@: (M,*) ~ (N,*)

determines

~@: ~M ~ ~N

in the

obvious way. In addition to being a topological *)

~M

space (pointed by the constant

admits a continuous multiplication f(2t)

O < t < 1/2

g(2t-l)

1/2 < t < 1

=

,

(f.g)(t) It is homotopy associative,

and

e,

cocommutative

algebra with diagonal More generally,

@ H*(~M;~);

let

E ~ B

~M

C

~M

at

.

the structure of a graded

has finite

q-type,

then

in this case we get a (dual) commutative

H~(~M;~) ~ H*(~M;~)

consider the pullback diagram

e.

identity.

H.(~M;@)

Hopf algebra° On the other hand, if

~) = H*(~M;~)

f,g

acts as a homotopy

If we pass to rational homology we get in

H*(~M ~ M ;

loop

~M × ~M ~ ~M:

@ H*(~M;~)

Hopf

arising from the multiplication.

be any fibration with fibre

F

over

* E B

and

22

F

~E i BPB

e,

~ PB

E ~ B

it gives a homotopy equivalence is a fibration with fibre (4.2)

(up to homotopy)

F-~EXBPB.

a continuous map (x,f)~. x'f .

Up to homotopy this is an action of E

~B

on

F (x'fg ~ (x.f).g

is itself the path space fibration the resulting map

homotopy equivalent Finally,

to the multiplication

H,(F;Q)

of the Hopf algebra

and

x.e, ~ x)

~B x ~B ~ B

is

defined above.

if we pass to rational homology

(4.3)

EXBPB ~ B

~--F x ~ B

F x~B ~ F ;

and when

On the other hand, the composite

F × ~B; thus

F-~E×BPB

defines

;

in (4.2) we get an action

® H,(~B;~) ~ H,(F;~)

H,(~B;~)

on

H,(F;~)

.

This action is the central object of study in the article

[F-T] of Felix-Thomas

in these proceedings.

4.4

Algebra.

Im g

In analogy with (4.1) we consider an augmented

generating

k. The path space fibration corresponds

h-DGF, ~: R ~ ~

with

to a free model

R<X> ~ k for

g

and its fibre is just the

As observed of quisms

h-DGA, k<X> = ~ ®RR<X>

whose homology

is

Tor(~,~).

in (3.12), a second free model leads to a unique homotopy class (rel ~) ~<X>---~<X'>;

4.5

Definition.

for

R . To understand

The

we can therefore make the

h-DGF

, ~<X>, will be written

~R

and called the loop DGA

the analogue of loop space multiplication

and of loop space action

on a fibre, we note that the analogue of the pullback diagram in (4.1) is, ef. (2.15), k ~-R<X>~----R

k ~.... where

R i s On t h e

a free

other

R<X> ~ ~<X>

hand

R<X>

model the

of

~ ~:

inclusion

a n d we n o t e

R R ~ S

and

R<X> = R @RR.

F x ~ B ~ EXBPB

corresponds

to

the

projection

23

(as

~<X> = k @ ~ < X >

DGF 's).

Thus altogether we get (4.6)

~ +Jl R<X>---~k O ~<X>

in analogy with (4.2). Here (cf. (2.14))

~

is the homotopy fibre of

~ . Observe as well that a

second choice of free models leads to a second version of (4.6); the two versions are then connected by unique homotopy classes of quisms (rel R or

~)

making the resul-

ting diagram homotopy conm~ute. Passing to homology in (4.5) yields the standard (4.7)

TorR(s~k) ~ TorR(s,~) @ TorR(~,~);

when

S = k

TorR(~,~)

this is an associative comultiplication in into a

F-Hopf algebra. For general

Hopf algebra on the

F-algebra

Dualizing with respect to for

S

TorR(k,~)

which makes

(4.7) defines a co-action of this

TorR(s,~). ~

converts

TorR(s,~)

into

ExtR(S,~ ) - this works

DGA,'s as well as for rings. In particular, the diagonal above dualizes to a

graded algebra structure in generally, the map (4. 7)

EXtR(~,~)

and this is eaxctly the Yoneda algebra. More

dualizes to make

EXtR(S;k)

(The Yoneda algebra is defined as follows:

into a module over

HomR(R<X> , R<X>)

by composition - passing to homology defines a product in = EXtR(~,k). ) When H(R)

is piecewise noetherian then

TorR(k,~)

in each degree and in this case the multiplication in multiplication in

4.8

Remark.

If

EXtR(k,~).

has a product given

H(HOmR(R<X> , R<X>)) =

will be finite dimensional

TorR(k,k)

dualizes to a co-

EXtR(k,k) - which is then a Hopf algebra.

R

is a supplemented

a unique homotopy (rel k) inverse

k-algebra then the quism

R<X>~k

k ~ R<X>. In this case we get an

has

h-DGF.

morphism (4.9)

k ~ k @ k<X>

whose (rel k) homotopy class is unique. When

S = ~

the resulting diagonal is homo-

topy associative (rel k); in general it is a homotopy co-action (rel ~). In this case, howewer, the bar construction can be used to provide a strictly associative version of (4.9).

4.10

Th_e dictionary, again.

We have

24

Loop space

B

<

> Loop

Homology algebra, H.(~B;~)

~

> Yoneda algebra,

Action of H.(~B;~) on H.(F;~); F the homotopy fibre of E ~ B

~......~ Action of

4.11

Warning.

space) and

(i)

The graded spaces

EXtR(S;~)

H.(M;~)

DGA ~R = ~<X>, where

R<X>-~

k

ExtR(S,~)

.

EXtR(~,k)

EXtR(k,~)

on

~ational homology of a topological

are two of the examples where we grade downstairs in topology

and upstairs in algebra. (ii)

Although

EXtR(k,k )

sponds to

H*(~B;~)

it is important to observe that

H*

is the dual of

corresponds to

H.(~B;~)

where

Ext

TorR(~,~)

is the dual of

correTor

and

H.. This distinction is necessary when these graded spaces do not

have finite type because then products do not dualize to coproducts. H*{~B;Q)

is a Hopf algebra;

EXtR(~,~)

is only an algebra

H*(~B;~)

is only an algebra;

TorR(~,~)

is a

In this case

and

(iii)

If

M

is~,a simply connected topological space and

integer such that and if

kR ~ 1

F-Hopf algebra.

HM(M;~)

# O

then

~M

= kM-l"

is the first integer such that

The observations

If

R

kM ~ 1

is an

HkR(R) # 0

then

is the first

h-DGF, ~R

with

Ho(R)--~k

= kR +I.

4.11 (ii) and 4.11 (iii) are examples in which simple minded

translation between the two sides fails. It is because of this kind of phenomenon that the translator needs to be very careful.

§ 5 THE HOMOTOPY LIE ALGEBRA

In this chapter

M

q-type while

will always denote an

R

with residue field

will always denote a simply connected topological

Now the rational homotopy groups

via the canonical

ture (the Samelson product);

H.(~M;~)

space of finite

is local (cf. (1.9))

subspace (of elements

~i (M) ® 9, equipped with the Whitehead product

In fact, when this product is transferred to

(degree-l) isomorphism it does give a graded Lie strucM.

h-DGF.'s we need two further identifications.

is a graded Hopf algebra. The Cartan-Serre theorem asserts that

the Hurewicz homomorphism

~.(~M) @ ~ ~ H.(~M;~)

~ ~ ~ ~ 1 + 1 @ ~

is an isomorphism onto the primitive

under the diagonal). Almost by definition

this is Lie isomorphism when the primitive subspace [~,B] = ~ -(-I) deg~degB ~ . Under the duality between Q* = H+/H+-H +

H(R)

the result is the r ati0na! homot0py Lie algebra of

To translate this to rings and Recall that

such that

k.

almost define a graded Lie algebra. ~,(~M) ® ~

h-DGF,

H.(~M;~)

of indecomposables.

and

P,

H*(~M;~)

The diagonal in

is given the Lie bracket

P,

H*(~M;@)

is dual to the space yields a Lie comulti-

25

plication in

Q*

dual to the bracket in

To translate this to

R

P,.

we recall (§4) that

which has the natural structure of a connected F-algebra

A

we will let

I(~ p)

H*(fM;~)

corresponds to

TorR(k,~),

F-Hopf algebra. Now for any connected

denote the linear span of the elements in

Im yP

and call QFA = A+/I ; the space of

I = A+'A+ +

F-indecomposables for

~ I(~ p) pZ2

A. In particular we set

n,(R) = QF(TorR(k,k))

;

to corresponds to the space of indecomposables

Q*(H*(~M;~)).

Dualizing we get a graded subspace ~*(R) ~ EXtR(~,k ) corresponding to space

P,

of

w,(~2M) @ ~. Now

EXtR(~,~ )

w*(R)

is always a subspace of the primitive sub-

and is in fact a sub Lie algebra,

diagonally induced Lie comultiplication in have

(with bracket dual to the

w*). However, when

char k > O

we usually

7, # P, . One justification for this choice resides in the following theorem (cf. Milnor-

Moore [M-M], Andr~ [Al],Sjbdin [Sj]):

5.1

Theorem

(i)

Tile Hopf algebra

H,(~M;~)

is the universal enveloping algebra of

~,(~M) ® ~. (ii)

The Hopf algebra

EXtR(~,~ )

For this reason we call then the numbers

5.2 E ~ B

Fibration sequences. (B

w*(R)

gi = dim ~i(R)

Let

is the universal enveloping algebra of

the homotopy Lie algebra of

are called the deviations of

F -~ E

R. When

R = R

~*(R). o

R.

be the homotopy fibre of a continuous map

path connected); then the sequence

~i(B) ÷ ~i(E) ÷ ~i(F)

is exact. Trans-

ferring to loop spaces yields an exact sequence of Lie algebra maps, (5.3)

~,(~B) ÷ W,(~E) ~- W,(fF). Now suppose

residue fields (5.4)

R ~ S ~

is an

~ , and let

h-DGF, S ~ F

morphism in which

H(R), H(S)

are local with

be a homotopy fibre. Then ([Av2])

w*(R) ~k~ ~ w*(S) ~ ~*(F)

is also an exact sequence of graded Lie algebras. For topological spaces one gets the long exact homotopy sequence from (5.3) as follows. Recall the action (4.2) of

fB

on

F. This restricts to a map

fiB ~ F

which

26

turns out to be the homotopy fibre of (5.5)

B ~+-E i F

+ ~B c ~

F ÷ E. Thus we get an infinite sequence ~E <~J

~F ÷...

in which each map is the homotopy fibre of the map to its left. Passing to homotopy and using

~i(B) = ~i_l(~B)

(5.6)

yields the long exact sequence

~i(B) ÷ ~i(E) ~ ~i(F) ÷ ~i+l(B) ÷ . . . . In algebra one can mimic everything except the last step. Let

and replace it by a free model

R ~ R. If

mentation then the homotopy fibre of

~

R<X>-~k

R ~---*S be as above

is a free model of the aug-

is represented by

j: R ~ R<X>

and

the co-action (4.6) yields in an obvious way R<X> ~ %<X> = ~R ® ~ as the homotopy fibre of (5.7)

j. This leads to the analogue

R ~ R ~ R ~ ~R ® ~

~>~S

~ ~F

...

of (5.5) in which each morphism is the homotopy fibre of the one before. The last step of the analogy fails, however. There is a natural surjection ~(~R)

~ ~*-I(R)

which is an isomorphism if

char ~ = O; when

char ~ > 0

the ker-

nel can easily be infinite dimensional. Thus "usually" there is no analogue for (5.6), although (cf. [AVl]) there are important and useful occasions when such an analogue can be established.

REFERENCES

[A 1 ]

M. Andre, Hopf algebras with divided powers, J. Algebra 18 (1971), 19-50.

[A 2]

M. Andre, Homologie des Alg~bres Commutatives, Springer, Berlin 1974.

[A-M] M.F. Atiyah and I.G. Macdonald, Introduction to Con~nutative Algebra, AddisonWesley, 1969. [Av I] L.L. Avramov, Local algebra and rational homotopy, in Homotopie Alg~brique et Alg~bre Locale, Ast~risque 113/114 (1984), 15-43. [Av 2] L.L. Avramov, Homotopy Lie algebras for commutative rings and DG algebras, to appear. [B-G] A.K. Bousfield and V.K A.M. Gugenheim, ~D the PL De Rham Theory and rational homotopy type, Memoirs Amer. Math. Soc. 179 (1976).

[c]

L. Carroll, Through the Looking Glass and What Alice Found There, Macmillan 1871.

[F-T] Y. Felix and J.-C. Thomas, Sur l'op~ration de l'holonomie rationnelle, these proceedings.

27

[G-L]

T.H. Gulliksen and G. Levin, Homology of Local R~Dgs , Queen's papers in Pure and Applied Mathematics - No. 2_~0Queen's University, Kingston, Ontario, 1969.

[G-M]

V.K.A.M. Gugenheim and J.P. May, On the The0ry and Application of Differential Torsion products, Memoirs of the Amer. Math. Soc. 142 (1974).

[Ha]

S~ Halperin, Lectures on Minimal Models, M~n. de la Soc. Math. de France 9/10, 1983.

[Ha-Le] S. Halperin and J.-M. Lemaire, Suites inertes dans les alg~bres de Lie gradu~es, preprint. (To appear in Math. Scand.) [L-Av]

G. Levin and L.L. Avramov, Factoring out the socle of a local Gorenstein ring, J. Algebra 55 (1978), 74-83.

[M]

J.C. Moore, Alggbre homologique et homologie des espaces elassifiants, S~m~naire H. Cartan, ~cole Normale Sup6rieure, 1959-1960, Expos~ 7, Secretariat Math., Paris, 1961.

[Ma]

H. Matsumura, Commutative Algebra 2nd ed., Benjamin/Cummings, Reading, Mass. 1980.

[M-M]

J.W. Milnor and J.Co Moore, On the structure of Hopf algebras, Annals Math. 81 (1965), 211-264.

[Q1] [Q2 ]

D. Quillen, Homotopical Algebra , Lecture Notes in Math. 43 (1967), Springer Verlag. D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math. (17), Amer. Math. Soc. 1970, 65-87.

[R]

J.-E. Roos, Relations between the Poincar~-Betti series of loop spaces and of local rings, Lecture Notes in Math. 740, 285-322, Springer Verlag, Berlin 1979.

[Sj]

G. SjSdin, Hopf algebras and derivations, J. Algebra 64 (1980), 218-229.

[Su]

D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES 47 (1978), 269-331.

[~]

J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-27.

[w]

G. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1981.

L. Avramov Institute of Mathematics University of Sofia ul. "Akad. G. Bon~ev" BI. 8 1113 Sofia, Bulgaria

S, Halperin Department of Mathematics University of Toronto Toronto Canada M5S IAI

HILBERT SERIES OF FINITELY PRESENTED ALGEBRAS by David A-NICK and Clas LOFWALL

Summary

Let /

denote the collection

of all Hilbert

connected graded algebras over a field This paper addresses

series of finitely presented

k . What can we say about the s e t / ?

itself to that question.

In 1974 Govorov

conjectured that only rational power series b e l o n g e d t o ~ . was first disproved by Shearer

[Go-2]

This conjecture

[Sh] , using methods which we will generalize

and extend in this paper. We will also show that the set

/

is countable

and derive some of its properties.

Definitions

Let direct

k

denote any field. A graded vector space (over

k) is any countable

sum

A = @ A of finite-dimensional k-spaces. A graded vector space n n=0 is a (connected) graded al~ebr a (over k) if there is an isomorphism

a: A 0 for each

~

k

m

and there are associative and

n

and if

~n0 o (I @ s-l): A n @ k

pairings

~mn: A

m

~0n o (~ I 8 I): k 8 A n ~ An

~ A An

~ A

n

A

m+n

and

agree with scalar multiplication.

Viewing

the A ' s as subspaces of A , we say that a non-zero element x~A is n _ degree ~ , written homo6eneous _of Ixl = n ~ if x E A n . The elements of positive

degree are written

A + . The Hilbert

series of a graded space

A = @ A is denoted A(z) and is defined to be the formal power series n=0 n A(z) B rank(An)zn . If A and B are graded vector spaces, then A n=0 is a graded vector space where (A @ B) n = . ¢. A i @ Bj . The Hilbert series l+j=n A 8 B(z) is equal to A(z)B(z) . If A and B are graded algebras, A 8 B is a graded algebra by the rule

(a I 8 bl)(a 2 ~ b 2) = a]a 2 8 blb 2

paper we have no need for the usually the product).

If

~

is a finite subset of

independent homogeneous which is a polynomial the two-sided space and directly

ideal of

A/ from

A .

introduced

elements,

in

z . If A

~(z) A

A

consisting

denotes

is a graded algebra,

~

. In this case,

of

of linearly

span(~)(z) ~=

is also a graded algebra,

generated by

(in this

sign in the definition

<~>

Z z lwl , w~ < ~ > denotes is a graded

both of them inheriting their gradations

33

For a finite set

S ,

S . If each element of k< S >

k< S > S

denotes the free associative k-algebra on

is assigned a positive integral degree, then

becomes a graded algebra in a natural way. An algebra

a quotient of such a

k< S >

finitely 6enerated and if

by a two-sided ideal

I = < ~>

and

~

I

A

is said to be

is finite, we say

A

finitely presented (henceforth abbreviated "f.p."). Furthermore, be chosen to consist only of degree-one generators, we say ~enerated and if in addition each relation a one-two algebra. Let ~ ( k ) ~

wee

which is

A

is if

is de6ree-one

~

and

~i2

=

(over

k

will be taken as fixed, and

. Likewise

degree-one generated algebras one-two algebras

(over

}

~I

= ~-I/~k)

k ) and

k ). Lastly,

{A(z)IA~ ~ 2

A

denote the collection of all f.p. algebras

k . Except in theorem I,

be shortened simply to

can

has degree two, we call l

over

S

J/=

~12

\

(~k)

will

is the collection of all = g~(k)_.12 consist~ of all

{A(z)IA~(~}

and

~I

= {A(z)IA~ ~ I }-

are the corresponding collections of Hilbert

series.

Local rings

Let us briefly mention why it is interesting to study

~12

' In the theory

of local rings, the Yoneda Ext-algebra is defined. This is a graded algebra, but in general not finitely generated

FRo]

. But if the local ring is graded,

the subalgebra generated by the one-dimensional

elements is an object in

~2

And if moreover the cube of the maximal ideal of the ring is zero, then this subalgebra determines the whole Ext-algebra is more than an object in ~ 1 2 the Hopf-algebras

in

ILl, lEo3. Now, this subalgebra

' it is also a Hopf algebra. Let ~ 1 2

(~12 ' and let

~3

denote

denote local algebras with the cube

of the maximal ideal equal to zero. The construction above gives a bijective correspondence between map

(~!2

' ~12

~3

and

2~12 . Now Anick CAn -I]

has defined a

which transforms the Hilbert series in a certain

exponential manner (see theorem 5(a) in this paper) and in theorem 6 in this paper we define a map

(~

~12

Hilbert series. The composite map

which in a sense does not change the (~

~3

was used by Jacobsson [Ja] to

disprove a conjecture by Lemaire.

Properties of

As mentioned, we are concerned in this paper with describing the set ~7~ . It seems unlikely that there is any easy analytic way to characterize the elements of ~

, for we shall see in theorem 5 that J

rather complicated operations.

is closed under some

Of special interest, however, are the rates of

"

34

growth of the sequences analytic properties exponentially,

{rank(An)}n20

of the series

, and these rates are reflected in the

A(z)

. Such sequences generally grow

and the radii of convergence

of their Hilbert

series give us

our crudest measure of their rates of growth. A more subtle measure is in the nature of the singularity at the radius of convergence toward examining this singularity proved that

{rank(An)rn}

infinity, as

z

We first prove that that

/

is taken in JAn-2, thm 4] , where it is

cannot approach

approaches ~

r . A first step

zero and that

A(z)

goes to

r , at least as fast as a first order pole. is countable.

In our next four results~ we show

is closed under certain simple operations,

as well as certain

complicated ones ihvolving infinite products.

Theorem

I

The set ~ j ( k )

, the union taken over all fields

k , is

countable. Proof

We construct

a countable

set which maps surjectively o n t o ~

Firstly, the prime fields, Zp , p prime, and the rationals~are many countable fields. Let of

F

F

is either of the form

F[x]/(p)

where

p

F(x)

, x

is an irreducible

and all these fields are countable.

Now, again, let

F

is given by positive

transcendent polynomial.

over

Since

of

F

F , or of the form F[x~

Repeating this we find that the set

be any countable field. A finite presentation integers

n,dl,...,d n

A map u/~

~(k)

, ]til = d i . Since

over

over

F F . Hence

let

A(z)E

in the presentation

a f.p. algebra over

F ; FE~}.

is defined by taking the Hilbert

of

[J~(k)

k0

rankk(A n) = rankko(B n)

where

series of the algebra

In order to prove that this map is A

is f.p. over a field

A , only a finite number of elements in

involved, there is a subfield and

k0c k0

k

such that

is isomorphic

the Hilbert series for

series for a f.p. algebra over surjective.

F

set is countable: = {finite presentations

derived from the finite presentation. surjective,

over

:and a finite set of homogeneous

F

is countable there are countably many finite presentations

~/~

is countable,

of this standard form

in standard form of the prime fields} is countable.

elements of the graded algebra

the following

countably

be any countable field, then a simple extension

there are only countably many simple extensions

= {finite extensions

(k)

F

A = B 8ko k to a field A

k . Since k

are

where F ~ ~

B

is

. Since

is equal to the Hilbert

and hence the map v ~

~ U . ~ (k)

is

35

Theorem 2

Let

A(z)B(z)(~

is replaced Proof finite

, B(z)(~

by

~I

A = k< S >/< ~ >

sets and

S

and

CO ~ k

and

T

are disjoint.

(this follows

given below on page 6). If product

of

A

and

B

if

S

are degree-one

and

T

product space which

For homogeneous {s @ t E A - B

and y

subsets

A-B

(A'B)(z) Hadamard

Theorem

(or

~12

, TCB

3

Let

belong

to

D

has the presentation

then

of the general

E ~ A~

as to

B

Write

method

is the free

~I

generators~

A =

and

~12

'

then C, D, and E If

~

and

8

" @ An n>O

and

and defined by we also define A

B =

~ Bn n>0

(A'B) n

S°T

and

in which multiplication

by ~I

. Then

A'B ~ I

B

is a

An ~ B n

=

as are graded algebras

is defined

as for

A @ B .

" Furthermore,

(or

~ 1 2 ), then

of convergence

A = k<S>/<~> generators.Assign

of

,

of

if

A(z) = ~ an zn n=0

~ anbn zn n=0 A(z)

(A.B)(z)

is

B = k/< ~ > every element

is an obvious map of graded algebras

is the

" The same result holds

42

). If the radii of convergence

of degree-one

,

A(z) =

A, B ~ (~I

throughout

C:

is given by the formula

C, D, E ( ~ ] 2

A.B

t ~ T}

C = k<SUT>/<~U~Uy>

are in degree two.

spaces

are

C(z) = i(z) + B(z) - I .

). Finally,

@

S, ~, T, B

~ a z n and B(z) = ~ b z n , then n n n=O n=O = ~ a b z n . I.e., the Hilbert series of a Segre product n n n=O product of the Hilbert series.

then the radius

Proof

and

Isl = itl). If both

is a graded algebra

is replaced ~ b nz n n=0

SeA

t~T,

. If

hence

of degree-one and

is denoted by

; s~S,

~ where

y = {st,ts I s ~ S ,

series

(see ELe]

of two graded

Note that if we write

=

Let

E = k<SUT>/~U~>

are in degree two as well, then The ~

"

e.g. by an application

consist

generated

The same results hold if

t(T}_Ck<S~T>

and its Hilbert

E(Z) -I = A(Z) -] + B(Z) -I - 1 note that

~.

, B = k< T >/< ~ >

6 = {st-ts I s ~ S ,

U @>

A(z) + B ( z ) - 1 ( ~ ,

J12

Cj ~ A. @ B. for j>0 J J D = A ~ B , then D(z) = A(z)B(z)

Letting

then

or by

and

k<8~T>/<~U~

graded

. Then

(A(z) -I + B(z) -I - I)-]~

throughout

Write

k<SUT> then

A(z)(~

, and

k<S × T>

and

belongs B(z)

and to

are

if

~I

B(z) = ~1

rA

and

rB ,

rAr B .

, where of

S × T

S, T

are finite

sets

to degree one. There

* k<S>.k

.

86

This is an isomorphism of the form by

a ~ b,

its inverse

where

a

and

~(a ~ b) = (s,t)@(a' ~ b')

The surjections

WA: k<S>---+

@ b

where A

is defined inductively on elements are words in s~S,

and

t~T

~B: £

S

and

T

and

laI=Ib[

and

a=sa ', b=tb ' .

~ B

induces a surjection

,

k<S>.k ----+ A-B , this is an algebra map and its kernel is <~>.k

+ k<S>. . This ideal is generated by the finite set

e.WTL) Ws.B

where

k<S>.k and

WT, W S

are homogeneous k-basis for

is freely one-generated, it followa that

B

are concentrated in degree two,

degree two and hence

A.B~ ~12

~-W T

k, k<S> . Since A'B~ ~ I

and

. Also, if

WS'B

are in

"

The remaining claims of theorem 2 are dispensed with easily. The fact that __~I (or--~12) is closed under Hadamard products is an immediate consequence of the closure of

~I

(or

~ 1 2 ) under Segre products. Lastly, it follows

from EAn-2, see lemma I and proof of thm 2] that whenever

lim{cnl/n}

exists and equals rc1 n-~oo C(z) . Application of this fact to deduce

if

~ cnzn~ ~I n=0 is the radius of convergence of

rC

C(z) =

A(z), B(z) and (A'B)(z) I/n -I B = lim(anbn )I/n = ( l i m a n )(lim bnl/n)~ = r A -I r B-I rA. n-~o n-~o n-~o

Remark

It is essential to assume that

= k<s,t> , Is l = 2 , It l = I , then {tsn 8 snt ; n~0}

A'B

allows us to

A, B E (~I " Indeed, if

A = B =

is not finitely generated, since

is a set of indecomposable elements.

To illustrate the idea that a wide range of series operations are encompassed by Hadamard products, we have the following corollary.

Theorem 4

Proof

Let

Writing

A(z)~ ~ I

A(z) =

(or

/ 1 2 ). Then

~ (n+1)z n = B(z) , where n=0

is a free c o m m u t a t i v e p o l y n o m i a l B ~ 2

•

~12).

~ a zn+1 = ~ (n+1)an zn n n=0 n=0

In view of theorem 3, we need only show that ~ 1 2 ). But

(or

~ a zn , we have n=0 n

d/dz(zA(z)) = d/dz

(or

d/dz(zA(z))& ~ I

ring

~ (n+1)z n n=0

belongs to

~I

B = k[x,y] = k<x,y>/<xy-yx>

on two v a r i a b l e s

o f d e g r e e one a n d

'

37

We now want to discuss a general method of eomputing Hilbert series of f.p. algebras, which will he used in the proof of theorem 5. The method applies also to the problem of deciding whether an algebra has a certain presentation. Let GC~]~. We want to find

G(z)

. There are two ways to attack this problem.

The first is to find a spar~ning linearly independent and hence

set

S~ G

and then prove that

G(z) = S(z)

. That

S

spans

S

is

G

is

usually proved by an easy induction. The proof of independence is however normally harder to get directly. The other way to find

G(z)

defining a graded vector space

S(z)) and a

multiplication on

S

S (with a known series

such that

Then one proves that

S

S

consists of

becomes a graded associative algebra.

has the same presentation as

G . This, again,

consists of two parts, one easy and one hard. The easy one is to find generators for

S

and a set of obvious relations, which, hopefully, constitute

all the relations in relations in

G . The hard part is to prove that there are no more

S . We now are very lucky, because we can combine the two

methods and just

do the easy parts of them and yet be done. In fact the ~coeffieJ_ent-wiseJ easy part of the first method g i v e s ~ i n e q u ~ y S(z) ~ G(z) while the easy part of the second method gives

helpful seen

when

the

second

method

as a generalization

Propositionl

Let

of the

A, B

a map of(graded)vector B 8 A

~

A 8 B

above

spaces. The map

T B @ k T

@

. The following propositions are . ..(in~roa3~sition lJ The eons%ruc~lon~0e--------"

out.

semi-tensor

and

product

for

@: B + @ A +

~

,A@B N@I k 8 B-----+ A 8 B q

is the natural embedding.

stands for multiplication):

(p

1 8ll A

1~ A

(~ J, 8 A

8 B

+

¢ DAB 1 .........:"

1@ (A 8 B) 8 (A 8 B) A 8 B

B+ 8 B+ 8 A+

~ B+ 8 A +

]

A 8 B

Then an associative multiplication on

making

be

A@k

B+ 8 A+ 8 A+

8A

A @ B

may be extended to a map

is the natural twisting isomorphism and

8B

algebras.

¢) by means of the maps

Suppose the following diagrams are commutative

A

~

Hopf

1~q *

and

of

be(graded)algebras

T

where

notion

(also called

kSA

S(z) J G(z)

is carried

¢~

A ~ B

8B

A

8B

@B

may be defined as

1 ~ A ~ A ~ B ~ B

to a(graded)algebra.

B+SA

PA 8 UV ~ A 8 B

pB ~ I B + 8 A +

¢ I 8 pB ,ASB

38

Moreover

A

and

B

may be considered

as subalgebras

of

A ® B

natural way and the resulting left A-module and right B-module A @ B

in the structures

of

are the natural ones.

Proof

The last statement

product.

The first

the product.

statement

Firstly,

follows directly from the definition

means that the composition

rows of the following two commutative

18~818181

diagrams

of the maps in the top

are equal.

~AS~B8181

> ASASBSBSA@B

181818¢81

18~81

~ ASBSASB

1818HB8181 ~ ASASBSASB

ASBSASBSASB

of

the diagrams will commute also after we have removed the

plus signs. Associativity

A@B@ASBSA@B

of the

follows if we can prove the associativity

1818¢~1 HA818181/18~AS~81 .....~ ASASASBSB 18(~81

1818~A8#= ASBSASASBSB

1818~A8181 ],-

5' ASBSASB

18,{{181

'+ A S A S B S B

~ ASB

UASUB t ~ ASASBSB UAS!J~ g ASB

181818PB/1818PB81 .....>'

ASBSASBSB

~ASp

' ASA@BSB

ASASBSBSB

PASPB t ~

ASASBSB .

Going by the lower rows of the diagrams instead, we see that the first A and the last B is u n c h a n g e d (except for the last map). Hence it is sufficient to prove that the composition of the maps of the top rows of the two diagrams b e l o w are equal. From the assumptions commutative

it follows that the diagrams

are

and since the lower rows are equal the proof is finished.

BSASBSA

> ASBSBSA

18]JB81 , ASBSA

1®¢~1

18(~ PA@I , A@A@B , ASB

1818]-1Bt

ASBSASB

'

t ~pA ~ 1 B@ASB@A

~ B@A@A®B

(~ 1

"* B@A@B

t~(~®1 ASB@ASB

UA~PB//¢

ASASBSB

1~PB>

-~ ASB@B

PA81~1 t '

/

A@B

D~ASPB/

AeA@B@B

39

It may be hard in general to check that the conditions of Proposition

I

are satisfied. The next proposition considers a case when this could be done by a machine (if the algebras involved are finitely presented).

Before stating

the theorem we introduce some conventions. If

S

and

given by

T

a

r

~

e

D(x 8 y) = xy

its image under

B

k-linear map

~: k<S> 8 k

is injective and we will identify

---+ k<S U T> k<S> 8 k

as a subspace of k<S U T>. In the same way

identified with a subspace of

k<S U T>. Also

k<S>

and

with

k 8 k<S>

k

is

are identified

with subspaees of k<S> 8 k.

Proposition 2

Suppose

S

and

T

d

%9: T x S

-+

k<S> 8 k

is a map such that im(%9)

c

1 8 k

im(~)

c

k<S> 8 1

@

span(S) 8 k

(1) or

k<S> 8 span(T)

(span(S) means the k-space spanned by @: k (as

@ k<S>

in Proposition

homogeneous let

--+

I).

elements

A = k<S>/<~>

induces

Suppose

making also

elements

B = k/<~>

8 k

~ c k<S> + of

: A 8 B

--+

S

and

. The

and T

to

an

having ~:

extension

associative

B c k +

imbedding

G = k<SUT>/<

is surjective, Moreover

¢(b @ s)

k<S>

a unique

degree k<S>

are one)

algebra sets

of

and

@ k

--~

k<S U T>

map

U ~

@ k

(the and

a k-linear

The map

k<S>

has

S). Then

and

¢(t @ a)

5

a U B U {ts - ~(t,s); s 6 S , t 6 T} > .

is injective if and o~ly if

are contained in (2)

<~> @ k

+

k<S> 8

If this is satisfied, then

@

for all

induces a map

to an algebra (as in Proposition

a 6 a, b 6 8, s 6 S, t 6 T .

B @ A

I), and the map

morphism.

The proof makes use of the following two lemmas.

~

--*

A @ B

making

A 8 B

above to an algebra iso-

40

Lemma

I

Suppose

satisfies map

condition

@: k<S U T>

Moreover,

and

T

are disjoint

(I) of Proposition k<S U T>

(a)

~x)

(b)

~ts)

(C)

~(XyZ) = ~ x ~ ( y ) z )

~

sets and

which satisfies

= x

if

>

k<S> 8 k

the following properties

x 6 k<S> 8 k

= ~(t,s)

is

~: T x S

2. Then there is a unique k-linear

--+

the image of

+

k

S

for

s 6 S , t 6 T for all

k<S> 8 k

in

x,y,z

, the restriction

k<S U T>

of

@

.

to

+

@ k<S>

hence

@

Proof

k<S> 8 k

satisfies the conditions

defines an associative

This is an application

im(~) E <SUT>

--*

algebra

of Bergman's

k<S> @ (k<S> ~ Span(T)), ( = the set of monomials

may be factored uniquely as Assign to

w

structure

Diamond lemma

on

SUT

) as follows.

w = O U l U 2 . . . u q , where

+ length(Ul).~

The induced partial order on

k<S> ~

k

I, and

.

[Be]. In the case

we may define a semigroup partial order on

a degree as the ordinal number length(o)

on

of Proposition

<S U T>

Any monomial o C<S>

and

on

SUT

u i 6 T<S>

.

(~ means the first infinite ordinal

+ length(u2)'~ 2 + ... + l e n g t h ( u q ) . ~ q is easily seen to be compatible

D

with

the semigroup structure and also compatible with the reduction rules defined by

~

, since the degree of

of monomials

ts

is

2.~

while

each of which has a degree of at most

There are no inclusion or overlap ambiguities so by [Be]

~(t,s)

~

among

(a) - (c) follow from

algebra im(~) E

a + ~

where

a

is finite.

{ts; (t,s) g T x S} ,

extends uniquely to a reduction rule on the entire free algebra

k<S U T> . The reduced words are obviously precisely

diagrams

is a linear combination

[Be]

of Proposition structure on

k<S> @ k

. At the same time property

I commute,

proving that

~

(c) shows that the

defines an associative

k<S> 8 k, which also follows from [Be]

k ~ (Span(S)

~ k)

may be h a n d l e d

, so properties

similarly.

. The case

41 + Lemma 2

Suppose

S, T, ~

and

@

are as in Lemmal and suppose

a c k<S>

+ and

~ ~ k

are sets of h o m o g e n e o u s elements

h a v i n g degree one) a n d let as a map

k < T > ~ k<S>

B 8 A

--*

--*

A ~ B

A = k<S>/<~>

and

(the elements of

B = k/

k<S> ~ k . Then

~

if and only if condition

S

and

. Consider

T

@

induces a map (2) of P r o p o s i t i o n 2 is

satisfied.

Proof

The "only if" part is obvious.

only the case w h e r e

~

case is similar). Let

gives that

I

(2) is satisfied. We c o n s i d e r

satisfies the first part of condition (I) (the other p

be the natural p r o j e c t i o n

and c o n s i d e r the set l = ( x can prove that

Suppose

C k<SDT>;

k<S> ~ k < T >

--*

A ~ B

p o ¢(x) = 0} . The l e m m a follows if we

is a t w o - s i d e d ideal, since

a c I

and

S c I

then also

k 8 <~> + <8> ~ k<S> c I, which is a r e f o r m u l a t i o n of the claim.

First observe that

x E I

~

sx E I

and

xt E I

for

s E S

and

t E T ,

also by (2) we h a v e

BS c I , T a c I . Next we c l a i m that it is enough to prove

that

S c I . Because,

T c I

and

by induction on the length of

x C

if this is true, then it is p r o v e d that

x < a > ~ I . Indeed, suppose

x<~> c I , then @(tx) = @ ( t ~ x < ~ > ) ) c <~>k + k < S > < ~ >

~

~t(<~>k

+ k<S><~>)) ~

.

By a similar argument it is p r o v e d that

N o w c o n s i d e r the f i l t r a t i o n of Fr<SUT> The m a p Claim:

@

@(t<~>)k + @(tk<S>)

= (w E < S U T > ;

k<S> c I

k<S U T>

provided

S c I .

induced by

w contains at most

r

elements from

p r e s e r v e s this filtration.

I n FIk<s U T>

Proof of claim: Suppose

is a left

k < T > - module.

x C I n FIk<s U T>

and suppose

t E T . Then

S} .

42

@(x)

6 FIk<sU

since

~

T> 0 (<~> 8 k < T >

consists

~(x)

where Now,

a. 6 i

5

~(tx)

Now

= ~(tai)b

we a r e a b l e

to p r o v e with p o

= ~a. ~ b. 1 1

by

that

s 6 S,

The next

Fr

Suppose

b 6 k

and

short

~

for

and

ha 6 I

step

T<5> So s u p p o s e

T<e>

is to p r o v e ~

N F r-1 c I

a i,a'i 6 < ~ > N F r-1 ¢(ta) By

= Z ¢ ( @ ( t s i ) a i)

(2)i,

p o

¢(ta")

= 0

8

,

.

claim

then

r

= 0

since

: 0 .

above

also

6 FIk<sUT>

and

for any

it is e n o u g h

btS c I

. But,

, hence

p o @(bS)

= 0

by assumption.

,

A Fr c

+ k<S>

where

I

a 6 <~> N F r @ )

+

since,

that

f o r all

T<5> and

+ = 0

, then @ k

+ Frk<S>

, b. 6 k < T > 1

8

•

, a! 6 k < S > i

,

p

and

= Fr<~>

a. 6 < 5 > n F r 1

Z@(tai)b i

to z e r o b y

A F r-1 c I

bS c I

for all

t 6 T

= @(t¢(ba))=

is m a p p e d

last

+ Za~b! iI for

FIk<s>

F r k < S U T>).

= Za.b. ii . Then

+

hence,

p o ~(tx)

I . By t h e

@(ts)

k<~>

@(ba) b~ 6 < $ > 1

and

that

b! 6 < ~ > i

= p o @(ta~).p(bl)

and

p o @()

I

~ k

F1 c 5

and hence

<8>S c

, t 6 T

6 F r A (<5> 8 k < T >

and this

where

that

¢(b@(ts))

since

But

and

p(¢(ta~)b~)

¢(ba)

@(tba)

The

=

N Fr c

(we u s e

Hence

~(ta~)b~

is t o p r o v e

T<5> as

+

b 6 <8>

= 0

step

i

= FI<5>

~a~ 8 b! 1 1

, a~ 6 k < S > i

(2) a n d

@(bts)

@(b@(ts))

+

to p r o v e

if

8 )

homogeneous e l e m e n t s .

of

, b. 6 k < T > 1

p o @(ta i) = 0

+ k<S>

A Fr c

Z@(ta~)b~ by assumption,

b[ 6 < 8 > 1

•

I .

si,s I 6 S

. Then

and

ZqS(@(ta~)s~) if

and

r

a 6 <~> N F r

and

ta. 6 1 1

a = Zs.a. + a's'.. + a" 1 I 1 1

a" 6 S p a n ( 5 ) . +

qb(ta")

@(ts i) = Zxjbj

Then

with

t 6 T

E k<S>

, bj

[ k

. , xj

,

43

then

@(@(tsi)a i) = Exj@(bjai)

step

p o ¢(bja i) = 0 . Also

and hence

since we have proved that

ta~ 6 1

by assumption

from which it follows that

S c I .

proof is completed by observing that

T<~> =

and from the previous

@(ta~)s~ 6 1 , since

@(ta~) 6 <~> ~ k + k<S> 8

@(ta~)s~ 6 1

Be

and by assumption

T<~> N F 0 = 0

and

U T<~> N F r . r=1

Proof of Proposition Lemma

2.

I proves the first part of the proposition.

it follows that

x = @(x)

in

Consider t h e k-linear map projection

k<S> ~ k

G

for all

p o @: k < S U T > --*

@(x) = x

. Hence in

of each other, b

and

s

~

ker(p o @)

but

~ ( p o ~(hs)) = bs = 0

G

p

is the

(2) is satisfied.

-+

~

A ~ B

(resp.

and

In

ideal

and since

are mutually

a

G

is surjective.

(2) is false.

(or there is

in

is a two-sided

induces a k-linear map

p o @(bs) # 0

~

A ~ B , where

condition

Suppose on the other hand that

inverses

Then there is t

such that

~ ( p o ~(ta)) = ta = 0 )

is not in~ective.

The last assertion

Remark

--*

it is obvious that this map and

such that

p o ~(ta) ~ 0) so

G

p o ~

ts =@(ts)

and hence

A ~ B . Suppose

the proof of Lemma 2 it is proved that in k < S U T >

x

Since

follows from Lemma

I

and

2 .

The proof is valid under the weakened assumption

consists of homogeneous condition

elements

(I) is satisfied.

if the first

(resp.

that

second)

~

(resp0

row of

B )

44

The following corollary to the proposition will be useful in the proof of theorem 6.

Corollary

Suppose

L, R

trivial multiplication ~: R + @ L +

~

A

of the zero map of a map and on R

L+

Moreover,

@ A

if

L @ A ~

and

L ~ A

L @ A @ R

L @ A @ R

@ ~ I ~A ~ A @ A-----~ A c

A

L

and

R

have

is an algebra by means

is an algebra by means

defined as zero on

R + @ (k @ A +)

as the composition

GL, G R and G A

respectively and

are algebras such that

of graded vector spaces. Then

R + ~ (L @ A) +

~

A

(L+) 2 = (R+) 2 = 0). Suppose also given a map

A + @ L +---+

R + @ (L + ~ A) +

and

(i.e.

~

L @ A ~ R .

are minimal generating sets for

has the presentation

k/<~ > , then

L, R and

A

L @ A ~ R

has

the presentation k < G L U G A Y GR>/<~ U {al, ii', rr', ra, rl - @(rSl); i,i'~ GL,aeGA,r,r'eGR}~

Proof

A trivial application of Proposition 2 shows that

L 8 A

is an

algebra with presentation k / < ~ U Moreover

R

(ii', al ; I,I' C G L , a C GA}>.

has the presentation

k/< rr'

may be restricted and lifted to a map

~ :

; r,r' E G R >. The map G R x G[j - +

k

which

defines a map :

G R x (G L U G A)

--+

k

by sending

GR x G A

t~ the map

~ . The claim of the corollary follows from the fact that the

extension

~

of

~

to zero and using

satisfies condition

~

on

G R x G L . We apply Proposition 2

(2) of Proposition 2. This again

follows by an explicit computation in a few eases.

All the operations on

~

we have discussed so far have the property that

if we start with rational Hilbert series,we end up with a rational series. In theorem 5 we discuss some operations for which this is not the case. In each of the constructions of theorem 5 , we obtain a Hilbert series which equals a rational function times a transcendental power of the original series.

infinite product, possibly times a

45

Theorem 5 and

Let

G~ ~

and write

~ = {Ul,...,UM } . Let

There exist

A, B, C ( ~

G = k/<~ > , where

gn = rank(Gn) and if

, so that

G( ~

T = {tl,...,t N}

G(z) = n=~0g n z n 6 ~ "

there exist

D, E E ~ I

such that

oo

(a)

A(z):~

if char(k) = 2

~(z)n%11(1 - zn)-gn }~[f(z)K(1+z2n-1)g2n-I/(1-z2n) g2n n=1

where

f(z) = (I - T(z) - z-IT(z)2)-1(1

if char(k) # 2

- z - T(z)) -I

is rational-

oa

(b )

B ( ~ ) = ( I -- Z 2 )- I ( i I z )- I G ( z )2 ~

( ] + zns(z) )

n= I (e)

C(z)

= (t

-

z2)-1(1

- z)-lG(z)2"~(1 n=l

- znG(z)) -I

(d)

D(z) = (O'G)(z2)G(z)fi(1 n=O

+ gn zn+1)

(e)

E(z) = ( G ' G ) ( z 2 ) G ( z ) ~ ( I n=O

- gnzn+]) -I

\ ay Proof N'See [An-l, prop. 8.4] above. In this case and

~

A

. It is also possible to use the general method

is the enveloping algebra of a graded Lie algebra

may be seen as an abelian extension of basic Lie algebras.

(b). This is a generalization proof" 2 . Let

of Shearer's example

G' = k/< ~'>

be a copy of

T' = {t~ .... ,t~}

is a set of generators

i.e.,

for

It~l =]ti[

set of relations as

disjoint from but identical to

i = I ..... N . Likewise, ~ , but among the

[Sh, see "note added in

G . This means that

{t~}

ix' = {u~ ..... u~} instead of the

{t.} . The

1

algebras

G'

and

G

The desired algebra

T ,

is the same 1

are obviously isomorphic. B

has a presentation

as

B = k/<~U~'UB> where

lal = Ibl = 1

and

Icl = 2

and

8 = {ae-ca, bc-aba, b 2} U

U {ati-tia , at~-t~a, eti-tic , ct[-t~c . t.t.-t.t., . . . .bt.-t.b . -

We c o m p u t e t h e k-basis A

for

spanning

a where

series

G set

with for

B(z)

1 G W and let B

is

1

j

l, J

l-

by means o f t h e g e n e r a l W'

be t h e

f o u n d by i n d u c t i o n

1

1

, I < i' < N}.

,U~7

methodf~-Le~

corresponding t o be a l l

W be a

k-basis

for

words of the form

qo p qr c w'w. baqlw, baq2w . . . . ba w. 10 11 12 ir

q1>q2>...>qr~>O

and

P'qO

are arbitrary, w'~ W' , w.lo,...,WirE W .

It is easy to see that this set has the series we look for. We now apply Proposition

2 to define an algebra

is a quotient of

B . Put

L ~ R

with the right series and which

G'

46

R = k<{w(i);

w~W,

lw(i>I = lwl

where

+

i

i~1}>/<{w](i)w2(J) ; i!j , W l , W 2 e W } > ,

L = k/

and put ~ G ~ G'

laI = I , IcI = 2 . The algebra structure

where

tensor product of algebras denote the element

on

L

(defined in "Definitions").

ZX.w.(i) J 0

in

R

where

is defined as the g EG

For

g = Z~.w. OJ

with

let

X. E k J

g(i)

, w:~ W . J

A map {w(i); w 6 W ,

--*

i_>]} × ( { a , c } U T U T ' )

k<(a,c} U T U T ' >

~ k<{w(i);

w6W,

iz1}>

is defined by

This map satisfies

w(i+1) a ® w(i+1)

(w(i), c)

,

(w(i), t)

~ (wt)(i)

(w(i), t')

~

L

since

(t'6T' corresponds

t @ w(i)

(I) of Proposition

condition

control of 12 equations, and

,

condition

extended map satisfies

of relations

(w(i), a)

2 (in fact both cases).

(2). The verification R

has 4 groups

The

of this involves a

has one group of generators of generators

to t~T).

and one group

and 8 groups of relations.

We give 3 of them below. If

i~j, w(i)w(j)

If

uE~,

tET

@ c

w(i) @ u

and

is mapped to

is mapped to

t' E T '

I ~ (wu)(i)

which is zero in

We may now apply Proposition

and

T'

to

which is zero in

and

c

G'cL).

in

L, b

L @ R.

L ~ R . If is mapped to

L ~ E .

2 and get an algebra

now easy to verify that this algebra a

which is zero in

correspond to each other, w(i) ~ (ct' - t'c)

(at - ta) ~ w(i+1)

mapped to

I @ w(i+1)w(j+1)

structure on

is a quotient

is mapped to

Since also the series

of

B ( a

I(I)4 R, T L ~ R(z)

L ~ R . It is and

c

are

is mapped to

is the right one,

GcL (b)

is proved.

(c). The algebra is replaced by

C b2a

is nearly identical

to

B , except that the relation

. We obtain the same generating

set for

C

as for

B

b2

47

except that now words are allowed with

q1~q2 ~ "'" ~qr " The algebra

R

is changed to R = k<{w(i); w ~ W ,

i~1}>/<{w1(i)w2(J) ; i<j , Wl,W2~ W}>

but otherwise the definition of

L @ R

obtain the stated product formula for

is unchanged. By similar reasoning we C(z) .

(d). This is a somewhat different generalization of the same example of Shearer. Let

G °p

be the algebra which is equal to

G

as a graded vector space, but

with reversed multiplication. A presentation of an obvious definition of the set G'G°P~ ~ I G°PE ~ I

m °p

since, by assumption,

(e.g. G ~ ~I

B

k/< o p >

with

(t]t2t~)°P~ 3 = t3t2tl )" By theorem 3 , and then also

~

k/< B >

is a finite set and the elements of t

Let

is

. From the proof of theorem 3 we have (W is a k-basis for G with I~W)

G.G °p ~ k.k/<~.WVW.~°P> where

G °p

(T x T) (2)

(rep.

T x T

are of degree one.

T x T

(resp. ~

%

~2)

) denote the set

elements of degree two. Let also

b

) with the

be a variable of degree one. Put

2) D=~(/<~UB(2)V{(tl,t21t-t(tl,t2~

) b 2, b(tl,t2~-)tlbt2 ; t , t l , t ~ T } >

op Since G'G Op m k/< B > , there is a m a p ( ) ! G'G ~re~.~ algebras which doubles t h e ~ A spannlng set for D

D

((~)is a map of

as a k-space consists

of all elements of the form

(wj1

•w. ~?~w. bw. bwi2.., J2 lO 11

bwi

r

where w.j1,w.j2, Wio ..... Wir ~ W , lw1! = !w21 , lwill > ... > lwir -> 0 and r>_O. This is easily seen by induction. Also, this set of elements has a series which is less than or equal to the indicated series. For the second step of the "general method" define

R = k< ~/<{ ~I ~2; Iwll<-lw21}> where

~=

{ ~; w E W

I~l

and

=

lwl + I}

and let

L = (G'G°P) (2) O G where

(G.G°P) (2)

is

G,G °p

but the degree is doubled.

is defined by

( { , (tl,t2)(2~ (T , t])

--*

t I ~ t2~

-~

I m wt I

•

48

(If in

G , g = Eliw i , I i 6 k

Proposition

w i6W

, then

g means ~'

2 applies and an algebra structure

on

ZI.~.

L 8 R

).

is induced,

since

the extended map satisfies the following.

If

lwll ~

lw21

W l W~2.[ t l , t 2 ~~2) - +

then and

~i~2 t

--*

"~ I )(t72) 1 8 (wlt I ¢ w1(w2t)

From the proof of Theorem 3 we get the presentation

k<(T x T)(2)>/<{(a.x)

(2)

a~C,

We have for

and

, ( x - a ° P ) (2)

xCk

~(a.x) (2)

-'+

o (2) w(x.a P) ,"

Finally,

if

mapped to

Hence

a 8 x~ --*

~a

and

--~

tl,t2,t 3 C T ,

; a£~

IaI

in

L~R

= 0

in

LSR

of (G'G°P) (2)

, xCk

as

, lal = I x l } >

.

L 8 R

x ~ aw

= 0 in

L @ R

I ~ wa

= 0 in

L ~ R .

~(tl,t2)(2)t 3

and

w~t3(tl,t2 )(2)

are both

t t 8 t2"~rt 3 .

L @ R

is an algebra and the series

and

b

? 6 R ) and this ends the second step and proves

A k-generating

E

is like

set for

E

--+

)--~ ( t 1 •t

is a quotient

The algebra

D ( t

is the right one

L 8 R --~

of

L 8 R(z)

and

(e).

.

= lxI,

= 0 in ""

= 0

D , with

consists

t 6 L , (t 1,t 2

b2

replaced by

EL

(d).

{b2t; t ~ T }

.

of words of the same form as for

D ,

except that we allow _ wi2 T>-..> _ _ lWir I _> Iwil l>I The product

formula for this set follows.

Also

0 .

L ~ R

is unchanged except

that

R, = k< ~ > / < { The product Remark

formula for

Let

r

is less than E(z)

~1 w2; Iwll < Iw21 >

L 8 R(z)

follows and

be the radius of convergence r , since

are (usually)

f(z)

has a pole for

smaller than

r . However

hence the growth of the coefficients .

(1+gnZ 1

n+1~

L 8 R

(r.c.) of

is a quotient

of

G(z). The r.c. of

z
is r.c. of

B(z)

in the series for H (1+znG(z))

; reflects on the growth of the coefficients

E . A(z)

C(z)

and

and

D(z),

and

of B(z) and O(z) respectively.

49

Four of the five constructions or generators

in theorem 5 explicitly

involve a generator

of degree two. Thus they leave open the question of whether or

not similar constructions

exist in

O p]

whether theorem 3, which only holds in

or in /]

~2

" Likewise we may wonder

, can be generalized to all of ~ W.

That such constructions and generalizations do exist is a consequence of our next theorem. Theorem 6 will show, at least as far as rates of growth are concerned that the properties by a series belonging to

Theorem 6

(a).

P1(z), P2(z)

Let

of any series occuring in

OWI

Op

are approximated

and~ less closely, by a member of

A((~.

Then there exists

with non-negative

B ( ~]

integer coefficients

~O~ 2

o

and polynomials

such that

B(z) = p1(z)A(z) + p2(~) . (b).

For any

non-negative

B ~ ~]

, there exists

integer coefficients

C ~12

and a polynomial

Q(z)

with

such that

B(z) S c(z) S Q(~)B(z) inequality being coefficient-wise

Proof

(a).

This is shown in [An-2, proof of thm 4] . A brief outline of

the proof is as follows. Write let

T

for these power series.

A = k<S>/

consist of the degree-one generators

There is a natural injection

@: k<S>

= ti1"...'tiisi I . The induced map is also one-to-one, algebra

B

A

~: A

G

property that

by A

{tij ; i=I .... ,N , j=1 ..... Isil).

k ~

given by

G , where

@(s i) = G = k/<@(a)>

may be viewed as a subalgebra of

monomials

in

G . The

T , which is minimal with respect to the

Proof is by induction on the "complexity"

to be

co( 8 ) =

generated algebra presentation.

B

8

B . Define the complexity

E (lwI - 2) . Let the complexity w~B be

co(B) = co( 8 )

This is well-defined,

co(B) ~ 0

of

in a free degree-one generated algebra

co(B) = We have

and

embeds in it as a subalgebra.

of a finite set of relations k

S = {s],...,s N}

with the desired Hilbert series is obtained as being a quotient

algebra of

(b).

hence

~

where

and

co(B) = 0

where

of a f.p. degreee-one

B = k/< 8 >

is a minimal

indeed we have ~ (i - 2)rank(Tot ~,i(k, k )) . i=3 if and only if

B E(]L12 . ~ u s

complexity

is a measure of how much an algebra deviates from being a one-two algebra and theorem 6(b) is trivially true if Suppose now in

~I

co(B) = N > 0

whose complexity

co(B) = 0 .

and that theorem 6(b) is true for algebras

is smaller than

N . We will show that there is an

50

algebra

D~(~I

with

integer coefficients

co(D) < N

and a polynomial

~(z) i D(z) 2 coefficient-wise. an algebra

Q1(z)

with non-negative

such that

Since our inductive

C @ (~12

QI (z)B(z)

assumption

applied to

D

gives us

with D(z) ~ C(z) ~ Q(z)D(z)

for a suitable polynomial

Q(z)

, we obtain

B(z) ~ C(z) ~

(Q(z)Q1(z))B(z)

as desired. Let

B = k/< B >

Itil = i

be a minimal

and with co(B) = c o ( B )

and let

B = k/< B - {y]>

presentation

and let

~

D

of degree two and a single

Define algebras LI

is

L, R

{uij ; 1~i,j<~)

is defined by

y'

as

is obtained

L = k ¢ LI

and a basis for

from

and RI

is

y'

where

in

,

B , so

of degree

a basis for

{v} (the algebra

structure

L 2I = R 2I = 0). We apply the Corollary to the map

In the same way

G = L ~ B ~ R

is an algebra and

y = ZcTt T

e Ck

where

For T = (T I .... ,T I)

where

,

f(v,uij)

= tit j

and

tT

denotes

let

T'

denote

to get an algebra

O = G/

. Since,

D is the image of

in =

y'

~l

G/
(T 3 ..... T I)

G ,

v.y' =

G = L ~ B ~ R .

tTitT2 .. .tTl when~-- (T1.. .,T I)

y' = ~CTU T T 2 ~ tT, I

y'

y

and ~3

t.t. all have a common ij y by deleting this factor.

R = k @ RI

B

where

relation

terms

~: R + @ L + ---+

Now, put

be of degree

, we shall adjoin new variables to

IYl - I . This is done such that the quadratic left factor and the relation

T = {tl,...,t p)

Y~ B

denote the image of

B/ = B . To obtain the promised algebra , along with relations

with

= N > 0 . Let

y'~ G

as

•

= y

and

~l<~,y,>

under the map

and define

I<Ti<_P •

=

G/<~> = G , we have

oI<~'>

G---+ G . Hence

D(z) ! O(z) = (I + p2z)(1 + z)B(z) .

51

Claim:

co(D) ! N-I

.

From Corollary we get a presentation

k<{u..}

{v} U T > / < (

and hence

co(G)

~-

= co(B)

of

as

{ y } ) U { v 2, vti, u..u lO mn , tiUmn, vuij - titj}> = N -

(IYl

-

2)

co(D) i N -

([Yl-

2) + ( I y ' l -

2) : N -

1

and the claim is proved. It remains to show that inequality

B(z) J D(z)

B

embeds in

. We have ker(B

where

B

D , which will imply the remaining

~

D

= G/)

is identified with the subalgebra

k ~ B ® k

~'E L+ @ B @ R

and this is a right ideal in

vy' = ~

and

in

G

G : G/),

so in

~B

G

= nB of

G . But

G , and G + ~ '

it holds that

= 0

(since

= ~'G

•

Hence

C L+ @ B ® R ~ k @ B ~ k = 0

and this completes the proof.

A few words are in order on the interpretation must unfortunately

of theorem 6(b). The result

be left in terms of an inequality rather than an explicit

formula like 6(a), because the explicit

formula involves the Hilbert series of

a certain right ideal. In special cases it may be possible to obtain an exact formula, and in general it is possible to complete the construction introducing

as many new variables

as we used here. For example,

without

see L6fwall's

reduction of Shearer's algebra to a one-two algebra in the appendix of [Ja] . In spite of the inequality we may still use 6(b) to make some strong claims about the relative rates of growth of B(z) =

Z b zn n n=0

assumption that

and

C(z) = n=0

{b n}

B(z)

and

C(z)

. For example,

c zn , and we allow the weak additional n

is monotonic

or even that

inf{bn+]/b n}

then we deduce at once that there is some fixed constant bn_< Cn j ~ n

for all

n .

if

q

is positive,

such that

52

,T,h,e set of radii of convergence We conclude with a few observations about the possible exponential rates of growth for the coefficients of a Hilbert series. As noted, the radius of convergence

(r.c.) of

A(z) =

growth of the sequence A(z)~ ~ }

Proof Let

{a n } . Let

and likewise for

Theorem 7

The sets

Clearly

~

~12C

where

~I

~IC

B(z)~ ~ I

~

= {r ~ (0,1]; r

is the r.c. of some

~12

and

~12

are equal.

~

so we need only show that

A(z)~ ~

with r.c. equal to

~ith

P3(z) = P1(z) + P2(z)

theorem 6(b) we obtain

gives the least subtle measure of the

and

' ~I

r E 6~ and choose any

we may choose

~ a zn n n=0

~

C_ ~ 1 2

"

r . By theorem 6(a)

A(z) S B(z) = P1(z)A(z) + P2(z) S P3(z)A(z) , has non-negative

C(z)E~2

integer coefficients.

with coefficient-wise

By

inequality

A(z) S B(z) ~ C(z) S Q(z)B(z) S P(z)A(z) where

P(z) = P3(z)Q(z)

has non-negative

inequalities hold coefficient-wise and

A(z)

converges for

integer coefficients.

and all coefficients

[z I < r , C(z)

are non-negative

must converge for

z £ [0,r) , and

consequently for all

Izl < r . Hence,

of

A(z) ~ C(z) , we also get the reversed inequality. The

C(z). But, since

r.c. of

C(z)

Theorem 8

is therefore exactly

~

rl/dE ~

r ~2]

is a divisible submonoid of

(0,1]

We must show that if

is less than or equal to the r.c.

r , and

a countable dense subset of

Proof

r

Since the

r E ~

under multiplication

and

(0,1]

~ d

' as desired.

under multiplication

and

in the usual topology.

is closed under multiplication and that is a positive integer. That

follows from theorem 3. Now let

~

d > I

= ~I and let

is closed r

53

be the r.c. of to

~ a zn( J. Consider an algebra B which is identical n=O n except that all generators (and hence all relations) have their degrees

A

A(z) :

th d . The n - - graded component

increased uniformly by a factor of be zero unless

d

divides

n , and when it does, B n ~ An/d

B(z) =

That

~

is countable

B

will

. It follows that

Z n = ~ a z dn = A(z d) d|nan/d z n= 0 n

which has radius of convergence

that

of

r

follows

I/d

immediately

from theorem

I. For density,

note

6~

tion

contains every e , where a is the smallest positive root of an equa2 d 1-elz-e2z -...-e,z = 0 with e.>0 integers. To see this choose A =

= k

with

2-P/q~ ~ dense in

~eiz i=I

T(z) =

i , then

for any positive

l-- = 1 / ( 1 - T ( z ) ) . A(z)

rational

p/q

For

T(z) =

2Pz q

we get

and these points are themselves

(0, I]

As to how Hilbert

series behave near their smallest

singularity,

we have the

following~

Theorem 9

Let

the r.c. of

Proof

rE~

C(z)

Since

Shearer's

and

r ~ ~

has r.c. equal to shown that,

• Then there is an algebra

if

= ~I

is an essential

, we may choose

r . In JAn-2, see lemma

A(z) =

algebra,

r

a zn n=0 n

as described

, then

a

CE ~2

singularity

some algebra

such that

of

r

C(z).

A~ ~I

such that

I and proof of t h e o r e m ~

> r -n n --

for each

in [Sh, see "note added"~

is

n . Let . This

it is H

H

A(z)

be has the

property that its r.c. is unity but hence the singularity a degree-one

B~(~I

Z gn zn ~ we have n=O

for each

n

I

generated algebra

By t h e o r e m 3, =

at

lim (I - z)dH(z) = ~ for any d , z÷1is essential. Using theorem 6(a) we construct

•

For

G

with this same property.

and the r.e. of B(z) =

z ~ [O,r)

Z angnZ n=O

B(z)

is also

B(z) > _

B = A.G

r . Writing

and inequalities

it follows that

Let

.

G(z) =

ang n ~ gn r

~ gn (zr-1) n , so n=O

54

substituting

X = zr

lim (r z÷r-

, we obtain

z)dB(z) > lim rd(1 ~÷I-

for any fixed the r.c. of

-I

l)d ~ gn n=0

rdlim (l - I)dG(x) ~+1-

d . Using theorem 6(b) we get an algebra C(z)

also equal to

r

and

C (~12

' with

lim (r - z)dc(z) = ~

for any

z+r

fixed

d . Thus

C(z)

has an essential

singularity at

r .

The reverse question to theorem 9, whether or not for every is an

A(z)~ J

converging

for

Izl < r

r ~ ~

and with a simple pole at

there r ,

remains open. If true, it would follow from theorem 4 that there are series with poles of any desired order at We close by mentioning set

~

one more open question about the set

~

contain any algebraic number which is not the reciprocal

algebraic

integer?1~n

particular,

motivated by the observation form

P(z)/Q(z)

Consequently Q(z)

r .

, where

, is the reciprocal

2/3

that when

Q(z)

the r.c. of

does

A(z)

A(z)

belong to ~ ?

. Does the of an

This question

is rational,

is

it always has the

has constant coefficient unity and

P, Q ~ Z [ ~

.

, which coincides with the smallest root of

of an algebraic

mean an algebra in which the sequence

integer. A r.c. of

{rank(An)}

2/3

grows like

It would be of interest to see how closely the coefficients

would

{(3/2) n} .

could approximate

such a sequence. REFERENCES [An-l]

ANICK, D., A counterexample to a conjecture of Serre, Ann. Math. 1-33. Correction: Ann. Math., 116, 1983, 661.

[An-2]

ANICK, D., The smallest 1982, 35-44.

[Be]

BERGMAN, G.M., The diamond lemma for ring theory, Advances 1978, 178-218.

[Go-l]

GOVOROV, V.E., Graded algebras, Math. Notes of the Acad. Sc. of the USSR, 12, 1972, 552-556.

[Go-2]

GOVOROV, V.E., On the dimension of ~raded al~ebras, Math. Notes of the Acad. Sc. of the USSR, 14, 1973, 678-682.

singularity of a Hilbert

1)The answer is now known to be yes (added in proof).

115, 1982

series, Math. Scand., 51, in Math., 29,

55

[Ja]

JACOBSSON, C., On the double Poincar$ series of the envelopin~ al~ebras of certain ~raded Lie al~ebras, Math. Scand. 51, 1982, 45-58.

[Le]

LEMAIRE, J.-M., Al~bres connexes et homolo~ie des es~aces de lacets, Lecture Notes in Mathematics, 422, 1974, Springer-Verlag, Berlin, Heidelberg, New York.

[L6]

L~FWALL, C., 0n the subal~ebra ~enerated by the one-dimensional elements in the Yoneda Ext-al~ebra, these proceedings.

[Ro]

ROOS, J.-E., Relations between the Poincar@-Betti series of loop spaces and of local rin~s, Lecture Notes in Mathematics, 740, 1979, 285-322, SpringerVerlag, Berlin, Heidelberg, New York.

[Sh]

SHEARER, J.B., A ~raded al~ebra with non-rational Hilbert series, Journ. of Algebra, 62, 1980, 228-231.

David ANICK Department of Mathematics Mass. Institute of Technology Cambridge, Mass. 02139

(USA)

Clas L~FWALL Department of Mathematics University of Stockholm Box 6701 s-113 85 STOCKHOLM

(SWSnSN)

ON E N D O M O R P H I S M RINGS OF C A N O N I C A L MODULES (joint work with Shiro @oto) Yoichi Aoyama Department

of M a t h e m a t i c s

Faculty of Science Ehime U n i v e r s i t y Matsuyama,

The purpose

of this note

790 Japan

is to show the main result

of the paper

[3]. A ring will always mean a commutative n o e t h e r i a n ring with unit. Le$ R be a ring, a an ideal cf R and T an R-module. injective envelope

ER(T) denotes an

of T and ~Ij(T) is the i-th local cohomology module

of T with respect to a. We denote by ^ the J a c o b s o n radical adic completion over a semi-local ring.

Q(R) denotes the total quotient ring of

R and we define dimR0 to be -~. First we recall the d e f i n i t i o n of the canonical module. Definition

1([5, D e f i n i t i o n 5.6]).

with m a x i m a l ideal n. An R-module if C ® R ~

Let R be an n - d i m e n s i o n a l

local ring

C is called the canonical module of R

HomR(H~(R),ER(R/~)).

When R is complete, module which represents

the canonical module the functor HomR(H~(~

(H~(M),ER(R/n)) ~ HomR(M,C)

(functorial)

2]~. For e l e m e n t a r y p r o p e r t i e s reader to[4,

C of R exists and is the ),ER(R/n)) , that is, HornR

f o ~ any R-module M ([5, Satz 5.

of the canonical module, we refer the

§6], [5, 5 und 6 Vortr~ge]

and [2, ~I]. If R is a h o m o m o r -

phio image of a G o r e n s t e i n ring, R has the canonical module C and it is well known that Cp is the canonical module of Rp for every p in SuppR(C) ([5, Korollar 5.25]).

On the other hand,

as was--shown by Ogoma [7,

~6],

there exists a local ring with canonical module and n o n - G o r e n s t e i n formal fibre, hence not a homomorphic general,

image of a G o r e n s t e i n ring. But,

in

the following fact holds.

Theorem 2([2, Corollary

4.3]).

Let R be a local ring with canonical

module C and let E be in SuppR(C).

Then C

is the canonical module

of R .

Let R be a rings M a finitely g e n e r a t e d R-module and t an integer. We say that M is (S t ) if depth M (M). maximal

Throughout

~ min ( t , dim M E ) for every ~ in Supp R

this note A denotes

a d-dimensional

ideal m and canonical module K, H = EndA(K)

ural map from A to H. We put U A ~ q

local ring with

and let h be the nat-

where ~ runs through all the primary

57

components

of the

zero ideal

in A such that

dim A / ~ = d. We have a n n A ( K )

= u i (cf. [2, (1.8)~. Lemma

3([7,

Lemma

4.1]

and

[3]).

If A is

($2) , t h e n d i m A / E = d for

every p in Ass(A). (Proof)

We p r o c e e d

obvious.

Let d ~ 3 and let

the zero

ideal

s and b = q s + l ~ . . . ~ q t

in S u D P A ( K ). T h e n U A

the c a n o n i c a l

module

= 0 by the

of A ~.p Since

b. S u p p o s e

that

tradiction

f r o m the exact

Proposition

on d. If d ~ 2, then the a s s e r t i o n

(0) = ql A - - - ~ q _ t

in A such that d i m A/qi

We put a = ~ l ~ . . . a q deal

by i n d u c t i o n

be a p r i m a r y

= d if and only

UA

hypothesis

= (UA) p by [2,

s < t. T h e n a + b is a ~ m - p r i m a r y

4([1],

sequence

[7, P r o p o s i t i o n

4.2] and

prime

because

(1.9)],

[31).

i-

K

is

we h a v e _P

ideal and we have

0 + A ~ A/a@A/~

of

if i s s (i s s ~ t).

. Let p be a n o n - m a x i m a l induction

is

decomposition

a con-

÷ A / a + b ÷ 0. The f o l l o w i n g

(q.e.d.) are

equivalent: a) The map h is an i s o m o r p h i s m .

b) ~ is (S2). c) A is (S2). Proof) We show

(a)~(b)

follows

(c)~(a)

and the a s s e r t i o n hypothesis

is k n o w n

and T h e o r e m

m" By L e m m a

from

by i n d u c t i o n

3, we h a v e

Assume

(S2)~.locus

6([3]).

0 ~ A + H ÷ Coker(h)

prime

÷ 0.

ideal

(q.e.d.)

that d i m A / R = d for e v e r y m i n i m a l

( p ~ S p e c ( A ) I A E is

Let R be an A - a l g e b r a

T h e n the f o l l o w i n g

Let d > 2. By the i n d u c t i o n

Ker(h) = a n n A ( K ) =--UA = 0. H e n c e we have C o k e r ( h )

Corollary

Theorem

b)----~(c) is well known.

If d ~ 2, then A is C o h e n - M a c a u l a y

([5~ 6 V o r t r a g ] ) .

sequence

p. Then the

(i. I0)] and

2, we have C o k e r ( h p ) = 0 for e v e r y

= 0 from the exact 5([3]).

[2, on d.

($2)}

with

prime

ideal

is open in Spec(A).

structure

homomorphism

f.

are e q u i v a l e n t :

(a) R ~ H as A - a l g e b r a s . (b) R s a t i s f i e s ( i ) R As

the f o l l o w i n g

conditions

(S 2) and a f i n i t e l y

(ii) For e w ~ r y m a x i m a l

generated

A-module,

ideal n of R, d i m R n = d,

(i~) d i m A C o k e r ( f ) S d - 2 and d i m A K e r ( f ) ~ d---l. (Proof)

(a)~(b):([2,

K is the c a n o n i c a l

Theorem

module

3.2])

Since

of A/U A by

[2,

H= EndA(K) =EndA/UA(K)

(1.8)],

we m a y a s s u m e

and

that

A

is u n m i x e d . (It is o b v i o u s that d i m A U A < d.) Let Ass(A) = { ~ l , . . . , ~ t ) and t S = A\i~_~l~i . Since K is t o r s i o n free ([2, (1.7)]), H is a l s o t o r s i o n free and the n a t u r a l m a p EA(A/Pi)

by

= Q(A).

Hence

[2, L e m m a

f r o m H to S-IH is i n j e c t i v e .

3.1],

we have

H is c o m m u t a t i v e .

Since

S-IH~ HomA(S'IK,s'IK)~

Since

K is a f i n i t e l y

S - I K ~ .$, l=i I i~iAp. ~ S- A

generated--~S 2) A-

58

module,

the c o n d i t ! o n

(i) is satisfied.

integral e x t e n s i o n of A contained proven by virtue of [6,

(34.6)].

sume that A is complete,

Since A !s u n i m i x e d and H is an

in Q(A), the condition

To show dimACoker(h) ~ d - 2, we may as-

Let p be a prime ideal of heght

is the canonical module of A

(ii) can be

([5, Satz 5.22]) and A

one. Then K

is Cohen-Macaulay.

P R Hence we have Coker(hp) = 0, ~hat is, dimACoker(h) ~ d - 2. (b)~--~(a):([3, TheoremS]

By the conditions, we can prove Ker(f) = U A. We

may assume that U A = 0 and f is injective because K is the c a n o n i c a l m o d ule of A/U A and H = EndA/UA(K). nonical module of R Since d i m A R / A S d -

We put L = H o m A ( R , K )_ Then L n is the ca-

for every maximal

ideal n of R by [5,--Satz 5.12]. n 2~ H o m A ( R / A , K ) = 0 and Ext~(R/A,K) = 0 by [2, (I.I0)].

Hence we have an i s o m o r p h i s m L = H o m A ( R , K ) ~ + H O m A ( A , K ) ~ K from the exact sequence 0 + A ÷ R ÷ R/A ÷ 0. From this isomorphism, gebra i s o m o r p h i s m from H to EndA(L).

we obtain an A-al-

It is obvious that E n d A ( L ) = EndR(

L). Since R is ($2) , R ~ EndR(L) by P r o p o s i t i o n 4. Hence we have R ~ H as A-algebras.

(q.e.d.)

For a r e l a t i o n between H and idea~ transforms, layness of H, we refer the reader to the paper Finally we note the following facts.

and the C o h e n - M a c a u -

[3].

They cad be proven by using

our T h e o r e m 2. Theorem 7-

Assume that Hi(A)

homomorphic

image of a G o F e n s t e i n ring.

is of finite length for i # d. Then A is a

m

Corollary 8.

If a B u c h s n a u m local ring has the canonical module,

it is a homomorphic P r o p o s i t i o n 9.

then

image of a G o r e n s t e i n ring.

If d = 2 and dim A / p = 2 for every minimal prime

then A is a homomorphic Acknowledgement.

ideal p,

image of a G o r e n s t e i n ring. The author was p a r t i a l l y

supported by G r a n t - i n -

Aid for C o - o p e r a t i v e Research.

References [I] Y. Aoyama, On the depth and the p r o j e c t i v e d i m e n s i o n of the canonical module, Japan. J. Math., 6(1980), 6 1 ~ 66. [2] Y. Aoy~ma, Some basic results on canonical modules, Univ., 23(1983), 8 5 - 94. [3] Y. A o y ~ m a and S. Goto, Preprint. [4] A. Grothendieck, Verlag, 1967.

J. Math.

Kyoto

On e n d o m o r p h i s m rings of canonical modules,

Local cohomology,

Lect. Notes in Math.

41, Springer

[5] J. Herzog, E. Kunz et al., Der k a n o n i s c h e Modul eines C o h e n - M a c a u l a y -Rings, Lect. Notes in Math. 238, Springer Verlag, 1971. [6] M. Nagata,

Local rings,

Interscience,

1962.

[7] T. Ogoma, Existence of d u a l i z i n g complexes, 24(1984), 27 - 48.

J. Math.

Kyoto Univ.,

GOLOD H O ~ M O R P H I S M S

by Luchezar L. Avramov (*)

There is a growing understanding homotopy

category

that theorems and constructions

reflect and are reflected by the homological

perties of local noetherian rings. Accordingly, have been perceived as the topological a closer inspection reveals

from the rational

and homotopical

pro-

maps which fibre as wedges of spheres

ghosts of local Golod homomorphisms.

a rather unsatisfactory

materialization

However,

on the algebraic

side of the looking-glass. Indeed,

in topology such a fibre

racterized by either its homotopy, of its loop space. According

is up to rational

to anybody's dictionary,

perties displayed by the homotopy, generally,

F

or its cohomology,

homology,

DG algebra with divided powers),

one should look for similar pro-

imposing such conditions

f: R ~ S

(2.3),

criteria

of

F . In this paper we

involving only the fibre. In

(3.4) and (4.6), one has:

be a local homomorphism,

and let

X

be a DG algebra with di-

vided powers~ which also is a free resolution of the residue field that the homology

H(F) = TorR(S,k)

on the fibre,

assumption on the map it-

self, and often this is harder to verify than the properties make the extra work unnecessary, by establishing

Theorem. Let

(or, more

which arises from a local homomorphism.

to make some additional

fact, as a particular case of theorems

ring structure

or Yoneda algebra of some ring,

The trouble comes from the fact that besides algebraists have found it necessary

equivalence uniquely cha-

or the Pontrjagin

of the fibre

F = S ~R X

k

o__ff R . Suppose

has length S # 2.

Then

the following are equivalent: (I)

TorR(S,k)

(2)

the homotopy Lie algebra

(3)

the Poincar~ P

s

(t)

has trivial Massey products of all orders

series of

~*(F) R

and PR (t)

> 2 ;

is free; S

are connected by:

=

+t - I lengthsTor~(S,k)ti+iDl i>0

As an immediate corollary one obtains: For a local ring

dimension

< I

R , the universal

if and only if

R

enveloping algebra

U

of

~2(R)

has global

is a Golod ring.

(~) During the preparation of this paper the author was a G.A. Miller Visiting Scholar at the University of Illinois (Urbana), partially supported by the National Science Foundation of the United States; and a Visiting Professor at the University of Toronto, supported by the National Science and Engineering Research Council of Canada.

60 This represents

a first step in answering ~ question of Roos

III]: he proved that if the completion through a sequence of

n

of

R

[Ro, §10, P r o b l e m

can be reached from a regular ring

surjective Golod maps,

then

gl dim U < n

[ibid, T h e o r e m

5], and has asked whether the converse holds. The first two sections needed for Golod's

contain mostly

construction;

definitions

and the yoga of Massey products,

for the reader's and the author's

messy part has not been skipped.

convenience,

Section 3 puts the Golod conditions

the

in the perspective

of the h o m o t o p y Lie Algebra theory of [Av4]. The fourth section deals with Golod h o m o morphisms remarks.

proper,

and this is followed by a last section containing

The reader should be warned

different

from that found in previous

included as Remarks

(2.5),

some miscellaneous

that our use of the "Golod lexicon" publications.

At the referee's

(4.7). and (5.4) detailed

comparisons

is somewhat

request,

I have

of the different

notions. I should like to thank Steve Halperin, interest

in this paper.

by the Universities

I.

SO~

of Illinois

(Urbana)

We shall need to manipulate

are either non-negative

case the standard

rules

n +~

"series"

integers,

that only a finite number of negative

with

and of Toronto,

Coefficientwise ~ < ~

comparison,

and

(1.1.1)

If

V~ = 0 = V'f

for

= ~ dim~Vitl

n < ~ V'

or the symbol

i

(1.1.2)

denoted

V"

acknowledged.

V'

More generally,

for the one involving (1.2)

n > 0

of "series"

and

In this

m > I, along

in the usual fashion. with the understanding

small

then the Hilbert = Hilbv,(t)

is a subfactor Hilbert

"series"

of

"series"

~ , such that

Hilbv,(t)

=

+ Hilbv,,(t), Hilbv,~v,,(t)

V", then

=

Hilbv,(t) ~ Hilbv,,(t) .

can be defined by means of length func-

ring. The formulas

above

still hold, except

tensor products.

Differential

non-negatively

for

coefficients.

n C IN.

and Hilbv,@v,,(t)

. If

restricted by the condition

~ , can also be attempted,

for graded modules over an arbitrary

algebra

provided

t , whose coeffi-

are graded vector spaces over some field

sufficiently

is defined,

=,

degrees occur with n o n - z e r o

= = + ~ = m .~ = ~ -~ = ~

for any

and

= Hilbv,(t)'Hilbv,,(t)

tions,

are gratefully

in an indeter~inate

0 . = = 0 , allow addition and m u l t i p l i c a t i o n

that

working conditions,

DEFINITIONS

(1.1) cients

Gerson Levin, and Jan-Erik Roos for their

The financial help and excellent

graded,

graded

(= DG)

skew-commutative,

algebras will be, unless with differentials

specified,

of degree

considered

-I. A graded

is said to be piecewise noetherian, if F is a noetherian ring, and for o F. is a finitely-generated F -module. A DG algebra F is augmented, if a l o surjective h o m o m o r p h i s m e to a field % is fixed, such that e d I = 0 ; we set

each

F

i

IF = Ker e . D G modules the sense ef Eilenberg

are non-negatively and Moore

graded,

and torsion products

[Mo]. In particular,

if

X

is a

are taken in

F-module,

such that

61

X#

(= X

of

F-modules

with trivial differential)

is

(i.e. the induced map

F#-free, and

H(X) -~ H(M)

X ~ M

is a quasi-isomorphism

is an isomorphism), then for any

N, H(X @F N) ~ TorF(M,N) canonically. Observe that, if (1.2.~) and only if

H(F)

is piecewise noetherian, then:

The Hilbert "series" H (F)

lilenghtHo(F)Hi(F)t i

has integer coefficients if

is an artinian ring; and

O

(I .2.2)

If

g: F -+ % •

is an augmentation, F

~

the Poincar~ series

"

PF(t) = ~ dlm~T°ri( ,~)t I has integer coefficients. (I .3) let

If

~ C X°

ex: X -~ ~

is a quasi-isomorphism of

be an element such that

augmentation zero, then

F-modules, with

gX(R) = I C ~. If

z E IZ(F)

z ]i 6 IZ(X) , hence there exists an

X#

F#-free,

is a cycle with

y C IX

such that

dy = z.

The assignment [z] ~

[y ~ I] c x ~F~

gives rise to a degree one map of

Ho(F)-modules

oF: IH(F) -~ TorF(k,k) , called the suspension. It is well-defined, and natural: for details cf. e.g. [GM, (3.6)]. (1.4.1) If there exist

h I .... ,hn

aij• 6 IF

are classes in

(I < i < j < n)

with

IH(F) , their Massey product is defined if J -dai, i = 0 , [ai, i] = hi' daij =v=1 I a.iv a vj•

_

(where

n

a = (-I) deg(a)+1

and (i,j) # (1,n)) . Then 5-a ' v = I Ivavn and all classes defined by such cycles form the Massey product (1.4.2)

Let

B = {h }

is a cycle in

IF

' c IH(F) .

be a subset (finite of infinite) of

IH(F) . It is said

e

to admit a trivial Massey operation disjoint union

=o

T/i=IB

i

to

(t.M.o.), if there exists a function

Y

from the

IF , such that

y(h e) = z e 6 IZ(F) , with n

[z e] = he;

and

dy(hel .... 'han ) = v=15-Y(h~1, .... hav) " Y(hev+1 '''''h~n ) ' (1.4.3) in

Ordinary Massey products have been generalized to operations on matrices

[Ma], to which we refer for definitions. We shall only need to know the set of all

matric Massey products in a

H (F)-submodule in

IH(F) , denoted by

MH(F) , and the

O

following result: (1.4.4)

[GM, (5.12)]. For a piecewise noetherian

MH(F) = Ker o (I .5)

F , there is equality:

F

The DG algebras

F

and

F'

are said to be (homology) equivalent, if there

exists a chain of quasi-isomorphisms F ~ G (I) ~ F (I) ~ G (2) ~

...

~ F (n) ~ G (n)

-+

F'

°

In the augmented case, one furthermore requires all these algebras to map to the

62

same field

2.

£ , and the quasl-isomorphisms

GOLOD ALGEBRAS

For an augmented

I +t Z (2.0) algebra,

F , denote by

~ .

GF(t)

its Golod

"series":

[ length= (IF)iti) -I i>0 ~o side denotes the "series" (IF) i t i + It ~ length

length

In this section,

such that

(2.1)

on

= (I - t

the right-hand

where

the identity

VIA HOMOLOGY

algebra

GF(t)

to induce

Lepta.

H (F) o

g: F ~

contains

For any such

%

(iF)iti] 2 + . . . .

denotes

a piecewise

noetherian

augmented

DG

some field.

F ,

the inequality

PF (t) < GH(F)(t) holds. Proof.

~ %) does not change the left-hand side, and can only o decrease the right-hand one, hence we can assume F local with maximal ideal m . o Completing in the m-adic topology leaves both sides unchanged, hence we shall moreover

assume

Localizing

F

at

Ker(s:

complete.

o

(2.1.1)

F

In this context

[Av4].

then it is equivalent

If

F

we have:

is an augmented

to a supplemented

DG

DG algebra

k-algebra

F'

with (i.e.

~': F' ~ ~ is the identity on %) , with F' complete and local. o o Both sides of the inequality being preserved by equivalences, supplemented, construction (2.1.2) Filtering sequence,

and replace of Eilenberg TorF(%,~)

B(F)

length Ho(F) and MacLane

by

TorF(%,%)

one obtains

EI = (BH(F)) = (siN(F) ®p) P,q P,q q

(2.1.4)

E2 = TorH(F)(%,%) P,q P,q

(2.2) upper bound

=

is reached

(For a comparison section.)

inequality

of "series"

• ~ i < ~ ( ~ dlm%Ep,q)t i>0 p+q=i

Definition.

we can assume

denoting

F

the reduced bar-

An algebra

F

in the previous with Levin's

an (Eilenberg-Moore)

spectral

, such that

(2.1.3)

PF(t)

B

= HB(F) .

to

Now the required

With

complete and local o % c F' , and o

[Ca], one has

by "the number of bars",

converging

dim~ .

F

follows

from:

I i = GH(F) (t) . ~ ( ~ dim Ep,q)t i>O p+q=i

as in (2.0)

is called

a Golod algebra,

lemma,

if

= GH(F)(t)

definition

i.e.

PF(t)

of a Golod algebra,

if the

.

cf. the end of this

63 (2.3)

Theorem.

Let

F

satisfy

the conditions

of (2.0). Then the following

are

equivalent: (I)

F

is Golod;

(2)

Ker o F = 0 ;

(3)

MH(F)

= 0 ;

(4)

H(F)

has trivial Masse y products,

hl,...,h n

(h.l 6 IH(F)),

i.e. for every

the Massey product

and every set

n ~ 2

is defined and contains



o n l y zero; (4 ~) there exists a set of generators

(5)

B = {h } a

every Nassey product



every set of elements

of

(5') some set of generators (6)(i)

(Ker g)-IH(F)

(ii) IH(F)

of

IH(F)

is defined

H (F) , such that o

over

for all

n > 0 ;

admits a trivial Massey operation;

IH(F)

over

H (F) o

admits a

t.M.o.;

= 0 ;

has an

k-basis

(iii) taking a free zero map

IH(F)

of --

{h }a6 A

F ° -module ~

V * IH(F)

V

w h i c h admits a t.M.o. Y;

w i t h basis

{va}a6 A

and the surjective

degree

(v~ * h a) , set:

d(1 @ SVal @ ... @ sv n) = and extend this map to

~ (hal,...,h) @ sv @ ... @ sv i=I i ai+1 an X = F @ T(sV) by r e q u i r i n g additivity and

= df @ v - ~ @ dv

v = SVal ~ ... @ sv~

d(f @ v) =

Fo then

d

for

is a differential

(iv) the natural

augmentation

T(sV) ;

i n the tensor algebra n

on X ;

e(f ~ v) = g(f)e(v)

induces an isomorphism

H(E): H(X) ~ ~; (6') there exists a DG (i)

X ~ ~F #

F-module

T(sV)

jective

X

with the following

V

is a free graded

properties:

F -module equipped with a suro ....... v * IH(F) , which induces an isomorphism V @F L ~ IH(F);

where

@Fo Fo-linear ms p

(ii) dX c (IF)X ;

o

(iii) the augmentation (7)

(Ker g) IH(F) = 0 , and

(7') F Note.

is equivalent If

W

W 2 = 0 , and Proof.

(I) ~

so is (2) by F

E(f @ v) = e(f)E(v)

contains

is an

F

is equivalent

to the trivial

£

induces an isomorphism

vector space,

to the trivial

extension

of

£

£ ~ W = % @ W

H(X) ~ % .

extension

by some

£ ~ IH(F) ;

~-vector space.

as vector spaces,

~

is a subring,

dW = 0 . (2). Condition [Ma, Theorem

(I) clearly

since

oF

under homology

equivalences,

1.5]. Hence we can, as in the proof of Lemma

L . Now the equality

(2.1.3). However, F this implies o

is invariant

(I) implies

that

is the map induced

is injective.

E I = E°°

(2.1), assume

in the spectral

in homology by

and

sequence

IF 9 x ~ x 6 BI,,(F) ,

64

(2) ~=~ (3) is Gugenheim (3)

~

(4). ~ s s e y

hence are always

defined,

assume any product this holds

for

Definition

1.2],

contains

and May's

products

of

of

theorem,

quoted

2 elements

and by the assumption

< n

elements



for



in (1.4.4).

are (up to sign) are trivial.

Inductively,

is defined and contains

only zero.

I J j - i J n - 2 . According

is defined,

ordinary

and the equality

to [ ~ , MH(F)

products, one can

In particular,

Lemma

= 0

1.3 and

implies

it

only zero.

(4) ~ (5). Let for an arbitrary

B = {h a}

be a set of classes

argument

to

shows

of the cycle

z

in

IH(F) , and set

y(h e) = z e

h . By induction, one assumes e n - 1IB i . Then by (1.4.1) the class y already defined on Ei= z = l y ( h I ,...,h a1'''''en ~i x y(h e ,...,h a ) belongs to < h ...,h a >, hence by the assumption it is aboundary. i+I n el' n It follows that for each sequence (el,...,e n) one can choose Y(h~1'''''he. n ) 6 IF , which bounds

choice

in its class

z~ I, • .. e ' hence one can extend the t.M.o. Y to H i=I n B l . The same 'n that (4') implies (5'), while (4') and (5') follow trivially from (4)

and (5) respectively. (5') ~ (6). Let defined,

and denote by

is a maximal

ideal

i = 1,...,m

and all

basis

of

choose

B = {h }

IM(F):

that map

be the system of generators

h 1,...,h m

~

of

those of degree zero.

Ho(F) , the fact that

a , shows

that

the restriction

Since

¥(hi)Y(h

¥

still

is a t.M.o.

d2 = 0

on

{v } . e is verified through

is

IH(F) ° = Ker(Ho(F) ~ ~)

) = dY(hi,h e)

n IH (F) = 0 . One can now cut

of

y

B

for down to a

It is now clear how to

V ~ IM(F) , and the basis

The fact that

X

makes use of the definition

(1.4.2)

the inclusion

leads

of

on which a t.M.o,

F

in

X

of a t.M.o.

to an exact

an instant

To show that

computation,

which

H(X) = ~ , note that

sequence

0 ~ F ~ X ~ X @F sV ~ 0 o where

the tensor product

is of DG modules,

with

d(sV)

= 0 . It yields

an exact

se-

quence . . . . i+j=n+1~ Hi(X) Obviously, n . However,

Bn+1 -->

®Fo Vj-I Ho(X)

working

= ~ , and

Hn(F)

~ Hn(X) ~i+j=n @ Hi(X)

3n+i(I

@ v e) = h a , hence

®Fo Vj-I . . . . . ~n

is surjective

for all

back from the sequence

0 ~ HI(X) ~ Ho(X)

@ V°

61> HI(F) ~ 0

I @ H I (F) one easily

sees that this implies

(6) = (6') is trivial. (cf.

(1.2)),

whence

H.(X) = 0 for i > I o I To deduce (I) from (6'), note that

while the assumptions

on

X

imply:

H(X @F ~) = X @F ~ = T~(sV @F %) = T~(sIH(F)) o the equality of power series.

,

TorF(%,~)

= H(X @F ~)

65

(7) ~ (7') needs no proof, while (7') implies (I) because the equality to be established is invariant under homology equivalences, and one has isomorphisms of vector spaces: Tor £ ~W(£,~) = H(B(~ ~W)) = B ( £ ~ W )

~ T(sW)

.

In order to complete the proof we show: (4) ~ (7). By the preceding, one can assume piecewise finite-dimensional Furthermore,

let

(sW) v

graded Lie algebra on on f

be the graded (sW) y

and let

s-IL(sw)V; G

= @ G n where r r=q_ n q (sL(sw)V) ®n for which

on

Gn q

h-dual of G

E(a)

sW , let

L(sW) v

be the free

denote the algebra of alternating cochains

is the set of degree

f(sv I ~... ® s v n) = s(~)f(sv (I) .... ,sv (n)), with

~ c F , and one knows IH(F) is a 2 (IH(F)) = 0 ; set W = IH(F) .

R-vector space with

v i C L(sW) v,

q-n

linear functionals

Z deg sv i = -q ,

standing for the usual ("Koszul") sign involved in a permutation of homo-

geneous symbols. Note that, by construction, on the vector space

G

is a free skew-commutative

(s~ (sw)V) v = s-1@L(sW)V) v. Furthermore,

G

R-algebra

has a differential

defined by df(SVl .... 'SVn) =i<jZ ±f(s[vi,v j], sv 1,...,svi,.^ ..,s~j,...,sv n) . (For the correct signs and a more detailed description, we refer the reader to the excellent discussion in

[Ta, Chapter I].)

Clearly, the construction of algebra, and its homology

H~(G)

G

can be performed starting from any (graded) Lie

is, by a definition (appropriately stretched for

the occasion), the (bi-) graded cohomology of L = L (sW) v, H~(G) = £,

L . In particular, for the free algebra

H~(G) ~ Der£0L(sW)V,%) ~ (sW) vv = sW , and

Hi(G), = 0

for

i > I . It follows the total homology to tion

H,(G) = @ Hn(G) is isomorphic as an algebra grp=, q £ ~W . Furthermore, this isomorphism is easlly seen to be induced by the projecG ~ £~ W , which extends the surjection

dualizing the canonical inclusion

(sL(sw)V) v ~ (s(sw)V) v= W

On the other hand dualizing the inclusion of surjection

algebra u tion

L(sW) v

into

j: T(sW) = (T(sW)V) v ~ (L(sw)V) v. Thus, a basis

system of generators G.

{u~1"''~n = s-lj(sw~| '''''swan )}

of

By the remarks above, the map which sends (n > 2)

obtained by

(sw)Vc-~L(sW) v. T(sW) v, one gets a

{w }

of

W

defines a

(~(sw)V) v, hence of the

u

C G

to

w

C W,

and

to zero, is a quasi-isomorphism. Also, a straightforward computa-

[Ta, (1.4.2)], shows the differential

d

of

G

can be expressed by the formula:

n =-~u u du~1...~ n i=i ~i...~i ai+1...~n Assuming (4), it is now easy to construct a quasi-isomorphism will establish (7). In fact, choose for each

e

a cycle

z

in

g: G ~ F , which IF

which maps to

66

w~

under

IZ(F) ~ IH(F) ~ ~ W , and set G
now a linearly independent

set among the

mula above shows the Massey product

<-u

g(l~1" for some

of

g(u ) = z . By induction,

defined on the subalgebra

G , generated by the

a.. = u 13 ~i" " "~j

products

form a defining

system for the

i ,... , nan> , hence

""

~i u

~i+I "''~n

C F , so that

Y~l"''~n

[Gu],

g

u i...~ n. For any one among them, the for-

(I < i < j < n) ---

) = <[z

~I

] .... , [z n] > = dY~1

g(u 1...~n) =

(2.4). Remarks. In the l i t e r a t u r e [Go],

one can assume

{u 1.1.~mlm < n} . Choose

"''~n

extends

Y~I"

g

G <(n*1) .

dealing with homology of l o c a l r i n g s , e . g .

[LeVl,2] , [AVl,2] , [Ro], the goal has been to connect

structure of

to

H(F) , for some special choice of

the trivial ~ s s e y

F , to the homology

of the

ring

F . This has led to the interference of delicate questions from linear algebra o (e.g: proving a certain complex is minimal) with the essentially quite simple homolo-

gical problems arising in a context with trivial Massey products. we have taken is that the trivial ~ s s e y intrinsic to the algebra.

their companion versions) resolution

X of

F

is borrowed

The point of view

should be analyzed

in terms

is made for the arguments

the equivalence of (I) with (4),

is due to Golod

in (6) (with X

F

This said, no claim of originality

presented above. In particular,

acyclicity

structure of

[Go] who introduced

(5) and (6) (and

the construction

the Koszul complex of a local ring). The proof of the from Levin

[Lev2]. The introduction

of the Gugenheim-May

theory in the guise of (2) and (3) was done in [AVl], The very useful condition seems to have been overlooked to Steve Halperin

(2.5.1). Remarks. in a non-equivalent (i)

F°

(iii) (iv)

i > 0, --

to denote a DG algebra

local ring with maximal F.

i

(7)

I am grateful

it out to me, with a somewhat different proof.

The term "Golod algebra" has been employed in the notes

context,

is a noetherian

(ii) for every

in algebra, but well known to topologists.

for pointing

of the

is a free

F

[Lev 2]

such that:

ideal

m ;

F -module of finite rank; o

dF c m F ; every set of elements of

IH(F)

admits a t.M.o, with values

in

mF.

H (F) contains a field, (2.3.5) shows our definition (2.2) is much more o general than Levin's. However, it is not for generality's sake that we have adopted

Thus,

if

the new framework.

The point is that the equicharacteristic

assumption will come in

for free for the DG algebras which arise in the main applications (ef. Sections 4 and 5), while in the same context to verify the last part of condition (2.5.2). A Golod algebra to be of the form

~

W

F

of Theorem (2.3)

it will be particularly

difficult

(iv) above.

which is a ring (i.e.

for some finite-dimensional

F = F ) has, by (2.3.7), o k-vector space. This is a very

67

particular approach

3.

instance of the "Golod rings" of local algebra: how they fit in the present

is made explicit

in (5.4).

GOLOD ALGEBRAS VIA HOMOTOPY

An algebra

F

is called a

defined on elements e.g.

x

F-algebra,

if it has a system of divided powers

of even positive degree, and satisfying

[Ca, Expos~ 7]; if moreover

F

is a DG algebra whose differential

dyi(x) = (dx)yi-1(x) , it is called a DG

Next we give a very brief account of the results of

(3.1). For any piecewise noetherian

with the Hopf diagonal, (cf.

augmented DG

P-algebra.

F-algebra structure on

[Av 4] (partly announced

F-algebra

If moreover,

HB(F) , where

I (2)

F ~ £ , TorF(~,%)

% c F , then this coincides

B(F)

for the ideal generated by

is of even positive degree and

i > 2.

Then the

in

in this section.

has the shuffle product,

and divided powers, defined in the work of Eilenberg-~cLane

[Ca]) . Write

satisfies

F-algebra.

[Av3]) , which are needed in order to go on with the exposition

has a natural structure of Hopf

{yi}i> 0 ,

the usual axioms:

12

and all

and Cartan

yi(x) , when

x

~-vector space

(ITorF(k,k)/l (2) TorF(k,k)) v is a graded Lie algebra, If F

F ~ G ~ F' and

F'

are DG

to note that

called the homotopy Lie algebra of

are quasi-~somorphisms

G

F-algebras,

then

need not be a

of DG algebras ~*(F) ~ ~*(F')

H (F) o Recall that for a graded Lie algebra

by all

F-algebra,

[a,b]

where

K

of

(cf.

L

such that

(a, b C L)

in characteristic

is denoted by

[Av3],

operator

~*(F) .

~) , and if both

in a natural way (it is important

denotes a piecewise notherian

contains a field. L , the graded Lie subalgebra, [L,L]

generated

(and called the commutator of

[L,L]

also all

ideal);

m(a) ,

L 2i+I ~ L 4i+2 , which is part of the structure

[Av 4] for details).

(3.3). Lemma. The suspension

graded

c: F ~ %

2, we include among the generators

is the quadratic

to

F-algebra).

(3.2). For the rest of this section, augmented DG

F , and denoted

(augmented

o

F

of (|.3) defines a natural degree zero map of

%-vector spaces: TF: z*(F)/[~*(F),

Moreover,

~*(F)] ~ Hom F (slH(F),~) . o

Im T F = Im(oF) v .

Proof. Compose

the degree

IH(F) ~ ITorF(£,£)

-I

isomorphism

(deg o F = I)

slH(F) ~ IH(F)

with the suspension

and follow this by the projection

F o :

68

ITorF(~,~) ~ i Tor F (~,~)/I (2) Tor F (~,~) . Since

HOmFo(,%)

= Hom~(,%)

yields a degree zero map

for vector spaces, dualization of this composition

~: ~*(F) ~ Hom F (sIH(F),%). It remains to show •

o

m [~*(F),~*(F)], which can also be wrltten as

[~*(F),~*(F)].Im o

F

Ker T m

= 0 . Since commu-

tators and images of the quadratic operator are decomposable in the universal envelope

(TorF(~,~)) v

of

~*(F) , it suffices to show (ITorF(%,~)v)2.Im o F = 0 . This F Im o in the ~-dual of the indecomposables of

is equivalent to the inclusion of the graded algebra of

TorF(~,~) v, which is canonically

TorF(~,~) . We shall show

oF[z]

identified with the primitives

is primitive for any

z C IH(F) . As at the

beginning of the proof of (2.~), one can for this purpose replace mented DG

k-algebra

F', with

,

~

H(F ) = H(F) @F Fo ' and •

Tor

F'

F

by a supple, F

(%,~) = Tor (%,~) ,

O

•

•

both equalities being provided by serles of DG algebra maps. However, w~th the IdenF' F' tification Tor (%,%) = HB(F') , and for z' C IZ,(F') , o [z'] = Ix] , where x is the element shows

x

z'

in

BI,,(F) ~ IF . The definition of the diagonal

~

of

B(F')

is primitive, whence the claim.

For the final statement, it suffices to note that since

~*(F)¢-~ (TorF(%,~)) v

is the inclusion of the Lie algebra into its universal envelope, it induces a canonical isomorphism of graded vector spaces ~*(F)/[~*(F),~*(F)] ~ (ITorF(~,~)v)/(ITorF(~,%)v) 2 . (3.4). Theorem.

For a DG

F-algebra

F , which satisfies the conditions of (3.2),

the following are equialent: (I) (2)

F

is a Golod alg.ebra;

IH(F)

is a vector space (through the augmentation

~: F ° ~ ~) , and there is

an isomorphism of graded Lie algebras ; ~: ~*(F) ~ L(slH(F)) v (= L) where

L

denotes the free Lie algebra functor; furthermore

cal map in the following commutative diasram:

~ * (F) / [ ~ * ( F ) , ~ * ( F ) ]

(sIH(F)) v

L/[L!] in which (3)

TF

(4)

~*(F)

p

is the canonical isomorphism;

is surjeetive; is a free Lie algebra.

~

induces the verti-

69

Proof. F

(I) ~ (2). It is easily seen that

to its completion

equivalence

F

in the

(2.1.1), which links

by using only homomorphisms

~*(F)

F

of DG

to a supplemented F-algebras

remarks at the beginning of this section, r-algebra

F'

over

F-algebra

(with

the homology

F', can be achieved

[Av4]). Hence, by the

can be replaced by a supplemented DG

y1(x) = 0

i > 2) , one can further replace

Furthermore,

%-algebra

(for details cf.

F

Z . Since the trivial extension

a natural way a DG and every

does not change when one passes from

Ker (F ° ~ %)-adic topology.

% ~W

(cf.

for every

F'

by

x

k~W

(2.3), Note))

is in

of even positive degree

. But now all the claims of

(2) are obvious. It is clear that (2) implies both (3) and (4). Noting that (3) is equivalent Ker o F = 0 ,

(3) implies

Now we assume

(I) by Theorem

to

(2.3)•

(4) and shall prove that

H(F)

is a Golod algebra.

Consider first

the homomorphism

f: F ~ F = H (F) of DG F-algebras, and the induced homomorphism o f*: ~*(F) ~ ~*(F) of graded Lie ~Igebras over L . Since ~I(F) is naturally isomorphic

to

(~/~2)v, where

~ = Ker(F ~ Z)

(cf.

[Av4]) ,

fl

is an isomorphism.

Subalgebras of free Lie algebras being free [Lem, Proposition A 1.10], it follows that I (F) generates a free Lie subalgebra in ~*(F) , hence I ( ~ ) generates a free subalgebra

L

of

The ring

~*(F) . ~

being equicharacteristic,

we can after localization

and completion

assume it is the homomorphic an ideal

a

there exist

minimally

image of the formal power series ring %[XI,...,Xn] by 2 generated ~y a 1,...,a r with a.l 6 n , n = (XI,..,X n) . Hence

~ • 6 Z , such that lj

ah -

_h n3 I a•.X.X. 6 i
h = 1,2 ...,r .

We now recall an equality established by SjSdin dimzExt2(Z,Z)_ F where

Ext (i,Z)

(ToJ(£,£)) v

is the universal

dim~(Ext1(£,£)) 2 _ F On the other hand,

+ r - rank a~ij)

is considered with the Yoneda product,

with the product on (TorF(z,Z)) v

= dimz(Ext1(%,%)) 2 _ F

[Sj, p. 8]:

The three equalities

induced by the diagonal

envelope of

of

(up to signs) coincides TorF(z,Z) . Since

~*(F) , the freeness of

L

yields

= n2

it is well k n o ~

dim~(Tor2(Z,~)) = _ F

which

that

(~) + r .

produce the expression

rank (-aij) = But this means _ I . 2 a = n by Nakayama. As remarked by the referee,

, whence

a +n 3 = n

, ~nd finally

a shorter proof of this equality

70

can be obtained Summing

by using

some results

up, we have proved

of LSfwall

~2 = 0

[LS].)

for the maximal

ideal

~

--

for any

i , Hi(F)

the

(local)

H(F)

= H(F)

tive) DG

DG

F-algebra

h-algebra

construction

of

F

over

) , and clearly i>0(iF- @i.v ponding

spectral

product

~ . As a vector Hi~p(IF@i)v

sequence

o

there we can replace

we can consider

(B(F)) v

is a filtration

F

by a

the (non-commuta-

is the dual of the diagonal

space,

length over

as in the proof of (I) ~ (2), given

~ . In particular,

(B(F)) v, whose over

H (F) . Since,

Ho(F)-module , it has finite

ring H (F) . In particular, with F o ^ @FoFo = H(F) , hence by the arguments

supplemented

of

--

is a finitely-generated

in the bar-

is the direct

by DG ideals.

product

In the corres-

one has

EP,* = (IF@P) v ; 0 E~'* = (IH(F)®P) v (Note that for the second over

Ho(F) , which

equality

implies

We have now a first quadrant differential

the already

dim%Hi(F)

< ~

spectral

established

for all

sequence

d : E p'q ~ E p+r'q-r+1 r r

and first

i,

finite

length of

is of crucial

Hi(F)

importance.)

{E**} of bigraded DG algebras, r term defined by the isomorphism

with of

algebras E~ * = T(E I '*) E II,* = (IH(F)) v. For standard

with

sense to

this sequence

converges

in the strong(est)

H(B(F)) v = (TorF(%,£)) v E II,* = E I~ * . Assuming

Next we set to show E I1,q # E1,q

for which = EI~ q

reasons,

for

Denote by

The pattern

r > q +I. Hence C**

cI,s

there

the subspace

of

E 1's r dl,q Ker r 0

=

By the choice of

is a largest

E~'*(c E l ' * )

for

s < q

for

s = q

for

s > q

C , its elements

i.e. they are in the image of the map v:

( I H B ( F ) , ) v -~ E I'*~ ~

Since composed

with the isomorphism

the dual of the suspension to elements

of

free Lie subalgebra the tensor algebra

L' T

of

choose

the smallest

guarantees

r

integer

q

that

E 1'q = r drE1'qr # 0 .

r , for which

, d e f i n e d by t h e c o n d i t i o n s

are permanent

cycles

in the spectral

sequence,

El'* . E II,* ~ (IH(F),)v , v (3.3)

form a basis

By our assumption

d

(= edge homomorphism)

o F , Lemma

z*(F) , which

this Lie algebra.

inequality,

of the action of

and

for the generators

[Lem, Proposition

~*(F) , hence

on the graded vector

gives up to a degree

shows that a basis

of

of degree

A 1.10],

(TorF(%,Z)) v = Uz*(F) space associated

C

to

shift,

can be lifted ~ q +I

of

they generate D UL' , which

C . Moreover,

a is

the

71

inclusion

is an equality

in degrees

dim Hq+2(B(F))

= dim E l 'q+1 + dim T q+2 .

Let now that

denote

the bigraded

dE l ' q c D . T h i s i s r q' < q , and by the

with If

D

r = 1

and

< q +I , while

clear

subalgebra

when

definition

a E E l'q

in degree

of

q +2

it yields

E** , generated by r in this case dE l ' q

r > 2 , since -of D one has

E

= D

C . We claim ~ Ep ' ' q '

in these

dimensions.

write

da = I c~ ~ c? + b i i •

>2

wzth

b E El-- '

linearly

*

1,0

independent.

dc~ = 0 , 1

Since

-~2 = 0 ,

Comparing

filtration

degrees,

c'~ E K e r 1,q = cl,q 1 dl " With T as above,

the

formula

same degrees

of some

among the

one sees that

a

r is

the

map

) # O.

r+l

which c o n t r a d i c t s Having forms

On t h e

of

the

r+1

equality

of

EI

established

= E

other

hand,

it

guarantees

the

exis-

q +2

'

generators

of

+ dim

Tq+2

earlier.

, it follows the

immediately

k-algebra

that

E I = E~

E 1. Now we h a v e ,

since

in view of

the

(H i B ( F ) ) t i = K ( K dim EP'q)t i = (I - K d i m IH.(F)ti+1) -I i p+q=i ~ i 1 (2.2)

of a Goiod algebra.

(3.5).

Remark.

Theorem

Theorem

(1.4) and Corollary

the results

(3.4)

(and most of the arguments)

is the basic assumption

(see below)

dc'~ = 0 , i.e. i

H.(F): I

definition

F-algebra

hence

s,t El,q+1 K dim Er+ I < dim ~ s+t=q+2 s>2

particular:

DG

shows

that

= dim E I'q+I~ +

that

a system

K dim i

context,

B(F)

d (a) # 0 , which produces a relation of degree r < q + I of E*'* hence in degree q +2

This means

the

proved

g-finiteness

is

of

ci

T ~ E*'* , defined by the inclusion r+1 E*'*r = E*'*~ in total degree _< q + I , and

shows that

a E E 1'q with r of degree

dim H q + 2 B ( F )

which

differential

,

the

a:

an isomorphism.

generators

(T ~ E

El'*

the

C'. @dc': = 0, i i

--

Ker

for

choosing

= db C ~ 3 , *

consider

C c Rr+ I . The choice of

tence

v

= (n) , and furthermore

hence

c! @ dc'~ = d(da +b) i i

in the

--

, c~ E E~ 'q , e~ E E,

resolution

of

looks

[Av2,

(and is) similar (1.6)].

of the previous (1.3)]

F ° . Indeed,

that we want to deduce

to results

The crucial

that

F

[Av2;

in

which makes

paper unusable

in the present

is a subalgebra

of the miniraal

it is precisely

from the properties

in

difference,

of

this condition TorR(s,K)

•

on

F = S~R X

72

4.

LOCAL HOMOMORPHISMS

In this section

f: (R,m,k) ~ (S,n,~)

denotes a homomorphism of local (noetherian

and commutative) rings, such that

f(m) c n . Furthermore,

F-algebra over

has finite

R , such that

by a natural augmentation The DG

F-algebra

Xi

e(= gX ) , commuting with that of

F = S ®R X , augmented to

the (homotopy) fibre of

X

denotes a free DG

R-rank for any

%

by

f , and the canonical inclusion

i, and

H,(X) ~ k

R(~ R/m = k) .

(S ~ S/~ = %) @ s , is called S ~ S ®R X

is denoted by

g .

The fundamental importance of the fibre in the study of the homology of the map f

is given by the next result proved in [Av4]: (4.1) There is a natural exact sequence of graded Lie algebras over (4.1.1)

~ @k~*(R)

f+---*~*(S)

g+-~*~*(F) +~ C(Coker f*) - - 0

where for any piecewise finite-dimensional graded vector space with

F~

denoting the free

Moreover,

Im 6

F-algebra of

is central in

Wv

over

W , C_*(W) = ~*(F%wv),

~ .

~*(F) .

If furthermore the flat dimension in odd degrees, its dimension is

~:

fdRS

is finite, then Coker f*

j fdRS + edim (S/mS) , and

(Here and below we use the notation

is concentrated

~i(Coker f) ~ Coker(fi-1).

edim R = dimk(m/m2)) .

(4.2). Lemma. For any local homomorphism, there is an inequality of power "series": Ps(t) < PR(t)'C~(t) where

G~(t) = (I - I lengths(ITor~(S,k))ti+1)-1

Proof. Setting

ei( ) = dim i (

)

is the Golod "series" of

a. = dim (Coker fl)

H(F) .

b. = dim Cl(Coker f*)

the exact sequence (4.1) yields (4.2.1)

e.(S) + a. + b. = e.(R) + e.(F) . I l i I l

Recall that for the universal envelope of a graded Lie algebra

L , the

PBW

theorem

gives the equality of formal power series

dim (UL)it i =

dim L I 3 dim L 3 (I +t) (I + t ) ... t2) dim L2 4 dim L 4 (I -

Write

A(t)

)

. . .

for the Hilbert series of the free

for that of the universal envelope of (4.2.2).

(I - t

Ps(t) < Ps(t)A(t)B(t)

F-algebra on Coker f*, and

C(Coker f*) . One now has: because

A(t) ~ 0,

= PR(t)PF(t)

by (4.2.1);

< PR(f)G~(t)

by (2.1) .

B(t) ~ 0 ;

B(t)

73

It should be emphasized quasi-isomorphic is uniquely property

DG

defined

to

(4.3). Definition. is a Golod algebra

f , i.e.

if

homomorphisms.

ITorR(S,k)

is necessarily

terion of flatness,

products

Golod.

is relegated

TorR(S,k)

~n(F)

is

Ker g* # 0 ;

(b)

Coker f* # 0 ;

(c)

Coker fl # 0 .

(4.4.2) Moreover,

hence

. Moreover,

for

become

0 # Ker (f1:

£ @k m / m 2

sequence

= I . This means

to

(trivially)

of

Tor I

~.

trivial

Tor-s,

implies,

hence

i = 0 , or

i = I.

is a quasi-isomorphism, ~*(S/mS)

I (these are very I

by the local cri-

in (4.4.1)

S/mS ~ ~ ~ % , hence

is the free

special

~ ~ ~ 72 . According

cases of (3.4)).

to (4.1) the

in the easily checked

form:

~ ~/22) "

I J dim£Ker

from

one-dimensional

sequence

are

the last inequality

fl = dim%Coker

hold throughout. (4.1)

Asstmle both conditions

(abelian)

is an exact

F

equivalent:

= I , hence equalities

(4.4.3).

if

products).

i and isomorphic

F = S @RX ~ S/mS

n # I , 2 , with

note that in this case

dim%~2(F)

the next

F .

it presents some deviations from theusualpat-

of all higher

R-flat,

£ , one can express

= 0 , the homotopy

denote

= 0

conditions

(a)

over

F

look at this simple situation.

Lie algebra on a single generator of degree

Dualizing

of

to (4.7) below.)

lengths(ITorR(s,%))

that since the vanishing

~*(F) ~ ~*(S/mS)

following

produce

. In view of this,

has trivial Massey

in degree

of

However,

the triviality

i = 0 , S

Accordingly,

f

structure

is called a Golod homomorphism

Suppose

is concentrated

First of all note,

so that

the homology

i , such that

tern, hence we have to give a closer

When

f

TorR(S,k)

with earlier definitions

In this case the Massey such a map

= TorR(S,k))

A local homomorphism

(4.4). Exceptional

l

H(F)

in particular

it does not depend on the choice of

(equivalently:

(A comparison

(4.4)..

so that

(note also that

is intrinsic

there exists an

that any two constructions of a fibre for

F-algebras,

reduces

(4.4) and

Lie algebras,

fl = dim~C2(Coker

C1(coker f*)

Since by construction

to the following (4.4.2)

hold,

concentrated

f,) <

statement:

and let in degree

A

and

B

I . Then there

of Lie algebras f*

0 +- A ~- £ @k~*(R) +---- g*(S) ~- B ~ 0 In particular,

Ps(t)

We now treat generated

the case

i = I . According

by a single element

~2(F) ~ ~ . As before, hence

= PR(t) -

in this context

(4.1)

of degree shows

one obtains:

to (3.4),

2 , hence

in this case

dim~n(F)

= 0

for

~*(F)

is free,

n # 2 ,

Ker g* # 0 ~==~ Coker f* # 0"=~ Coker fl # 0 ,

74

(4.4.4). Assume both conditions one-dimensional

abelian Li e algebra,

(4.4) I and (4.4.2) hold, and let concentrated

in degree

A

denote a

Then there is an exac_t

sequence of Lie algebras f* A ~ £ ~k~*(R)* ...... ~*(S)+--0 .

0~ In particular,

Ps(t) = PR(t)(I +t) -I .

We shall call a homomorphism holds. To vindicate

exceptional,

the terminology,

if length S Tor~(S,k)

we note the following

= 2

statements,

and (4.4.2) whose easy

proofs are left to the reader. (4.5). Examples. in

not

2

(a) The projection

is an exceptional

(b) Let

X 2 + aX + b

R ~ R/(b) , with

homomorphism with

be a polynomial

in

b C m

(ITorR(S,k))i

R[X]

with

a nonzero divisor

= 0

a C m,

for

i # I

b C m,

b ~ 2.

Then the composition R r--~R[X] ~ R [ X ] / ( X 2 + aX + b) i s an e x c e p t i o n a l (c)

If

extension

S

homomorphism w i t h is a finitely

k ¢ ~

(ITorR(S,k)),

= 0 for i # 0 . 1 R-module via f , a nd t h e r e s i d u e

generated

is trivial,

then every exceptional

homomorphism

field

is obtained either

as in (a) or as in (b) above. A homomorphism

which is not exceptional

is said to be standard.

We have now come to the main result: (4.6). Theorem. fibre

The following are equivalent, for a local homomorPh~ism

f

with

F :

(I)

f

is Golod and standard;

(2)

Ps(t) = PR(t)G~(t)

(3) (i)

;

S/mS = ~ ~ N , where

(ii)

the natural hence

V

is an

~-vector space of dimension V = ITorR(S,k)

edim S - e dim R ;

satisfies

nV = 0 ,

is in a canonical way a piecew.is e finitely dimensional graded

vector .space over (iii) ~*(F)

N

S-module structure on

~ ;

is the free . Lie algebra on

V v, and the sequence

f* 0 ~ ~Qk~*(R)

~

g* ~*(S) ~

. L ( V )+--0

is exact. Proof.

(I) ~ (3). The fibre

from Theorem

(3.4). When

F

being a Golod algebra, assertions

lengthsITorR(S,k)

vial or free non-abelian.

In both cases,

by (4.1) and this implies

Coker f* = 0

prove

g*

is injective assuming

(4.4) showed in this case

# I , the algebra

~*(F)

is either tri-

it has a trivial center, hence by the construction

lengthsITorR(S,k)

Ker g* # 0

(i) and (ii) follow

of

= I . However,

if and only if

f

Ker g* = 0

C* . It remains to the discussion

is special, hence we are

in

75

through. (3) ~ (2) is trivial. (2) ~ (I). The power series equality (2) shows

G~(t)

has (non-negative) in-

teger coefficients. Using this and comparing (2) with (4.2.2), one gets the equalities: Ps(t) = Ps(t)A(t)B(t) = PR(t)PF(t) = PR(t)G~(t) which imply

PF(t) = G~(t)

and

A(t)B(t) = I . By the definition of

and by Theorem (2.3), the first equality means hand,

A(t)

and

B(t)

possible only with means

f*

F

G~(t)

in (4.1),

is a Golod algebra. On the other

having non-negative coefficients, the second equality is

A(t) = I = B(t) . In view of the definition of these series, this

is surjective, hence (4.4.3) and (4.4.4) show

f

cannot be exceptional.

(4.7). Remarks. Golod homomorphisms were used implicitly by Golod [Go] for regular

R , and by Gulliksen [Gu] in a more general context. Their

explicit introduc-

tion, for surjective maps, is due to Levin, who started a systematic study in [Levl]. In [Av 2] the notion was extended to maps which induce the identity on residue fields. In particular, it was proved there that either set of conditions characterizes such homomorphisms: Ps(t) = PR(t)G~(t) TorR(S,k) That for a surjective

and

nITorR(S,k)

= 0 ;

has trivial Massey products and f

f*

is surjective.

one recovers the original notion was shown in [Lev2], where

several alternate characterizations can also be found. It follows from Theorem (4.6) above, that the a priori much broader notion of Golod homomorphism adopted in (4.3) coincides, once exceptional maps are dropped, with the earlier concepts, whenever these are defined. The structural result for Golod maps, contained in (4.6.3) is known in the residually trivial case from [Av I ] and [LS]. All the papers just quoted provide numerous examples.

5.

FINAL REMARKS

(5.1). Let to

~

f: R -~ S

be a homomorphism of DG

F-algebras, augmented to

respectively. In this case one can still define the fibre

k

and

F = S @R X , and

(4.1) holds with the change from flat dimension to formal dimension, i.e. setting fdRS = {max dlHd(F) # 0) : (details are given in [Av 4]; when

R

and

S

are rings,

the formal and flat dimensions coincide). The definition given in (4.3) for a Golod homomorphism continues to make sense in the DG

F-algebra setup, so we use it to define Golod maps there.

What does change somewhat is the discussion in (4.4) of the exceptional homo-

76

morphisms:

with the local criterion of flatness unavailable,

satisfied for every ~*(F)

i > 0 . However,

condition (4.4) can be

it is still easily shown that, for any

is the free Lie algebra generated by a single element of

i,

degree (i + I) , and

the analysis of the exceptional cases, although slightly more involved,

is not hard.

We shall not pursue it further at this point. (5.2). Example. For a DG

F-algebra

R , with

R

local, denote by

R

the one

O

obtained by adjoining a (minimal) set of tors of

degree I variables, which kill the genera-

m = Ker (Ho(R) ~ £) . (Thus, when

Koszul complex).

In any case,

R

shown in [Av3], [Av 4] that the inclusion

is a local ring,

R~-~ R

is "its" and it is

induces the exact sequence

,

~*(R) = ~ 2 ( R ) Suppose now

, in a canonical way.

f: R ~ S

for some ideal with

is a map, for which

Jo C_m 2) . Then

S @RR

f]

is bijective

is a good choice for

with the naturality of the sequence (4.1), one sees that f: R ~ ~

R

~(R)~-~*(R)~-~*(~)~-0

0~

hence

R = R°

is uniquely defined up to isomorphism,

being the canonical map and

(5.2.1).

f

F

is Golod if and only if

(e.g.

S = R/J

S , and playing

v*(F) ~ ~*(~) , with

denoting its fibre. Hence: ~

is.

This observation can be useful in some computations.

For example, consider the

spectral sequence with (5.2.2).

E 2p,q = TorH(S)(~,~)p,q ~ Tor~+q(£,~)

It yields the formal power series inequality

i

P~(t)s ~ PH(~)(t) , which becomes an

equality precisely when the sequence degenerates. (5.2.3).

E P,q 2 = TorH(R~H(S),~) ~ Tor~+q (~,~) P,q

the inequality

Similarly,

the spectral sequence

produces

_H(R) R G ~ ~ G~ . Combining this with (4.2) one sees that: H(S) S " " H(R)(t) PH(~)~t)GH(~)

~

P~(t)G~(t) V

PH(~) (t) Since

f

is not exceptional,

~

P~(t)s

these inequalities can be used in conjunction with

(4.6) and (5.2.1) to obtain the following sufficient condition for Golod homomorphisms: (5.2.4). degenerates,

If then

H(~) f

is a Golod homomorphism, andthe spectral sequence (5.2.2)

is Golod,

77

The preceding assumption

statement has been proved in [Ba, Lemma 5] under the additional

that the spectral

using the inequalities

sequence

above,

(5.2.2)

for

R

also degenerates.

this is seen to be a consequence

(5.2.4), which also imply the degeneracy of the sequence to the referee for drawing my attention

to Backelin's

However,

of the hypothesesiin

in (5.2.3).

I am grateful

result.

(5.3). The inequality of Lemma (4.2), although giving a natural upper bound on the Poincar~ series of

S,

is of little practical

artinian ring. However,

passing as above from

F

interest when to

F,

S/mS

is not an

one gets the inequality

edim H (F)

Ps(t ) < pR(t )

.

(I + t )

o

co

I

-

Z dim H. (F)t z+1 i= I i

which has the advantage of involving only series with integer coefficients. (5.4). The preceding formula also brings us hack to the origins of the entire circle of ideas considered

in this paper.

of the same embedding dimension as Koszul complex of

Indeed, when

S , and

f

R

is a regular local ring

is surjective,

S = F = F

S , and the formula reduces to the well-known

is the

Serre inequality:

(I + t)edim S PS (t)

co

I - Z dim H. (S)t i+I i>I i The upper bound is reached precisely when that this condition nition

is equivalent

to

S

is a Golod ring [Go], and it is clear

'S being a Golod algebra

in the sense of defi-

(2.2).

(5.5). The results of this paper completely

characterize

those homomorphisms

of

local rings whose fibre has a free homotopy Lie algebra. At the other end of the spectrum, interesting

one has the abelian Lie algebras,

to ask for them the corresponding

question.

and it is

The answer here turns out to

be much more subtle. On the one hand, any

f

with

f*

such homomorphisms

exist. In fact,

injective has this property.

it follows from (4.1) that

Examples of such maps can be found in

[Lev3] , where Levin studies them under the name "large hombmorphisms". On the other hand, under restriction (or, for that matter, nilpotent),

only if

follows from joint work with Halperin,

that f

fdRS < o~ , is a complete

~*(F)

can be abelian

intersection map: this

which is being prepared for publication.

REFERENCES

[Av 1 ]

L . L . Avramov,

[Av 2 ]

L.L. Avramov, Free Lie subalgebras of the cohomology of local rings. Trans. Amer. Math. Soc. 270 (1982), 589-608.

Small homomorphisms

of local rings. J .

Algebra 50 ( 1 9 7 8 ) ,

400-453.

78

[Av3 ]

L.L. Avramov, Local algebra and rational homotopy, in Proceedings of the conference "Mgthodes d'alg~bre homotopique en topologie", In Homotopie Algebrique et Alg~bre Locale, Ast~risque 113/114 (1984), 15-43.

[Av4 ]

L.L. Avramov, Homotopy Lie algebras for commutative rings and DG algebras. To appear.

[Ba]

J. Backelin, Golod attached rings with few relations (III). Reports, Dept. of Math., Univ. of Stockholm, No. 13, 1983.

[Ca]

H. Cartan, Alggbres d'Eilenberg-MacLane (S~minaire ENS, 1954-1955, Exposes 2 11), in Oeuvres, Volume III 1309-1394, Springer-Verlag, Berlin, 1979.

[Go]

E.S. Golod, On the homologies of certain local rings. Dokl. Akad. Nauk SSSR 144 (1962), 479-482 (in Russian); English translation: Soviet Math. Dokl. 3 (1962), 745-748.

[Gu]

T.H. Gulliksen, Massey operations and the homology of certain local rings. J. Algebra 22 (1972), 223-232.

[GM]

V.K.A.M. Gugenheim and J.P. May, On the theory and applications of differential torsion products. Memoirs Amer. ~ t h . Soc. 142 (1974).

[Lem]

J.-M. Lemaire, Alg~bres connexes et homologie des ~spaces de lacets. Lecture Notes in Math. 422, Springer-Verlag, Berlin, 1974.

[Lev I ] G. Levin, Local rings and Golod homomorphisms. J. Algebra 37 (1975), 266-289. [Lev2 ] G. Levin, Lectures on Golod homomorphisms. Reports, Dept. Math., Univ. of Stockholm, No. 15, 1976. [Lev 3] G. Levin, Large homomorphisms of local rings. Math. Scand. 46 (1980), 209-215. [es]

[Ms] [No]

C. LSfwall, On the subalgebra generated by the one-dimensional elements of the Yoneda Ext-algebra. These proceedings. J.P. May, Matric Massey products. J. Algebra 12 (1969), 533-568. J.C. Moore, Alggbre homologique et homologie des ~spaces classifiants. S~minaire H. Cartan, ENS 1959-1960, Exposg 7, Secretariat Math., Paris, 1961.

[Ro]

J.-E. Roos, Homology of loop spaces and of local rings. 18th Scandinavian Congress of Mathematics, Proceedings 1980, 441-468, Birkhguser, Basel, 1981.

[Ta]

D. Tanr~, Homotopie rationnelle: ModUles de Chen, Quillen, Sullivan. Lecture Notes in Math. 1025, Springer-Verlag, Berlin, 1983.

L.L.

Avramov

Institute for Algebraic Meditation Department of Mathematics University of Toronto Toronto, Canada M5S IA] and Institute of Mathematics University of Sofia ul. "Akad. G. Bon~ev" BI. 8 1113 Sofia, Bulgaria

ON THE RATES OF GROWTH OF THE HOMOLOGIES OF VERONESE SUBRINGS J~RGEN BACKELIN

Let R be an (associative, non-negatively graded) connected algebra, generated by RI, over a field k. In [5] Ralf FrSberg and Z proved that if the homogeneous minimal relations of R appear only in degrees ~ r ~ ~ , then similar limitations hold for the Veronese subrings R (d) of R (d = 2,3,...), with r decreasing as d increases till eventually all minimal relations are quadratic. (This generalizes a result of D. Mumford:

[13, thm I].) Likewise (generalizing a result of S. Bar-

canescu and N. Manolache:

[6, thm 2.1]) we proved that if R is a Koszul algebra,

then so are all R (d). (See [5, prop 3 and thm 4].) In this note both these results are generalized, by proving that if Tor~(k,k) is concentrated in degrees c(p-1)+1 for p = 2,..., n

(as in figure I below), then a similar assertion

holds for R (d) , with a new rate of growth [c/d] replacing c. Eventually we get the rate I. (The precise general result is given in theorem 3 below.) The "rate of growth"-result is particulary interesting in the case where ~ p are concerned (theorem I). In case R is commutative and finitely generated, we also get a "limit algebra" type of result, namely that R (d) is a Koszul algebra for d ~ 0

(theorem 2). Figure I.

~P 0 1 . . .

Tor R

P,q

n

(k,k)#0 only in dotted area

80

In section I the main results are given and are proved to follow from theorem 3. The main ideas of the proof of theorem 3 are outlined in section 2, and the details fill the succeeding three sections.

In the last section possible improve-

ments of the results are discussed.

i, DEFINITIONS AND MAIN RESULTS, In this article,

k, V, I, P and R

space over k, the augmentation T(V) ( = k @ V @ V O V

will denote a fixed field, a fixed vector

ideal in the (naturally graded) tensor slgebra

~) . . . ), a fixed homogeneous two-sided T(V)-ideal contained

in 12 , and the graded residue class ring T(V)/P, respectively

(*).

d will denote an integer such that d ~ 2.

Vggon~e sub~ng R (d) = ~ R ( . d) is defined by

The

R(#) = Rjd J

for j = 0,1,2

k i s a graded R-modtLle (by t h e augmentation R - ~ R / I ~ k )~ whence t h e homology spaces R

Torp(k,k)

, p= 0,1,2

are graded:

Tor~(k,k)

=

If Tor~(k,k) c~[-B~

PR (x) =

~Tor

R

q20

P'q

(k,k)

.

is finite-dimensional

s~gs

as a k - v e c t o r

s p a c e f o r a l l p, t h e n t h e Poin-

o f R i s d e f i n e d t o be

~ xPdimkTor~(k,k)

p~O

[C] denotes the integer part of c, while [c I ( = - [ - c ] )

denotes the smallest in-

teger not less than c.

Roughly following D. Anick ([I, section 4]), put

qO(P,R) = s u p { q I T o r ~ , q ( k , k ) # O

o r q=p } ;

(*) V is not assumed necessarily to be finite-dimensional.

Sometimes in what

follows results are quoted and applied generally, though in the sources finite dimension was assumed.

In these cases, the proofs of the quoted

results make no use of this assumption.

B1

the extra case q=p will only affect q0(P,R) if the global homological dimension gldim R < p , since by e.g. [9, first inequality in lemma 2, formula (9)] Tor~,q(k,k) = 0 Clearly

for q < P

p£q0(P,R) ~

Formally, let the rate rate(R) = sup { Clearly

of growth of the homology of R be

(qO(P,R)-1)/(p-1) I p= 2,3,4 .... ]" . R is called a (homogeneous) Koszg~l algebra or a Fr$berg

I ~ rate(R) ~ ~.

ring if rate(R)= I . (This generalizes the ordinary definition,where in addition V is assumed to be finite-dimensional;

cf.

e.g. [5, 1.16.].)

In several interesting cases rate(R) < ~ , e.g. if dimkV < ~ and R is commutative ([I, thm 4.2]), or if q0(2,R) < ~ and R fulfills a certain distributivity condition (cf. [3, prop 2.5 (b)]). In these cases, theorem I and its corollary apply.

THEOREMI.

-

~s~e

th~

rate(R)=c<~

, and t h a t d>2 i s an i n t e g e r . Then

rate(R (d)) S

COROLLARY, - I f

d ~ rate(R),

then R (d) is a Ko~zul algebra.

As is well known, commutative finitely generated Koszul algebras have rational Poincar@-Betti series; hence and from the results mentioned above we get

THEOREM2, If R is (as above and) commutative and f i n i t e l y generated, then R(d) ~ a Koszul algebra for d>>o. In p a r g i c u l ~ , PR(d)(X) /s r a ~ i o n ~ (for -

t h o s e d). Theorem I follows easily from

THEOREMB, (I)

-

~sume t ~

q0(P,R)- 1 S

[j/(n-1)J(~1)d , p= 2 ..... n

q0(n,R(d))-1

~

Then (2)

for some i n t e g e r s j and n (j~n~2) we have

j

B2 For, assume that rate(R) = c < ~ and fix an n ~ 2 . We want to prove that q0(n,R)-1S

[c/d](n-1) . However, if j = [c/d](n-1) , then [j/(n-1)]= [c/d] ~ c / d ,

whence for

p=2,...,n

qO(P,R)-I ~ c(p-1) S [jl(n-l)J(p-1)d

,

whence indeed by (2) q0(P,R(d))-1S j .

2. FUNDAMENTAL IDEAS OF THE PROOF OF THEOREM 3, The following lemma is proved by Govorov in a more general form ([9, lemma I]). (He however assumes but does not use that di~ V < ~ .)

~MMA I, - For s = T ( V ) / Q , where Q is any proper homogeneo~ 2-sided T ( V ) - i d e ~ , we have the following i s o m o r p ~ m s of graded k-vector s p a c e : S

Tor2m+1(k,k) ~

(IQmAQml)/(IQmI+Qm+1)

S m-1 m m m Tor2m(k,k) ~ (IQ IAQ )/(IQ +Q I)

(m = 0,1,2 .... ),

and

(m= 1,2 .... )

Note that we do not have to assume that Q E 12. Thus, we may apply lemma ] not only for R but also for R (d) (3)

R (d) =

T(v(d))/P (d) ,

where V (d) = Id, and where p(d) Hence we may rewrite the For (4)

(for all i) = PidC lid = I(d) l

assumptions (I) in theorem 3 as follows:

i~ 2+[j/(n-1)](p-1)d and p = 2r+I odd (Ipr)iA (prl)i =

while for (5)

if we make the following natural identifications:

(iprl)i+(pr+1)i ,

i~ 2+[j/(n-1)](p-1)d and p=2r even

(ipr-Ii)iN (Pr)i =

(ipr)i+(pri) i .

Equally, the conclusion (2) may be rewritten thus, if n = 2m+I is odd: For t ~ 2+j (6)

(l(d)(p(d))m)tN ((P(d))ml(d)) t

= (l(d)(p(d))ml(d))g+((P(d))m+1)g

and thus, if n= 2m is even: For t ~ 2+j (7)

(z(d)(p(d))m-11(d)) t N ((P(d))m) t = ( l ( d ) ( p ( d ) ) m ) t + ( ( P ( d ) ) m l ( d ) ) t

;

83 Clearly it is sufficient to prove (6) and (7) for t = ]+2. By the identifications we may regard (6) and (7) as statements concerning subspaces of l(]+2)d . To be concrete, we have e.g. (Pm)i = ~ PbiPb2" " "Pbm (sum over b= (b I .....b m) such that

S[ bs = i ) ,

while by (3) (P(d))~+2 = ~~ P b l P b 2 . . .Pb m (sum over b = (bl,...,b m) such that

bs=

(j+2)d

and that dlbs

for

all

S)

.

S

mhus ((p(d) )m)]+2E_ (Pm)(j+2)d, and more precisely a term Pbl'''Pbm in (Pm)(j+2)d m is one of the terms defining ((p~d~));+2j if and only if there is no "disallowed

k

border" ak=

s[IbS=,

not divisible by d, "between" the factors Pbk and Pbk+1 for

some k6 {I,...,m-I] ; cf. the figure

below.

Figure 2. at

General term:

a2

, ,a3 ~

am-1

I pb I 0

3

........ ~'d

y+~)d

(j+2)~

a Term

in ((P(d))m)j+2: 0

Allowed borders:

Pb d

I ii2 Pb

2-d

"

• • • .... +

T-I "

0 121

+

Pb

"

<j+ )d

1

~j )~'-+2"-

<j+2>d

The situation is parallel for (I(d)(p(d))m)j+2 (c(IPm)(j+2)d) ,

((P(d))ml(d))j+ 2 (E (pmI)(j+2)d),and (l(d)(p(d))ml(d))j+ 2 (~ (Ipml)(j+2)d)• Thus, assuming n = 2m+I odd and applying (4) for p = n, we get

(I(d)(p(d))m)j+2 n ((P(d))mI(d))j+ 2 (~ (IPm)(j+s)dn (Pml)(j+2)d)) E (iPmi)(j+2)d+ (Pm+1)(]+s)d • and we want to prove that the assumptions imply that the inclusion remains true

84

even if we omit the terms with "forbidden borders" on the right side. In order to achieve this we "forbid" these borders one by one. The order in which the borders are forbidden is determined, in the first place by their residues modulo d, and in the second place by their own order. Thus the next border to be forbidden in the situation of figure 3 is

gd+e-1 .

Figure 3.

gd+e- 1 4.

I ....

.....I .......

ooo + (j+2)d

d+e

(j+1)d+e-1

(The sum of the terms in pm (ipm, etc. ) whose borders are compatible with

• m,d,d,e

figure 3 will be denoted ~P

)(j+2)d,g-1

( (

IP

m,d,d,e

)(j+2)d,g-1

" etc.); cf.

definitions (22) to (25).)

In order to eliminate this border, we use the conditions (4) and (5) and some technical generalizations of these. I.e., we have to make induction with respect to the claim that (4) and (5) remain true if on both sides of the equalities we forbid all borders having too small residues modulo d. This is the content of lemma 2 in section 3, which is proved in that section, up to the crucial part of the induction step. Before the induction hypotheses

can be applied, some rather

lengthy calculations must be done. Then only elementary properties of the modular lattices ({W},+,N) of subspaces of the k-vector spaces Ii (given in section 4) are used. I fear that these manipulations (since they take too much place and are very technical, due to the lack of a good theory for the logical interrelations of different distributive conditions) may make the reader miss the actual points of the proof. However, the induction and thus the proof of theorem 3 is completed in section 5.

3, A

DEFINITIONS.

-

TECHNICAL

GENERALIZATION,

For any positive integers e, £ and f, such that

(8)

(IpO)~ "d'e = (pOl)~'d'e = (lpOl)~'d'e = I£ ;

(9)

(p1)/f,d,e = Pi ;

e=f
85 and for m > 1

f,d,e :=a~ pal pa2-al ...pam-am-IPi-am

(10)

(pm+1)/

(11)

(Ipm)~ ' d ' e

= ~ I a

(12)

.f,d,e

P

...P

al a2-al

P.

;

%n-am_t ~ - a m

= ~ p p ...p a al a~-al am-am-11i-am ; and (for m ~ 2 )

(Pml)i

(13) ( I p m - l I ) ~ ' d ' e = ~ I 0

P ...P I. ; al a2-al %n-am-1 ~-a m

where in each case the sum is taken over all m-tuples of integers a = (al,...,am), such that 0 < a I < ... < a m < i

, and that

a r - f @ 1 , 2 , . .... e-l(mod d) For

X = p m , i p m , p m l o r Ipml

( r = 1 . . . . . m)

we have

x~'d'e~ x,, but in ~f,d,e,.

ders we want to get rid of are forbidden

(14)

x d( j,+d2, d) d = x j+2 (d)

(t5)

X~ ' d ' l

in general some of the borIn fact,

, while

= X.

(independently of the value of

f). Thus, theorem 3 is a special case of the

following lemma:

LKF~A 2, - Let n, d, e, i and f be p o s i t i v e i g t e g e r s such t h a t d>2 ; e f ;

j:=[(i-f-1)/d]>n-1 ; and (16)

qO(p,a)-1<[j/(n-1)](pi1)d

(if

n>1)

for p = 1

..... n .

Then• i f n= 2m+I / s odd we have (17)

(Ipm)f'd'eN (pml)f'd'e

and i f n = 2 m i s (18)

=

(ipml)f'd'e÷ (pm+1)f,d,e

even we have

(ipm-11)f,d,en(pm)f,d,e = (ipm)f,d,e+(pml)if , d , e

(Note that condition (16) always holds for p=1.)

86

The l e m m a is p r o v e d by induction,

first on n, then on e, and then on £.

L(n',e',i',f') be the claim that (n,e,i,f)= (n',e',i',f').

Let

the implications

in the lemma are true

whenever

L(1,e',i',f')

follows directly from the definitions

(8) and (9).

Thus we may henceforth assume that (19)

n ~ 2 ~d

L(n',g',i',f')

integers e',

for

n'=1 ..... n-1

(and for all positive

i' and f').

If e = I, then we note that if the assumptions

of the le~ma are fulfilled, we in

particular have q 0 ( n , R ) < 2 + [ j / ( n - 1 ) J ( n - 1 ) d S 2 + j d s i , Tor~ /(k,k)

whence

: 0 ,

whence by lemma I and by (15) the conclusion of lemma 2 holds. Thus we may h e n c e f o r t h assume that (20)

If

e~2 , and L ( n , e ' , i ' , f ' )

i = I,

then

j

(since

< 0<

n-l)

for

e,=l .....

e-1

the assumptions

.

of the lemma are not fulfilled,

whence indeed L(n,e,l,f'). Thus we may henceforth

(21)

assume that

t h e ~ s u m p t i o n s of lemma 2 are f u l f i l l e d , for i ' = 1 ..... / - 1 .

(n,e,i,f) is a L(n,e,i,f) is proved,

and L ( n , e , i ' , f ' )

From n o w on

fixed set of integers fulfilling

When

then so is lemma 2.

DEFINITIONS. (22)

(23)

(24)

(25)

- For any integer h such that 0 <

(19), (20) and (21).

f+hd+~
and for any m ~ I, let

(pm+1,f,d,e = ~ p p ...p p, • )i,h a al a2-az am-am- I r - a m .

m,f,d,e

p, am-am_ I ~ - a ~

,

"pmI'f'd'e = ~ P P ...P I, )i,h a al a~-al am-am_ I ~-a m

,

tiP

~i,h

=

~ laIP

a

a2-al

...P

~

~ f , dh, e = (Ipm-11Ji,

~ I al Pa2-al . . . P

a

and (for m~2)

am-am~11i-am

where in each case the sum is taken over all m-tuples of integers a : such that

(al,...,a m)

87

0
at- f

... < a m < i , @ I . . . . . e - 2 ( m o d d)

(r=

1.....

m) , and

at- f # e-1(mod d) for r such that ar
(27)

if in addition f+hd+e-1 < i, let

,(pm+l~f,d,e = ~ p p Ji,h

e

al a2-al

...P.

c

g-a m -

(pm+l.f,d,e )i,h

, et cetera,

where the sums are taken over Q such that (26) is fulfilled and that in addition

for some r (I ~ r S m) we have (28)

at- f = hd+e-1 .

Thus, putting h I := [(d-f-g+1)/dJ

and

h 2 := [(i+d-f-e)/d]

,

and taking X=P m, =IP m, =PmI, or =IPmI, (for some relevant m,) we have: (x)f: d'e is defined iff h = h I ..... h 2

, r~~i ,f h, d , e (29)

i s defined i f f h= h I , . . . , h 2 - I

;

;

r~f'd'e

(X) f'd'£-1

= ~li,hl

(3o)

• .f,d,e ,..f,d,e . .f,d,e ~x)i, h = ~x)i, h + <x)i,h+ I f o r h= h I . . . . . h2-I ; and

(31)

rx~f ,d,e (x)f,d,g ~ li,h2 =

Assume henceforth that n = 2m+I is odd; then we have to prove (17). (The even case is handled similarly. )

By the definitions,

by L(n,e-l,i,f) and by (29)

(IPm)~ ' d ' e n (pmI)f'd'e-r

_c (Ipm)~'d'e-I B (pmI)~'d'e-1=

= (Ipmi)i,d,e-1 + (pm+1)if,d,e-1 = (Ipml)f,d,e i,hl + (pm~1,f,d,e )i, hl

'

whence

(32)

(Ipm)~'d'eN (pmI)~'d'e = (Ipm)i'd'eN (pmI)~'d'eN((IpmI)~e+(pm+ 1~f,d,e~ Ji,hl /"

In section 5 it is proved that for h= h I .... ,h2-I

88

(ipm)f,d,eN (pmz)f,d,e N ((Tpmi)f:d,e+ (pm+l,f,d,e~ (33)

: (ipm)f,d,eN (pmi)f,d,e N (( m .f,d,e+ (pm+l.f,d,e, Tp t)i,h+t ~i,h+l] "

By (32), (33) and (31)

(ipm)f,d,en(pml)if , d , e = (ipm)f,d,en(pml)f,d,en ((ipml)f,d,e+ (pm+1)f,d,e) whence indeed

(Ipm)f'd'eN (pml)f'd'e c (Ipml)f'd'e + (pm+1)if,d,e . Since the opposit inclusion in (17) is trivial,

(17) indeed follows.

Thus theorem 3 indeed is meduced to (33) (and to a similar

omitted even

counterpart).

4,

SOME

LATTICE-THEORETICAL

PREPARATIONS,

For a general reference on lattice theory, see e.g. [10]. Recall that in general a lattice (L,v,A) is called modular if (34)

xA(yvz) = (XAy)VZ for all x,y,z C L such that x~z (the

or, equivalently, (35)

modular

law)

if

xa(yvz) =XA(yV(ZA(XVy)))

for all x , y , z E L

(the

shearing property).

In the rest of this article, all lattices considered are assumed to be modular.

It is well known that for any k-vector space W, ({subspaces of W),+,N) forms a (modular) lattice; in particular, we may take W=Ii, (0 S£' Si). We shall also be concerned with some well-known relations between elements in the lattices of subspaces of W' , W", and W ' ~ W " , for two k-spaces W' and W":

[EMMA 3, - Let V~ . . . . . t i v ~ g y . Then we have (36)

V~@W"NW'@V r

v/~ and v'~ . . . . .

I'' = V I'~V I'' s

(37) (rv,~ef ~v'd uj:

, and

v s be subspaces of w' and of w", r e s p e c -

89

We also need

LE~IA 4, - Assume t h a t (L,v,^) is a modular l a t t i c e , t e g e r , that a I ..... as,b I ..... b s g L ,

that s is a positive in-

and that a I < a 2 < ... < a s and

b I ~ b 2 > ... > b s. Then S

S-I

Proof by induction on (If S ~ 2 then ^(rest = ^(rest

S and by the modular

law (34).

of factors) ^ (aS_iVb s) ^ a s =

of factors) a (a6_ I V (bsAas))

= ( ^(rest of f a c t o r s ) ^ a s _ I )

v (%Ab s)

.

)

C0ROLLARYI - Assume t h a t t > 1, t h a t a 1 ..... a2t,b 1 ..... b2t E L and t h a t a I <. . . <. a2t . . and . . b.I .>

> b2t • Then

(39) (yl(a2U_IAb2u--1))A

(a2uAb2u)

=

= V (auAbu+I)

=

U= 1

For, ( ~ (a2u_IAb2u_I)) A ( ~ (a2uAb2~)) = t-1 t-] = blA(u61= ( a 2 u - l V b 2 u + l ) ) A a 2 t - l A b 2 A ( A ( a 2 u V b 2 u + 2 ) ) A a 2 t

=

12t-2 = b2Atug~= I ( a u v b ~ + 2 ) ) A a 2 t - 1

=

~(a~Abu+1)

•

5, PROOF OF (33), By (30) and by the shearing property

(35),

,__m_l ),i ,f h,d,e +tF ,_m+l,f,d,e~ (IPm)~'d'eN (pmI)~'d'en (tiF )i,h ' = ,h+l

= (zPm)fi'd'en(Fmz)f'd'en

"~*

'i,h+l

~

~

~'i,h

-

<~

'i,h

.. mTJi,h+l+tP . f , d , e . m+l.f,d,e ttzP Ji,h+l +

+((, (ipmT)f :d,e+, (pm+l)f"d'er,n )N((( IPm)f,'d'en(r pmI )fr ,d,e )+ (TpmT)fx.,n+]'d't+(pm+1,,i,h+lf,d,e,,,,).

90

Thus

(33) follows

from

( (ipmi,f,d,e , pm+1,f,d,e, (40)

(')i,h + ( f,d,e)Ji'h j+n(ipml"f'd'e (pm+1,f,d,e, (((Ipm)~'d'en(pmI)i )i,h+1+ )i,h+1) ~

(ipml,f,d,e pm+1~f,d,e )i,h+1 + ( )i,h+1 " (40) may be regarded as an inequality subspaces

concerning

elements

in the lattice

of all

to I i = li, ®li, , , where

i' :=

f+hd+e-1

the border to be eliminated,

and where

i" := i - i' Also, let

d-e+1 .

f" :=

The idea of the proof of (40) is to reformulate

the inequality

in terms of sub-

spaces v., x., u. and w. of li, and li,, , and then to apply the inductive hypotheses on these. Each term

lal P a2-al ...P am+1-a m I,~-am+ 1 in the sum which defines

by (28) has a border a r = i' and thus is contained

~f,d,efpm-r+11~f",d,e-1 ~ Ji"

(Ipr-lji '

for some r such that

(41)

I ~ r ~ m+1 . Hence and simi-

~ (IP -I )f~d,e,_m-r+1~ l)i,,d,e-1 r=I f",,,d,e-1 = m~ (Iprli~d'e(pm-rl)i r=O

smd )i,h For

= r=1

r=O,...,m , let

us write

'e

in

larly we get

'

'(Ipml)~

91

(pm-rI~f"'d'e-1

(43)

u2r+l

(44)

V2r+1 = (IP)i'

, and

r,f,d,e ;

and for r = 1,...,m , let us write e-1

(46)

V2r =

(pr)f,d,e

Then (recalling that n = 2 m + 1

(47)

'(Ipml)

and by lerr~na 3) we may summarize

~i[~,e+,(p. m+l,f,d,e )i,h

=

n

[ Lus S=I

=

nlii,Us aL I i

~ S=

(41) and (42)thus:

''

Next, let T = P a l P a 2 _ a l...Pi_am be a term in the sum defining

(pm+1.f,d,e)i,h+1 " Since

i' is not a border in T, either there is an r (with I
ar
< at+ I , or 0 < / ' < a I , or a m < / ' < i . Thus either

..

"Par-a~_11ar+1-arP ar+2-ar+

_

.

,

. I

] f ,d ,e( ipm-r ] f" ,d ,e- 1

"'Pi-sm -c (prI'i ,

~0 f,d,ecvom~f",d,e-1 . . . . i" ,

TCIaiPa2_a I ..P _am_C(~ I)/, TmP

al

--

...p

"

"i"

, or

or

(pml)~;d,e(ipO)~" ,d,e-1 am-am_11i-amc"

Henc e (48)

(pm+1.f,d,e)i,h+1 c_ m[ (Bri)i,fsd,e( ipm-r)~i',',,d,e-1 }L:0

Similarly c_

(IPmI)f,',d'e

c

m~ ~{Ipr-II~f'd'e{Ipm-rI~f"'d'e-1"i' ' 'i"

+ li,(IPmI)~,::'d'£-1< + (IPmI)~:d'£I/''

r= I Ii,(IpmI)/,,f"'d'e-1

c__(IP°I)f'd'e
(IPmI)~:d'eIi,,_c (Ipm-II)~.~d'e(IpOI)f',', 'd'e-1~ ~ (49)

(IPmI)~'~+l c --

m

~

r = 1

~

and , we have

(Ipr-li)f~d'e(Ipm-rI)~/j'd'e-1 '~

" Since

92

For r = 0,...,m , let us write

(50)

W2r+l

(T~m-r~f",d,e -I , and . . . . i"

(51)

X2r+1 =

(prI)~?d'e

=

;

and for r = 1,...,m , let us write

(52)

W2r =

(53)

X2r =

( i p m - r z ) ~ " , d , e -1 ,,

, and

(Ipr-II)~Y 'e

Then we may summarize (48) and (49) thus:

. (54)

m .f,d,e

. m+l.f,d,e

)i,h+l

tiP T)i,h+I+/P

n

n

= Z=l ~

xtwt = t~=lzi'wtnxtli''

Also note that (55)

u1~_u2~_...CUn

,

(56)

viD v2~...~Vn

,

(57)

w1_~w2_~...KWn

, and

(58)

Xl~X2~...

,

~Xn

and correspondingly

for the li,Us, Vsli, ,, li,w t and xtli,, .

Furthermore, by similar arguments m

(zPm)~"d'e

m

~1zi'w2r+lnx2rzi"

and

S r =~l X2rW2r+1 = r = m

m

(pml)~'d'e ~ r=1 x2r-lw2r~ = r~1 li'w2rNx2r-11i''

'

whence by the corollary to lemma 4 2m

(59)

_

2m [ Xr:Z,,n:z,w r = ~ x w r=2 r=2 r r

Thus by (47), (54) and (59)

(60)

(right side of the inequality (40)) E

(!iv us) !ixt) Nt£

S

93

By lemma 3

= s~ tX (vsnxt)(usn~t)

(6~) (~Vsus)n(~ xtwt)s

•

Below I prove that for each 3 and t

(60)~ (61) and (62) imply (40), indeed.

Proof of

(62): Fix S and t. We get three cases and two suhcases in each case,

depending on whether S < t, S > t or S =t,

and whether S is odd or even, respec-

tively.

i) ia)

If

S
for

S:2~+I

oddxs+lUs=X2(r+1)U2r+1= (Iprl)~d'e(Pm-rl)~:;'d'e-1

Now, if lalPa2_a ~ ...P at+l-at I., ~ -ar+1 is one of the terms defining

(IPrl)~ d'e, and

if P

.,P .. P I, is one of the ar+2-~ ar+3-ar+ 2 • am+1_am ~-am+ I

{~m-rT~f",d,e-1 ~Ji" , then

terms defining ~ I

P ...li, P .,...I. c al a2-al -at+ I ar+2-g ~-am+ I --

IaiPa2_a1...Par+2_ar+1...Ii_am+1 , which

(ipmlf,d,e )i,h+1 " Thus

is one of the terms defining

(and by lemma 3) indeed

Similarly,

it)

for S = 2r even =

ii)

If s > t ,

then

,~.

(VsaXt)(usnwt)~_VsWt~ Vt+lWt

K ~

)i,h+l

; proceed as in case i).

Thus we have finally exhausted the elementary properties of the involved lattices, and reduced the proof to a Case where the inductive hypotheses of lemma 2 may be applied:

94

iii) Assume that s = t . In this case some different

subdivisions

are necessary.

Let S" := n-S+1 . and that h" := j-h-1 = [(/"-f"-1)/dJ

Note that h = L(i'-f-1)/d],

. Furthermore we

have

CLAIM,

-

If 2 ~ s S n - 1 ,

then

[j/(n-1)] S max ( [h/(s-1)] , [h"/(s"-l)]).

PROOF,

-

Note that ( S - 1 ) + ( S " - I ) = ( n - l ) ,

while h+h"= j - 1 .

If j @ O(mod n-l), then [j/(n-1)] = L(j-l)/(n-1)] whence the claim follows the fact that

(j-1)/(n-1)= (h+h")/(s-1+s"-1)%max (h/(s-1),h"/(s"-1)

Assume that j = / ( n - l ) ~ 0(mod n-l) . Then h+h"= l(s-1)+l(s"-1)-1, both h and h" are integers,

~'~ l(s"-l).

2<=S<=n-I 2gS~n-I

iiib)

).

and since

we must have either h ~ l(s-1) or

In either case the claim follows.

Thus we get the following iiia)

from

six subcases:

and [j/(n-l)J<[h/(s-l)]

,

and [j/(n-l)J<=[h"/(S"-l)]

iiic) s = n

and h = j

iiid) S = n

and h_< j-I

iiie) S = I and h = - 1

, , , and

iiif) S = I and h > 0

In case iiia) we have q0(P,R)-l<[j/(n-l)J(p-l)d<[h/(s-l)J(p-l)d and by the inductive hypothesis Thus lemma 2 applies:

;

(19) we have L(S,e,i',f).

If S = 2r+I is odd, then by (17)

= Similarly

for p=1 ..... s

VsNXs=Vs+I+xs+I

=

)/,

= Vs+1+Xs+1 •

by (18) if S is even. Thus in either case (and by

lemma 3)

(%nx s) (%nws) : (%+i+%+i)(usn %) = Vs+~(usnw s) + %.1(usnVs ) ~_ c Vs+lWs+xs+lus~_(lPml)f,'d'e~ r,n+1 ÷ (pm+l,f,d,e )i,h+1 " -

indeed.

95 Similarly,

in case iiib) we have

qO(p,R)-I ~ [h"/(s"-l)j(p-1)d whence by

for p=l . . . . . S",

L(S",e-l,i",f")

usOWs= uS_I+Ws_I

•

whence

c" pm .f,d,e+.pm+t.f,d,e

(VsnXs)(UsnWs)~VsWs-1+XsUs-1 - ~ I In case iiic) we must have h 2 = J + 1 ,

I)i,h+l

[

)i,h+l

"

and by (16) and L(n,e,i',f)

(cf. (21)) we

have

(VnOXn)(UnaWn) = ((Ipm)~;d'eN (pmI)~;d'e)(Ii,,N Ii,,)= = ( i p m i ) ~ d , e l i . + (pm+l.f,d,e)i, Ii"-c(IpmI)~'d'e + (pm+l )if'd'e ( = (ipmi)f,{,%+

~,J+1

Similarly,

(pm+1,f,d,e )i,j+1 ) "

in case iiie) we have h" = j and

L(n,g-1,i",f")

(ef. ( 2 0 ) ) w e

and by (16) and

I£,)((pmI)f',','d'e-ln (Ipm)f',:'d'e-1) = . m .f",d,e-1 +Ii,(Pm+1)i, f",d,e-1 c(ipmi) f:d,e + (pm+1 )f ,d ,e Ii,~IP I ) i , ,

(viNxl)(UlnWl)=(li, =

i' = f-d+e-1 < f,

have

n

--

~0

In case iiid), like in case iiic) we have

(VnnXn)(UnnWn) = ((IPm)~d'eN

(pml)~d'e)l£,, c

Since hS j-l, we have an "allowed border" Ii, = IP

If+jd_i, Ii_f_jd.

m,f,d,e )i'

(Ipm)~d'eli,, .

f+jd> f+hd+e-1 = i',

and

Thus~ if IaiPa2_al...P/,_a m is a term in the sum defining

, then Ial...Pi,_amli,, = Ial

" " P i ' _amlf+jd_i,li_f_jd -c

.f,d,e c Ial ...Pf+jd_amli_f_jd E (IPml) ~ ,d,e _c (I pmI)i,h+ I .

_

Case iiif), finally, is handled analogously.

Thus indeed (62) and hence (40), (33), lemma 2 and theorem 3 are proved.

96

6, POSSIBLE IMPROVEMENTS? Some of the conditions sure the consequence

(I) are slightly stronger than necessary in order to en-

(2) claimed in theorem 3.

of the CLAIM in the preceeding qo(2,R)-1S[J/23]d

l

section,

E.g., for n = 4 ,

by a refinement

if

,

qo(3,R)-1 <[( j 2)/3]d , and

q0(4,R)-1 =<#d ,

then indeed q0(4,R (d))-1 -<_j . I do not know what the best possible conditions that there are but minor improvements

(in general) are, but I suspect

of (I). One reason for this is the fol-

lowing example, where actually R has global homological

dimension 2.

(If V has a basis (T I,T2,...), then we write k
~(~kE,

c - Let R=k~TI,T2~/(TIT2)

for some positive integer c. Then q0(2,R)-1 =c,

while for all other p q0(P,R)-1:p-1.

However, if d2_2 and c' = rc/d], then

T(V (d)) = k(Tf, TId-IT2,T Id-2T2T I ,.. .,T2~ (2 d "variables") generated by some monomia~s in these variables,

and p(d) is minimally . ~d-1 - d-c'-1 d-1 includlng ~2 TI "~T2) "T2 TI"

whence by [2, thm 2] gldim R (d) :~, and indeed q0(n,R(d))-1 =c'(n-1)

Thus for p = 2 there can be no greater improvement [I + (c-1)/dJ In particular,

for all n.

of (I) than replacing

rc/d] by

(which amounts to nothing if c is an integer). proposition

3 a) in [5] can not be improved. However, there we

also gave a roughly "twice as good" result for the case where R is C o ~ o J ~ v e , namely that then (63)

q0(2,R)-1 ~
in particular,

(64) qo(2,R(d))-l=l

if

d_->~([cJ+l)

It is conceivable that in this case (I) in general

(and thus the corollary to

theorem 1 ) may be improved similarly.

Let us finally discuss how Anick's theorem ([I, thm 4.2]) on the finiteness of rate(R) for R finitely generated and commutative might be improved. Let n = edim(R)= dimkV < ~, and let r = q0(2,R)

. (Since R is commutative

and n <

97

we have r < ~.) By analysing two proofs of An{~k's theorem we get

CLAIM, - T h ~ e i ~

an i n t e g ~

C= C(n,r)

(only depending on n and r) such t h ~

rate(R) £ C;

and we get some ideas on how we might get some good estimates for C.

Anick's proof goes roughly as follows: 1. Choose an ordered basis (X 1,...,Xn) for V. Then R ~ / a ac~

:=k[X1,...,Xn]

for some ideal

(the commutative polynomial ring). By a classical method of

Macaulay, this uniquely determines an associated ideal b c ~ ,

generated by monom-

ials in X I,..., X n , and thus an associated "monomial ring" A = ~/b to R. Anick constructs a spectral sequence (see [I , lemma 4. I]) A Tor.,,(k,k)

R Tor,,,(k,k)

whence in particular rate(R) -<_rate(A)

Tor~,q(k,k) # 0 ~ Torp,q(k,k) # 0 ; thus .

2. Since by [4] the double P o i n c a ~ - B e t g i s e ~ i 6 s

(k'k)i~ PA(X 'Y) : [ [ xPYqdimkT°rA,q Pq is rational, and indeed PA(x,y) = (1+xy)n/(1 + a

P,q

[ a ,qxPyq) p,q>1 p

where the integers

are 0 for almost all (p,q), we have rate(A) Smax ( I ; (q-1)/(p-1)

: ap,q~ 0 )
L.L. Avramov has given an alternative proof, using the bigraded version of the Serre inequality (65)

( [8, formula (I) ] ) :

PR(X,y) _-< (1+xy)n/(1-

where ep,q = dim k Tor~,q(R,k),_

~ c r xP+ly q) p,q>O '~'~ ( = dim k ( H p ~ ) q

(term-wise inequality)

.

, where K R is the Koszul complex

of R), whence (similary to step 2 above) (66)

rate(R) Smax ( I ; (q-1)/p : Cp,q#O ) <°°

The Avramov approach may be combined with a result deduced in the proof of [7, prop 1], namely (67)

There is an N = N(n,r) such that c

P,q

= o for all q > N;

98

in order to yield (68)

rate(R) ~ max(t-1 ,(N(n,r)-1 )/2 ) <

Thus the claim is proved. However, the (implicitly) given boundaries for N(n,r) in [7] are ridicuously large even for small n and r. In fact, FrSberg, Gulliksen and LSfwall use the limits calculated by G. Hermann in [11] in order to find an

N' = Nl(n,r) which bounds the q:s such that ci, q # 0 ; and then (roughly) bounds the q:s such that c2, q # 0 by N'(n,N'(n,r)), et cetera• Since already the Hermann bound (even as improved by D. Lazard in [12]) is very large, N(n,r)

is quite out

of (effective) control. There is however possibly a more promising approach• Note, to begin with, that the existence proof in Anick's step 2 may be replaced by

CLAIM, (69)

- If B iS a "monomial r i n g " , then

rate(B)

= q0(2,B)-1

I have proved this (unpublished) by a rather lengthy argument, involving a generalization of the Taylor resolution (cf. [ 14] ). For our discussion, however, a weaker but short Avramov-type argument will suffice: The Taylor resolution of B is an K-free resolution

•..--->Fp--->Fp_ i--->...--->F0--->B--> 0 , where each F

is generated (as a graded K-module) by elements of degrees < P < pq0(2,B). Tensoring with k and taking homology, we get

Cp,q

0

for q>pq0(2,B)

,

whence by (66) rate(B) < q0(2,B) . Thus, by Anick's argument rate(R) < q0(2,A)

(and actually rate(R)
and it is "enough" to find a bound B = B(n,r) for rain (q0(2,A) : A is an associated monomial ideal to k ' ~ R

for some field k'm__k ) .

As far as I know, the existence of such a bound is unproved, but (as FrSberg has suggested) it is plausible that a Hermann-type argument would yield one. One might hope that in fact there is an estimate of the type suggested by Lazard in [12]; i.e. that B(n,r)~n(r-1)

.

99

Note added in proof:

D. Anick and R. Fr~berg have remarked that indeed (in the notation from above) B(n,r) ~

. One way to prove this is the following:

By the results in [7], for fixed n and r there are only finitely many possible Hilbert series. Furthermore, there are only finitely many non-isomorphic "monomial rings" with a given fixed Hilbert series.

The latter statement, and indeed a direct bound on the highest degree of a relation, may be based on the ideas in the proof of Macaulay's theorem on feasible Hilbert series. If we adopt the terminology of the proof of this theorem given in R. Stanley,

algebras,

Hilbert functions of graded

Adv. in Math. 28 (1978), 57-83 (Theorem 2.2), we see that

if (hi)i> 0 is a given Hilbert series, ideal of monomials'

if M is the corresponding 'order

(taking lexicographically first monomials in the

appropriate amounts), and if j is the maximal degree of relations in the "monomial ring" whose non-killed monomials are exactly M, then hi+1 = h i~i>

for all i ~ j ,

Hilbert series (h.)

whence no (commutative) algebra with the

can have relations in degrees > j .

i

Note however that this gives a [7J-type, not a Hermann-type of estimate for B(n~r), whence it does not directly yield any improvements on the bounds C(~,r) for the rates of growth.

Also note that David Bayer and Michael Stillman

computing syzygie~,

(On the complexity of

manuscript~ March 1985; the authors are at

Columbia University and at University of Chicago, respectively) have shown that the trae bound for q0(2,R) is high - of "double exponential" type - l i k e G. Hermann's own bound. Thus C(n,r) is also at least "doubly exponential".

100

REFERENCES, D.J. Anick, On the homology of associative algebras, Trans. ~mer. Math. Soc. (to appear). J . B a c k e l i n , La s ~ i e

fi~

de Poincar~-Betli d'une algebre gradu~e de type ~ une relation est rationelle,

C.R. Acad. Sc. Paris 287 (1978), 843-846. J. Backelin, A i~t~butiveness

property of augmented algebras, and some related homological results, part of thesis, Stockholm, 1982.

J . B a c k e l i n , L~ anneaux locaux ~ relations monomiales ont des s ~ t £ ~ de P o i n ~ - F ~ 2 2 i ~ o n n e g £ ~ , C. R. Acad. Sc. Paris 295 (1982), S6rie I, 607-610. J. Backelin and R. Fr6berg, Koszul aggebr~, V 6 ~ n ~ e subrings and rings with linear reffutions, Rev. Roumaine Math. Pures Appl. 30 (1985), 85-97. S. Barcanescu and N. Manolache, Betti numbers of Segre-Veronese singu£~es, Re. Roumaine Math. Pures Appl. 26 (1981), 549-565. R. FrSberg, T.H. Gulliksen and C. L6fwall, Flat families of local, artinian algebr~ with an i n f i n i t e number of Poincar~ ~ r i e s , these proceedings. E. S. Golod, 0 FOaO~Or~x H~KOT-op~;x IO~n~H~x ~o~e~j Doklady Akademii Nauk SSSR 144, 1962, 479-482.

V.E. Sovorov, ~)ag~e~o~T~ ~ wp~,~o~ rpamympos~H~V~X, Sihirsk Math. J. 14 (1973), 1200-1206. (Translation: p. 840-845.) I0

O. Grgtzer, Generag l a t t i c e theory, Birkh~user Verlag, Basel und Stuttgart, 1978.

11

G. Hermann, Die Frage der endlich uicgen Schrigte in der Theo~e der Polynomide~e, Math. Ann. 95 (1926), 736-788.

12

D. Lazard, Alg~bre g i n ~ i r e sur K[X I ,... ,X], Bull Soc. Math. France 105 (1977) , 165-190.

13

D. Mumford, Varieties defined by quadratic e q u a t i o n , in Questions on algebraic varieties, edited by E. Marchionna, C.I.M.E. IK Ciclo, Varenna, 1969, Roma 1970, 29-100.

14

D. Taylor, Ideals generated by monomia~ in an R-sequence, Ph.D. Thesis, University of Chicago, 1966. Department of Mathematics University of Stockholm Box 6701 s- 113 8 5 ST0CK~0~M (SWEDm~)

~t ggimination,

WHEN IS THE DOUBLE YONEDA EXT-ALGEBRA OF A LOCAL NOETHERIAN RING AGAIN NOETHERIAN ? by JSrgen BACKELIN and Jan-Erik ROOS

§ O. INTRODUCTION. Let (R,m) be a local commutative noetherian ring with maximal ideal m and residue field k = R/~. Consider the graded vector space

(I)

E~t~O

which is a graded algebra under the Yoneda product [13],[19],[24]:

Indeed, Ext~(k,k) is even a graded, eoeommutative Hopf algebra over k [13] ,[24]. Now iterate this Ext construction and form (2)

Ext* (k,k) Ext~(k,k)

which is again a graded algebra (but not a Hopf algebra) under the Yoneda product. Note that (2) has an extra grading, coming from the grading in Ext~(k,k). Note also that while Ext~(k,k) is highly non-commutative in general (indeed, if Ext~(k,k) is graded commutative, it follows from [5] that R is a local complete intersection of a special type), it is always true that (2) is ~omm~tative in the bigraded sense, i.e. if x and y have bidegrees (r,s) and (r~,s~), then xy = (-I) rr#+ss~ yx. This follows from the fact that E = Ext~(k,k) is a Hcpf algebra, and the fact that in this case two "natural" products on Ext~(k~k) coincide (c_~f. [26], page 15-13). It is very rare that E = Ext~(k,k) is ncetherian (left noetherian is equivalent to right noetherian, since E is a Hopf algebra). Indeed, in these proceedings, B~gvad and Halperin [7] prow~ that E is noetherian precisely when R is a local complete intersection (and then E is even generated as an algebra by a finite number of generators of degree I and 2). It happens more often that (3)

Ext* (k,k) Ext~(k,k)

is noetherian

and when this happens we can deduce interesting results about Ext~(k,k) and R. Thus, for instance, we will prove in § I that the following assertions follow from (3): (i) PR(Z) = ~ dimk(EXti(k'k))'zin i>O

= ~ di~k(TOr~(k,k))'zi~ i>O

is a rational function of Z.

More ~recisely, i_~f(3) holds, then there is a finite set of algebra ~enerator { {~i}in1.=

of (2) with bidegree(~i) =(mi,n i) (contrary to the convention in [26] we let both

102

degrees take non-negative values) and n m. n. PR(Z) = H'(I-(-I) iZ l)/po](Z) i=I 2

where

2

.

means that we take the product over those i:s such that m.+n. Is even,char k#2,

and where pol(Z) is a ~olynomial in Z.[ In particular, algebra (2) are of bidegree

if the generators of the

(1,1), it follows that PR(Z) = (1+z)n/pol(Z).]

(ii) EXtR(k,k) is finitely ~enerated as an al~ebra. (iii) When we present~ usin~ (ii), EXtR(k,k) as a ~uotient of a finitely ~enerated free ~raded alsebra, it is also true that the (twosided) kernel ideal is ~enerated (as a twosided ideal) by a finite set of ~enerators.

[We say that ExtR(k,k)

is a

finitely presentad algebra.] (iv) Furthermore, there are only a finite number of relations between the relations in (iii) etc. The

first example where (ii) is false is given in [24], p.314 , using ideas from

J.-M. Lemaire's thesis [15]. Using similar ideas, one can construct examples (R,~), where (ii) is true, but (iii) is violated etc. Later Anick [I] gave many examples (R,m), where PR(Z) is non rational, thus violating

(i).

Thus there are many examples where (3) is not satisfied. However, in [23], one of us proved that if (R,m) is a Golod ring (examples: R = ~/~s

, s~1, where (R,~) is

a regular local noetherian ring, or R = R/x.a , where x6~ and ~ is a proper ideal of R...), then (3) holds, and so does even the following more precise assertion [(3) follows from (4), if we take M = R]: If M is a finitely senerated R-module, then the (left) Ext*

-(~)

(k,k)-module

Ext~(k,k)

(*)

Ext*

(Ext~(M,k),k)

Ext~(k,k) is n o e t h e r i a n . Note that here the Yoneda product Ext~(k,k) ~ Ext~(M,k)

> Ext~+J(M,k)

gives Ext~(M,k) the structure of a left Ext~(k,k)-module, (~) becomes a left Ext* (k,k)-module. Ext~(k,k)

and that in a similar way

Using a reasoning, similar to that in

[23], it follows that if R comes from a local complete intersection S by a Golod map S

> R, then (4) and (~ fortiori)

this, cf.

§ 2

intersection,

(3) are true. For a complete proof of

below. In particular,

(3) and (4) are true if R is a local complete

a fact which can also be deduced from [12].

In [6],one of us proved among other things, that if k is a field and R is the quotient

103

of a formal power series ring: R = k[[X1, .... Xn]]/(M I .... ,Mr) , where the Mi:s are monomials in the X.:s, then PR(Z) is rational of the form (1+z)n/pol(Z). It was 0 therefore very natural to ask whether the stronger statement (3) could hold for these rings with monomial relations. This is indeed the case, and that is one of the main results of the present paper (Theorem 5 of § 3). The proof is a combination of the ideas from our papers [6] and [23]. It follows from the theory of § i~ below, combined with results of C. Jacobsson [14], that the stronger statement hold in general for these rings with monomial relations.

(4) can not

The fact that (3) does hold

for these rings has already been used by Anick [2] in his study of the homology of certain loop spaces. Indeed he proves in [2] that H,(O~X,Q) is finitely presented if X is a generalized fat wedge. The paper by us, cited as [4] in [2], has been incorporated in the present paper. We study commutative and not anti-commutative R:s here, but the methods are the same. The plan of the present paper is as follows: In § I we deduce general consequences of the validity of (3) and (4) for a local ring. In § 2 and § 3 we prove the main theorems, using general results about the (co)homology spectral sequences of extension~ of Hopf algebras. Finally, in § 4 some open problems are mentioned.

§ I. THE DOUBLE Ext-ALGEBRA AND THE DOUBLE Ext-MODULES.

THEOREM I.- Let (R,m) be a local, commutative noetherian ring such that Ext*

(k,k) is a (bigraded) noetherian ring. Then:

Ext~ (k ,k ) (a) EXt*xt~(k,k)(k'k)E has a finite set of algebra generators

{~i)i=~n with

bidegree(~i )= (mi'ni) ' and we have the following formula for the corresponding double Hilbert series : (5)

~ dia_(ExtP, q ~ \ (k,k)).Xpyq = p, q~0 Ext R(k ,k )

PoI(X,Y) H ~ ( I-xmiy ni ) 1
where " means that we can restrict the product in (5) to those i:s such that 2 2 • m.+n. is even, if char(k)#2, and where P(X,Y) is a polynomial in X,Y with

int e~ral c oeffic ient s. (b) Assume furthermore, that M is a finitely generated R-module such that Ext* , (EXtR(M,k),k) EXtR(k ,k)

is a (bigraded) finitely generated (thus noetherian)

module over Ext* (k,k). Then we have the following formula for the Ext R (k ,k ) correspondin G double Hilbert series:

(6) x di (ExtP'q p,q~O

K\

ExtR(k,k )

IExt I , l, )). PYn= P°IMIX'Y) H'(I-X I
m i n. Y l)

104

where the numerator is a polynomial in X,Y (integral coefficients),

depending

on M, and where the denominator is the same as in (5). (c) When (a) is satisfied it follows also that the algebra Ext~(k,k) has a finite number of ~enerators, that all relationsbetween ~enerated by finitely many relations

a minimal set of generators are

(we say that Ext~(k,k)

is finitely presented),

that there are only "finitely many relations between thes e relations" etc. Furthermore, (7)

in this case (notations as in (5)): H" ~1-(-1)mizni) 1
i/0

Pol(-1,z)

(d) Z~nen the stronger condi%ion (b) is satisfied, it follows also that the left Ext~(k,k)-module

Ext~(M,k) is finitely generated, even finitely presented, and

even that it has a minimal, free Ext~(k,k)-resolution, generators in each ste~. Furthermore,

• (8)

havin5 a finite number of

in this case (notations as in (5) and (6)):

PolM(-1 ,z)

PM(z) = ~ dimk(TOr~(M,k))'Z I =

i>0

Pol (- I ,~)

PROOF: We will need the general observation

"

[15], Chapitre I, that if A is a graded

connected algebra over k, then a minimal set of graded generators of A corresponds to a basis of the graded vector space Tor~(k,k) = I(A)/I(A) 2 , where I(A) is the augmentation

ideal of A. In the same way, a minimal set of graded relations for A A corresponds to a basis of the graded vector space Tor2(k,k) , etc. Furthermore, i the graded vector spaces EXtA(k,k) are dual to the Tor~(k,k). We will use this below for A = Ext~(k,k). But let us first recall that we have supposed that the bigraded algebra B = Ext~(k,k) is noetherian.

~us

I(B)/I(B) 2 is finite-dimensional

and

therefore B has a finite set of generators{~.} n as in (a) of Theorem I. Since B i i=I is the Ext-algebra of the Hopf algebra A, it follows from [26], Th@or~me I, p.15-13, m-m-+n.n. that ~$~j = (-I) I $ i ~ j ~ i " We will now use the following bigraded variant of Theorem 11.1 in [3]: LEMMA.- Let B be a bigraded connected algebra over a field k. Suppose that B ha___~s a finite set of algebra generators (*)

{~i}1
~i~j = (_1)mimj+ninj

~i~j

Let V be a finitely generated bigraded module over B, and let V(X,Y) = [ dimk(Vp,q).xPYq p,q~O be the two-variable Hilbert series of V. Then V(X,Y) = PoI(X,Y) m.n. H'(I-X ~Y ~) 1
=(mi,n i) and

105

where PoI(X,Y) is a polynomial in X,Y (integral coefficients) and where " m e a n s that we can restrict ourselves to take the product for those i:s such that 2+ 2 m~ n. is even (if char(k)~2). ll SKETCH OF THE PROOF OF THE LEMN~: We will use induction over the number n of

generators ~i of B. If n=0, then V is a finite-dimensional bigraded vector space and V(X,Y) is a polynomial. Suppose n > 0. The ~.:s "almost" commute and therefore 3 B is noetherian. Furthermore, if char(k)#2 and m~+n~ is odd, it follows from (*) that ~ = 0 .

We then have an exact sequence

0

> ~i v

> V

> V/~iV

> 0

and ~i V and V/~iV are finitely generated and bigraded modules over B/(~i) which has n-] generators. ~uus V(X,Y) = (~iV)(X,Y) + (V/~iV)(X,Y) 22. and we can use the induction and ignore such an i. For a ~j such that mj+n.$ is even, we consider, as in [3] the (9)

0

b ~ r a d e d exact sequence: > V P,q

> Kp,q

> Vp+mj ,q+nj

> L p+mj,q+nj

--> 0

where the finitely generated B/(~j)-modules K and L are defined by (9). The alternating sum of the dimensions of the vector spaces in (9) is O. Multiplying this sum by XP+mJY q+nj and summing over p,q we obtain that (I-xmJy~0)V(X,Y)~ can be m~n.

expressed as

L(X,Y) - X ~Y JK(X,Y) + a polynomial in X,Y . The result now follows

from the inductive hypothesis. Now both (a) and (b) of Theorem I follow immediately from this lem~a. Using a minor variant of Proposition A.2.4 of the Appendix of Lemaire's thesis [15] (cf. also [11]) we can now also deduce the formulae (7) and (8) of (c) and (d) from the formulae (5) and (6). Indeed, if N is any graded module over a connected graded algebra A overk (assume that A and N are finite-dimensional in each degree), then (10)

N(Z) = A(Z)PN(-I,Z)

Here N(Z) and A(Z) are the Hilbert series of N and A and PN(X,Y)

= ~ di~k(TOrlq(k,N))'XPYq p,q~O

The proof of (10) follows (as in [15]) from taking a minimal A-free resolution of N and by taking alternating sums of dimensions in each degree of corresponding bigraded modules. Apply now (10) first to A = EXtR(k,k) and N = k (this gives (7) as * a consequence of (5)) and then to A = EXtR(k,k) and N = Ext~(M,k) (this gives (8) as a consequence of (6), using (7)).[We have used that the vector space duals of the Tor~,q(k,N)~ are the Ext~'q(N,k).]_ The remaining assertions of (e) and (d) now follow from the explicit formulae (5) and (6) , which give that for each fixed p, the graded * Ext~(k,k) vector spaces Tor EXtR(k'k)(k,k) and Tor (Ext~(M,k),k) have total P,* p,* finite dimension. Thus Theorem I is completely proved.

106

§ 2. SOME L O C k

RINGS WHOSE DOUBLE Ext-ALGEB~

IS NOETHERI~.

The following theorem contains Theorem 2 of [23] and gives a generalization

that

in a sense is the best possible one (for the definition of e.g. Golod maps, we refer e.g. to § 3 of [25]): THEOREM 2.- a) Let R be a local com~01ete intersection, R and M a finitely 5enerated R1-module. Ext* EXtR1

Then

(Ext R (M,k),k) is a noetherian

(k ,k )

> R I a Golod map

Ext*

(k,k)-module

.

ExtR1 (k,k)

1

In particular Ext* (k,k) is a noetherian algebra (take M = R I ). E xt_(k,k) * b) Suppose further that R I

> R 2 ~S a second Golod map. Then Ext* (k~k) Ext,* 2~ (k,k)

is asain noetherian. c) If R I is as in a), then EXtR1(k,k ) is a (graded) coherent al~ebra [25]. Remark.- Theorem 2 is the best possible in the following sense: i) There exist local (commutative, noetherian)

rings R 3 that come by three Golod

maps from a local complete intersection (even a regular ring) R : Golo_____~d > RI Golod > R2 Golod > R3 such that Ext* (k,k) is not noetherian. EXtR3(k ,k ) ii) We also have a situation (R regular): R Golod> R] Golod > R2

and a finitely

generated module M over R 2 such that: Ext

(Ext*2(M'k)'k)n Ext,2 (k ,k )

is not a noetherian Ext* (k,k)-module. ~ EXtR2 (k ,k )

Indeed, if the modules in i)-ii) were always noetherian,

it would follow from Theorem]

of § I (formulae (7) and (8)) , that PR (Z) and pM (Z) were always rational. But in [14], Theorem 2.1 (and Corollaries 2.2 3and 2.3) C. 2 Jacobsson gives examples where this is not the case.(In these examples R 2 is even a ring with quadratic monomial relations : ) In order to prove Theorem 2 we will now first make a general analysis of the behaviour

of double Ext-algebras with respect to a Golod map. Thus let R

> S

be any Golod map ([25], § 3) and recall (loc.cit.) that we have an "exact sequence of Hopf algebras":

(11)

k

>

T ....

>

Ext~(k,k)

>

Ext~(k,k)

> k

where T is the free associative graded algebra on the graded vector space { Ext

(S,k) }i>2 " Thus the global homological dimension of T is

1 and the diagonal

is the natural one. We want to compare the ring theoretical properties of the

107

algeNras Ext~(k,k)_ and EXtR(k,k) (and also for later) purpose(s)

(in particular their Ext-algebras)

and for this

it is convenient to study first a very general

situation of extensions of graded, connected, cocommutative Hopf algebras: (12)

k

Recall [22],[13],[9] if

B --~"

i.e.

i

> B

0

> C

......... > k

that we say that (12) is an exact sequence of Hopf algebras

> C is onto, and if i is the kernel of j in the category of coalgebras,

i induces an isomorphism A ---~

of coalgebras k @A B

> A

~

> k D C B , where Q is the cotensor product

[20]. Then it follows that we have also an algebra isomorphism

> C ( and

B @~

N

> C ). Now a cocommutative Hopf algebra can be

considered as a group in the category of cocommutative coalgebras. very natural to expect that corresponding to the extension a Hochschild-Serre

It is therefore

(12) there should be

spectral sequence, relating the (co)homology of B with the

cohomology of A and C. This is indeed the case, and the spectral sequence has already been studied in [15],[23],[22],[21]

and [18]. Our aim now is just to state all these

results in a form that is the most convenient one for our applications

J

the next section: Consider the ring map B left B-module. situation (13)

> C

in this and

in (12) and let V be a graded

Consider the change of rings spectral sequence associated to this

([8], Chap.XVl):

E~ 'q = Ext~(Tor~(C,V),k)

~

Ext~(V,k)

(of course there are many other variants of this spectral sequence). Since C = k @A B and B is flat (even free) as a left A-module ([20], prop. 4.4) it follows that (14)

Tor~(C,V) = Tor~(k ~AB,V) ~ ~Tor~(k,V)

which, combined with (13) gives the following "Hochschild-Serre

k)

spectral sequence":

xt IVk)

Note that we have suppressed the extra grading coming from the grading in A,B,C and V in (13)-(15). On what follows we will need the following more precise results about the operations of Yoneda Ext-algebras

on the spectral sequences

(13) and (15).

The proofs of these results can be found in Ming [21]. First recall that the Yoneda @

product makes EXtB(V,k) into a left module over Ext~(k,k). induces a ring map Ext~(k,k)~ left Extc(k,k)-module. .

A

Extc(TOrq(k,V),k)

> Ext<(k,k)~

Since B - ~ - - >

C

, it follows that Ext~(V,k) is also a

In the spectral sequence (15) (or (13)...) the E2-term

is also a left Ext~(k,k)-module by means of the Yoneda product.

How is the spectral sequence (15) (or (]3)...) related to these Ext~(k,k)-module structures? Here is the answer (Ming [21] Theorem 2.7, p. 238): There are structure maps: Ext~(k,k) @ EP'qr

@ r

>

EP+S'qr

108

such that: i)

02

is exactly the Yoneda product Ext~(k,k) 8 Ext~(Tor~(k,V)~k) .....

> Ext~+S(Tor~(k,V),k)

ii) The differential dr of (15) is a left graded Ext~(k,k)-module homomorphism, and @r+1 is induced from Qr by passing to cohomology. iii) Let { FPExt~(V,k) }p~O

be the decreasing filtration of Ext~(V,k), corresponding

to the spectral sequence (15), and let pp be the natural quotient map defined by: 0p 0 > FP+IExt~(V,k) > FPExt~(V,k) > Ep'n-p~ > O With these notations the following diagram is commutative: yFP+SExt~+S(v,k

Ext~(k,k) ~ FPExt~(V,k)

i inclusion

Id 8 inclusion

.> E~+s,n-P

Extc(k,k) ~ Ep'n-p

Here Y" is induced by the Yoneda product and the natural algebra map

j*

Ext~(k,k)

> Ext~(k,k) , so that the following diagram is commutative: .8

J ~ Id Ext~(k,k) ~ E x t ~ ( V , k ) - - ~

Yoneda > Ext~+S(v,k)

Ext~(k,k) ~ Ext~(V,k)

Id ~ inclusion

inclusion

y" Ext~(k,k) ~ FPExt~(V,k)

>

FP+S~ _n+s~ ~ ~x~ B Iv,K)

Summing up, one can say that the spectral sequence (15) is compatible with all the *

left Extc(k,k)-module structures in sight. We will use this in some special cases: THEOREM 3.- Let k

>A

>B

>C

>k

be an extensi0n of cocommutative 6raded connected Hopf al~ebras such that A is is a free al~ebra (i.e. gldim A = I). Then, for each ~raded left B-module V, we have an exact sequence of left Ext~(k,k)-modules: B * .-I(TorB(c,V),k ) --> .. (16)..--> Ext .-2 c (Tor](C,V),k) --> Extc(C@BV,k) --> Ext~(V,k) --> Ext c . PROOF: The isomorphism (14) gives that the Ep'q of (13) (or (15)) are zero for q > 1 2 if gldim A = I. Therefore, in this case the spectral sequence (13) degenerates into a long exact sequence (16). The assertions about the left Ext$(k,k)-module structure are just reformulations of Ming's results, quoted above, in this special case.

109

Remark.- Theorem 3 can be applied to the Hopf algebra extension a Golod map R

> S. In this case A =

graded vector space { EXtR-l(s,k)

(I I ), coming from

T = the free associative graded algebra on the

)i>2 [ This graded vector space will be henceforth

be denoted by s-l~TR(S,k ) , i.e. the "suspension" of the elements of degree > 0 in ExtR(S,k).] , *

R = Ext~(k,k) and C = EXtR(k,k).Furthermore , for V = k, the isomorphism

(14) becomes: (17)

TorIB(C,k) --~ TorA(k,k) ~ s - 1 ~ ( S , k )

and here the left C = ExtR(k~k)-module

structure on Tor (C,k) corresponds to the

left EXtR(k,k)-modul e structure on s - 1 ~

(S,k), defined by Yoneda product

[18].

Therefore we have: COROLLARY.- Let R left Ext* E xtR(k,k) *

q) > S be a Golod map. Then we have a lon~ exact sequence of (k ,k)-modules :

> Ext*-2 (s-1~(S,k),k) Ext~(k,k)

> Ext* (k~k) Ext~(k,k)

)

(~8) <

> Ext* * Exts(k,k

where map

a 1 1

(k,k) - - >

t h e module s t r u c t u r e s

Ext *-I (s -l-----* EXtR(S,k),k ) - - > Ext~(k,k)

...

a r e d e f i n e d by Yoneda p r o d u c t s and by t h e a l ~ e b r n

Here is another application of the spectral sequence

(13) (or (15))and the Ming theory:

THEOREM 4.- Let (19)

k

> A

> B

~ C

> k

be an extension of cocommutative graded connected Hopf algebras such that gldim A =N<~. , B Suppose that V is a ~raded left R-module such that the Extc(Tort(C,V),k ) are

noetherian

(coherent) Extc(k,k)-modules

Yoneda product).

for 0
Then Ext~(V,k) is a noetherian

the al~ebra map Ext~(k,k)

(coherent) Ext~(k,k)-module,

via

> Ext~(k,k) and the Yoneda product.

Remark.- Recall that a
(graded) submodule is finitely presented.

It follows that

modules are coherent, and that the kernels, cokernels and

images of module maps between coherent modules are again coherent. A ring is called coherent if it is coherent as a module over itself. For more details, cf. e.g. [25] and the literature sited there.

110

PROOF OF THEOREM 4: We consider the spectral sequence (13), whose E~'q-term is Ext~(Tor~(C,V),k).

Since gldim A = N < ~, it follows from (14) that E~ 'q = 0 for

q > N. From the Ming theory (i)-(iii) above, it now follows that for each t (0 < t < N), the

( Fl-tExtB (V'k) }i>O (put F s = F ° if s < O) are sub-Extc(k,k)-

modules of Ext~(V,k). Denote these submodules by F*-tExt~(V,k). F*-tExt~(V,k)/F*-(t-1)Ext~(V,k)

= E~ -t't

for 0
Then we have

(Note that F*-NExt~ = Ext~

and that F*+IExt~ = 0.) It is therefore sufficient to prove that each E *-t't < . . (O~t~N) is a noether~an (coherent) EXtc(k,k)-module. To prove this, it is sufficient .-t,t .-t,t *-t,t to prove that each E 2 (note that EN+ 2 = EN+ 3 = ... = E~ -t't) is a noetherian (coherent) Ext$(k,k)-module.

But this is exactly the hypothesis

of Theorem 4 (modulo

a translation of degrees)~ and the Theorem 4 is proved. Here are some applications of Theorem 4 (and T%eorem 3): COROLLARY

I.- Let B = Ext~(k,k) where R is a local complete intersection.

Let V be

a finitely ~enerated sraded left B-module. Then Ext~(V,k) is a noetherian left Ext~(k,k)-module.

(In particular Ext~(k,k) is a noetherian

(bi)graded algebra.)

PROOF: It is known (cf. e.g. [13] or [4]) that B sits in the middle of an exact sequence (19), where C is an exterior algebra (on a finite number of generators of degree I) and where A is a commutative polynomial algebra on a finite number of generators of degree 2. Thus gldim A < ~, and we are in the situation of Theorem 4. We start by proving that the Ext~(k,k)-modules

Ext~(Tor~(C~V),k)

are noetherian.

But sinc~

C is an exterior algebra on finitely many generators of degree I, it follows that Ext$(k,k)

is a bigraded commutative polynomial algebra on finitely many variables

of bidegree

(1,1). Therefore Ext~(k,k)

is noetherian.

Every graded C-module L that

is of finite length (or finitely generated - that is the same thing here) over C is an iterated extension of a finite number of trivial C-modules k (placed in different degrees). The exact sequence of Ext,( noetherian Ext~(k,k)-module.

,k), therefore gives that Ext~(L,k)

is a

Now V is a finitely generated B-module and therefore

also a finitely generated A-module (this is a general assertion about exact sequences •

(19), where C is finite-dimensional).

a flnite-dimensional

Therefore Tor~(C,V) ~ Tor~(k,V) is B

•

vector space, so that Tort(C,V) •

*

•

•

ms a C-module of flnlte length, B

and therefore, by what we have just proved, Extc(Tort(C,V),k)

.

is noetherlan over

Ext~(k,k). Now Theorem 4 implies that Ext~(V,k) is a noetherian bigraded module over Ext~(k,k) via the ring map Ext$(k,k) - - >

Ext~(k,k) and the Yoneda product. But this

assertion is even stronger than the assertion in Corollary

I which is therefore proved.

Remark.- The fact, just proved, that Ext~(V,k) is a finitely generated bigraded module W

over Extc(k,k) , implies (Use the Lemma of § I!) that the double Hilbert series of Ext~(V,k) is equal to POlv(X,Y)/(I-xy)N

.

111

J

COROLLARY 2.- Let R be a local complete intersection, and let R

> S be a

Golod map. Let V be a left finitely presented graded Ext~(k,k)-module. TTlen Ext* (V,k) is a noetherian left Ext* (k,k)-module, via the natural Ext~(k,k) Ext~(k,k) j** ring map Ext* (k,k) > Ext* (k,k) and the Yoneda product.

Ext~(k,k)

Ext~(k,~)

PROOF: We have an exact sequence of Hopf algebras:

(2o)

k

> T

> Ext~(k,k)

> ExtR(k,k)

> k

where T is a free algebra. Apply Theorem 3 to this sequence (20)! Using at some places the short notation B =

Ext~(k,k)and

C = Ext~(k,k) we obtain (a part of (16)) the

following exact sequence of Ext* (k,k)-modules: Ext~(k ,k) (21) ..--> Ext* (C~BV,k) --> Ext* (V,k) --> Ext *-I (Tor~(C,V),k) -->. Ext~ (k,k) Ext ~(k,k ) Ext~(k ,k ) Now V is a finitely presented left B-module and therefore we have an exact sequence (22)

0

> W

> F

----> V

- - >

0

of left B-modules, where F is a finitely generated and free B-module and where W is a finitely generated B-module. Tensor (22) with C! We obtain an exact sequence of left C-modules: (23)

0

> Tor~(C,V)

> CSBW

> CSBF

> C@BV

> 0

Here CSBF is a finitely generated free C-module. Therefore V ° = CSBV is also a finitely generated C-module. I claim that V I = Tor~(C,V) is also a finitely generated C-module. Since W is a finitely generated B-module, it follows as before that CSBW is a finitely generated C-module. But since C = Ext~(k,k) is noetherian (R is a local complete intersection) we have that C@BW is noetherian, and therefore its submodule Tor~(C,V) (use(23)!) is finitely generated.

Applying Corollary I to the

finitely generated EXtR(k,k)-modules V ° and VI, we obtain that EXt:xt~(k,k)(Vi,k) * (i=0,I) are both noetherian Ext* (k,k)-modules, and therefore the middle term Ext ~(k ,k ) of (21) is also so, and the Corollary 2 is proved. COROLLARY 3.- Using the notations and hypotheses of Corollary 2, we have that Ext~(k,k] is a (graded) coherent al@e~or_aa. (Left coherence and right coherence are equivalent since Exts(k,k) is a Hopf algebra.) Furthermore, for each finitely 6enerated S-module M, we have that the left Ext~(k,k)-module Ext S* (M ,k ) is coherent. PROOF: Put B = Exts(k,k). Recall that we proved in Corollary 2, that if V was a finitely presented left B-module [ i.e. if ,limk(TOr~(k,V)) < ~, 0~i~I, or , equivalently, if the dual vector spaces Ext~(V,k) had finite dimension for 0
Ext~(k,k)

(k,k)-module. But

112 this last (bigraded) algebra has a finite number of generators, complete intersection.

since R is a local

From this one easily sees that dimkExt~(V,k)

too, so that Ext~(k,k) is coherent

< ~ for all i>2

[25]. The last part of Corollary 3 now follows

from Theorem I of [23], and therefore Corollary 3 is completely proved. Remark.- Taking M = S in Corollary 3, we obtain that k is a coherent presented)

Ext~(k,k)-module.

(thus finitely

Therefore Corollary 2 for V = k shows in particular

that Ext* (k,k) is a finitely generated Ext~(k,k)

(bigraded) algebra. This will be

applied in: COROLLARY 4.- Let S be a local ring that comes from a local complete intersection by a Golod map, and let S

~P > S"

be a second Golod map. Then Ext*

..... a noetherian

(k,k) i_~s

Ext~.(k,k)

(bigraded) algebra.

PROOF: Apply the middle part of the exact sequence

(18) of the Corollary of Theorem 3 .

to the Golod map S

> S ". We obtain an exact sequence of Ext

(k~k)-modules Ex't~(k,k)

(24)-,--> Ext*

Ext~(k,k)

(k,k)

~** > Ext*

(k,k) - - >

Ext~.(k,k)

Ext *-I

(s -I=-7,* EXts(S ,k), k )-- > .

Ext~(k,k)

Now according to Corollary 3 and the Remark following it, both Ext~(S'~k) and k are coherent Ext~(k,k)-modules.

Thus s - 1 ~ ( S ~ , k ) "

is also a coherent Ext~ (k ,k )-module,and

therefore Corollary 2 implies that Ext* (s-IE--x~(S~,k),k) is a finitely Ext~(k,k) generated Ext*

(k,k)-module.

This last ring is noetherian

(Remark following

Ext~(k,k) Corollary 3) and therefore Ext* (k,k) sits between two noetherian modules in Ext~(k,k) the exact sequence (24). Thus Ext* (k,k) is a noetherian Ext* (k,k) Ext~.(k~k) Ext~(k,k) module, and ~ fortiori Corollary 4 is proved. PROOF OF THEOREM 2: This is now immediate:

a) and c) follow from Corollary 2 and

Corollary 3 of Theorem 4 and b) Zollows from Corollary 4 of the same Theorem 4. § 3. RINGS WITH MONOMIAL RELATIONS. THEOREM 5.- Let k be a field, k[X],...,X n] the (commutative) polynomial rin 6 in n variables, let MI,...,M r be monomials in the Xi:s , and let R = k[XI,...,Xn]/(MI,...,M r) Then Ext* (k,k) is a (bigraded commutative) noetherian ring. Ext~(k,k) Remark I.-

The R of Theorem 5 is not local in general, but it has the same Ext-

algebra as the corresponding local ring k[[XI,...,Xn]]/(MI,...,M~) , and so the preceding theory can be applied to R and related rings.

113

Remark 2.- Since each variable X. defines a grading on R, it follows that R is 1 n-graded and that Ext* (k,k) is (n+2)-graded. In the course of the proof Ext~(k,k) of Theorem 5 we will obtain a more precise result about how the finite set of generators of Ext*

(k,k) can be chosen.

EXtR(k,k) PROOF OF THEOREM 5: Consider first the case where all the M.:s are squarefree. Fix a I t, I < t < n, and consider S t = R/(X t) = k[X I ~''" ,Xt,...,Xn]/(those

M.:s, where X t does not occur) j

(" means that the corresponding variable is omitted). For those Mi:s , where X t does occur, we write M i = XtM E (note that there is no X t in M E , since all Ms:S are squarefree).

Let --at be the ideal in St, generated by the images of these M E. Then

R = St[Xt]/a_tXtSt[Xt].

Of course we can suppose that all Mi:s have degree ~ 2, and

then ~t is generated by elements of degree ~ ]. Writing for simplicity S = St, ~ = ~t and X = Xt, we therefore have a Golod map: (z5)

six]

six]

> ~.x.s[x]

= R

(of course (25) is also a Golod map for more general S:s and ~:s). We now apply the Corollary of Theorem 3 to the Golod map (25)~ and we obtain from (18) the following long exact sequence of Ext*

(k,k)-modules:

Exts[x](k,k)

.... Ext

q0** (k,k) -->Ext* (k,k) -->Ext *-I (s-IE-~-* (R,k) ,k) EXts[x] (k,k) EXtR(k ,k) EXts[x] (k,k) S[X]

(26)

<

!

>Ext *+I

(k,k)

>..-

~Ex~[X] (k ,k ) But EXts[x]* (k,k) = Exts(k,k) variable T of degree I

(27)

(T 2 = 0). rl~erefore

Ext* *

(k,k) = Ext*

EXts[x] (k,k) and ExtE(T)(k,k) bidegree

~kE(T), where E(T) is the exterior algebra on one

(k,k) ik

Exts(k,k)

*

(k,k)

ExtE(T)

= k[V], i.e. the commutative polynomial ring on one variable V of

(1,1). Thus (26) is an exact sequence of graded modules over

EXt~xts(k,k)(k,k)

~kk[V]. In particular

(26) inherits a grading from X and the

operations of V are compatible with this grading.We claim that V operates on the four modules surrounding Ext* (k,k) in (26)~ as it does on (27), i.e. Ext R (k ,k )

114

I) multiplication 2) each element of X-degree

U > I is a multiple

by V of an element

of the module

U-I.

Of course this

of (27)

by V is a monomorphism, of X-degree

is clear for Ext* (k,k) and Ext *+I (k,k), Ext~rx1(k,k) * (k,k) bL ] EXts[x]

in view

(we may even take U > I in (27)). But, since

EXts[x]~x._--~.S[xI,k)

~ s

Exts[x]tX-~-S[X],k)

this is also true for Ext *-I ,

~ s -1 Ext~(~,k)

8kHOmk[x](X'k[X],k)

-* (s - I -Exts[x](R,k),k) , and Ext*-2(s -I

,

...

Exts[x](k,k) Now it follows v = ~**(V),

easily,

using

I) and 2) and the exact

then every element

~ of X-degree

form ~ = v.~ ~ ~ where ~" has X-degree Returning

to the old notations

u-1. Furthermore

result

(n+2)-multihomogeneous

{ 6 Ext* *(k,k) ~xtR( k ,k )

products

of v.:s

I is at the verify

[ each v. has

( + 2 ) nd place

result

(n+2)-multidegree

] with elements

nonzero

algebra

is unique.(Use

the 5-1emma!)

the notation

is a linear combination

(1,1,0..,I,..,,0),

space,

of the bar resolution

of

where the last (al,...,a n)

spanned by these last elements

is an easy consequence

of the classical

- that if B is a non negatively

over k, then the graded vector

elements

~

whose last n ~mltide@rees

This last assertion

- proved by means

connected

if we put

for all t (I < t < n) we obtain that each

0 < a. < I (I < i < n). But the linear

is finite-dimensional.

(26), that

X t = X, S t = S and ~t = a, introducing

v t = v, and using the previous element

sequence

u > I in Ext* (k,k) is of the Ext~(k,k)

of degree < p. Thus Ext* EXtR(k,k)

graded,

spaces TorB(k,k) can contain no P (k,k) has a finite number of

generators,

if the M.:s are squarefree. Note that if one of the variables X.:s J J in all M.:s, then we do not need any extra elements above ~ith last n

absent

is

1

multidegrees

(al,...,a.j_1,1,aj+ I , . . . .,an)

We now pass to the general Xi:s

(we assume,

of course,

there are still canonical of multidegree

1

is at the

independently

due to FrSberg

(i+2) nd place

and Weyman

to a ring with lower m I . We can assume

iI that M I = X I M~ . . . . .

X I does not occur in Mk+1, =

-the last

(corresponding

to X i) (~ust use

Let m. = the maximal exponent of X. among the M.:s. l i j I, then we are in the preceding situation. Assume therefore m I > I.

reduce ourselves

R"

v. 6 Extl (k,k) l Ext~ (k ,k )

of Extl).

Using a procedure,

necessary!)

elements

(1,1~0,...,I,...,0)

the interpretation If all m i ~

case, when some of the M.:s might contain squares of the J that all deg(Mj) ~ 2). First of all it is clear that

, i I > 0 .....

... ' Mr • Now introduce ii-I ik-1

k[Xo,X 1 . . . . . Xn]/(XoX 1

Now Xo-X I is a non-zero

ik Mk = X I ~

divisor

M~. . . . .

XoX1

([10], p. 30), we will

(renumber the M:s if

~'

ik > 0 , but that

a new variable

~+1 . . . . .

X o and put

Mr)

in R" and R~/(Xo-X I ) ~ R . Thus we have a "large" map

115

R" ~'j > R in the sense of Levin [17]. It follows that the Hopf algebra map Ext~(k,k) - - >

EXtR.(k,k) is a monomorphism, and that (of. loc. cit. p. 212)

we have an isomorphism of left Ext~,(k,k)-modules: (28)

Ext~.(k,k) @

k ~

EXtR.(R,k)

mt~(~ ,k) NOW Ext~.(R,k) = 0, i > I and Ext~.(R,k) ~ k, 0 < i < I. Therefore we have an exact sequence of EXtR.(k,k)-modules (with trivial operations on k and s Ik): 0 -->

s-lk

> Ext~,(k,k) 'Z

> k

> 0

which gives rise to a long exact sequence of Ext* (k,k)-modules: Ext~.(k,k) .-- Ext * (k,k) w--->Ext* (s-Ik ,k) (Ext~(R,k),k)-->Ext* Ext~.(k,k) Ext~.(k,k) Ext~.(k,k)

(29)

<

-

6 > Ext *+I

Ext~.(k,k)

(k,k)-->" • "

Inview of the formula (28) and the fact that Ext~.(k,k) is Ext~(k,k)-free, we obtain that the map w* can be identified with the ring map (30)

Ext* (k,k) EXtR.(k,k)

J-->

Suppose now, inductively, that Ext*

Ext~.(k,k)

(29 ~) that Ext*

*

EXtR.(k,k )

Ext*

Ext~(k,k)

(k,k)

(k,k) is noetherian. It then follows from

(Ext~(R',k),k) is a noetherian Ext*

Ext~.(k,k)

(k,k)-module, since

it is an extension of two such modules. Using the identification of w* with j** , we now obtain that Ext*

ExtR(k,k)

(k,k) is a noetherian Ext* (k,k)-module, and Ext~.(k,k)

therefore, ~ fortiori, it is a noetherian ring. Thus Theorem 5 is ~roved[ However, we wish to continue and obtain a more precise result about where the generators of Ext*

(k,k) are situated. Here is the result we are aiming at:

EXtR(k,k) THEOREM 5".- Let R = k[Xl,...,Xn]/(MI,...,M r)

where the M.:s are monomials in the

X.:s (of degree > 2) let v. be the element of multidegree (1,1,0,...,I ...,0) in i -~ i ....................... ' -Ext I (k,k) (corresponding to X. and defined in general above) mud let • 1 EXtR(k,k) m i be the maximal exponent of X i in the M :s. Then: a) Th___~e<(n+2)-multi6raded algebra Ext*

Ext~(k,k)

(k,k) is generated by Vl,...,v n and

(necessarily) finite number of elements having last multidegrees (al,...,an) , with 0 J a i j m i (I < i < n).

116

b) Multil ~ t i 0 n

by v i i__nExt* (k,k) ~ives an isomorphism of the space EXtR(k ,k)

of elemenlts of de~ree (s,t,bl,...,bi,...,b n) onto the space of elements of de~ree (s+1,t+1,b I

,bi+1,.

,bn)

provided b.>m.

Therefore the (n+2)

-

variable Hilbert series of our (n+2)-~raded double Ext-al~ebra is: n p(ZI~Z2,YI~...~Yn)/ ~(I-Z~Z~Y.) where p(ZI~Z2~Y I ,Yn ) is a polynomial i=I I z i ' '" " " n+2 variables with non-negative

integral coefficients,

in

where furthermore

for any monomial ..... -IziIzl2vJ-2 -I I . . .y~n

with non-vanishin~ coefficient n j]<ml .... ,Jn<mn and max(il,i 2) < Z Ji " i=I

we have

PROOF OF THEOREM 5~: The theorem follows already from the proof of Theorem 5 if all m i < I. Let us assume m I > I etc. as above and consider R ~ Clearly R ~ is (n+1)-graded by grades grading) and the ring map R ~

> R = R'/(Xo-X~)

(bo,bl,...,b n) (we use Xo too, to define the

> R is n-multihomogeneous

(bo+bl,b2,...~b n) on R ~. Those elements

in Ext I Ext R.

if we use the n-grading (k,k) that are analogous to

(k ,k)

the v.:s for R will be denoted by v~. (0 < i < n). The long exact sequence J (n+2)-multihomogeneous, zero

scalar multiple

and it is easily seen that 6 is the multiplication

of v ° - v~

.

(29) is by a non-

If we c o u l d p r o v e that v" - v~ w e r e a n o n - z e r o o

divisor in Ext*

(k,k), it would follow from (29) that we have an (n+2)-graded

Ext R.(~,k) exact sequence of Ext* (k,k)-modules: EXtR~ (k ,k ) v °

(31) 0 - >

-

v~

Ext *-I (s-lk,k) - - ~ EXtR~ (k ,k )

and thus all assertions that m~ = I, m~ = m I

.

Ext Ext R. (k ,k)

(k,k) -->Ext* (k,k)-->O EXtR( k ,k )

in Theorem 5 p would follow by induction from (31) (recall I,

m E = mi, i > I ).

We therefore now endthe proof of Theorem 5 p by proving that v~ - v~ divisor. Assume the contrary,

i.e. that there is a non-zero c 6 Ext*

such that (32>

is a non-zero(k,k)

Ext~(k,k) (v~ - v ~ ) . c

We decompose c into homogeneous

= 0

components with respect to the degree

(denoted by

N

dego(

Z c~J )) defined by X ° : c = j=O

( CN # 0 ) "

'Fnus dego(Cj) = j ' dego(V~)=1 '

deg(v~) = 0 and it now follows by taking the deg ° -component of degree N+I of (32) that v~.c N = 0. But m~=1 and if N > O, it would follow from the inductive hypothesis o ~ (about multiplication by v~ ) that c N = O, which is impossible. Therefore N = 0, i.e. c = c o , and now (32) gives v~.c = 0 (take components of deg ° -degree 0 in (32)). But this is an equality of elements, whose deg ° -grading is zero. Now use the following trick of multihomogeneous

algebra:

Let R ~ o

be the subring of R', where

117

deg ° = O. We have (recall the notations

in the proof of Theorem 5 !):

R S = k[X I ..... Xn]/(Mk+ I ..... Mr) , where the Mk+ I ..... M r do not contain XI, so that R~o ~ A[XI]' where A = k[X2,...,Xn]/(Mk+ I~...,M r ). Clearly Ext* . (k,k) Ext R ~ (k ,k ) part of Ext

*

.

(k,k), where deg ° is zero. Since the e ~ a l l t y Ext~,(k,k)

o

= that

v~-c = 0 takes

place in this last part, and since R~ = A[XI]~ Ext* (k,k) = Ext* (k,k)[v~] Ext~(k,k) ExtX(k,k) o (v I is a polynomial variable), it follows that c = O, which is a contradiction and the Theorem 5 ~ is completely proved. Remark.- We had to work rather hard to get that 6 was a monomorphism or, equivalently, that the ring map (30) was an epimorphism.

There are reasons for that. Indeed, in

general, if (R',m ~) is a local commutative noetherian ring, x~C m_~ ~ ( ~)2 a non-zerodivisor, then R ~ j

J

> R'/(x ~) = R is still large [17], but it is not true that

zs an epimorphism or, equivalently that

(33)

Ext~ (k ,k ) Tor. (k,k) - - >

E x t ~ (k ,k ) Tot. (k,k)

is a monomorphism.

Here is a eounterexample,

due to Clas L6fwall and reproduced here

with his permission:

R" = k [ [ X , Y , Z ] ] / ( X Z -

y3)

,

Clearly X is a non-zerodivisor intersections,

R = R'/(X) = k[[Y,Z]]/(Y 3)

in R p. Furthermore, both R p and R are local complete

and their Ext-algebras

are generated by elements of degree I and

elements of degree I and 2 respectively

(34)

(thus x'= X).

Ext~(k,k) --

(cf,~.~.

[27]). Consider the inclusion:

> Ext~.(k,k)

and take an indecomposable generator T of degree 2 of EXtR(k,k). The image of T .

under (34) must be a decomposable of degree I. ~ u s ,

element, since EXtR~(k,k)

is generated by elements

already on the Tor I -level, the map (33) is not a monomorphism.

§ 4. FINAL REMARKS. OPEN PROBLEMS. It might be interesting to try to find other classes of local commutative noetherian rings R, for which Ext

(35 )

(EXtR(M,k),k) EXtR(k ,k )

is a noetherian Ext* (k ,k )-module for all finitely EXtR(k ,k ) generated modules M

PROBLEM I.- Could we "classify" those rings that satisfy (35) ? It is also possible to study the right EXtR(k,k)-modules *

EXtR(k,N)

(one of us did so

in [23] and Lescot also did so in [16]. This was applied to "Bass series" IN(z) = E dimk(EXt~(k,N))'Z i ~ i>O I

in [23] and [16]. In particular, Lescot has proved [16]

118

that the Bass series IR(z) is rational for any ring R with monomial relations as in § 3. He has proved this as a consequence of a very general theorem about rationality of P~(Z):s for multigraded M:s over such R:s, and indeed he has rationality of the multigraded version of PRM

(n+2 variables) too. There is probably a "double-Ext"

-

version of this, corresponding to the theory of § 3. Finally it should be remarked that the rings studied in § 3 contain the "StanleyReisner" rings (or "face" rings [28]) associated to a finite simplicial complex A . PROBLEM 2.- Give a combinatorial-geometrical interpretation of the coefficients in the rational function of n+2

variables in Theorem 5" for the case when R is the

Stanley-Reisner ring associated to a finite simplicial complex A. BIBLIOGRAPHY [I] ANICK, D.J.,A counterexample to a conjecture of Serre, Ann. Math., 115, 1982, 1-33. Correction: Ann. Math., 116, 1983, 661. [2] ANICK, D.J., Connections between Yoneda and Pontrjagin algebras, Lecture Notes in Mathematics, 1051, 1984, 331-350, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo. [3] ATIY~, M.F. and MACDONALD, ~.~.,T ~ Introduction to Commutative Algebra, AddisonWesley, Reading, Mass., 1969. [4] AVRAMOV, L., Local algebra and rational homoto#y, Ast@risque, 113-114, 1984, 1543. [5] AVRAMOV, L., Differential graded models for ~°cal rings, RIMS Kokyuroku, 446,1981~ 80-88, Kyoto Research Institute for Mathematical Sciences, Kyoto, Japan. [6] BACKELIN, J., Les anneaux locaux ~ relations monomiales ont des s@ries de Poincar@-Betti rationnelles, Comptes rendus Acad. Sc. Paris, 295, S@rie I, 1982, 607-610. [7] B~GVAD,R. and HALPERIN, S., On a conjecture of Roos, These Proceedings. [8] CART#~ H. and EILENBERG, S., Homological Algebra, Princeton Univ. Press, Princeton, 1956. [9] COHEN, F.R., MOORE, J.C. and NEISENDORFER, J.A., Torsion in homotopy groups, Ann. Math., I09~ 1979, 121-168. [10] FR~BERG, R., A study of graded extremal rings and of monomial r i n ~ , Math. Scand., 51, 1982, 22-34. [11] GOVOROV, V.E., Dimension and multiplicity of graded algebras, Siberian Math. J., 14, 1973, 840-845. [12] GULLIKSEN, T.H., A change of ring theorem with applications to Poincar@ series -and intersection multi P licit , Math. Scand. 34, 1974, 167-183.

119

[13] GULLIKSEN, T.H. and LEVIN, G., Homolo@y of local rings, Queen's Papers in Pure Appl. Math., n ° 20, Queen's Univ., Kingston, Ontario, 1969. [14] JACOBSSON, C., Finitely presented ~raded Lie al@ebras and homomorphisms of local rin~s,

J. Pure Appl. Algebra, 38, 1985, 243-253.

[15] LEMAIRE, J.-M., Al~bres connexes et homolo@ie des espaces de lacets, Lecture Notes in Mathematics, 422, 1974, Springer-Verlag, Berlin, Heidelberg, New York. [16] LESCOT, Th~se, Caen 1985

and letter from J. LESCOT to J.-E. ROOS, June 14, ]985.

[17] LEVIN, G., Large homomorphisms of local rin~s, Math. Scand., 46, 1980, 209-215. [18] LEVIN, G., Finitely 6enerated Ext ~ e b r a s ,

Math. Scand., 49, 1981, 161-180.

[19] MACLANE, S., Homolo@y, Springer-Verlag, Berlin, Heidelberg, New York, 1963. [20] MILNOR, J. ~ud MOORE, J., On the structure of Hopf a l~ebras, Ann. Math., 81, 1965 211-264. [21] MING, R., Yoneda products in the Cartan-Eilenber @ change of rin~s spectral sequence with applications to BP,(BO(n)), Trans. Amer. Math. Soc., 219, 1976, 235-252. [22] MOORE, J.C. and SMITH, L., Hopf algebras and multiplicative fibrations I-II, Amer. J. Math., 90, 1968~ 752-780 and 1113-1150. [23] ROOS, J.-E., Sur l~alg~bre Ext de Yoneda d'um anneau local de Golod, Comptes rendus Acad. Sc. Paris, 286, s6rie A, 1978, 9-12. [24] ROOS, J.-E., Relations between the Poincar6-Betti series of loop spaces and local rin6s, Lecture Notes in Mathematics, 740, 1979, 285-322, Springer-Verlag, Berlin, Heidelberg, New York. [25] ROOS, J.-E., On th e use of ~rade d Lie algebras in the theory of local rin~s, London Math. Soc. Lecture Notes Series, 72, 1982, 204-230, Cambridge University Press, Cambridge. [26] S6minaire H. CARTAN, 11 e ann6e 1958/59, Invariant de Hopf et op6rations cohomolo@iques s6condaires, Paris, Seer. Math., 11 rue Pierre Curie, Paris 5,1959. (Has also been published by Benjamin, New York.) [27] SJODIN,G., A set of ~enerators for EXtR(k,k) , Math. Scand. 38, 1976, 1-12. [28] STANLEY, R.P., Combinatorics and Commutative Al6ebra,Progress in Mathematics, vol. 41, 1983, Birkh[user, Boston, Basel, Stuttgart. Department of Mathematics University of Stockholm Box 6701 S-113 85 STOCKHOLM (SWEDEN)

ON A C O N J E C T U R E OF ROOS

by

Rikard B~gvad and Stephen H a l p e r i n

i.

Introduction.

T h e o r e m A:

In this paper we prove the f o l l o w i n g two theorems:

Let R be a local c o m m u t a t i v e

is noetherian. T h e o r e m B:

ring whose Y o n e d a E x t - a l g e b r a

Then R is a c o m p l e t e intersection.

Let S be a 1-connected finite CW c o m p l e x and suppose the

P o n t r j a g i n algebra H,(~S;~)

is noetherian.

all but finitely m a n y degrees.

Then ~,(S)Q~ vanishes

(S is e l l i p t i c - cf.

T h e o r e m A was a q u e s t i o n of Roos

[14].

in

[8].)

T h e o r e m B is its trans-

lation to t o p o l o g y via the standard d i c t i o n a r y

([3],[4]),

and was

posed by Roos in [13]. The main tool in the proof is Sullivan's notion minimal models,

([12],[15])

d e f i n e d by him for the study of t o p o l o g i c a l spaces,

and adapted by Avramov

([3]) for the study of local rings.

The first key ingredient

is the n o t i o n of "category" of a m i n i m a l

model.

This was i n t r o d u c e d by F e l i x - H a l p e r i n in [7] for S u l l i v a n

models,

and shown to c o i n c i d e w i t h the c l a s s i c a l d e f i n i t i o n of

L u s t e r n i k - S c h n i r e l m a n n category. as well

of

(in sec.

Here we adapt it to A v r a m o v ' s models

2).

The second i n g r e d i e n t is the fact that a m i n i m a l m o d e l d e t e r m i n e s a g r a d e d Lie algebra whose u n i v e r s a l e n v e l o p i n g algebra is c l o s e l y related to the E x t - a l g e b r a or P o n t r y a g i n algebra the n e c e s s a r y facts in sec. In sec.

([3],[1]).

We recall

3.

4 we combine these ingredients to prove a single t h e o r e m

about models of w h i c h both T h e o r e m s A and B are corollaries.

In sec.

5 we deduce a t h e o r e m on graded Lie algebras. We thank L. Avramov and C. L~fwall for m a n y helpful discussions.

2.

The c a t e g o r y of a m i n i m a l model.

Let X = p ~ p

space over a field k

(possibly of c h a r a c t e r i s t i c

d e g x = p and

For n~0 we put X(n)=!pT>_nX p-

Ixl=Ipl.

be a g r a d e d vector >0).

If x~X

P

we say

By AX we shall mean the tensor p r o d u c t of the e x t e r i o r algebra on Xod d w i t h the s y m m e t r i c algebra on Xeven. where

APx=xA...AX

Then AX= • APx; w h e r e p!0

(p factors).

We shall use m i n i m a l model to m e a n a DGA of the form which

(AX,d)

in

121

(i)

X=X>0 or X=X<0 , and each Xp has finite dimension.

(ii)

d is a d e r i v a t i o n

(iii)

There is a homogeneous

(iv)

and such that dxiEA{x I ..... Xi_l}. ImdcA!2X.

The basis

of degree-l,

and d2=0.

basis,

{xi} , of X with

{xi} will be called a KS basis

Given such a model and some mll,

for

IXi_ll~Ixil

(AX,d).

the projection ~m:AX÷AX/A

endows the quotient with a differential

d.

It is straight

>m

X

forward to

factor ~m as the composite of homomorphisms (2.1) (AX,d) i~(AX@AY,D) p.....(AX/A . >m X,d), where Y is a graded space and (i) (ii)

Y=Y>0 (resp. Y=Y<0 ) if X=X>0 (resp. X=X<0). i is the obvious inclusion; D is a degree-I derivation; D2=0.

(iii)

Y admits a homogeneous

(iv)

AX®A{Y 1 .... ,Yi_l } • H(p) is an isomorphism.

Following 2.2

basis

{yi } with DYiE

[7] we make the

Definition.

The category,

cat(AX,d),

of a minimal model

least m (possibly ~) such that there is a sequence

is the

(2.1) admitting

a

morphism r : (AXeAY,D) ~

(AX,d)

with ri=id. 2.3

Lemma.

factors)=0,

(i)

If cat(AX,d)<_m then H+(AX).....H+(AX)

(ii) Proof.

If Hi(AX,d)=0

(i) is immediate.

For

for

iil>n then cat(AX,d)!n.

(ii) we note that there is a d stable

acyclic ideal IcAX such that Ii=(AX) i for ~:AX+AX/I

(m+l

and any cycle in A>mX is a boundary.

then factors through

In"

lil>n.

Decomposing

The projection In as in (2.1) we

factor ~ as AX

AXeAY~AX/I Because H(~)

is an isomorphism

and ~ is surjective

an induction argu-

ment on the basis Yi of Y shows one can lift ~ through

0 to get

r : AX®AY~AX. For this paper we need an e l e m e n t a r y version of the mapping theorem

[7; T h e o r e m

of the p o s s i b i l i t y (AX,d)

5.1].

The proof of

that char k>0.

[7] needs m o d i f i c a t i o n

because

To state the result we note that if

is a minimal model and if we divide by the ideal generated by

elements xCX with

Ixl

122 (AX(p),d). 2.4

Proof. for

The result we need is

Proposition.

cat(AX,d)

~ cat(AX(p),d)

Let Z be the span of the xi(i!2),

(AX,d).

(AZ,d).

Thus AX=A(Xl)®AZ

It is clearly

Let

Let

F(u) denote

by x I gives a minimal

to prove cat(AX,d)

(u) be a one dimensional

degxl+l.

where {xi}i> 1 is a KS basis

and dividing

sufficient

for all p.

vector

the exterior

model

~ cat(AZ,d).

space with basis u and degu=

algebra

on u if deg u is odd;

otherwise F(u) is the graded/, algebra with basis {7Pu}p!0 such that 71u=u, 7 P u .T q u = [P~q) TP+qu and degyPu=pdegu. Extend ( A X , d ) t o 7°u=l, a DGA

(AX®F(u),6)

(AZ,d)

factors

by setting'" ~(~Pu)=xl®TP-lu.

(AX,d)+

as (AX,d)

with

The projection

¢(7Pu)=O,

p>0.

~ (AX®r(u),~)-L

Because

(A(Xl)@F(U),6)

CAZ,d) is acyclic,

H(¢)

is an

isomorphism. Now suppose

cat(AX,d)=m.

Then for a suitable

factorlzation

(2.1)

of ~m we get morphisms (AX, d) -i--+(A X®AY, D) r-~ (AX, d) (AX/a>mX,d) with

H(p)

an i s o m o r p h i s m

®AxI(AXI@F(U),~)

and r i = i d .

,

P u t A=AX/A>mx a n d a p p l y

to get

(AX®F(u),6)

i' ;(AX®AY®r(u),D ') r' )(AX®F(U),6

J o'

(2.5)

A®r(u),6'), Clearly

Let s P = ~ o ( ¥ J u ) . A simple induction on p then 3: shows that each restriction (AX@AY®S p D') ~ (A®sP,6 ') of ~' gives an isomorphism

r'i'=id.

in homology.

Let IcA®F(u) Z, x[ and XlU if

Hence H(p')

IXll is even.

definition

of minimal

by I gives

a DGA with homology

degrees.

is an isomorphism.

be the ideal generated models

Thus the inclusion

by Z if IXll

In either

implies

is odd,

case condition

that 6'(I)cI.

~ in degree

Iek ~ A®F(U)

(iv)

Moreover,

and by in the division

zero and zero in the other induces

an isomorphism

in

homology. Similarly

if we set J=(p')-~(I) J@~ ~ AX®AY®F(u)

are homology

a morphism phism).

the composite

isomorphism.

we obtain

that

~ AX®F(U)

¢ : K~k ~ A X ® F ( u ) 9 ~ A Z

By induction

o : AZ + K~k such that ¢o=id

Thus

K~

isomorphisms.

In particular, tive homology

and K=(i')-1(J) and

(2.5) yields

is a surjec-

on a KS basis of AZ we obtain (and hence H(o)

the DGA diagram

is an isomor-

123

AZ' io ~J@k Cr' p,,

@ " the restriction of jective,

p'

Again H(p")

,AZ

is an isomorphism,

p" is sur-

and (~r')o(io)=id.

But by c o n s t r u c t i o n

I.I ..... I (m+l factors)=0.

tors through the p r o j e c t i o n AZ~AZ@AW÷AZ/A>mZ

AZ~AZ/A>mZ.

Thus p"i'a

If this is d e c o m p o s e d

as in (2.1) then the induced map AZ@AW+ISk

through

p" to a DGA m o r p h i s m AZ®AW+J~k.

desired

retraction

facas

lifts

Composing with ~r' yields

the

AZ@AW+AZ. O

Next observe that the proof of in our context 2.6

2.1] applies verbatim

to imply

Proposition.

dimX=~,

[8; T h e o r e m

If (AX,d)

is a m i n i m a l model with cat(AX,d)<~

and

then there is a sequence of odd integers n i such that d i m X n i ~

as i ~ . There follows an o b s e r v a t i o n 2.7

Corollary.

dimX=~,

If (f~,d)

of F e l i x - H a l p e r i n - T h o m a s

(unpublished).

is a minimal model with cat(AX,d) <~ and

then for some p, H,(AX(p),d)

is non-zero

in infinitely m a n y

degrees. Proof:

Put m=cat(AX,d).

and dimXk~m.

Now set p=Ikl+l. many degrees

By (2.6) we may choose k so that k is odd

Let Zl,...,z r (rAm) be a basis of X k. Suppose H,=H,(AX(p),d)

is zero in only finitely

and choose n so Hn~0 and H£=0 for

is represented

by ~E[AX(p)] n then ZlA...AZrA~

is not a boundary.

This contradicts

(2.3

I£I I nl.

If ~ 0

is a cycle

in H n

in AX which

(i)). O

3.

The Lie algebra of a minimal model.

Then d=d2+d3+.., particular,

Let

(AX,d

where the d i are d e r i v a t i o n s

be a minimal model.

m a p p l n g X to AIX.

In

(AX,d 2) is itself a minimal model.

The gradation AX=@APx P a bigraded algebra

is inherited by H(AX,d 2) which thus becomes HP(Ax,d~). p,q q z On the other hand (cf. [3],[1]) (AX,d 2) determines algebra L = ~L i as follows. Li=Hom(X_i_l;~). linear functions <xAy;~,B> The Lie bracket

a graded Lie

As a graded space L is given by

Thus the elements

of A2X may be interpreted

as bi-

in L via = <x,B>

+ (-i)IYl IXl<x,~>,x,yeX,~,S~L.

in L is then defined by <x;[~,8]>

By [3; Theorem 4.2],

= (-l)l$1.

(L,[,])

the homotopy Lie algebra of

is a graded Lie algebra: (AX,d).

(3.1) it is called

124

The grading of L induces a grading algebra,

UL.

In particular,

in the universal

enveloping

there is a resolution d

M.(i)

i 7 M.(i-l)

of the UL-module k in which each M,(i)

....

-+k

is a free graded UL-module,

and

d

is homogeneous of degree zero. This induces in turn a gradation in 1 each TorUL(~,~) which we write as Tor~L(~,k) : ~TorU L (~,~). 1 ' p l,p 3.2 Proposition. If L is the homotopy Lie algebra of a minimal model (AX,d),

then H~(AX,d 2) = Hom(Tor~L_p_q(~,~);~)

Proof: 280].

When L is ungraded

this is e s s e n t i a l l y

[6; Theorem

7.1, p.

The proof here is a modification. Denote by A

be the graded dual

the graded dual of AX ([16])

graded dual of left m u l t i p l i c a t i o n Ixl > lql.

: Aq P = Hom(APX)q;~).

Let

of d2, and for xsX let i(x):Ap+Ap_ 1 be the

Thus if xl~X,

by x.

Then i(x) vanishes

in A q if

~x~L are dual bases we can define D : UL®Ap~

UL@Ap_1 by D(zSa)

: (-I) IZlzesa - [(-I) 1z!IxXl ~ "z®i(xA)a. X We shall show that ---~UL®A D)UL®A D ...----+UL D E-E-+k p p-i is a resolution. The p r o p o s i t i o n is then immediate. A straightforward

computation

shows that D2=0.

(3.3)

Now write

(3.3)

in the form E : (ULQA,D)+k.

Denote by UicUL the linear span of words

of the form

s~i).

filtration

81.....8s

(~jsL,

Filter ULQA by the increasing

F,:

Fi(UL~Ap) = Ui_p@ApFinally,

notice that

A : x~-~x®l+l®x makes

the dual algebra structure

£(AI), on the graded space A I. identifies

the associated

AX into a coalgebra;

in A is the free divided powers

algebra,

Thus the P o i n c a r e - B i r k o f f - W i t t

graded algebra

(for the filtration,

theorem F,) of

UL®A as AL®F(AI). But by definition,

A~=L_q_I.

The filtration

respect to D and the associated differential, unique d e r i v a t i o n particular,

such that ~I(L)=0

H(AL®F(AI),6)=~

and 61(TPa)

and so (3.3)

F, is stable with

6, in A L Q ? ( A I) is the = a~TP-la,

aEA I.

In

is exact. O

4. 4.1

The proofs of Theorems Theorem.

Let

A and B.

We shall deduce both from

(AX,d) be a minimal model with homotopy Lie algebra

L such that (i)

cat(AX,d)

(ii)

UL is noetherian.

is finite,

Then X is finite dimensional.

and

125

Proof:

We suppose dimX infinite,

observe that both (AX(k),d),

(i) and

and deduce a contradiction.

First

(ii) also hold for the q u o t i e n t m o d e l s

because of P r o p o s i t i o n

(2.4) and the fact that the corres-

p o n d i n g Lie algebra is a s u b a l g e b r a of L.

In view of C o r o l l a r y 2.7 we

m a y thus suppose that H(AX,d)

has infinite dimension. >p (AX,d) by the ideals A-- X to produce a spectral

Now filter

sequence c o n v e r g e n t to H(AX,d).

In view of P r o p o s i t i o n

3.2 the columns

of the E 2 - t e r m can be i d e n t i f i e d w i t h the g r a d e d duals of the spaces Tor~,(~,k).

Since UL is noetherian,

each c o l u m n has finite total

dimension. But Lemma 2.3

(i) shows that

m+l columns c o n t r i b u t e to E ~. total dimension.

(since cat(AX,d)=m<~)

only the first

Hence E ~ (and so H(AX,d))

has finite

This is the d e s i r e d contradiction. []

Proof of T h e o r e m A:

Let K R be the Koszul c o m p l e x of R.

A v r a m o v associates w i t h K R a m i n i m a l m o d e l field of R such that H(AX,d)=H(KR). b e d d i n g d i m e n s i o n - depth)(R),

In [3]

(AX,d) over the residue

Because Hi(KR)=0 unless 0
Lemma 2.3

(ii) shows that cat(AX,d)

is

finite. On the other hand Lie algebra of

[3; I.I,

1.2,

3.1 and 4.2] if L is the h o m o t o p y

(AX,d) then L is a sub Lie algebra of a graded Lie

algebra E for w h i c h UE is the Ext-algebra;

m o r e o v e r dim(E/L)

is

finite. By hypothesis,

UE is noetherian.

Thus so is UL and so T h e o r e m

4.1 implies that L (and hence E) is finite dimensional. of G u l l i k s e n

Now a result

[I0] shows that R is a c o m p l e t e intersection.

Proof of T h e o r e m B:

Let

(AX,d) be the S u l l i v a n m i n i m a l m o d e l of S

(over ~); i d e n t i f y XI=X_i.

Then H(AX,d)

so that cat(AX,d)

(Lemma 2.3

is finite

On the other hand

= H(S;~)

is finite dimensional,

(i)).

([i]), the h o m o t o p y Lie algebra of

given by L = ~,(~S)®~ with the S a m e l s o n product. w i t h the P o n t r j a g i n product,

(AX,d)

is

Thus UL = H,(~S;~)

and so we have only to apply T h e o r e m 4.1. []

5.

G r a d e d Lie algebras.

Let L be a g r a d e d Lie algebra such that

L<0=0 or L>_2=0 and each L i is finite d i m e n s i o n a l cf.

see that

(AX,d) w i t h d=d 2.

(3.1) defines a m i n i m a l model

[3].

(over some field ~,

p o s s i b l y of p o s i t i v e characteristic)

Then it is easy to It is an easy

e x e r c i s e to deduce that c a t ( A X , d 2) is the g r e a t e s t integer m for w h i c h H~,(Ax,d 2) is non-zero.

Thus

(cf.

[8]) it follows from P r o p o s i t i o n 3.2

that c a t ( A X , d 2) = g l . d i m UL. Now T h e o r e m 4.1 s p e c i a l i z e s to

126

Theorem C.

If the graded Lie algebra L has a noetherian universal

enveloping algebra of finite global dimension then L is finite dimensional and nilpotent. Q

(For the more or less pan-homological proof for groups see Bieri

[5].)

Department of Mathematics,

Department of Mathematics,

Massachusetts Institute of Technology,

University of Toronto,

Cambridge, Mass 02139

Toronto, Canada

U.S.A.

M5S IAI

Department of Mathematics, University of Stockholm, Box 6701 S-I 1385, Stockholm, Sweden

References [i]

P. Andrews and M. Arkowitz, Sullivan's minimal models and higher order Whitehead products, Canad. J. Math.

[2]

30 (1978) 961-982.

L.L. Avramov, Free Lie subalgebras of the cohomology of local rings, Trans. Amer. Math. Soc. 270 (1982), 589-608.

[3]

L.L. Avramov, Local algebra and rational homotopy, in Homotopie Alg~brique et Algebre Locale, Asterisque 113-114

[4]

(1984) 15-43.

L. Avramov and S. Halperin, Through the looking glass:

a

dictionary between local algebra and rational homotopy, these proceedings. [5]

R. Bieri, Homological Dimension of Discrete Groups, Math. Notes, Queen Mary College, London (1976).

[6]

H. Caftan and S. Eilenberg, Homological Algebra.

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Y. Felix and S. Halperin, Rational L.S. category and its appli-

University Press, Princeton, N.J.

Princeton

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cations, Trans. Amer. Math. Soc. 273 (1982) 1-37. [8]

Y. Felix, S. Halperin and J.-C. Thomas, The homotopy Lie algebra

[9]

Y. Felix and J.-C. Thomas, Characterization of spaces whose

[i0]

T.H. Gulliksen, A homological characterization of local complete

[ii]

T.H. Gulliksen and G. Levin, Homology of Local Rings, Queen's

for finite complexes, Publ. Math.

I.H.E.S. 56 (1982) 387-410.

rational LS category is two, to appear Ill. J. Math.

intersections, Composition Math.

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Papers in Pure and Applied Mathematics 20, Queen's university,

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Kingston, [12]

Canada

S. Halperin, de France,

[13]

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Lectures on Minimal Models, Mem. de la Soc. Math.

9/10, Paris

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J.-E. Roos, Relations between the Poincar~ - Betti series of loop spaces and of local rings. Springer-Verlag,

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Berlin,

Lect. Notes Math.

740, 285-322,

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J.-Eo Roos, Homology of loop spaces and of local rings, 18th Scand. Cong. of Math. ii, Birkhauser,

(1980) 441-468, Progress in Mathematics

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[15]

D. Sullivan,

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47 (1978)

in topology,

Publ. Math.

269-331.

Lect. Notes Math.

Modeles de Chen, Quillen,

1025, Springer Verlag,

Berlin,

1983.

Two

examples

non-zero

of

smooth

projective

that

purpose

there

rings

of

exist

example

the

varieties

will

be

able

different

us

recall

Massey

a cocycle

we

get

(1.1) We

of

is

it

the

reader ~ cohomology

sharp

contrast

give

to

connected directed

algebro-geometric and

(The in

any

we

the

a non-simply

a simply

primarily

prime

work

Massey

~,b,c2

have

an

products

2.

Using

the

that

there

is

3i-I.

for

as

will

to details.

consider

interested

reader

characteristic

and

ab=be=O

b

dy

= b'c'.

the

the

in

then

and we

Choose

a,


From

space

follows:

Varying

element

resp.

c and

define

(singular) eochains

"Fhen a ' y + ( - 1 ) i + I x c

choices

made

we

see

'

that

H~i-1(X,Z/i)/~{2i-]+H2i-l.c,

definition

we

immediately

see

then t3
evident

I will

that

lifting = a'b'

product.

=

a topological

property

dx

degree

tEHJ(x,z/£)

Finally, Massey

on

mod

first

is

numbers.

if X

product

defined

~a,tb,te

also

not

the

in

from

a fixed

could

the

s.t.

a well

triple if

time

we

a' ,b' ,c'6 z i ( x , ~ I )

is

that

that

have

x,y~C2i-1(X,~)

the

in

the

f r o m ~ .)

triple

cocyeles

this

complex

that

convince

I will

that

As

be

to

varieties, [I]).

some

the

try

products

show

I will

see

a,b,c~H1(X,X/Z) the

then

spend

over to

to

(ef.

constructed.

following

only

is

l~rojective

and

I will

Ekedahl

Massey

products

be

topologists

note

non-zero

example may

Let

this

smooth

Massey

connected

I.

of

some

rational

In

with

blasse Z p r o d u c t s . Torsten

The

varieties

assume

functorial known

property

that

there

for are

maps

between

spaces

with

spaces. non-zero

i=I.

classification a fibration

of

principal

K((Z/~)2,1)

fibrations > T

it

is

g> K((Z/~)3,1)

clear

129

such that f: Y

if

a triple

> K((~

there

is

some

the t r i p l e well

sequence

then f l i f t s

)7,1) space

a,b,c

defined

d,e,f~tIl(Y,~)

with

of

cohomology

elements

Massey

by a map

de=el=0.

degree

Hence,

as

I triple Massey

represented

product.

that T is a K ( ~ , I )

shows

represented

to T i f f

a non-zero

and n o n - z e r o

also

is

by g has

The l o n g

for

product, a

exact homotopy

some ~-group

~

of

order i 5 . Lemma

2.1

Any

connected

space

X with KIX -~

has

a non-zero


us

making

a fibre

sequences we

a classifying F with

h I : HI(T,Z/~)

~ HI(x,z/~)

H2(X,~Z)/h


It

group

complex

first

example

Theorem

~.I

There

surface

X with

The

~6

complex

embedding variety

a,b,c

Consider

X in p6

If we put U

to u s e

:=

~

such

lli+J(x,~Z)

that

and 112: I12(T,7./~)

and

thus

[4~,

(of. up

We have

then ~-I (U)

that

every

finite

connected

and

this w i t h L e m m a

2.1

we

connected a n d p r o j e c t i v e that

a blow-up

and b l o w

to g e t ~6.

~6\ X

shows

of a smooth,

ab=bc=O

to g e t

a n y X as i n Thin 3.1.

, ~6

spectral

~(T,Z/Z)/aHI+HIc

to S e r r e

a smooth,

6-space

a fibration

~ O.

Combining

HI(x,~)

projective

j: X

group

exists

i d e a is n o w

example.

due

surface.

get o u r

4.

This

is a m o n o m o r p h i s m

fact,

the f u n d a m e n t a l

projective

i

is a n i s o m o r p h i s m

= ~2
is a g e n e r a l

is

the S e r r e

a monomorphis~. Hence ~ 2

1(a) H 1 + H 1 h I(c)

> T into

= Hi(T,Hj(F,~Z) ~

= 0 f o r all E~i,I 2

have

, H2(x,~z)

11~. I n

fibration Ei,j 2

of this

therefore

zcIF =

m a p h: X

(cf.

a simply

It may be

E2:p.

and

173.~).

~O. connected

embedded Choose

such an

E3 :II,8:24] ) the

the f o l l o w i n g

in

sub-

situation

> U is a n i s o m o r p h i s m

and

130

~: X

~ X is i s o m o r p h i c

associated

to N,

(Note that, ~(N)

is

the c o n o r m a l

as is u s u a l l y

the b u n d l e

"~leorem 4.2

connected

it s u f f i c e s

fundamental

groups.

~6 As X h a s ~i U'

surjective.

Consider

<~

and


~'induces

that

in N.)

isomorphism

an

on

the f o l l o w i n g d i a g r a m :

U greater

than 2 in ~6

real e o d i m e n s i o n

This i m m e d i a t e l y

To c o n t i n u e

of

to s h o w

geometry,

and there are e l e m e n t s

is w e l l - k n o w n .

real c o d i m e n s i o n

and as X has

connected

bundle

of X in ~6.

and n o t l i n e s

a"b"=b"c"=0

The f i r s t p a r t

the p r o j e c t i v e

the case in a l g e b r a i c

~6 is s i m p l y s.t.

> X,

bundle

of h y p e r p l a n e s

a",b",c"~H3(~6,~£) Proof:

to ~(N)

2 in ~6

gives

~IU,

the d e s i r e d

the p r o o f let us recall

~iP6 = ~I U = ......>.. ~1 P6 is conclusion.

some f u r t h e r p r o p e r t i e s

(4.1). As X is of

(complex)

I is a l i n e b u n d l e linebundle

codimension

o n ~6. Its

~P(N)(I).

Let

I i n ~6 its d e f i n i n g

restriction

c16H2(X,Z/~)

to X is the t a u t o l o g i c a l

be the first

of Ir~. Then, as N i~ of ra~k 4, ~ ( i , ~ l ) free H ~ ( X , Z / l ) - m o d u l e step we n o w c o n s i d e r that < a ' , b ' , c ' ~ O

on g e n e r a t o r s a',b',c'

because

and a m o n o m o r p h i s m

~

in d e g r e e

oriented

i : H~(X,~) Apart

from the p r o j e c t i o n

need a well-known to p r e s e n t element

i

formula

a simple

class

I, ci, c21 and c~. As a f i r s t resp.

is an i s o m o r p h i s m

~c

and we find

in degree

I

2.

(real)

> Hm(~6,~Z)

Chern

is, t h r o u g h f ~, a

:= ~ a , ~ b

The next step is to p u s h a',b' are c a n o n i c a l l y

ideal

and c' into ~ 6

manifolds

of d e g r e e

formula f o r i~oi

As X and

~6

we h a v e a G y s i n - m a p

d i m ~ ~ 6 - d i m ~ ~ = 2.

x.i y = i (imx.y) . I will

p r o o f of it in the n e x t

we w i l l

take the o p p o r t u n i t y section.

The

131

element

i

(I)

is

the

class

As X is of c o d i m e n s i o n class

of

(4.2.1)

b"c"=0,

that

prepared

that

i

(1.1)

a",b",c"

and

(4.2.1)

we

-c13
independence -
of

a",

us v e r i f y

that

a"b"

this

that

i~i

and it o n l y pull

that

(I)

c 2 and

= a'x2+c'y 2 which

Put

a"b"=

(a')-b ') w h i c h b"c".

for

remains

back

are

there

This

to s h o w

to X. U s i n g

x,y~HS(~,~)

x = X o + C l X 1 + C 2Ix2 and

) and u s i n g

c~ we see

c".

= -c13(a',b' ,c'>

Writing

xi,YitHM(X,~

and

indeed

(i~i

i~
= cla'X+ClC'Y.

b"

similarly

we m a y

c' . A s s u m e

I, ci,

= i

= 0 and

is d e f i n e d

see

y = Y 0 + C l Y l + C l2Y 2 w i t h

Chern

is I -I . H e n c e

construct

formula

To do

HS(X,~Z)/cla'HS+HSCl s.t.

to

(-cla'b')

it is n o n - z e r o .

which

the first

x~H~(X~ ~ / ~ )

l~y the p r o j e c t i o n equals

equals

~ in H~(~6,~Z).

that

:= i a' ,i b' ,i c' . L e t

(4.2.1)

implies

linebundle

shows

now f u l l y

a",b",c"

class

(=) = -ClX

i"i

We a r e

by

(5.1)

the s u b v a r i e t y

I this

the a s s o c i a t e d

= - ci and

of

that

contradicts

the l i n e a r

x0=xl=Y0=Yl=0

the fact

and

that

~O. 5.

I will

following

end b y g i v i n g

5.1

Indeed,

equal.

The

Y]

equal

. *

> X be a c l o s e d

(~)

=

•

*

i"~"i

two m o r p h i s m s

projection mapping

to s h o w

H o m D ( X , R ) (R,i

d

in D(X,R)

a: H o m D ( X , R ) (i R,i

clearly

proof

of

the

immersion

(1).x

for

, i"i

( 1 ) " (--) : i

of

x~H"(Y,l~),

ring.

the n a t u r a l

sufficient

i: Y

Then

we h a v e

> i R[2codim

with

Let

~arieties.

a commutative

are

short

proposition.

Proposition smooth

the p r o m i s e d

that

and it s u f f i c e s

formula R

~ i R are

the m a p p i n g

R~2codim

RL2codim

shows

Y~)

Y]

their

equal.

that

R,i

R

)

mapping

The m a p p i n g

Ru~y(i

they

composites

It is thus

a: H o m D ( X , R ) ( i

is i n j e c t i v e .

to the n a t u r a l

that

to show

R

R~2codim

a is Y3)

132

RO<(i

in Y to o r d i n a r y support [I].

Y~)

Rf2codim

hypercohomology.

P. Deligne,

P. Griffiths, theory

29, p. 245-274 .

As i R[2codim

J. M o r g a n

support Y] has

Inv.

math.

(1975) Principles

Wiley-Interscience

}{. Hartshorne:

and D. Sullivan:

of K i h l e r manifolds.

and J. Harris:

P. G r i f f i t h s

geometry. [~ .

with

in Y, a is in fact an isomorphism.

Real h o m o t o p y

[~

from h y p e r c o h o m o ~ o g y

Algebraic

of a l g e b r a i c

1978

geometry.

Springer

Verlag

1977 [~ .

J.-P. en

Serre:

Sur la topologie

caract6ristique

Algebraica,

Mexico

des vari~t6s

p. S y m p o s i u m

alg4briques

de T o p o l o g i c a

(1956)

Department of Mathematics University of Stockholm Box 6701

s-113 85 S T O C ~ O L M (SWEDEN)

THE RADICAL OF

~,(~S)

®

by Yves Felix,

Let

S

Stephen Halperin,

be a simply connected

dimensional

topological

nal in each degree.

space whose rational homology

~,(~S) ® ~

(Samelson product)

is finite as well,

It can be identified with the Lie algebra of primitive

Denote by

cat (S) its rational Lusternik-Schnirelmann o cides with the classical L-S category of the localization

so

is also finite dimensio-

H,(~S;~) , and is called the rational homo topy Lie algebra of

complex.

Thomas

in each degree. This is then true for the rational homotopy

that the graded Lie algebra

in

Daniel Tanr~ & Jean-Claude

category S~

elements

S . IF-H]. It coin-

[T] when

S

is a

CW

The restriction

cat (S) < ~ imposes serious conditions on the rational o [FLIT]; in particular, with rare exceptions it will be infinite

homotopy Lie algebra dimensional.

On the other hand,

in general there is little nontrivial

dimensional Lie algebras.

For example,

Lie theory for infinite

the sum of solvable ideals may not be solvable.

By contrast we have Theorem

I. Suppose

S

is simply connected with finite rational homology

degree and assume that cat (S) is finite. The sum, o in ~,(~S) @ ~ is then solvable. Definition.

R

Theorem

is called the radical of

I is an immediate consequence

length of a solvable Lie algebra Theorem 2. Let ideal in

S

in each

R, of all the solvable

ideals

~,(~S) ® ~ . of Theorem 2 below. Recall that the solv

is the length of its derived series.

be as in Theorem

I with

cat (S) < m . Suppose I o are then equivalent:

is a graded

~,(~S) @ @ . The following conditions

(i)

Either

(ii)

I

dim ~ , ( ~ S ) @ ~ is finite, or for each 2k-I dim I2i < m . i=k

k

is solvable.

When these conditions hold then solv length Corollary.

If

(I) < 4 + 21og2m .

I is solvable then every finitely generated

subalgebra of

I

even

is

134 finite dimensional. Proof of the corollary.

Let

~1,...,~t

C I

have even degrees

and consider

a sequence

of elements Y~ = [~.~[~.I~_i[...[~i2,~ii]...] If

£ = max deg e.

then the degrees

I _< i~ _< t .

of the

Y~

lie in an interval

of the form

1

[k,2k-2]

if

£m < v j

Proof of Theorem asserts some

2. Suppose

that either

B CLod d ,

as follows derived

series,

(ad~)IB # 0

and

I

from

(i) that

(i) fails for

for

I'

all

i . In the latter

yv = O ,

I . Theorem

or that for some

e E I

case

8.1 of [FHT]. By induction

(k + 1 ) m . m

2.2 of

[FHT] then

of even degree and

(i) must

still fail for

I'

(i) fails for each term of the

that the

I (k+1) , of the derived

(k + 1)st term,

k + I ~ 4 + 21og2m . There are then homogeneous

deg ~ < deg B

We may, moreover

v ~

is not solvable.

assume

for some

such that

first that

(i) fails

from Proposition

Conversely, nonzero

(Z + 1)m . It follows

and

[~,~]

suppose

~(el,...,Ej)

series

elements

is

~, B C I (k)

# 0 .

that for each

E I (k-j) ,

ey = 0

I < j < k

we are given

2j

elements

I

or

for which = [e(0),~(1)]

and

~(e I .... ,sj) = [~(E? .... ,sj,0), Now note that finite dimension

L

is necessarily

we would have

This provides

2k

linearly

k > I + log2(m) . Hence Next observe then the sequence

L

the least

~(el,...,e n)

of the # O,

is infinite

dimensional.

2m

n

k)

were of

we have

elements

at least

m

of

(I)

L , which contradicts

of the

~(el, ....,e k)

even elements

such that (say y)

2m- I

of the

~(el,...,ek)

2n > 2m . Then amongst

is built up from elements

have odd degree

whose degrees

[deg B + I, deg B + deg e] . This violates

suppose at most

integer

L

sinde if

I < i < 2k . elements

(I) above contains

at least one

~(e1'''''e

independent

that if at least

On the other hand,

dimensional,

.

B]]...]

even and lie in the interval

Choose

m

=i,...,~2k

[~i,[~i+1,...[~2k,

infinite

.

([F-H]):

dim Lod d < dim Leven _< But for some ordering

~(e1,...,cj,1)]

the

are

(i). have odd degree.

2n

elements

~(E1,...,e k)

all of

which have even degree. As above we may expand

y

to a sequence

of independent

elements

of the form

135

[¥i' Yi+1''''[~2k-n-1'Y']]'''] ' all of which have even degrees in the interval 2n-1 < 2m,

2k-n-1 > 2k/sm > 8m2/8m = m

I < i < 2k-n-1 , [deg y' + 2, 2deg y'] . Since

and so (i) is still violated.

REFERENCES

[B]

N. Bourbaki, Groupes et alg~bres de Lie - Chap. I.

[F-H]

Y. Felix et S. Halperin, L.S. category and its applications. Trans. AMS 273 (1982), 1-38.

[F-H-T]

Y. Felix, S. Halperin et J.C. Thomas, The homotopy Lie algebra for finite complexes. Publ. I.H.E.S. n ° 56 (1982), 387-410.

[T]

G. Toomer, Lusternik-Schnirelmann category and the Milnor-Moore spectral sequence. Math. Z. 138 (1974), 123-143.

Y. Felix Universit~ de Louvain-la-Neuve 2, Chemin du Cyclotron 1348 - Louvain-la-Neuve, Belgique S. Halperin Department of Mathematics University of Toronto Toronto, Canada M5S IAI D. Tanr~ et J.-C. Thomas Universit~ de Lille I U.E.R. de Math~matiques Pures et Appliqu~es 59655 - Villeneuve d'Ascq Cedex, France

SUR

L'OPERATION Y.

D'HOLONOMIE

FELIX ~ et

J.C.

RATIONNELLE

THOMAS ~

Etant donn~ une fibration (au sens de Hurewicz)

F -m-+] E ~P--+ B,

la propri~t~ de rel~vement des homotopies d~finit (~ homotopie pros) l'op~ration (bien connue) du gique

F,

H-espace homotopiquement associatif

~B

sur l'espace topolo-

notre

:

QBx

F

~

F.

Dans le cas particulier de la fibration canonique

~B

+

PB

÷

B,

est simplement la multiplication des lacets. Pour chaque anneau

R,

commutatif unitaire,

gauche de l'anneau de Pontryagin

~

H~(QB;R)

sur le

: H (~B;R) ~ H (F;R)

~

~

induit une operation

R-module gradu@

H~(F;R)

H (F;R).

Par r~f~rence ~ la g~omgtrie diff~rentielle, nous appelons

repr~s~tion d'holonomie de la fibr~on.

v

une

En effet, si la fibration est un fibr~

diff~rentiable localement trivial, de base paracompacte, la donn~e d'une connexion deEhresmann fournit une application de diff~omorphismes de

F

ceaux. Les inclusions

et

u k : ~k(B) + G

~k(B)

o~

G

d~signe un sous-~roupe

l'espace des facets de classe

G ~--+ Diff(F) ~" , Hom~o(F) ~ > aut(F),

d~signe le monolde des "self equivalences" de

F,

o~

Ck

par mor-

aut(F)

fournissent alors le morphisme

par adjonction. L'op~ration si

V

est homotope

7 2 : ~B × F + F

(~)

Chercheur

est dite triviale (resp. ,

@-triviale, resp. H-triviale),

en tant qu'application continue, ~ la projection


qualifi~

au C . N . R . S .

~ ~2 "

resp.

H (~,@) = H (~2,~.

F.N.R.S.

(~)

E.R.A.

(~)

Nous ne demandons pas que les applications

n ° 07

590 %)t :

d~finies par l'homotopie, soient des operations de

~B×F÷F, ~B

Dour tout

t.

137

L'~tude de l'op~ration

~

a ~t~ abord~e notamment par Stasheff et

Gan~a qui d~montrent : •

ES~

•

rG a , ~

~

est une application fortement homotopiquement multiplicative (S.H.M.) ; v

est triviale si et seulement si l'injection

j :

F + E

est un

monomo~phisme dans la cat~gorie homotopique.

Dans cet article, nous nous int~ressons principalement ~ l'op~ration d'holonomie rationnelle

V ~ : H (~B;@) @ H (F;~)

---+ H (F;Q).

Les es~aces sont supposes connexes par arcs et ayant le type d'homotopie de C.W. complexes de type fini. La base

B

sera suppos~e l-connexe. Sauf mention

explicite, l'homologie et la cohomologie des espaces sont suppos~es ~ coefficients ratlonnels. La notation

H+

d6signe l'homologie rgduite.

D'apr~s les thgor~mes de Milnor-Moore et Cartan-Serre, tifie ~ l'alggbre enveloppante de l'alg~bre de Lie d'homotopie

H (~B)

~(~B)

@ 9.

s'idenDonc

une repr~sentatior du premier ~quivaut ~ une representation du second. En particulier, si sentation

e~ e~

note la representation de de

~ (~B) @ ~

~(u)(a)

~ (~B) ~ ~

darts H~(F)

dans

H (F),

alors une repre-

est d~finie par :

= (-I) deg u.deg a+lo (u)~(a) °

Elle iEduit une reprgsentation Nous verrons au § 3.2 que l'action

0~

T : H (~B) @ H~(F) + H~(F).

s'obtient par des d~rivations.

A l'aice de la thgorie des modules de Sullivan, nous donnons au § 3 une description calculatoire de

et de

~.

Ceci nous permet de d~montrer les

r~sultats ~nonc~s ci-apr~s. D'apr~s Bogvad (EBo]), si si et seulement sl

H (~B)

g~n~ralise ce r~sultat :

Cato(B) < ~,

alors

dim ~ ( ~ B )

~ ~ <

est une alg~bre noetherienne. Notre premier th~or~me

138

Th~or~me 4 . 1 . de dimension f i n i e ,

i) ii)

Si

et s i

H,(F)

~t

~,(gB) 0 ~

B et un

F sont

Sont d ~ espaces v e c t o r i e ~

l-connexes, alors

H(~B)-module noetherien ;

La s ~ r i e de P o i n c ~ forme

H~(E)

et

H(F)

de

P(t)/~(|_t2if2 i

avec

e s t une f r a c t i o n r a t ~ o n n ~ l e de l a P(t) e~[t]

~

~.: = dim ~.:(~B) ~ ~.

i Dans la correspondance topologie-alg~bre locale EA.H],

ce r~sultat cor-

respond ~ un th~or~me de Gulliksen (EGu,~). Une question surgit alors : si conditions

H (F)

est-il un

dim ~ (~B) @ @

est infini, sous quelles

H (~B)-module finiment engendr~ ? Si

quet de spheres, la seule condition

dim H~(E) < ~

B

est un bou-

suffit (§ 8). Plus g~n~ralement,

cette condition suffit-elle ? L'int~r~t de cette question est illustr~e par le th~or~me suivant :

Th~or~me 4 . 6 . - S i l e connectant de l a s u i t e longue d'homotopie r a t i o n n e l l e d'une f i b r a t i o n e s t ~ l l ' a l g ~ b r e de cohomologie

et si

H(F)

H~(F)

Dans de nombreux cas,

Th~or~me 5 . 1 . - S o i t homotopique de l a p r o j e c t i o n l i b r e engendr~ par

~t

un

H~(~B)-module f i n i m e ~ engendr~, alor~

e s t de nilpotence f i ~ e . H+(F)

i : A + X X ~ X/A,

est un

H~(~B)-module libre° En effet,

une c o f i b r a t i o n f ~ m ~ e ~ a/0rs

H+(F)

e s t un

F

la fibre

H~(XyA)-module

H+(A).

Ce r~sultat permet de retrouver l'homologie des espaces

~A,

~A • ~ .

II permet aussi le calcul de l'homologie des espaces de Gan~a et des produits r~duits relatifs de James. A l'antipode des actions libres, nous trouvons les actions triviales introduites par Gan~a (cf. plus haut). Leur ~tude apporte des pr~cisions sur la structure fibr~e.

139

Rappellons qu'un

O-cSne est un point et qu'un

rattachant un bouquet de cellules un

(non n~cessairement

n-cBne est obtenu en

toutes de m~me dimension)

(n-l)-cSne.

Proposition.-

Si

B ~t

un

[-c~ne, a~ors une f i b r a t i o n de base

B

est

homotopiquement t ~ i v i a l e s i l'op~rw~ion d'holonomie e s t t ~ i v i a l e . La suite de Wang qui est dgfinie quand dans

certains

cas

B

est un

1-cBne se g~n~ralise

:

Proposition 6 . 3 . dans £a f i b ~ o n

F ~

E

Si P-~ B

B

~t

un

2-cSne e t s i l ' o p ~ r a t ~ n d'holonomie

~tt~iviale,

alors i l e x f ~ t e une longue s ~ e

exacte Hn-l(F)

A

(H+(B)[9 H(F)) n

R),

j +

Hn(F)

A+

(~ coefficients

...

dans un

on voit que ~(H+(~B;R)

J~ ~ J~2'

puisque

Hn(E)

en homologie

Si on consid~re l'op~ration anneau

~

o~

~ H (F;R~C

~2

II en r~sulte

Ker(j~

: H (F;R) ---+ H (E;R))

d~signe la seconde projection

~B × F ÷ F.

(pour le cas ii) utiliser ~Go]).

Proposl~on.i)

Toute f i b r a t i o n T.N.C.Z.

i n j e c t i v e ) a une holonomie

r~

R

(i.e.

j ~ : H (F;R) + H~(E;R)

@~t

H-t~ivi~e.

i i ) S i la f i b r e d'une fibrat~on e s t une v ~ i ~ t ~ ferrule dont l a ca~ac£ ~ r i s t i q u e d ' E u l e r - P o i n c ~ e s t non n u l l e , a l o ~ l a classe fondamentale n ' a p p ~ t i e n £ paS ~

~

(H+(~B;R) 0 H~(F;R)).

A la fibration une suite spectrale

F ~

E

P>

Best

associ~e classiquement

(EE.M])

140

E 2 = EXtH~ (B) (H~(E),@)

=>

H (F)

appel~e suite spectrale d'Eilenberg-Moore. Or la multiplication de Yoneda fait de Ext

(~,@)

une alg~bre et de

Ext

H~(B)

(H~(E),~)

un

Ext

H (B) Th~or~me ~ . I . - S i l'appZ~ca£ion

l e s f o r m a l i s a t i o ~ de

B

e t de

Ext

(~,~)-module.

H~(B) p

est forma~able

(_~.S.]),

alors

E i n d u i s e n t d ~ ~omorphism~ (@~,Q) ~ H (f~B;Q)

H~(B) ~t

ext

(H~(E),~) ~ H (F;~). H~(B)

Le premier e s t un isomorphisme d'alg~bres, l e second un isomorph~me de modul~ S ~ ces a l g e b r a . En particulier, les suites spectrales d'Eilenberg-Moore d~g~n~rent (Iv]). A titre d'exemple, rappelons que toute application holomorphe entre vari~t~s k~lh~riennes est formalisable ([D.G.M.S~). Des th~or~mes 4.1 et 7.1, on tire imm~diatement un analogue d'un th~or~me de Levin ( [Le] ).

Coro£lai~e.- S i H-alg~bre gradu~e, ~ s£

H e s t une

EXtH(~,~)

~t

ExtH(H ,(~) est un

de Yoneda), a£ors

~-alg~bre gradu~e

l-connexe e t

H'

une

une alg~bre n o e t h ~ e n n e {po~ l e p r o ~ i t EXtH(~,~)-mod~e noethe~en.

De m~me, des th~or~mes 5.2 et 7.1, on tire une version gradu~e d'un r~sultat de Roos (ER]).

CoroZla~e 7 . 2 . - S o i t (H+) n # O

et

~ngendr~ par

(H+)n+l = O, 1

a/ors

H une

~-alg~bre gradu~e

ExtH(H / + n,~) t(~ )

est un

l-connexe. S i EXtH(~,~)-module libre

EXtH(H/(H+)n,¢ ) .

La representation d'holonomie fournit finalement une suite spectrale, appel~e suite spectrale d'holonomie qui ggn~ralise la suite spectrale de Milnor-

141

Moore d'un espace

EQ]"

Th~or~me 8 . 1 . - Pour chaque f i b r a t i o n , i l e x i s t e une s u i t e spect~ale du premier q u a ~ a n t v ~ r i f i a n t

E 2 = EXtH

Si

B ale

(~B)(~,H

(F))

-~-> H (E).

t y p e d'homotopie r ~ t i o n n ~ l l e d'une s u s p e ~ i o n ,

spect~ale d~g~n~reau t ~ m e

cette s~ite

~t on a un %~omorph%Sme d'espaces v e c t o r i e l s

E2

gradu~s ExtH

Le texte s'organise

÷

2.

Quelques points d'homotopie r a t i o n n e l l e .

3.

Calcul de l ' o p ~ r a t i o n d'holonomie r a t i o n n e l l e .

4.

Operation d'holonomLe noeth~rienne.

5.

Op~r~ion d ' h o l o n o ~ e f i b r e .

6.

Op~ra~ion d'holonom~e t r i v i a l e .

7.

Holonomie e t s u i t e

8.

S u i t e s p e c t r a l e d'holonomie.

PB

en

s p e c t r a l e d'Ei~enb~g-Moore.

de l ' o p ~ r a t i o n d'holonomie. F

÷

E

+

B

l'espace des chemins de

PB × E B

constant

:

D ~ f i ~ i t i o n de l ' o p ~ r a t i o n d'holonomie.

1 . 1 . - Soit

F

comme suit

~ H~(E).

I.

§ I - D~fi~on

Notons

(gB)(~,H~(F))

l'application b . o

topie pros par

F

envoyant

L'injection fournissant

d'holonomie de la fibration

une fibration. d'origine

B f

canonique

Supposons b

o

•

B

Dgsignons

polnte-~ en par

1

b • o

:

sur

(e b ,f) og cb dfisigne le chemin o o ~B x F ÷ PB × E se factorise ~ homoB

ainsi un morphisme

: ~B × F ÷ F

appel~

op~a~o~

142 1

\/

÷

PB×E B

$~B× F

Proposition

(~wJ).- L'op~ration d'holonomie u

H-~pace homotop~uement a s s o ~ i ~ i f Cette action

est s.h.m.

(~B,~)

est une operation du

sur l ' ~ p a c e

F.

([St]).

1 . 2 . - Example.

(I~

(2)

Si

f : E ~ B

est une application

fibration

homotopique

nomie

:

~

La fibration

canonique

l'injection

de base

d'holonomie

~B~

p

BS1

(Ef = B I × E) B ' par

b

la composition

L'op~ration

B

Ff ÷ Ef ÷ B

et

l'op~ration

la d'holo-

est d~finie

~B + PB ÷ B

du point

est simplement

(3)

associ~e

~B × Ff + Ff

continue

y

~tant dans

la fibration B,

o dans l'espace

= ~(l))

d'holonomie

des lacets.

dans la fibration

(p(~) = ~(0)

associ~e

l'op~ration

des lacets

libres

est la conjugaison

des

lacets -I ~(~,~') (4)

Le connectant

f~E

~P>

~

~B

de la suite de Baratt

>>

F

est par construction est compatible

D'apr~s S V T

Ganea +

~a

S × T

On v~rifie

= ~ o w' o w

J-~

ale

P>

B

la restrictiton

avec

~,

E

les operations

de

~

de

~ ~B

la fibre homotopique type d'homotopie

alors que l'op~ration

~B x {f }. o sur

F

x (~2S • ~T)

÷

F.

~(S) • ~(T).

est fournie

ration diagonale (~(S) x ~(T))

et sur

de l'inclusion

du joint

d'holonomie

~B

f~S • ~2T.

par l'op~-

143

§ 2 - Quelqu~ points d'homoto~e rationnet~e. Tousles

espaces consid~r~s sont suppos6s connexes par arcs et du type

d'homotopie faible d'un C.W.-complexe les espaces vectoriels

H~(S;@)

et

est la th6orie des modules minimaux. et proprigt6s 616mentaires.

de type fini. On notera H (S;@).

H~(S)

et

H~(S)

L'outil principal dans la suite

Nous rappelons ici quelques d6finitions

Pour plus de d6tails,

le lecteur est invit6 g se re-

porter ~ [Su, Ha, Ta].

2,1.- Le module miv~(mal de SulZivan. Tousles

espaces vectoriels et toutes les alg~bres sont suppos~es d6fi-

hies sur le corps

~.

Une alg~bre diffgrentk~£1e gradu@e commutative (a.d.g.c.) une alg~bre gradu6e commutative dA

de degr~

(xy = (_|)deg x.deg Yyx)

et de carr6 nul. Elle est dite libre si

+l

(A,d A)

est

munie d'une d~rivation A

est le produit tenso-

riel d'une alg~bre sym~trique sur un espace vectoriel

Y

par une alg~bre ext~rieure sur un espace vectoriel

concentr6 en degr~simpairs

on note

A = AX,

X

~tant la somme directe

vectoriel des mots de longueur

en

(x)~e A

de

X

Y @ Z.

Notons alors

A~X

dA(X) C A~2X

et s'il existe

:

l'espace

X.

est dire mi~im~e si

(A,d A) une base

i

Z

concentr~ en degr6s pairs

A = ]IX,

si

index6e par un ensemble bien ordonn@ tel que

dA(X ~) C AX< . Un morphisme

morph~me si Si

d'a.d.g.c.

~ : (A,d A) ÷ (B,d B)

: H (A,d A) + H (B,dB) (A,d A)

est un isomorphisme.

est une a.d.g.c, v~rifiant

H°(A,d A) = ~,

unique (~ isomorphisme pros) a.d.g.e, minimale (AX,d) (/iX,d)

+

(A,dA).

Le foncteur associe g chaque espace (ApL(S),ds)

(AX,d)

est appel6 un quaSi-iso-

munie d'un quasi-isomorphisme

s'appelle le module minimal de

PL-formes construit par Sullivan ([S~), S

il existe une

une a.d.g.c.

not6

ApL

Le modgle minimal de

(ApL(S),ds).

est par d~finition le module minimal de

(A,dA).

S.

144

2.2.- L'alg~bre de Lie d'un module minimal. Soit

L

une alg~bre de Lie gradu~e connexe (a.l.g) de type fini,

alors le complexe de Koszul (~Ta])

(C~(L),d)

est l'a.d.g.e.

(AX,d)

l-connexe

d~finie par (i)

X = Hom(sL,~)

(ii)

d : X

(iii)

~ A2X

= (-l) deg V<x,sEu,v]>.

C'est un modgle minimal. Par eontre, si

(I~,d)

est un module minimal, posons

= Hom(s(A+X/A~2x)P+I,@). Les ~l~ments u de L d~finissent des fonctions P su : A + X ÷ ~ par la formule = (-l) deg U<sb,u>.

L

L'espace

A2sL

s'interprgte alors comme

Hom(A~2X/A~3X;~)

= = (_])deg v<~,s~u,v]> ' d~finit ~le structure d'alg~bre de Lie gradu~e sur L D~composons v~rifiant

(AX,d).

d

o~

d.(X)~l AIX'

ont m~me alg~bre de Lie

d = d 2 + d 3 + ...

alors

(kX,d 2)

L

._(Ax,d 2) ~ C~(L).

et

," (AX,d)

et

(AX,d 2)

L du module minimal dtun

1-conne×e ~ti~omorphe ~ l'alg~bre de Lie d~homotopie r a t i o n n ~ l e

~ (~S) e ~

de

2.3.Soient et

d.l d[signe la d~rivation

est une a.d.g.e.

Th~or~me (ESu~, EA.A~).- L'alg~bre de Lie espace

~ e A+X,

L.

s'appelle alors £'alg~bre de Lie de sous la forme

et la formule

(A,dA)

s.

K.S.-extensions. (A,dA)

est augment~e par

chaque homomorphisme commutatif

et

:

(B,dB) gA

d'a.d.g.c,

sur

des a.d.g.c, telles que @.

Dans

H°(A) = H°(B) =

(EHa~), S. Halperin montre que

f : (A,d A) ~ (B,d B)

est associ~ un diagramme

145

f

+

(A,d A)

(B,dB)

(A OAX,d)

~

~

( ~ , d)

EAOid o~

: 1)

g

eat un quasi-isomorphisme

;

2)

i

eat l'injectiom

canonique

;

3)

Ii existe une base

(x)~e K

de

dx

~ < B =>

s'appelle un K.S.

Ix [ @ IxBI ,

eat alors unique ~ isomorphisme

la

pros

B

l-connexe

Th~or~me.-

(EG~)

Si

de f a ¢ o n ~ 6 1 p o t e n t e s u r m i n i m a l de

~ :

K.S.-extension

de

f.

eat dite minimale.

r~side dana le th~or~me

F ~

E

P-L B

et si

Elle

suivant d~ ~ Grivel

o~

e a t une f i b r a t i o n

f = ApL(p),

d'holonomie

Dana ce paragraphe, B

a/ors

(EH~).

~I(B)

(AX,d)

op~re

e a t un m o d u l e

fiB x F + F.

nous supposerons

Nous explicitons

H (~B) 8 H~(F)

6 7r (CLB) 8 ~

3. I . -

÷

rationnelle.

un C.W.-complexe

connexes par arcs. Nous calculons

H~(F)

toujours que

~ : F ~

E ~

l-connexe de type fini et

tout d'abord le modgle minimal

de

ensuite l'op~ration

d@finie dana l'introduction

Le m o d u l e m i n i m a l de en homotopie

B

E, F

et montrons

que pour

on a :T(~) 6 Der H~(F).

Traduisons

au § 1.1.

(Koszul-Sullivan)-mod~le

F.

eat une fibration avec

T :

tel que

et g S. Halperin dams le cas g~n~ral

H~(F),

§ 3 - Calcul de l'op~ra2ion

:

K

([H~).

L'utilit~ de la construction dana le cas

et un bon ordre sur

e A ~ [~<.

Un tel diagramme Si en outre

x

~ : ~B × F ÷ F.

rationnelle

la construction

de

~)

faite

146

D6signons par fibration PB ÷ B.

~.

(A,d) i_~ (A @ AY,d) ~

Formons le K.S.-mod~le acyclique

II r@sulte alors de

(A 0 ~

F

et de

un K.S.-mod~le de la

(A 0 AX,D)

de la fibration

([H~ , § 2 O) que

8 AY,D) = (A 0 AX) 0 (A 8 AV) d@f A

inclusions de

~B × F

ql : (A 8 AX 8 AY,D) ÷ (AY,d) o

(AY,d)

dans

est un module de PB × E B

q|.

De plus, les

ont pour modules les projections

et q2 : (A ~ ~

une section du quasi-isomorphisme

PB x E. B

@ AY,D) ÷ (~X,O) 8 (AY,d).

Un module

~

de

v

Notons

est alors donn@

par la compos@e q2° ~.

3.2.- La repr~se~on Pour chaque a.d.g.e. des d~rivations de par

Der (A) P

A

(A,dA) ,

de degr@

O H~(F) --+ H~(F).

~ : H(~B)

0

d@signons par

v@rifiant

Od A + dAO

l'espace des d@rivations de degr~

La somme directe

Der(A) =

Dero(A)

l'espace vectoriel

= O

et, si

p > O,

-p.

@ Der (A), p~O P

munie du crochet

v

D,O~

=

@.6' - (-I) deg 8.deg

08'.@ et de la diff~rentielle

a.l.d.g. "Consid@rons la K.S.-extension module de la fibration pour la bigraduation

O~

de

Der(AY)

~, X =

(~X,d)

Notons

d

de

est une

k > (AY,d) A+X

homog~ne

d@finit des @l@ments

par la formule

De la relation

0 = I db

(b)~c A

La diff@rentielle

d]Ay = 1 0 d +

(~)

i + (AX 0 AY,d)

et choisissons une base @ (APx) q. P,q

~8 = 2 A , O ]

0 8~ + I

L

d 2 = O,

(_)Ib~lbR

~ b ~A

0 e~ .

on tire alors :

I

0 ~8 ~ + ~

l'alg~bre de Lie de

~

(t~,d)

(-l)

IbalbabB

et

cation lin@aire d@finie par ~(u) = ~ 8 ~ .

O [e$,O~.

:

L -+ Der(AY)

l'appli-

147

La premiere ~tape de la construction de

T

repose sur le lemme

suivant :

Lem~e. I)

~@ = 0

2)

L'applicatio~

;

~

: L ÷ H (Der(AY),%)

i n d u i t e par

@ ~t

un

homomorphisme d'alg~bres de Lie. D~monstration du le~me : I) La formule (~) montre que si comme

= O,

2) Rappelons

~Ta]

b

e AIx,

il est clair que

~@~ = O,

alors ~

:

o.

qua

= (-I) deg V<x,s~u,v~>.

Un simple calcul donne alors le r6sultat.

Composons maintenant H (Der(AY),~) ÷ Der(H ~ (AY,d)). dans

H~(AY,d)

~

avec le morphisme canonique

UL

Le morphlsme

: H(aB;~) 8 H~(F) + H~(F)

restre/~ tio~

da~

op6ration

L

de

:

: UL 8 H~(AY,d)

Th~or~me 3 . 2 . -

de

Nous obtenons ainsi une representation

par des d~rivations. Ceci se prolonge en une

l'alg~bre enveloppante

T

•

.... H~(AY,d).

~

coincide avec l e morphisme

d~fini da~ l'introduction.

En p a r t i c u l i e r ,

~(~B) e ~ d ~ f i n i t une a c t i o n de c e t t e alg~bre de Lie par d~rivaH~(F).

D~monstration du th~or~me 3.2 : Puisque l'alg~bre de Hopf

H~(~B)

est

T

T

^

primitivement engendr~e par

~ (~B) 8 ~ m L

et puisque les operations

sont naturelles, il suffit de faire la d~monstration lorsque

B

et

est une sphgre

a) C~. d'une s~h~re ~ p a i r e . Dans ce cas, la K.S.-extension de la fibration prend la forme

148

(Ab,O)

i

(Ab @ AY,d)

k > (AY,d),

Pour construire du quasi-isomorphisme

l'action

~

ne contenant

Soit maintenant

u

dans

avec

~

Puisque

cocycle

de

Db = b,

(AY,d)

tout d'abord

(AY,d)

une section

(Db = b)

par la formule

la composition

d

= I.

d = 1 @ d + b @ ~.

n!

pour



par

ensuite

÷

avec

(-i) n ~n 0 on(~).

n~O

On effectue

impair,

on construit

ql : (Ab ~ Ab ~ AY,D)

0(~) =

La formule

deg b

et

L

pas de

(alg~bre

on ~crit

b,

on peut

de Lie de

b = sb

identifier

(Ab,O))

o

~

l'~l~ment

et on a alors

¢. d~fini

= I

et

~ e H (F).

b) Cas d'une sphere paire. Notons

de la fibration de

(Ab/b 2 ,O)

:

d@finie

Un morphisme

~

par

ql

sur

induisant

o = id + by I + by 2 + cy 3 + y

on d~duit

v

÷

(AY,d)

un

(Ab~h 2 ~ A(b,~)

K.S.-mod~le

@ AY,D)

l'extension

DE = -bb.

est alors

~ AY,D) ÷ (A(b,~)

construit ~ AY,D')

en composant

la

avec une section

O

(AY,d).

Une telle section

y :

Soit

Db = b,

q2 : (Ab/b2 ~ A(b,~)

de la projection

et oh

(Ab~b 2 O AY,d)

d = I ~ d + b @ 0.

(Ab/b 2 ~ AY,d)

projection

÷

AY + (b ~ A+(b,c)

~

est n@cessairement oh les

Yi

de la forme

sont des endomorphismes

~ AY) @ (A~2(b,~)

lin~aires

~ AY). De l'~quation

:

@ + YI = ¥2 d - dY2"

de

od = Dd,

AY

149

En particulier, si cocycle

~

de

AY

u e L

et tout

a

satisfait ~

de

H (F),

alors pour tout

on a

=

(_|)deg u.deg ~+]<~],v

=

(_l)deg u.deg ~ + 1 < ~ ( D ] ) , u

=

(_1)deg u.deg ~+I<~ e ¥i~, u @ ~>

=

(_l)deg u.deg ~+l(_])deg u.(deg ~+l)<%l~,~><~,u>

=

<@(~),~> = <~(u 8 ~]),~>

3 . 3 . - Seconde construction de Notons de nouveau de la fibration

= I,

~,

et

(AX,d)

~ ~>

car

= -I. •

T.

1_j_+ (AXe AY,D) ÷ (AY,d)

DerAx(AX ~ AX) C Der(AX ~ AX)

des d~rivations s'annulant sur

une action par d~rivations

P

la sous-a.l.d.g, form~e

DerAX(AX @ AX) + DerAxsAy(AX 8 AX @ AY) de

H (DerAx(AX @ AX))

D'autre part, l'alg~bre de Lie B

de KoS.-mod~le

AX.

L'injection canonique

au module de Quillen de

(u @ e)>

(~H~),

sur

Der~,~(AX @ 5X)

induit

H~(AX ~ AX ~ AY) =H~(AY). ~tant quasi-lsomorphe

on a un isomorphisme d'alg~bres de Lie.

p~ : H~(DerAx(AX @ AX))

:~-+ L.

Ceci est donn~ explicitement par <~,p~([e])> = (-I)deg @+lee(x), l'augmentation canonique de

~

e ~X.

Proposition 3 . 3 . - L'isomorp~sme ration

T :

~ (~B) e H~(F) ÷ H~(F)

D~mo~£TLcug£on

Les operations

~

et

:

~

d~signant

i)~ i d ~ n t i f i e l ' o p ~ r a t i o n

~

~ l'op~-

d ~ f i n i e dans £ ' i n t r o d u c t i o n .

II suffit de montrer que

~ = ~

(l'op~ration de 3.2).

~tant naturelles, il suffit de nouveau de faire la

150

d@monstration lorsque la base est une sphere.

a) C~dlune sphere impasse. Notons

(Ab,o) ÷ (Ab ~ AY,d) ÷ (AY,b)

d = I ~ d + b @ 0. = I.

Soit

L

l'alg~bre de Lie de

La d~rivation

un cycle v@rifiant

du

O~([d~)

Notons alors

un K.S.-mod~le de la fibration :

de

= u.

ql

et

DerAb(Ab @ Ab) Soit

o

(Ab,O)

Du

et

u

d@finie par

son image dans

dans

L

avec

du(b) = -I

est

DerAb@Ay(Ab~AbOAY).

la projection et section construites en 3.2.a.

a(¢) =

X ~,)nbnen(¢) • n~O

Alors

~(u)([~])=

~iDu~(~)]

=

= [q1( ~ ~(-I)n ~n@n+l(,))~

[O(~)] d@__ff ~(u,[~>]).

b) Cas d'une sphere paire. Notons fibration

(d = ] @ d + b @ O)

Soit

En par

u

particulier, du(b) = -I

Notons

Ab/b2 de

du

(Ab/b2,O) + (Ab~b 2

dans

avec

= -I. et

et +

~ AY

AY,d) + (AY,d)

un K.S.-mod~le de la

l'alg~bre de Lie de

= I.

La d@rivation

du(~) = - b e s t

ql

@ A(b,c) @ AY dans

L

et L

0

(Ab~b 2 ,0).

On reprend les notations de 3.2 b. du

de

DerAb(Ab/b 2 0 A(b,e))

un cycle v~rifiant

d~finie

p~([du]) = u.

la projection canonique et une section de celle-ci. D@signons par

DerAb/b2~Ay(Ab/b 2 ~ A(b,c) 0 AY).

Alors,

= [qlDu(¢ + b'Zl(¢) + bY2(¢) + cY3(•) + y(¢)]

=-[yl(¢)] = [0(¢)] d@f ~(u,[¢J).

,

Du

l'image

151

§ 4 - Operation d'holonomie noeth~6enne, Th~or~me 4 , 1 , - S o i t 1-connex~, S i

F + E ÷ B

dim ~(B) 0 ~ < ~

e s t un

H(F)

(2)

et

P(t)

dim H(E) < ~

a/0rs

H(~B)-module noeth~rien ;

La s ~ r i e de P o i n c ~ forme

une fibration e n ~ e espac~

de

/~(1-t 2i)

~t

H(F)

une f r a c t i o n rat~onnelle de l a

dim ~2i(~B) O ~

o~

P(t)

d~signe un polyn$me

coefficients entiers, Avant d'entreprendre d~finitions

extraites de EGu l , ~ .

Soit P(t)

la d~monstration de ce rgsultat, rappelons quelques

~(t) ~ £Et]

est dite

un polynSme v~rifiant

~-rationnelle

~(0) = ± I.

s'il existe un polynSme

R(t)

dans

Une s~rie formelle £Et]

tel que

P(t) = R(t)/~(t). Si

H

est un espace vectoriel gradu~ v~rifiant

on appelle s~rie de Hilbertde

H

IHI(t)

H

G =

$ G p~O p

est dit a) b)

Ps(t)

de

IHI

Un

G-module

si

est d~finie

Pour tout sous

S,

S.

un anneau connexe gradu~ inf~rieurement.

~-rationnel

La s~rie

~ dim Hi.t i i~O

d~signe la cohomologie d'un espace topologique

d~signe la s~rie de Poincar~ Soit

gradu~

H

i,

la s~rie formelle

IHl(t) =

Ainsi, lorsque

dim Hi< ~ pour chaque

G-module

; N

de

H,

la s~rie

INl(t)

est

q-ration-

nelle. Dans les l e w e s rieurement

(M =

Lemme I

I M ). p~O P

~u ~,-

1~ 2, 3 suivants,

les modules sont supposes gradu~s inf~-

Le lemme 2 est le dual de l'~nonc~ de Gulliksen.

sort

gradu~s, H e s t n o ~ h ~ r i e n e t

o ~.+ H' ~ H ~ H" + 0

une s u i t e e x a ~ e de

G-modules

~ - r a t i o n n e l s i ~ seulement s i H' #~i H" l e sont.

152

Lemme 2 gradu~s avec de

I

IGu 21.-

H' J-~ H'

Soit

de degr~ z ~ o t e l que

G de degr~ s t ~ c t e m e n t p o s i t i f .

~-rationnel, ~'(t)

H'

e s t un

J

I

H une s u i t e exacte de

G-modules

s o i t la mult~plic~gion par un ~l~ment

Alors, s i

G-mo~le n o e t h ~ e n

H

~

e s t un

G-module noeth~rien

~'-~onnel

avec

= (I - t deg g)n(t).

Lemme 3.- S o i t finie,

UL ~ t

alors

une

L

un

~-alg~bre de Lie gradu~e et connexe de dimension

UL-module noeth~rien e t

~L-rationnel avec

dim L2i ~L(t)

= ~ (I - t 2i) i

D~mo~tration du lemme 3 : Le r~sultat et soit

est ~vident

g e L

l'hypoth~se noetherien

pour

un ~l~ment

de r~currence et

Proc~dons

dim L = O,1.

~L/g-rationnel.

est un

Si

g

UL-modules

O + U(L).g ÷ U(L) + U(L/g)

Si

g

le r~sultat

UL-modules

UL

×g'~

UL

>

L

~B + F + E

E 2 = H~(E)

~ H (~B), et convergente Comme

H (F),

H (E)

est noetherien

UL.g ~ U(L/g)

avec le lemme

du lermae 2 appliqu~

spectrale vers le

est de dimension ~-rationnel.

Corollalre 4 . 2 . - S i

s

de L.

n > O

Par

UL-module) et la suite exacte

] fournit

le r~sultat.

~ la suite exacte

de

UL/g.

est une suite

et

de dimension

(et donc

D~monstration du th~or~me 4.1. : La suite spectrale fibration

sur la dimension

qui est donc dans le centre.

est impair,

provient

donc

U(L/g)-module

de

est pair,

Supposons

de degr~ maximal, U(L/g)

par r~currence

de

H~(~B)-modules

H (~B)-modu!e

finie,

de Serre de la avec

H (F).

le lemme 3 montre

que

E2

et donc

•

e s t un espace topologique v ~ r i f i a n t

a)

dim H~(S;~)

< ~ ;

b)

~(~S) ~ ~

c o ~ t i e n t une s o u s - ~ g ~ b r e de Lie l i b r e de c o d ~ e n s i o ~

f i n i e , alors la s ~ r i e de Poincar~ de

~s

est rat~nnelle.

153

De~o~Ybk~o~ ~

~>p(~S)

tel que

@ @

de Postnikov de

S.

c o r o ~ l ~ e 4.2. : Notons

soit libre. Notons alors La fibre homotopique

F

Sp

p le

de

le plus petit entier p~ @tage de la tour

~ : S ÷ S

est un bouquet P

de sphgres et d o n c : ~(~F)

D'apr~s Comme

~ ~ = ~(V)

R~a~que 4.5.-

(I - (P(F) - 1)) -I.

dim ~(B) @ ~ < ~

En effet, d'aprgs le r~sultat de Bogvad H (~B)

H (~B)

P(~S)

est rationnel.

dens le th~or~me 4.1. (EBo]), si

est un anneau noeth~rien

est de dimensi3n finie.

QB ~ PB + B,

P(~F) =

le th~or~me 4.1. montre que

L'hypothgse

C.W. eomplexe fini, alors ~ (B) 8 ~

P(F) = t. IV ] + I.

la form~le de Hilton-Steer,

P(~S) = P(~F).P(~So) ,

est n~cessaire.

avec

B

est un

si et seulement si

Ii en rgsulte que dens la fibration

ne oeut ~tre un

H (QB)-medule noeth~rien

que si

~(B)

@

est de dimension finie.

Question 4.4.- si avec

dim H~(E) < ~,

H (F)

Ex~ple~ 4.5.- Si spheres et si

H~(E)

F ÷ E ÷ B

est une fibration entre espaces

]-eonnexes

est-il un

H (~B)-module

?

F * E * B

finiment

engendr~

est une fibration de base un bouquet de

est de dimension finie, alors

H (F)

est un

H (~B)-module

finiment engendr~. En effet, dens ce cas un K.S.-mod~le

de la fibration est de la forme

(H~(B),O) ÷ (H~(B) @ AY,d) ~ (AY,d). La diff@rentielle

d

s'~crit donc de la forme

d = I 0 d + ~ b. @ @ i , i I

o~

bi

pareourt une base de

H+(B).

Si

plus ~lev~ que la dimension cohomologique d~ # O.

Ii existe donc

bi

avec

Di(~

~

est un cocycle de

de

E,

# O.

alors dens

(AY,d)

(H~(B) @

de degr~ AY,d),

154

Th~or~me 4 . 6 . 1-connexes, t e l l e que

Si

H~(F)

F J

E

est une fibrat~on de f i b r e e t base

P~ B

s o i t un

H ( ~ B ) - m o d ~ e fi~iment engendr~, alors l e s

conditions suivantes sont ~ q u i v a l e ~

:

(I) L'alg~bre de cohomologie e s t de nilpotence f i n i e . un c e r t a i n

((H+) n = 0

n).

(2) L'application

@n : ~n+1 (B) 0 ~ + H n ( F ) 0 @

(compos~e du connectant

de l a f i b r a t l o n avec l'homomorph~sme d'Hurewicz} ~ t n

sup~eur

~ un c ~ t a i n

D~m0~p~trat~0n : Dans

(EOp]),

I)

-----> 2).

2)

-~---> I).

Il

est un

Soit

xnc

q

et p o ~ t o u t

Hn(F)

n

A(~ X n) n

Notons

sur

un sous-espace dual ~

Corm~e ~

est un morphisme de

H (~B)-module ainsi que les id~aux

R = H+(F)/Im(6

gendrg.

Les

sous

: H (~B) ÷ H (F)).

H (~B)-modules

J

n

de

R R

In

d~finis par

est un formgs

tousles

g~ngrateurs

de

R.

J

n

est

des

done

R

~l~ments

un hombre

fini

r

(H+(F)) r+l C 11

de g~n~rateurs.

et

n

H (f~B)-modules,

In = In-]'H+(F)"

et

I

Im 6~

On a d o n e

(H+(F)) (r+l)n° = O.

•

orthogonaux

pour lequel

o

o

Finalement, le r~sultat de Oprea montre que sur

est un

H (~B)-module finiment en-

forment une suite croissante. II existe, d'autre part, un

rieure

Im @n"

A(~ X n) + H~(F) -----+ H~(~B) n II en r~sulte clairement que

Im 6~.

I I = ker 6 ~.

Notons

contient

nul pour t o u t

pa~.

Oprea d~montre que le compos~

isomorphisme de

pour

n

I n

~

J

n

o

= Oo o

est l'alg~bre ext~-

155

§ 5 - O p ~ r ~ o n d)holonomie l i m e . 5.1.- Soit

Notons

h : F ÷ Y

quement trivial, k

f : X ÷ Y

la fibre homotopique de il existe une application

en une application

~Cf

sur

une application continue de cofibre

~Cf x X

k :

et sur

~Cf x X ÷ F F.

g.

Le compose

k : X + F

H (~Cf) O H+(X)

+

Cf

Notons alors

(K~ 0 AZ,d)

F.

D'apr~s

une

K~A~

H+(K~ ~ AZ)

@tant le compos6 module de

k,

dans

est

X.

Y

eSt 1-connexe

i n d u i t un isomorphisme de

K.S.-extension

(EHa] ' § 20),

et

H (~cf)-module~

--+

:

~0!

Ay

~

A X.

([Ha]).

acyclique minimale. Dans ce cas,

K~ 0 (K@ O AZ)

Ay 0 AZ

÷ 0

est un module

d'espaces vectoriels dif-

la suite exacte

AyOA~

@rant nul,

Ay • AZ

Ay

f :

K~ = (Ker ~ 0 @)

0 ÷ K@ + Ay + q

f6rentiels induit par tensorisation

÷

g o f

est alors fourni par l'a.d.g.c.

La courte suite exacte

0

~Cf x X ÷ X

: Choisissons un modale surjectif de

Un module de

pour

k

Prolongeons

H+(F).

~@m0~£A~ti0n

AZ = H~(~Cf).

hk ~ f.

n o t a t i o ~ pr~c~de~tes, s i

conne×e par arcs, l ' a p p l i c a t ~ o n

@tant homotopi-

compatible avec les actions ~ gauche de

homotope g l'inclusion de la fibre homotopique de

x

avec

La seconde projection

Th~or~me 5 . 1 . - Avec l ~

gf

g : Y ÷ Cf.

--~-+ ~ O A ~

H+(~)

+

O.

est un isomorphisme.

> AX 0 AZ

- -

D'autre part,

q

0 AZ et

)

une a p p l i c a t i o n continue,

@ ~ I

un

le r@sultat s'en d@duit aussitSt. • r

Corollaire

: Soit

j : Y ÷ z =

Y ~

@

( v

e

n.+l l

i=l

alors 1)homologie r ~ d u i t e de l a f i b r e homotopique de H (~z)-module au module l i b r e

Ceci g~n@ralise ~ route cofibration le point de Halperin-Lemaire

(EHL]).

e s t i s o m o r p h e comme

j

r n. H (~Z) 0 ( 0 H+(S l ) ) . i=l b

du th6or~me suivant

156

Th~or~me.- S o i t al

b)

j

: x

~+ Y = x ~ ( v

est surjectif

~ (j)

de Lie l i b r e

~(v).

Pans ce cas,

V

une a p p l i c a t Z o n c o n t i n u e .

e ~)

si et seulement si

e s t un

Ker ~ ( j )

~t

une a l g ~ b r e

H (~Y)-module l i b r e .

5.2. - Exemples. ])

Les espaees

G

dfisignons par

G

l'espace

constante. Si

G

et

et

Gn+ 1

0

n

la fibre

n

de Ganea. Soit

X

un espace topologique,

•

et

r @ d u i t ~ un ~ o i n t

f : X ÷ G n n

f

: X ÷ 0

O

C n

sont d~finis, posons

h o m o t o p i q u e de l ' i n c l u s i o n

Gn ÷ Cn.

l'application

G

la cofibre de

f n

Le thgor~me 5.1. montre

alors : X

2)

Consid~rons l'injection

T(S3,S3,S 3) F

de

i

est

I-connexe,

S3 V $3 V S3

-~

d~signe le "fat-wedge" des trois spheres

T($3,$3,$3), S3.

o~

La fibre homotopique

admet comme 7-squelette dans une d6composition homologique rationnelle (S5 V

L'application envoie les spheres les spheres et la sphere S 97 de

H+(G n) ~ H~(~Cn_ 1) 0 H+(X).

si

S~, 7 S9

S5

f : F

S 5 V S5) V ( V s~). 1~i~9 ÷

S3 V S3 V S3

est d~finie comme suit :

sur les repr@sentants d'une base de

i ~ 8,

f

w5(S 3 V S3 V S3) O Q,

sur les repr~sentants d'une base de WT(S 3 V S3 V S3) O Q

sur le point de base.

provient par la longue suite exacte d'homotopie de l'~Igment non nul

~8(T($3,$3,$3)) @ Q. N~anmoins,

correspondant g

7 S9

H7(F) ~ H2(~T(S3,S3,S3)) 0 H5(F). est g la fois dans

L'~l~ment de

H2(~T(S3,S3 S3~.H5(F) et dans

H7(~T(sB,sB,s3)).Ho(F).

3)

Consid~rons la fibration de Hopf g~n@ralisge f~ • 2X

--+ EgX

.>

X.

H7(F)

157

Si alors

X

est coformel et de categoric de Lusternik-Schnirelmann

H+(f~X • ~X)

est un

H (f~X)-module libre.

5.5.- D~signons par

F

la fibre homotopique de la projection

(S3a V S3)

q9

e8

-~

[a, Ea,@] p

2,

p :

3 Sb •

n'est p as la cofibre d'une application. N~anmoins,

H+(F) ~ H (~B) O (u,v) = T(x) 8 (u,v)

avec

deg x = 2, deg u = 3, deg v = 8.

Une d~composition homologique rationnelle de

F = ( V

S 3+2n)

n>.O

L'action de

H (~S~)

an

sur

F

qJ

( V

[al, an]

H+(F)

peut ~tre d~crite comme suit :

e8+2n).

n~O

est d~finie comme suit :

x.S 3+2n = $3+2(n+l)

x.e

8+2n

= e

8+2(n+|)

,

x e H2(QS

~)

•

B

5 . 4 . - Probl~me. D~terminer des conditions plus g~n~rales que celles mentionn~es dans le th~or~me ! sous lesquelles

H+(F)

est un

H (~B)-module fibre.

158

§ 6 - Operation d ' h o l Q n o ~ e t ~ i v i a l e . Soit D~signons

~ : F

par

la seconde

J~

-~P

~ : ~B x F ÷ F

projection.

module minimal

de

$

Notons et

D~inition.(resp.

E

~ ~ ~2'

L

~

B

une fibration

l'op~ration alors

de base

d'holonomie

et par

(AX,d) ~ (AX ~ AY,d)

l'alg~bre

de Lie de

estt~viale

(resp.

B

| -connexe. ~2

P

: f~B x F -~ F

(AY,d)

un

K.S.-

si

~ ~ ~2

(AX,d).

@-triviale,

H-triviale)

H (~) = H (z2)).

est donc : L ÷ Der H~(F)

H-triviale

est nul.

II en r~sulte

H-triviale

dans une fibration

1-connexe,

car Hans ce cas

6. I . -

si, avec les notations

ou la fibre

Der
du § 3.2,

que l'op~ration F

d'holonomie

est une vari~t~

(F) = O

le morphisme est toujours

kglh~rienne

compaete

(IMp).

~-trivialit~.

Lemme.- Toute f i b r a t i o n ~ holonomie

~ - t r i v i a l e de base une sphere ~ t

rationnelleme~ triviale. D~monstration enti~rement Sn-1 x F

d~finie

k > F.

ment homotope Si k-~

~(~xid),

: Toute fibration

F

÷

par la classe d'homotopie

La fibration

seconde

~ : S n-| -~ ~S n

est l'adjoint

÷

Sn

de base une sphere est

de son morphisme

est rationnellement

~ l'application

d'o~ le lemme.

E

triviale

si

d'embrayage k

est rationnelle-

projection. de l'identit~

de

S n,

alors

•

Th~or~me.-

L ~ propositions s u i v a n t ~ s o n t ~ q u i v a l e ~

I)

~J e s t

~-triviale ~

21

Le morpheme d'alg~bres de L / e

@

:

:

L -> H (Der(AY))

e s t nul. 3)

~

admet un module pour l e q u e l

dYc

AY @ (A>~2x @ AY).

d ~ f i n i en 3.2.

159

l)

=>

: Sn + B,

2).

Si

la fibration

est donc rationnellement

2) = > bigraduation

3).

]IX =

@ p,q

II s'ensuit maintenant

u~ e L

donc nul et morphisme

y

est

image r~ciproque

Choisissons

sur une base

(APx) q

;

~

Ii suffit

alors

et

J~2

par

AX

homog~ne

d = l ~ d +

e AIX,$@ ~ = O.

alors

y e Y,

: y(z)

j : F + E

~tant homotopes,

Elle

pour la

~ b ~eA

y(y)

Ceci est

D~finissons

= y +

~ @~.

Si

~(u B) = D ~ "

alors un

~ b i (y) b eAX

et

- (z + ~ b i ( z ) ) ~A~2X ~ AY

(AX @ AY,D)

par

l'inclusion

FAx ~ A(y(Y)),D_].

de la fibre.

il suffit de voir que pour tout

et pour tout couple d'applications

f|'f2

: X ÷ F

si

Jfl @ Jf2'

fl ~ f2"

Pour chaque a.d.g.c. (AX ~ AX 8 AX,D) Soient et

pour

alors de remplacer

~-triviale.

i B e Der(AY).

On a a l o r s

D~signons

ju

pour

en posant

pour

I).

b

e 6 L,

est nulle.

de

peut donc s'~crira

deg iB_ dJ~

multiplicativement.

X

b

= ~ B '

en p r o l o n g e a n t

espace

d

@(~)

(voir ~, § 3) que pour tout

y : AY ~ AX @ AY

Les applications

~ une holonomie

et la d~rivation

satisfait

3) = >

alors pour chaque

triviale

@B = iB ~ _ (-I)

z e AY.

@-triviale,

o~

Dx = dx,

alors

on a

la d~rivation

de degr~

s(x) = x,

s(x) = O,

D~signons ~(x)

= p(~) = O.

~'°

Par contre

par

Dx = ~

notons et

une homotopie

~IX@y = gp

-|

d~finie

s(1) = O,

p

et

(AX,d)

I

l'a.d.g.c.

D~ = O.

g, h : (AY,d) ÷ (AZ,D)

~ : (AX 8 AY,d) I + (AZ,D)

que, par d~finition,

(AX,d),

deux homomorphismes

([H~)

entre

gp

¢.eSd+ds IX@Y = hp,

et ok

s(y) = y,

AY I ÷ AX @ AY I

que

hp. s

Rappelons

d~signe

par : s(y) = O,

s(9) = O.

: (]IX ~ AY) I + AX @(AY) I la projection

Nous montrons

d'a.d.g.c.

~

se factorise

est un morphisme

g travers

d'adgc

p

d~finie

par

en un morphisme

et nous montrons

que

160

le composa

AY I + hx @ iY I ÷ AZ

a) D~signons par s(A2X)

et

X.

l'id~al de

(AX @ AY,d) I

Un petit calcul montre que

b) La relation

c) Pour tout d'une K.S. base Comme

J

est alors l'homotopie recherch~e entre

O(x) = 0

x e X,

(x)~e A

(sd)(x)

e I,

de

J

est stable par

x a X

montre que

~(1) = O. (f~,d).

d'aprgs (a)

par hypoth~se de r~currenee

pour

engendr~ par s

g

et

h.

A2X,

et par

d.

~[X = O.

Ceci se d~montre par r~currence le long

Supposons

~(iB) = O

pour

(sd)P(x) e I

pour tout

p ~ I.

~((sd)P(x )) = O,

V p ~ I.

~ < ~. On a donc,

Ii r~sulte alors de la

fo=ule [HaJ hp(x ) = gp(x ) + ~(x ) +

que

~(iC~) = O.

d) Puisque

X c l'id~al engendr~ par

La d~composition en somme directe d'~crire

d = d| + d2,

En particulier, Puisque dans

~. @2

Posons 0

=¢oe

J

alors

d'a.d.g,c,

y

de

d2(Y) e AY,

est stable par

X,

~(J) = O. @ (Ay)I

permet

Y :

dl(Y) = dl(Y) = d2(Y) = O,

d2(~) = ~.

et

s,

l'image de

@ 2 = sd2+d2s.

eI

On a alors

sd I + dls

est contenue

Im(en-@2 ) C J

et donc

.

~' : (AY) I ÷ AZ

~'d 2 = D~'

entre

et

Im d i C J .

@ = sd+ds

e) Soit (AY) I,

X

(/IX @ iY) I = E(AxI) + ~ (AY)~

avec pour tout

dl(Y) ¢ A~2X 0 AY,

¢oe

~ - L ~((sd)P(x )) p~l p!

g

et

d~finie en restreignant

et il r~sulte de h.

(d)

que

~'

~

g la sous-alg~bre

est une homotopie

m

6.2.- Classifiants. A chaque espace 1-connexe ~F

:

F

÷

EF ÷ BF

F

est associ~ une fibration universelle

(BF = B aut F, EGo_]). Les fibrations ~ fibre

F

et ~ base

161

|-connexe sont alors classifi~es par les classes d'homotopie d'applications continues de

B

dans le rev~tement universel D~signons par

de

Der(AY)

(AY,d)

d~finie oar

Der

-

BF

de

un module de = ~Der|

et

BF.

F

et par

Der

o

Der(AY)

= Der p

pour

la sous-a.l.d.g.

p > O

;

p

H (Der) = H+(Der).

BF = I _ ~ B~,

B~

~ (~BF) = @ ~

On a

une fibration de fibre ~

:

~(~B

B ~+ BB p

parcourant les sous-complexes finis de

F

(~B)

sur

B

) 0 ~ ÷ H D~(AY)).

.

L'injection

~ B~

B

BF.

d~finit par pull back

et donc (§ 3.1) un morphisme ea compatibilit~ des

~

avec les inclusions

d~finit par passage ~ la limite un morphisme d'alg~bres de Lie

: ~ (~BF)

H~(Der(~Y)).

O ~ ÷

Th~or~me 6 . 2 . - Avec l e s notations p r e c e d e n t s , I) P o ~ t o ~ e

fibration

~

de f i b r e

on a

e£ de base

F

B

l e diagramme

s ~ v a n t commute ~ (~BF)

o~

~

~

~

-

H (Der(AY))

'+

d~signe l e morphisme c o n s t r ~

f i a n t e de l a f i b r a t i o n 2)

0

en 3.2

~

1 'application clas-

~.

e s t un isomorphisme d'alg~bres de Lie.

D~mo~t~ation : I) provient de la construction de 2) Soit

~

Sn ÷ ~BF

:

L'application adjointe de base

Sn+! .

(Lemme 6.1) et

Comme ~ = O.

avec

et de la naturalit~ de

~.

p(~) = O.

~' : sn+l ÷ BF

~(~) = ~ ~ ( ~ ' = ) p

~

O,

est donc injective.

~

d~finit une fibration

est rationnellement triviale

162

La surjectivit@ de ~e = 0 base

et

[O] # O,

S n+l

avec

e

~

se voit comme suit : si

induit une fibration

[@] ¢ I m

~.

[0]

n'est jamais

que

F

(si

~n(F) O ~ # 0

Elle peut cependant ~tre

6.3.-

Si

dim w ( F )

< ~,

~>n(F) ~ ~ = O,

on a

H-triviale

elle

: supposons, par exemple, F

@tant F.

T.N.C.Z.

L'holonomie

H-triviale.

Fibratio~

de b ~ e

Proposition 3.- Si d'holonomie est

Q-t~iviale

un 2-c~ne.

~ : F J-~ E e t o~

B

P+ B

e s t un

e s t une f i b r a t ~ o n o~ l ' o p ~ r a t i o n

2-c~ne, a l o r s i l

existe

une l o n g u e

e x a c t e en cohomologie : Hn+l(F)

&n+l

D~mo~t~ation (A,d A) * (A

>

B

[H+(B) @ H~(F)]n

@tant un 2-cSne,

q~ (AY,d)

d(A I) C A2,

est envoy@ dans

supposer

:

AY,d)

0

AI.A 2 = A2.A 2 = O, A~2X

triviale de

l'op@ration d'holonomie

il en est de m&me de la fibration universelle de fibre

est darts ce cas

type

et

satisfait

Im ~ . .

soit une sphere paire, alors toute fibration de fibre

(ETh]),

suite

F -~ E~ + B~

H~(D---$T(AY)) + H+(Der AY)).

~-triviale

H (Der(AY)) # O). n

non rationnellement

appartient donc aussi ~

bans la fibration universelle est donc l'isomorphisme

~

~ ¢ Der (Y) n

A 2.

d(A 2) = O.

La

~-trivialit@

d(Y) C AY $ (A 2 ~ AY).

O

Comme

avec

+

~

Hn(E)

j~

Hn(F)

admet (EFT])

n ....

un module du

A = @ @ A 1 $ A2, AI.A I C A2, bans le module minimal

hX ÷ A,

de l'op@ration montre qu'on peut

Consid@rons alors la courte suite exacte

---+ Ker q

~ (A @ AY,d)

Ker q = (AI$ A 2) ~ AY

et

---+ (AY,d)

>

O.

d IKer q = d A O I = l @ d,

on obtient

la longue suite exacte +

oN

A

EH+(B)

0

est induit par

H~(F)] n

'

Hn(E) J---+ Hn(F) ~

d2 : Y ÷ A 20

Y. •

[H+(B)

0

H*(F)] n+1

. . . .

163

R~arque H~(B)

et de

:

H~(F).

Si

on a

est de dimension finie, il e n e s t

En effet, d~composons

H+(B) = HI(B) $ H2(B) HI(B) O H~(F)

H~(E)

sous la forme

~ partir de la d~composition

s'injecte alors dans

dim H~(F) < ~.

H+(B)

H~(E).

Ii en r~sulte que

de m~me de

A + = A 1 @ A 2.

Com~le dim H~(E) < ~

et

HI(B) # O,

dim H~(B) < ~.

Ceci soul~ve la question suivante

:

Ques£J.on : si

est une fibration avec holonomie

triviale et

F

dim H~(E) < ~,

J > E

-P-+ B

a-t-on

§ 7 - Operation d'holonomie ~

dim H~(F) < ~ ?

o p e r a t i o n de Yoneda.

7.1.- Pour toute fibration Eilenberg et Moore ~-M]

F -~

E --p-+ B

entre espaces l-connexes,

ont construit une suite spectrale v~rifiant

E 2 = ExtH~(B)(H

Rappelons que l'application

p

(E),~) =>

H (F;~).

est dit f o r m a ~ a b l e

(EL~)

s'il existe

un diagramme commutatif

H (B;~)

~

H (E;~)

"

(~E,dE)

P

(~B,dB)

avec

~I _

et

-@2

Si

p

d'Eilenberg-Moore

des quasi-isomorphismes

est formalisable,

alors,

"collapse" au terme

E2

et

p

(IVY),

un module minimal de

la suite spectrale

et l'on a

p.

164

Ext

(H (E),@) ~ H (F;~). H~(B)

Sous cette m~me hypothgse de formalisabilit~,

Ext H~(B)

Th~or~me 7 . 1 . -

Si

H (~B) @ H.(F) ~--~-+ H (F)

p

B

est formel et l'on a

(@,@) = H (~B;@) ~ .

e s t format%sable, l'op~ration d'holonomie

colncide, via les isomorphi~mes precedents, avec l'op~-

ration de Yoneda Ext

(~,@) 0 Ext H~(B)

~0~£/ta£210N H~(B)-module et si

P'

(H~(E),~) ~ Ext H~(B)

:

Si

P

est une r~solution projective de

est une r~solution projective de

alors la cooperation de Yoneda

(H~(E),~). H~(B)

A

sur les

Tor

~

H~(E)

comme

comme

H~(B)-module

est d~finie ([Le])

par la commu-

tativit~ du diagramme suivant :

A

Tor H~ (B) (H~(E),~)

~~l@g-1

H~(P ~@ ~) H (B)

o~

g

%

d~signe l'augmentation

Notons et

, H (P ~@ P') H (B)

÷

TorH~(B)(@,@) @ TorH~(B)(H~(E),~)

+

H~((p ~@ ~) H (B)

@ @

(P' ~@ @) H (B)

P' ÷ ~.

(TLX,d) -~ (AX @ AY,d) + (AY,d)

(AX,d) -~ (~X @ AX,D) ÷ (AX,O)

un K.S. module de la fibration

p

est un K.S.-mod~le de la fibration des chemins

PB÷B. Comme (H~(B) @ AY,d)

B et

est formel, il existe un quasi-isomorphisme (H~(B) @ AX,D)

(AX,d) ~

sont alors des r~solutions projectives

(H~(B),O). P

et

P'

165

de

H~(E)

et

~

comme

H~(B)-modules.

traduction de la d~finition

7. f.- C0ro//a/re

Soit +

a/0rs

H une

(3.1) de l'op~ration de l'holonomie, i

(version gradu~e d'un th~or~me de Roos (ER]).

~ - a l g ~ b r e gradui~e connexe. S i

ExtH(~(H+)m,Q) e s t an ~)[m0yL~t%~on

sh.sh' = 0,

sh.h' = O,

La projection un quasi-isomorphlsme.

0

montre alors que

Le diagramme ci-dessus devient alors la

:

Ext H(@,@) -module ~ b r e

Notons dh = O,

(H S s(H + )nl,d)

(H+) m # 0

(H+) m+! = O,

et

engend~ p~

ExtlH(H/(H+)m,@~).

ii a.d.g.c, d6finie par :

dsh = h.

q : (H @ s(H+) m) ÷ H/(H+)m_

de noyau

(H+) TM % s(H+) m

est

La eourte suite exacte :

÷

(H,O)

(H,O)

÷

(H $ s(H+)m,d)

i> (H/(H+)m,O)

÷

(s(H+)m,o)

+

0

est un module de la eofibre d'une appli-

cation. Ce module ~tant formel, l'homologie r6duite de sa fibre homotopique est + EXtH(H/(H+)m,~) , qui est donc (§ 5) un EXtH(~,~)-module fibre engendr~ par s(H + ) m .~ Ext HI ( H /(H+)m,~ ) .

7.3.gradu~ de

H,

•

C o r o l l a i r e . - S o i t H une @-a/g~bre gradu~e connexe, I un i d e a l + a/ors Extl(H,@) e s t un E x t ^ ( ~ , @ ) - m o d u l e l i b r e engendr~ par H/[ , I

D~monst~ation :

Ceci r~sulte du § 5 et de la courte suite exacte

0

÷

I

+

H

÷

H/I

÷

0.

166

§

8

-

S u i t e s p e c t r a l e d'holonomie d ' u n e ~ i b r a ~ i o n . Soit

~ : F

J~ E

un K.S.-modgle minimal et

L

P' B

une fibration (AX,d) i ~

l'alggbre de Lie de

En filtrant l'a.d.g.c.

(AX @ AY,d)

(AX N AY,d) ~

(AX,d).

par la longueur des mots en

on g~ngre une suite spectrale du premier quadrant v~rifiant

E~ 'q =

et convergeant vers

P,q El

s'~crit :

H~(flX ~ flY,d).

dl= dL + d~

avec

dL

d~finie par la representation

d@(~) = (xi)Ie I

X

iel

La diff~rentielle

@ u..~ 1

dl:

la diff~rentielle de : h ÷ Der HX(AX,d)

x.

,

(flY,d)

~

~

EAPx +

X, H~ (Ay,d~ p+q

E~ +l'q

(flX,dL) S C~(L)

et

d~

avec,

H~(AY,d).---

1

d~signe une base de

X

et

(u. g L)

la base duale :

i

<xi'suj> = ~ij " ([GHV~, § 5.25)

Ii r~sulte alors clairement de

E~ 'q

~ ~ q ( L ; H ~ ( F ) ~ p+q

que

Ext~q(@,H~(F)).

Nous pouvons donc ~noncer :

Proposition 8. I . - I1 e x i s t e une su~%e spectAale du premier quadrant

E p'q = E x t H ~ B ) ( C , H ~(F))

=>

H ~(E)

que n o ~ appelons s u i t e s p e c t r a l e d'holonomie de l a fibrat~on

~.

R~m~que • La formule de dualita de Cartan-Eilenberg construit une suite spectrale duale :

H(~B) E2 = Tor P,q P,q

Corollaire 7.- S i

(~,i (F))

F~

E~

~>

B

i (E).

e s t une f i b r a t i o n ,

et si

B ale

£ype

d'homotopie rat~onnelle d'un bouquet de spheres, alors l a s u i t e s p e c t r a l e d'holo-

167

nomie d~g~n~re au terme

E2 : H(~B) H(E)

Coro££~6~e

f i b r a t i o n avec r i e n , alors

2

H(E)

de finitude sur I

H~(E)

B

:

H~(F)

SoY~i

F

e s t un

÷

E

÷

B

une

H (~B)-mod~e no~h~-

Notons

(fiX,d) un module minimal de

montre qu'il existe un quasi-isomorphisme AX

(AXil~ @ AY,D)

v~rifie

E~ = O p,~

D'autre part, les espaces

(R~ciproque du th~or~me 4.1).-

e s t de dimension f i n i e .

est un ideal de

pour module

(H,(F),~).

B un C.W.-complexe f i n i . S i

D@mons£ra£ion

o~

= Tor

contenant = d~f

(~/l,d)

pour

H~(F)

~ : (AX,d) ÷ (AX/I,d)

pour un certain

@ (AX O AY,d) (AX,d)

L'hypoth~se

r.

E

admet alors

et le gradu~ associ~

p > r.

~tant un

ToriH~(~B)(H~(F),@)

est donc de m~me de chaque

A~rx

B.

H (~B)-module noeth~rien,

sont de dimension finie pour chaque

i.

II en

Ei, ~.

BI

BL I OGRAPHI

ANDREWS P.and ARKOWITZ M.

E

- Sullivan's minimal models and higher

order ~nitehead products. Can. J. of Math. 30, n ° 5 (1978), 961-982. AVRAMOV L. and HALPERIN S. - Through the looking glass : A dictionary between rational homotopy theory and local algebra (These proceedings). B~GVA D

R.

-

Graded Lie algebras in local algebra and rational homotopy. Thesis Stockholm (1983).

DGMS]

DELIGNE P.,GRIFFITHS

P.,

MORGAN

J.

and SULLIVAN

D.

-

Real homotopy theory

of K~hler manifolds. Invent. Math. 29 (1975), 245-274.

168

EILENBERG

S. a n d

MOORE

J.C.

Homology and fibrations

-

I. Coalgebras,

cotensor

product and its derived functors. Comment. Math. Helv. 40 (1966), FELIX Y. and

HALPERIN

S.

-

199-236.

Rational L.S. category and its applications.

Trans. A.M.S.

273 (1983),

|-37.

FELIX Y., HALPERIN S. et THOMAS J.C. - Sur certaines

alg~bres

de Lie de

d~rivations. Ann. Inst. Fourier, FELIX Y. et THOMAS

J.C.

GANEA T.

|43-150.

Sur la structure des espaces de cat~gorie

-

A para~tre

a-O

32, (1982),

2.

Ill. J. of Math.

- A generalization

of the homology and homotopy

suspension. Comment. Math. Helvet. 39 (1965), 295-322. GANEA T.

- On monomorphisms Topology,

EGo]

GOTTLIEB D.

in homotopy

Vol. 6, (1967),

theory.

149-152.

- On fiber spaces and the evaluation map. Ann. of Math. 87 (1968), 42-55.

FG.H.V

GREUB

W.,

HALPERIN

S. a n d

VANSTONE

R.

cohomology

-

Academic Press, GRIVEL

P.P.

-

Connections,

1976.

Formes diff~rentielles Ann. Inst. Fourier,

Su|]

GULLIKSEN T.

et suites spectrales.

29 (1979),

17-37.

- A change of ring theorem with applications Poincar~ Math.

~u-2~

curvature and

III.

GULLIKSEN T.

to

series and intersection multiplicity.

Scand. 34 (1974),

167-183.

- On the Hilbert series of the homology of differentiel graded algebras. Math.

HALPERIN

S.

-

Scand. 46 (1980),

15-22.

Lectures on minimal models. M~moire de la S.M.F. n ° 9/10 (1983).

HALPERIN

S.

et LEMAIRE

J.M.

-

Suites inertes dans les alg~bres de Lie.

Preprint

(1983), Nice.

(To appear in Math. Scand.)

169

LEMAIRE J.M. et SIGRIST

F .

-

Sur les invariants d'homotopie rationnelle

li@s ~ la L.S. cat@gorie. Comment. Math. Helv. 56 (1981), 103-122. [Le]

LEVIN G.

Finitely generated Ext-algebras.

-

Math. Scand. 49 (1981), 161-180. [Me]

MEIER W.

Some topological properties of K~hler manifolds

-

and homogeneous spaces. Math. Z. 183, (1983), 473-481.

[0p]

OPREA J

- Infinite implications in rational homotopy theory. To appear in Proceedings of A.M.S.

[Q]

QUILLEN D.

- Rational homotopy theory. Ann. of Math. 90 (1969), 205-295.

R]

ROOS J.E.

Homology of loop spaces and local rings.

-

Proc. of the 18 th

scand, congress Math.

Aarhus (1980).(Progress in Mathematics, n ° 11, Birhguser, 198].) [St]

STASHEFF J.

Parallel transport and classification of fibrations.

-

Lect. Notes in math. N ° 428, (1974). ES~

SULLIVAN D.

- Infinitesimal computations in topology. Publ. I.H.E.S. 47 (1977), 269-331.

TANR D.

Homotopie rationnelle : ModUles de Chen, Quillen,

-

Sullivan. Lect. Notes in Math. n o 1025 (1983), Springer Verlag.

IT@

3.c.

Rational homotopy of Serre fibrations.

-

Ann. Inst. Fourier 31 (1978), 71-90.

[v]

VIGU M

-

R~alisation de morphismes donn@s en cohomologie et suite spectrale d'Eilenberg~ioore. Trans. A.M.S. 265 (1981), 447-484.

[W_]

WHITEHEAD G.

- Elements of homotopy theory. Graduate texts in math.

Yves

F E L I X

(I 978), Springer Verlag. Jean-Claude

T H 0 MA

S

UNIVERSITE CATHOLIQUE DE LOUVAIN

UNIVERSITE DE LILLE I

1348 - LOUVAIN-LA-NEUVE

59655 - VILLENEUVE D'ASCQ CEDEX

(Belgique)

(France)

Flat families of local , artinian algebras with an infinite number of Poincar@ series

by

Ralf FrSberg, Tor Gulliksen and Clas LSfwall.

Introduction. For a local ring (R,m,k) let PR(Z) denote the Poincar@ series i~0dimkTor~k,k)z i. The origin of the present work is a question how Poincar6 series may vary in a flat family of local artinian k-algebras. In particular we were interested in knowing if such a family might have an infinite number of Poincar6 series. We will show that this is indeed the case by exhibiting a one-parameter family {Rl}16 Q of local artinian Q-algebras of length 85 such that the corresponding Poincar6 series form an infinite set. We also get as a bonus an example of an augmented 2-algebra A, free of rank 85 as 2-module, such that A/p are local rings and PA/p(Z) are different for all primes p, and also TorA(2,~) has p-torsion for all primes p.

It turns out that it is possible to construct families of local artinian q-algebras whose Poincar6 series vary quite vividly and depend on various algebraic and/or arithmetic properties of the parameters. For instance we show that there exists a family {RI}16 C and a power series f(z) such that PRI(Z)= f(z) if and only if I is transcendent over Q, and that there exists a family {$I}16Q2 such that the calculation of PS (z) for all I is equivalent to solving Fermat's equation n n 11 + I 2 = I for all n and 11,126~. It was natural for us to start looking at local k-algebras (R,m,k) with m 3 = 0. Let C be the class of such k-algebras and let B

be the

class of algebras of type k/(gl,...,gs) , where k p

is the free associative (non-commutative) algebra and the gi s are linear combinations of the elements T~,m I S i S n, and TiT j + TiT i ,

171

I ~ i < j ~ n. It follows from results of LSfwall [LS] that if we can construct a family {B l} in B with infinitely many Hilbert series Bl(z) = i>0Xdir~(B~).zIA i , we get a family {A I} in C with infinitely many Poincar@ series. Each element BEB is the universal enveloping algebra U(G) of a graded Lie algebra G. Anick and L6fwall-Roos, see [L6-Ro], have a construction, which to any graded (non-commutative) algebra N gives a graded Lie algebra G, such that U(G)(z) is determined by N(z). If N is generated by elements of degree one and has relations of degree two 0nly, then U(G)EB. Thus we have a construction available, which to any family of non-commutative algebras with generators of degree one and relations of degree two and with infinitely many Hilbert series gives a family of confutative local rings with m 3 = 0 and with infinitely many Poincar@ series. For this reason we were lead to the study of noncommutative graded algebras. Exhibiting a family in B with infinitely many Hilbert series also makesit possible for us to construct a family of topological spaces {XI} , in fact mapping cones of maps between wedges of spheres 6~S 3--~-~ yDS2, with 1 I infinitely many series Z ( d i m ~ w . ( X ~ ) z l, w. denoting homotopy groups. i~O ~ i ~ l

I. Poincar@ and Hilbert series of families of sraded k-alsebras. A graded algebra will in this paper mean an algebra which has a presentation k/l. Here k is the free associative (non-commutative) algebra in the variables TI,...,T n of degree one and I is a homogeneous two-sided ideal in k , k a field. Of special interest to us will be the case when I is generated by elements of degree two. We call such algebras 2-related. For a graded module M = ~ M. over a graded algebra we define the i~0 l Hilbert series of M to be i M(z) = i~odimkMi'z . The set of elements of positive degree in a graded module M will be + denoted M .

172 For an augmented k-algebra A (or a local ring (A,m,k)) we define the Poincar~ series of A to be

PA(Z) = Z di~Tor§(k,k)'z i i~0

~

l

By a famiily of k-algebras {A~}, ~ = (XI,...,~m)C k m, we will mean a set of k-algebras together with a finitely presented k[X]-algebra A, X = (XI,...,Xm) , such that AX =A/(X-X)

for all ~ E k m. The family {A~} is

called a flat family if A is k[X]-flat. We call the family ~raded if A has a presentation A=k[X]/(fl, .... fr ), Y = (YI,...,Yn), where the fi's are homogeneous in Y. Thus all A~ are graded k-algebras in a graded family {A~}. Finally we call the family commutative if A is commutative.

First we examine how Hilbert series A~(z) may vary in a commutative graded family {A~} of algebras. Claim. In a commutative graded family there are only finitely many Hilbert series. In fact a much more general statement is true as the following proposition shows. We note that in a graded family {A~} there is a uniform bound for the number of generators and the degree of the relations, namely if A can be presented as a free algebra in n variables over k[X] with relations of degree Sd in Y, then each A~ has a presentation k modulo forms of degree Sd.

Proposition

I. Let n and d be fixed integers. There are only finitely

many possibilities for A(z) when A belongs to the class of graded algebras of the form k[Xl,...,Xn]/(f I .... ,fr ) where k is a field and the fi's are forms of degree Sd. Proof. Let B=k[XI,...,Xn]

and let A be a graded factor ring of B. The

syzygy theorem of Hilbert states that A has a minimal graded resolution b bI 0-~ i~_~B[-ni,r ] --~...--~ i~=iB[-ni, i]~-~ B ~-~A.-~ 0 for some r S n, where the brackets stand for a shift in degree,

(i)

173

(B[-k]) d = B_k+d. To construct a step in this resolution is equivalent to solve some linear system of equations with coefficients which are forms in B. It is shown in [He] (also c.f. [Se] and [La]) that there is a bound M, only depending on n and the degrees of the coefficients in the system, such that all solutions can be generated by solutions of degree the resolution is of length

~ M. Since

S n, this gives a bound N = N(n,d) for all n.

Since the degrees of the syzygies are uniformly bounded it follows that the number b. of syzygies are uniformly bouude4. In each fixed degree the resol

lution (I) is an exact sequence of vector spaces and thus their alternating sum of dimensions is zero. Taking generating functions we get the formula b~Izni,1 + ~2zni,2 _ ... + (-I )r ~rzn i ,r)/(1 _z)n i=I i=I i=I and hence we see that there are only finitely many possibilities for A(z).

A(z)=(1-

We are interested in the following property of a commutative graded family {AI}:

(P) The set {PAl(Z)} of Poincar6 series is finite.

We will show that there exist flat families of local graded artinian k-algebras not satisfying (P). There is another, seemingly weaker, property for a commutative graded family {At}:

AI AI (P') There is a number N such that, if dimkTor i 1(k,k) = di~Tor i 2(k,k) for all i ~ N ,

then PA

(z) = P A 11

(z). 12

In fact (P) is equivalent to (P') for a family {A~}. Of course (P) implies (P'). But if {A~} is a graded family, then there are only finitely many AI possibilities for dimkTor i (k,k) for fixed i. This follows by the same reasoning as in the proof of proposition I: Let A be a graded k-algebra in

174

n variables and with relations of degree ~d. Constructing a step in a graded free A-resolution of k is equivalent to solving a system of linear equations over A. This can be lifted to B. By induction over i it follows from the theorem of Herrmann mentioned above that there are only finitely many possibilities for

N A zi ' Z dim. Tor. X(k,k) Hence (P') implies (P). i=O ~ i

In the study of non-commutative

families of graded algebras we will

be interested in the following two properties for a family {BI}: (H) The set {Bl(z )} of Hilbert series is finite. (H') There is a number N such that, if dimk(B~1 )i =dimk(Bl2)i then

B),I

(z )

= BI2

for all i ~ N ,

(z ).

The properties

(H) and (H') are equivalent for a family {B)). Of

course (H) implies (H'). If n is a bound for dimk(B~) I for all I, then dimk(Bl) i ~ n z, thus there are only finitely many possibilities for i=0

dimk(B~)izi

hence (H') implies (H).

In next section we will show that (H) is not satisfied for all families of non-commutative algebras.

2. Hilbert series of non-commutative graded families. In this section we give two methods of constructing graded algebras with badly varying Hilbert series.

Construction

I. Let A be a graded (non-confutative)

algebra and let

A L and AR be two graded vector subspaces of A +. Let T be the coproduct of A with k/(a2). If W = {I}UW + is a graded k-basis for A, then +

+

+

{WaW aW ...aW aW} is a graded k-basis for T. Let I~-T be the two-sided ideal generated by aA L and ARa and let ~ = T/I. We note that if A is 2-related and AL,ARCAI,

then ~ will be 2-related. Let W I (and W3, respectively) be a graded

has, for a complement to AA R (and ALA , respectively)

in A and let W 2 be a

graded basis for a complement to ALA + A A R in A +. Then a k-basis for A is WU{WIaW2aW2...W2aW 3) and hence A(z) =n~0W1(z)W3(z)zn+1(W2(z))n

+

A(z) =

175

(A(z)-AAR(Z))(A(z ) -ALA(Z))Z[I ~ ( A ( z ) - I - ( A L A + A ~ ) ( z ) ~ -I + A(z)

Example I. Let A=k/(bc-cb-lc2),

16k. As a k-vector space A

(2)

is generated i

by c i ,c i-lb, ...,cbi-l,bi hence A(z) S (l-z) -2. In fact we have equality since k/(bc-cb) :k[b,c] is made to a cyclic left A-module by b*cib j =cib j+1 +ilci+Ib j and c.cib j =ci+Ib j and hence A(z) ~ (l-z) -2.

Let AL: (c-b)k and AR : b . k ,

then

ALA(Z) : AAR(Z)=

z(l-z) -2

since, as is easily seen, (c-b)ci,(c-b)ci-lb,...,(c-b)b i (and cib,ci-lb.b .... , bm.b, respectively) are linearly independent. Finally we have (ALA+~&R)i+ I = Aib + (c-b)ci-k = Aib + (1-iX)c i+1.k. Thus if

~=0

or I-I~{1,2,...}

we have

(ALA+AAR)(Z) = (l-z) -2 - Io If chark = 0 and I-I : q6 {1,2,...} we have (ALA+AA R) (z) : (l-z) - 2 - I - zq+1. If chark = p and I -I = q 6 {1,2,...,p-I} we have

(A~A+AAR)(Z) = ( I-z)-2

I - zq+1(1-sP) -I .

Hence if I= 0 or I-I ~ {1,2,...}, then A. =k/(a2~ be-cb-lc2,ac-ab,ba) it

has Hilbert series (by formula (2)) At(z) = (1-z)-2(1+z). If chark = 0 and I-I = q 6 { 1 , 2 .... }, then

~1(z) = (1-z)-2(1+z-z

q+~ • ){1-zq+2) -I .

Finally, if chark = p and I-I = q 6{1,2,...,p-I}, then Ax(Z) = ( I-z)-2( 1+z-zP-zP+1-zq+~)(I-zP-zq+2) -I .

If we replace k by • and put ~ = 1 we get a ~-algebra B = ~/(a2,bc-cb-c2,ac-ab,ba)

(3)

176 such that the Hilbert series of B/pB are different for all primes p (and also different from the series of B ® g ~ ) .

This phenomenon can not occur

in the commutative case according to proposition I.

Example 2. Let A=@/(bc-~cb),

~E@

and let A L = A R = (c-b).~. This gives

A~= ~/(a2,bc-~cb,ba-ca,ab-ac). It is possible to compute ALA(Z) , AAR(Z) and (ALA+AAR)(Z) as in example I to get

~(z) = (1+z-z2)(1-z2)-1(1-z) -2

if In # ] for all n > 0 and Al(z) = (1+z-z2-zn+2)(1-z2-zn+2)-1(1-z) -2 if ~ is an n'th primitive root of unity.

Construction 2. (This is an alternative to construction I, it yields algebras with smaller Hilbert series but needs one more generator.) + Let as before A be a graded algebra and A L and A R be subspaces of A . Let T* be the coproduct of k/(L 2) and A and k/(R 2) modulo the twosided ideal (*L, R*)

where * stands for anything of positive degree. As a

graded vector space this algebra equals

k/(L2)~k Let I # be t h e t w o - s i d e d

A ~ k k/(R2). ideal

generated

by LAL and ARR. Then

I" = LALA + AARR + L(ALA+AAR)R and if ~ = T*/I" then ~(z) =A(z) + z ( 2 A ( z ) - A L A ( Z ) - A A R ( Z ) ) + z2(A(z)-(ALA+AAR)(Z)) •

(4)

We note that if A is 2-related and AL,ARCAI then A will be 2-related.

To be able to give simple descriptions of the spaces ALA, AAR and ALA+AA R we will put restrictions on the algebra A. If B is an algebra and M a B-bi-module, the trivial extension B U M

is B ~ M

as vector space and

has multiplication (b,m)(b',m') = (bb',bm'+mb'). From now on we put

A = B ~ ( V ~kB), where B = k<S> and S is a finite set of elements of degree one and V is a

177 k-vector space of finite dimension. We make V @ k B a B-bi-module in the following way. For each s6S there is given a linear transformation Js: V--~V. To each monomial B = SlS2...s n in B we consider the composite map JB= Js1°Js2 °'''~Jsn" For B = I we let JB denote the identity map. This defines V as a non-6raded left B-module by B.v = JB(v) and k-linear extension. For each s6S we define s(v@ b) = s v ~ s b which extends as above to an operation of B to the left on V ® k B. The operation of B to the right on V ® k B is the obvious (v® b ) b " = v e ( b b ' ) .

these two operations are compatible. To define V ~ k B module we let the degree of v ~ b

as a ~ B - b i -

be I + d e g b , that is we consider V as

concentrated in degree one. With this definition V ~ k B indeed becomes graded, since deg(b1(v @ b2) ) = deg(blVgblb2) = deg b I +deg b 2 + I = deg b I + deg(v~ b 2) and deg((v~ bl)b 2) = d e g ( v ~ b l b 2) = deg b I + deg b 2 + I = d e g ( v g b I ) + deg b 2 •

Proposition 2. Let E be a basis for V. The algebra B~<(V~kB)

has the

following presentation H=k<SUE>/((ee ~,se-J s(e)s; e,e'6E, s6S}) and hence B ~ < ( V ~ k B) is 2-related. Proof. First consider the surjection p: k<SUE>

>BW(V®kB)

defined by p(s) = (s,O) for s6S and p(e) = (O,e®1)

for e6E. Then

p(ee ~) = (O,el~ 1 ) ( 0 , e ~ ' ~ 1 ) = ( 0 , 0 ) and p(se-Js(e)s) =p(s)p(e) -p(Js(e))p(s) = = (s,O)(O,e~ I) - (O,Js(e)~1)(s,O) =

178 = ( 0 , s ( e @ 1 ) ) - (0,(Js(e)@ 1)s)= (0,Js(e)@ s)- (0,Js(e)@ s ) = 0 . Thus p factors through H and we get

H(z) ~ B~(V~kB)(z). Using the relations ee" = 0 only, we see that words of type Wlelw2e 2...wrerwr+1, ~here w.1 is a word in S and e.EEI for all i, generate H. Using also the relations se = Js(e)s we see that it suffices to take words w I and ew2, where Wl,W 2 are words in S and eEE, to generate H. Hence

H(z) S B(z) + zdimkV.B(z) = B ~ ( V ® k B ) ( z ) . We have shown t h a t i(z)

: B~(V@kB)(z)

which g i v e s H ~ B ~ ( V @ k B ) s i n c e B~(V(~.B)K i s a f a c t o r o f H. To be a b l e t o a p p l y t h e c o n s t r u c t i o n +

two subspaces A L and ~

1 ( o r 2) we need t o s p e c i f y

of A . We will choose these as subspaces of

VcAI and call them V L and V R respectively.

Let {~} be the k-basis of

monomials for B and let B~ = B'k. We have ALA = V L ~ B = ~ (VL~Bs) and AA R = im(B~VR--~ V ~ B ) = ~ J s ( V R) @kBs whence

A~A+~ R :~( (%(V R )+V~)':~kB~). Taking k-dimensions we get (~)

ALA(Z) = z.dimkVL.B(z) AAR(z)

= ~di~JB(VR).zdeg B +

(ALA+AAR)(Z) = ~dimk(Js(VR)+VL).zdeg B + I If J $ is an isomorphism for all sES

(7)

then (6) may be simplified to

~(z) = z.dimkVR.B(z) The formulas (5), (6) (or (6")) and (7) may be inserted in either epnstruction

(6)

I or 2, formulas (3) or (4), to get formulas for A(z) or

(6")

179

~(z) depending only (if Js is iso for all s) on dimkV L, dimkV R, the number of elements in S and the crucial term • ~d~mk(J~(V R)+VL)z1+deg We have ~(z) = (1+zl2B(z)(1+zdi%V)-

z2(di%VL.B(z) + ~di~Je(VR)Z aeg e) _

_ z3(~di~(JB(VR)+VL)zdeg B

Suppose now B = C<s>, the free algebra over ~ on one generator of degree one, and suppose V = ~2. Then there are two canonical forms for Js' namely

I ~2

and

. This motivates the first two examples

below. The properties of the rings in examples 3 and ~ are similar to those in examples I and 2. The set of algebras obtained in examples I and 2 have one generator less, they can not A=B~(V@k

however

be'obtained as A where

B) for any B and V.

Remark. We note that the above reasoning is also valid in the more general case when B is a graded monoid algebra, that is B = k<S> modulo a set of monomials and differences of monomials of equal degree. If the monoid algebra is 2-related, then A will be 2-related. In particular we could have 2 2 used B=k[Sl,S 2 .... ,sn] or B =k[sl,s2,...,sn]/(s~,s2,...,sr).

This of

course imposes restrictions on the maps J . s

Example 3. Let B=~<s>, V = @ 2, VR:(I,0).~, ~ ~n Then OTnf1~ s ~ =~n~n-1

0

r~t~ ~nn 1 I=

21

VL=(I,1)'@

and Js =(~

,hence dim~(J~(VR)+V L) = I if X = n > 0

~)" or

X = 0 and n > I~ and dim@(J~(VR)+V L) = 2 otherwise. Thus we have a family {Ax) with "generic" value of Ax(z) if X ~

and with A~(z) different for

all X@JN. (When ~ = 0, Js is not iso so the series AAR(z) has to be computed by means of (6).)

180

1 I,

~I I" Hence dim¢(Jns(VR)+VL) = 2 if

#I and

dim~(~s(VR)+V L) = I if In= I. This gives a family {Ax} with "generic" value of A}(z) if and only if ~ is not a root of unity.

Example 5

Let B=~<s>, V=¢3, VR •

and let J

= s

(I 3 1 3 ~ 3~'~ ="

I"

2'

VL

3 ~ ~'

(I,0,-I)~+ (0,I,-I)~ =

(~ 0 0 i 01 X 0 I Then Jns(VR)= (xn+3 ~n+3 >n+3~ 0 02 13 ! " ~ I ' 2 '- 3 ~.~ and

dimQ(jn(VR)+V L) = 2 if the determinant In+3

I

0

0 _~+3 _ I

I -I

xn1+3

1

n+3 + ~n+3 n+3 . n = 3 otherwise. = 11 2 - ~3 = 0 and dlm~(J s (V)+V_) R L

Moreover dimQ(J~(VR))= I if (~I,X2,~3) # (0,0,0). For 13= I this gives exceptional values of At(z) if and only if Xn+3+~n+3 I -2 = I for some n. We have one value of At(z) for I= (~i,~2)= (1,0) or (0,1), another value for X= (-1,0) or (0,-I). The statement that At(z) is independent of X for all other values of i is equivalent to Fermat's last theorem.

Example 6. Let {~0=I,~I,...,~N } be the set of monomials in {~I .... ,~m } of degree Sd, let S= {s0,...,SN,S~,...,s~} , let B=@<S>, V=@2, V L = V R = =(1,0)-¢ and let Js z= Ii

and J s.,= i

B b e a m o n o m i a l i n S, t h e n JB = l ( a )

-~i

I

for i=0,1,...,N. Let

where l(a)

is a linear

combi-

nation of ~O,...,~N with coefficients in g. Any such linear combination can be achieved by appropriate choice of B. Thus we have exceptional values for A~(z) if and only if i(~) = O, that is if and only if (~1,...,~m) satisfies a polynomial equation of degree Sd. If we restrict to m = I we get a family {A~} with exceptional values of A~(z) for algebraic numbers ~ of degree Sd, and if we restrict further and also let d = I, we get exceptional values for IC~.

181

Example 7. Let S= {So,S~,Sl,S2,S2,S3,S3,...,Sm,Sm} , let B=@<S>, V=C2, V L = V R = (0,I).~ and let • j

So= 0

1

I -I}

s~ =

I} ~ Jsi ={Ol~ ~} for i = 1,2 ..... m and Jsf=l 01

for i= 2,3,...,m. If B is a monomlal zn S, then JS = 0

" where

p(A) is a polynomial in {AI,A2,...,km,A -I 2 ,A-I 3 " " ' k m -. } and we have an exceptional value of AA(z) if p(A)= O. Claim. We can get any polynomial p(A) Cg[AI,...,X m] in this way. Proof. First we see that we can get any q(~)E~[~,...,A m] in this way. Let ~ be any monomial in A2,...,Am, then

,0

0lj~ Is obtained as an appropriate product of the Js. "s, it follows

Since (0

that there is a B such that J~ =' 0 Also if

j

i BI = 0

lJ"

and J@2 =110 Y) then j@182= 0I X l ~ " Hence any matrix of

the type (I0 q ( ~ ,

q(A) 61[~ 2 ..... Am] is obtainable as a J B. Suppose now

d • ,Xm) D(A) :Do(A2 ..... Am) +AIpI(A 2 ..... Am) + ... ÷ X1Pd(A2,..

then p

~

0

1

and the claim is proved.

In the presentation of A1 which follows from proposition 2 the only relations containing

,...

are the relations s e I = A

for i =

m

2,3,...,m. If we replace these relations with ~is~el =else, i =2,3,...,m, we get a family { ~ } with exceptional values of A~(z) for all ~ = (~1,...,~m) which satify some polynomial equation over Z at least if AI~2...A m # O. But if AIA2...A m= 0 we get, as is easily checked, a value of dimk(A~) 3 which does not agree with the generic

182

value. Hence the family {A~} has exceptional values of A~(z) if and only if ~= (~1,...,Xm) satisfies some polynomial equation over g. In particular when m = I we get a one-parameter family {A~}~@ with exceptional values of Ax(z) if and only if ~ is an algebraic number over ~.

3. Families of graded Hopf algebras with infinitely many Hilbert series. We now recall the construction in [L6-Ro] mentioned in the introduction. Let N=k/(gl,...,gt)

be a 2-related algebra. Let

ml,...,mn2_t be a set of monomials in {TI,...,T n} of degree two whose images in N constitutes a k-basis for N 2. To the algebra N we define a graded Lie algebra G in the following way. The Lie algebra G is generated by a set {T I ,... ,Tn,L I ,.. .,Ln,Y,R I ,...,Rn,Zml ,. "',Zmn2_ t} of variables of degree one and has the following relations (I)

T~=O,l I S i S n ,

(2)

[LI,T j ] = [Ti,Rj], 1 < i , j < n

(3)

~ c

i,j 10

and [Ti,Tj] =0 , I S i < j

[Li,Tj]= 0 if and only if

Z c..T.T.E i,j 1j i J (gl '" "''gt )

(4)

[Y,Ti] =0, I
(5)

[Z,T i ] =0,

(6)

[Li,R j ] =0, I ~
(7)

[Li,Z] =0, I Si_-
(8)

[Y,Ri] =0, ISi_-
(9)

[Y,Zmi ] = [Ljl,Tkl]

I
•

.

if m i=T.

~n

T

m I , " ' , Z m n 2 t}

,''',Zmn2_ t}

.

0 i k i

It follows from [L6-Ro] that G is an extension of graded Lie algebras + O--~Nab'

~ G--~F----)O.

+ denotes N + considered as an abelian Lie algebra and F = F I x F 2 Here Nab i~ a product of two free Lie algebras F I and F 2 generated in degree one. The ~enerators for F I are LI,...,L n and Y, and the generators for F 2 are

183

RI,...,R n and Zml,...,Zmn2_$. In particular

G(z) =~+(z) +F(z). For the enveloping algebras U(G), which are 2-related Hopf algebras and has a presentation as above (where the relations [A,B] is interpreted as AB+BA), this gives

u(O)(z) : U(N~b)(Z).U(F)(z).

(8)

We have U(F)(z) = (1-(n+1)z)-1(1-(n+n2-t)z) -I and if N(z)=

Z a.z I, then i>O i U(N~b)(Z) =i~1[(1+z2i-1)a2i-I/(1-z2i)a2i]

(9)

Let {AI} (or {~i}) be any family from section 2. We use the construction above with N =AI. As is seen by inspection, dimk(Al) i are independent of I for i = 1,2. Hence U(F)(z) is independent of I. From (8) and (9) follows that U(G)II(Z):U(G)x2(z) if and only if All(Z) = Al2(z). Thus we have constructed for each family from section 2 a family of 2-related Hopf algebras with infinitely many Hilbert series. Finally we list the relations in the case {AI] is the family from example I in section 2. We get a family {U(G)I}, where U(G)I =k<TI,T2,T3,LI,L2,L3,Y,RI,R2,R3,ZI2,Z22,Z31,Z32,Z33>/I where I is generated by

(i)

T~, I < i < 3

(s)

[Li,T j] - [Ti,Rj], I _-
(3)

[LI,TI], [L2,TI], [LI,T 3] - [L I,T2], [L2,T 3] - [L3,T2]- I[L3,T3]

(4)

[Y,Ti], I < i < 3

(~)

[Z,T i], I
(6)

[Li,Rj], 1
(7)

[Li,Z] , I Si <3 and all ZE {Z12,Z22,Z31,Z32,Z33}

and [Ti,T j] (=TiTj+TjT i) I S i < j <3

184

(8)

[Y,Ri] , I ~ i $ 3

(9)

[Y,Z 12 ]-[LIT2 ] , [Y,Z22]-[L 2,T2] , [Y,Z31]-[L 3,T I ], [Y,Z32]-[L 3,T2] , [Y,Z33]-[L3,T3].

We have thus constructed a family {U(G) X} of Hopf algebras with the following properties. We have dimk(U(G)~) i independent of ~ for i = 1,2. If char k =0 we have infinitely many values of U(G)~(z). Replacing k with in the definition of U(G)~ above and restricting to ~ = I, we get a graded ~-algebra C such that the Hopf algebras C/p have different values of C/p(z) for all primes p (also different from C ~ ( z ) ) .

Remark. The construction in [L6-Ro] presumes char k =0, but this is an unnecessary restriction.

4. Flat families of local k-al~ebras with infinitely many Poincar@ series. In this section we pass from Hopf algebras to commutative local rings. Let N be the isomorphism classes of (not necessarily commutative) graded kalgebras N with N 3 = 0. Let A be the isomorphism classes of 2-related not necessarily commutative algebras. Then, quoting [LS]: There is a I-I correspondence between N and A. If N E ~

corresponds to

A CA, then PN(Z) = zA(z)/(1 + z-A(z)(1 -dimkN1-z+dimkN2.z2))

(10)

We restrict the class N to the class C of commutative graded algebras C with C 3 = 0. Since artinian commutative graded k-algebras are local, all algebras in C are local. We get a I-I correspondence between C and B = the isomorphism classes of 2-related Hopf k-algebras, that is algebras of tyDe

185

k modulo relations which are linear combinations of T 2i, I =
Sn. Thus B contains the U(G)~s from sec-

tion 3. From (10) it follows that if BI,B 2 E B and B1(z) ~B2(z) , then PcI(Z) ~Pc2(Z) , where C i C C corresponds to B i for i = 1,2. Thus for each family (A~) (or (~X}) from section 2, we get a family (C~) of commutative local rings with m 3 = 0 and with infinitely many Poincar@ series. That the families {CI) are flat follows from the following well-known lemma. For the convenience of the reader we supply a proof.

Lemma 3. Let C be a commutative Noetherian domain and let M be a finitely generated C-module. Then the following conditions are equivalent: (i) M is C-flat. (ii) dim~Cp/pCp , _ M p'/~Mp is independent of p for all p C $pec C (iii) If Q is the fraction field of C, then dim_ . _ M /mM = d i m Q ( M ~ c Q )

Cm/mumm

m

for every maximal ideal m in C. If the Jacobson radical of C is zero these conditions are equivalent to:

(iv)

dim C /mC Mm/mMm i s i n d e p e n d e n t m-

o f m f o r e v e r y maximal i d e a l

m i n C.

m

Proof. (i) ~ (ii) : Let P E Spec C. A flat module over a local ring is free, hence Mp is free, M p = C n.p This implies that M @ c Q = Q n ,

n = dimcp/pCpJMp/~1 l

hence

is independent of p.

(ii)~(iii)

is trivial.

(iii) ~(i):

Since M is C-flat if and only if M m is Cm-flat for all maxi-

mal ideals m in C, we can assume C to be local. Suppose n = d i m c m / ~ M m / ~ m = m =dimQ(M@cQ). with

Cm/mCm.

Let Cn •f~M be a surjection which is an iso when tensored Then n 0 --~ Ker f @ c Q ---~Q ~

M @ c Q --~ 0

• n = d i m Q ( M @ c Q ) = n we have that is exact since Q is C-flat. Since dlmQQ is an iso and hence that Ker f @ c Q = 0. This implies that K e r f C-module. But K e r f ~ C n , hence K e r r

= 0 and M is free.

is a torsion

186 Now suppose that the Jacobson radical is zero. We prove (iv) ~ (ill): Consider U = (pC Spec C ;

dimcp/pCpMp/pM p ~dimQ(M@cQ)~.

The sets

S = ~ {p; p D A n n ( C o k e r f )} are closed. Hence U, which is the complement r CX~-~ M of S d i m Q ( M ~ c Q )

, is open. Furthermore U is non-empty since (0)E U. Since

the intersection of all maximal ideals is zero, U must contain a maximal ideal m. Thus

dimcm/mCmMm/mMm~ d i m Q ( M ~ c Q )

, hence the same inequality is

true for all maximal ideals. The opposite inequality is trivial and hence we have equality.

Let

(or ( ~ } )

be a family from section 2 and let (C~) be the

corresponding family of local rings. Since dimk(A~) i is independent of for i = 1,2 we have that d i ~ (C~) = I +dimk(C~) I +dimk(C~) 2 is independent of ~. To show that (C~) is flat, we can assume that k is algebraically closed. To see this, let A be the finitely presented k[X]-algebra from the definition of a family. Then A @ k k

is k[X]-flat if and only if

A is k[X]-flat since k[X] is a faithfully flat k[X]-algebra, where k is the algebraic closure of k. If k is algebraically closed all maximal ideals of k[X] are of type (X-~), ~ g k .

Hence (iv) in the lemma with

C = k[X] and M = A shows that all our families are flat.

The material in this section supplies a proof of the following theorem. To get an explicit example we use the family (U(G)~) from the end of section 3.

187

Theorem A. Consider the family {Cl} where Cl = kiT I ,T2,T3,L I ,L2,L3,Y,R I ,R2,R3,ZI2,Z22,Z31 ,Z32,Z33]/I}~ and Ii is generated by (T,L,Y,R,Z) 3 and (I) L.L. and L.Y and y2 i J m

,

I
=

<3 =

(2) R.R. and RiZ I and ZIZj, I < i < j i j

< 3 and l,JE {12,22,31,32,33}

(3) LIT2+LIT3+YZI2+TIR2+TIR3 , L2T2+T2R2+YZ22 , L3TI+T3R1+YZ31, L2T3+T2R3+L3T2+T3R2+YZ32 , XL_T_+IT2R 23 3 +L 3T3 +T 3R3 +YZ 33" Then {Cl} is a flat family of local k-algebras of length 85. If char k = 0, then {Cl} has infinitely many Poincar@ series.

Theorem B. If we replace k with g in the definition of Cl above, and put I = I, we get a graded Z-algebra D such that D is L-free of rank 85 and D/p are local rings with different Poincar@ series for all primes p. (They are also all different from PD@~(z).) Furthermore TorD(g,~) has p-torsion for each prime p.

Remark. Theorem B contrasts to [Mo, Cor.9] where it is shown that in some situations there can only be a finite number of Poincar@ series.

The proof of the first statement of Theorem B follows from the beginning of this section using the last sentence of section 3. The last statement will follow from the lemma below. That we could draw this conclusion about p-torsion was observed by L. Avramov who also supplied a proof of the lemma.

188

Lemma 4. as a

Let

D

be an augmented

~-module, and let

p

~-algebra, which is finitely generated

be a prime number. Then

p-torsion if and only if for all

X

Let

--* D

X/pX

X

--* D

is a

Hence

= dim~Tor~(~,~)

D-free resolution. Since

D-free resolution. Hence

--~ D/p~ ×Ipx

be a

is a

~Dlpn Dlp2

has no

i ,

dim~/p~Tor~/PD(D/pD,~/p~)

Proof.

TorD(~,~)

X/pX

.

D

is

D-free,

is acyclic and

D/pD - resolution. But

Dlpn) ~nlpn 21p2

= (X ~D

TorD/pD(~/p~,D/p~)

= H((X ~D ~) ~

= x ~n ~Ip~ = (X O n 2) ~

~IpD

~/pD) . Moreover we have

TorD(2,~) = H(X ~D ~) . By the universal coefficient formula, 0 --* H i ( X ~ D D ) ~ / p ~

--~ Hi((X~DD)e~7/p~ ) --~ Tort(Hi_ I

is an exact sequence, and hence from above the following sequence is exact, 0--* Tor~(2,2)@22/pD ~ Hence for all

Tor~/PD(~/p~,~/pD)l

--~ T°r~(T°r~-1(~'2)'7/P~)

i ,

dim~/p2(Tor~(2,~) with equality for all

i

On the other hand, for all dim~ T o r ~ ( ~ , ~ )

~

D/p~) ~ dim~/p~Tor~/PD(~/p~,~/p~)

if and only if

i

has no

p-torsion.

i ,

= dim~(Tor~(~,~) ~

with equality for all

TorD(~,~)

~) ~ dim~/p~(Tor~(~,D)

if and only if

TorD(~,~)

has no

~

~/p~) p-torsion.

Putting this together we get, dim~Tor~(~,~)

~ dimD/p~(Tor~(~,~)~

with equality in both places for all no

D/p~) ~ dim~/p~TOr~/PD(~/p~,~/p~) i

if and only if

TorD(~,D)

p-torsion.

5. Flat families of non-commutative algebras with infinitely many Poincar@ series. We note that we can construct families with infinitely many

has

--~ 0 .

189

Poincar@ series without the construction in section 3. However, these families will consist of non-confutative graded algebras.

Example 8. Let Bl=k/((ab+ac,

b 2, bc+cb, bc+Ic 2, ca)+ (a,b,c)3).

Then, in the correspondence of section 4, Bi will correspond to Al=k/(a 2, bc-cb-lc 2, ac-ab, ba), which is the algebras from example I of section 2. Then (10) of section 4 gives

P~ (z)=(1-3z) if X = 0

or ~-I~{1,2,...}.

-I

If c h a r k = 0

and X-I = q g {1,2,...} we have

PB (z)= (1+z+z q+2 )/(1-2z-3z2-2zq+2+5zq+3-zq+4) -

and if c h a r k = p

and ~-16 {1,2,...,p-I}, then

~ q+3+~zq+4 PB~ (z) = (1+z_zp_zp+1_zq~2)/(1_2z_3z2_zp+1+~ ~z p+2+~~z p+3+ z q+2 -~z p -z q+5~j.

Example 9. A graded algebra A is called a Koszul algebra if PA(Z)A(-z) = I. We can show that the family {A~} above generically consists of Koszul algebras. More precisely A~ is a Koszul algebra if and only if X = 0 or

~-1 ¢ {1,2 . . . .

}.

6. A family of topological spaces with ~nfinitely many Poincar@-Betti serie>. The third homotopy group of a wedge of 2-spheres

m 2 v S., is known i=I i

to be a free abelian group on m(m+1)/2 generators (see e.g. [Gr-Mo, 2 2 p. 240]). The generators are the Whitehead products [si,sj], iS j, where s 2 denotes the canonical element in ~2($2). Thus elements in

m~ z3(~S ) may be identified with elements of ~<TI, .... Tm> which are linear combinations of TiT j +TjTi, i~ j. Let fl,...,fr be elements of

g which are such l i n e a r combinations. Then f t , . . . , f r

gives

rise to a continous map r 3 m 2 v S:---@ v S.. i=I ± i=I i let X denote the mapping cone of f. It is implicitely proved by Lemaire f:

that there is a rational correspondence between the Hilbert series of

190

~/(f 1,...,fr ) and the homology algebra of the loop space X of X, which we denote by H~(~X;~)

(see [Le, p. 64] and [Le ~, p. 117]

and for an explicit formula [Ro, p. 450, formula (9)]). It is a wellknown fact that H~(~X;~) is the enveloping algebra of the graded Lie algebra ~ (wi+1(X)@~) i=I a i = dimQ(wi+1 (X)~Q)

defined by the Whitehead product. Hence if

then

H~(~X;~) = U (1+z2i-1)a2i-1(1-z2i) -a2i. i=I We now choose the Hopf algebras U(G)~ from the end of section 3. From these we get a family {X~} of topological spaces, namely X~ is the mapping cone of a map f~: oo

69 3 15~2 vS~-~¥~ , with the property that the series I 7

i=idzm~ ( ~" ~ i + 1 ( X ~ ) ~ ) z i takes infinitely many values when ~ varies in ~.

Remark. During the work on this paper we have been informed that D. Anick independently has constructed a family of graded noncommutative algebras over ~ with infinitely many Hilbert series. He also has a family which generically is of ~lobal dimension 2. This can be achived also by our methods. More precisely, consider the following slight variation of our example I: Let D~ = k/(ab-ac,bc-cb-~c2,bd). 2 if and only if X = 0

or ~-I ~ (1,2,...).

Then D~ has global dimension

191

References.

[He], Herrmann G., Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736-788.

[La], Lazard D., Alg~bre lin$aire sur K[X I .... ,Xn] , et @limination, Bull. Soc. Math. France 105 (1977) no. 2, 165-190.

[Le], Lemaire J.M., Alg~bres connexes et homologies des Espaces des Lacets, Lect. Notes in Math. 422 (1979), Springer.

[Le'],Lemaire J.M., A finite complex whose rational homotopy is not finitely generated, Lect. Notes in Math. 196 (1971), 114-120, Springer.

[L6], LSfwall C., Une alg~bre nilpotente dont la s@rie de Poinear@-Betti est non rationelle, C.R. Acad. Sc. Paris 288 (1979), s@r. A, 327-330.

[L6-Ro], LSfwall C. et Roos J.-E., Cohomologie des algSbres de Lie gradu@es et s@ries de Poincar@-Betti non rationelles, C.R. Acad. Sc. Paris 290 (1980), s$r. A, 733-736.

[Mo], Mount K.R., Fitting~s invariants and a theorem of Grauert, Math. Ann. 203 (1973), 289-294.

[Ro], Roos J.-E., Homology of loop spaces and of local rings, Proc. 18th Scand. Congr. of Math., Progress in Math. 11 (198~)~ 441-468.

[Se], Seidenberg A., Construction in algebra, Trans. Am. Math. Soe. 197 (1974), 273-313. Ralf FrSberg and Clas LSfwall Department of Mathematics University of Stockholm Box 6701, S-113 85 STOCKHOLM (SWEDEN)

Tor Gulliksen Department of Mathematics University of 0slo Blindern - OSLO 3 (NORWAY)

A N O T E ON I N T E R S E C T I O N

MULTIPLICITIES

by Tor H. G u l l i k s e n

Let

(R,_m)

denote

be

finitely

generated

R

is r e g u l a r we d e f i n e

a local

[3 ].

If

DEFINITION. surjective

R

The p u r p o s e R-modules ding"

is not r e g u l a r

Let

~

of this note M and N,

REMARK.

If

R

be computed the P o i n c a r ~

n

is the

=

R

of

as

follows:

We c h o o s e

R.

is a r e g u l a r

xR(M,N)

a

local

only depends

of m i n i m a l

intersection,

regular

then

on t h e

"embed-

xR(M,N)

in the f o l l o w i n g way.

can

Consider

[] ] t h a t

= lim P ~ ' N ( t ) ( ] - t 2 ) n - d i m R t÷-I

over

dimension

of

also

R, i.e.

the d i m e n s i o n

of

R/m.

One may ask if it is p o s s i b l e

TorR(M,N)

If

~ l e n g t h ( T o r ~ ( M , N ) ) ti i;0 4.2 in

embedding

m / m 2 as a v e c t o r s p a c e

of

N

series

from c o r o l l a r y

QUESTION.

and

length.

from the t h e o r e m below.

TorR(M,N)

xR(M,N) where

M

Then we put

follow

of

xR(M,N)

completion where

is to s h o w t h a t

p~'N(t)

It follows

R -+ ~

is a local c o m p l e t e

in terms

let

finite

multiplicity:

and n o t on the c h o i c e

R ÷ ~. T h i s w i l l

has

we will d e f i n e

d e n o t e the m - a d i c

dimension.

ring and

M~N

= Ei(-l)ilength(Tor~(M,N))

ring h o m o m o r p h i s m

ring of m i n i m a l

such that

the i n t e r s e c t i o n

xR(M,N) as in

(noetherian)

R-modules

for rings

R

to e x p r e s s

xR(M,N)

which

are not l o c a l c o m p l e t e

local

(noetherian)

in t e r m s inter-

sections.

THEOREM. and

N

Assume

Let

C

be an a r b i t r a r y

be C - m o d u l e s that

A ÷ C

of finite t y p e and

B ÷ C

are

ring, has

and let

such that

M®N

finite

surjective

ring h o m o m o r p h i s m s ,

M

length. A

193

and

B

being

dimension

regular

of

A

and

local that

rings

of

B

of m i n i m a l equal

the

dimension,

embedding

that

is the

dimension

of

C.

T h e n we h a v e xA(M,N)

PROOF.

We m a y

Hence

so is the

A

XcB

have

ass~ae

that

A,

fiber-product

is a h o m o m o r p h i c a commutative

= xB(M,N).

B

and

A

XcB.

diagram

--~

a: = K e r ( R of

c m

R.

and

By means of

÷ A)

b c m

of t h e s e

m/m 2 .

and

Since

B and

give

rise

maps

local

local

structure ring

rings. theorem

R, t h u s

we

ring homomorphisms

B

, C

b: = K e r ( R

A,

complete

By C o h e n ' s

of s u r j e c t i v e

A

ideal

are

image o f a r e g u l a r

R

Put

C

R

are

÷ B). all

Let

m

regular,

b e the m a x i m a l the

inclusions

to injections a/ma

÷ m/m 2

b/mb

÷ la/m 2

we will

consider

a/n~

and

b/n~

as s u b s p a c e s

Put s = dim R - dim A

Since

dim A = dim B

both

elements.

Let

a] ,...,a r

respectively

5] ..... '~r

representing mal

sets

a

and

(r(s)

~ow

are

minimally

be a basis

b] , . . . , b r

5] ' "'''~r"

b

for

be elements

extend

these

generated

a/ma in

two

by

N b/mb.

~

s

Let

respectively

sequences

to mini-

of g e n e r a t o r s a],...,ar,...,a s bT,...,br,...,b s

for

a

and

b

respectively.

For e a c h

i

(04i<s)

the e l e m e n t s

a | , . . . , a i, b i + I, .... b s represent

linearly

of a regular they

generate

system

independent

of p a r a m e t e r s

each

for

in

m / m 2. H e n c e

R. L e t

~i

they are

denote

the

part

ideal

and put A

Then

elements

Ai

is a r e g u l a r

A s = A.

In the

following

suffices

to prove

1

:= R / a .

--i

local

let

ring.

|(i(s.

Ai_ ] X

Observe To prove

Ai (M,N)

= X

CM,N)

that

A0 = B

the t h e o r ~ n

and it c l a r l y

194

Here

we w i l l

pose.

To

be the

use

a technique

simplify

ring

~ich

the n o t a t i o n

R/[

where

£

was

we p u t

is the

used P

ideal

in

:= A

[2] for i-] and

generated

a similar Q

:= A

pur-

i Let

L

by

a l, .... ai_ I , a i , b i , b i + I, .... b s. Observe

that

L

need

not be

regualar.

a. 0--~ P ~

where ly.

ai

From

and the

bi

P-~ L-~

0

0 - ~ Q bi~ Q - + L - ~

O

denote

sequences

We h a v e

multiplication

above

we o b t a i n

by

exact

ai

standard

sequences:

and

bi

spectral

TorL(~i,TorP(N,L))p q

=> T O r p ~ q ( M , N )

Tor~(M,Tor~(N,L))

=> T o r p ~ q ( M , N )

respectivesequences

and

where equals

TorP(N,L) q z e r o for

• ..TorL(M,N)

Tor[C. from w h i c h

it

and

TorQ(N,L) q q % 0,1. Hence ÷ T o r i P] (M,N)

÷ Tori% follows

equals we

N

obtain

for exact

q = 0,1 sequences

÷ ToriL| (t,i,N) ÷ ToriLl (M,N)

MN) ÷

or2+ ,

÷

and

orih

÷ ...

*

that xP(M,N)

= xQ(M,N).

REFE RE NCE S

[}]

T.H. G u l l i k s e n , A c h a n g e of ring P o i n c a r ~ s e r i e s and i n t e r s e c t i o n (1974) }67-183.

theorem with applications to m u l t i p l i c i t y , Math. Scand. 34

[2 ]

M.-P. M a l l i a v i n - Brameret, Une r e m a r q u e sur les a n n e a u x r ~ g u l i e r s , S e m i n a i r e D u b r e i l - P i s o t ( A l g ~ b r e et T h ~ o r i e Nombres), 1970/]971 no. 13.

[3]

J.P. Serre, A l g ~ b r e L o c a l e M u l t i p l i c i t ~ s , Mathematics, ]], S p r i n g e r - V e r l a g , Berlin,

Lecture 1965.

Notes

locaux des

in

Department of Mathematics University of Oslo Blindern - Oslo 3 (nORWAY)

195

REDUCING

THE

T O THE

POINCARE CASE

SERIES

OF L O C A L

OF Q U A D R A T I C

RINGS

RELATIONS

by Tor

Let

(R,m,k)

be a c o m m u t a t i v e ,

a finitely

generated

the

power

formal

a given

attempts cases.

M

have One

R-module.

tion

of the

over

two

been made

Poincar~

different

R(M)

Later

G.

series

series

of

modules

over

In the where

the

artinian

DEFINITION. quadratic

where

we

relations

say if

R0

is a field

generated

by

ideal

that

it has

closed

the m × m - m a t r i c e s with

of

M

M

be

we m e a n

respect

P RM

and m a n y

to h o p e f u l l y 2] w h e r e

to the case

we h a v e

reduction

cf.

R

showing

simpler

computaM = k, b u t

the

formula

by

M.

how

the P o i n c a r ~

of P o i n c a r ~

series

of

[2]. the

artinian

quadratic

a local the

of

in terms

reduce

ring

case

to the

case

relations.

is d e f i n e d

by

special

form

Xm, Y I . . . . .

or a c o m p l e t e

is g e n e r a t e d

the q u a d r a t i c

case

reduced

extension

by certain

i,j where

was

rings,

shall

is d e f i n e d

We w i l l

maximal

let

k (|_tp~)-] = PR"

ring

R0[[X I.....

whose

and

series

in [ ], t h e o r e m

expressed

local

following

ring

ring

to c o m p u t e

the g e n e r a l

presicely

a beautiful can be

difficult

M

More

trivial

made

of a m o d u l e

very

appeared

rings.

is the

Levin

By the P o i n c a r ~

to r e d u c e

reduction

k PR(N) where

local

[ dimkTor~(M,k)t q p) 0

it is u s u a l l y

such

noetherian

series P RM =

For

H. G u l l i k s e n

Ym] ] / I

regular

by a p r i m e

local

ring

number,

and

of d i m e n s i o n I

is an

I ideal

forms ~..X.Y. 13 i 3 '

(~ij)

~ij

£ R0

run t h r o u g h

to t r a n s p o s i t i o n .

a set of m a t r i c e s

which

is

196

THEOREM.

Let

generated R*

÷ R

tions

R

be an a r t i n i a n

R-module.

where

Then

R*

local

there

is a local

ring

defined

PROOF.

By C o h e n ' s

properties

÷ R

and

let

by

~,i be

a finitely

surjective

special

homomorphism

quadratic

rela-

= p R pM

R*

R0

and

a local,

and pM

with

ring

exists

induces

R.

Let

in

theorem

above,

an i s o m o r p h i s m

v] , .... v m

an b-module, entries

structure

described

and

R* R R

and

is an a l g e b r a

such

between

that

the

the

residue

b e a set of g e n e r a t o r s

consider

the

following

over

a ring

structure class

for

m,

R0

map

fields

of

R0

considered

set of m x m - m a t r i c e s

as

(aij)

with

RO: A =

{(aij) I [ = i j v i v 9 1,]

= 0}

N o w put R* = R0 [ IX] ..... X m , Y ] ..... Y m ] ]/I where

I

where

(aij)

in

R*

is the

and

structure minimal lifted in

ideal

6 A. let

map

f:R*

resolution

to an R * - f r e e

m * , the m a x i m a l For

as a free q differential

with

entries

the

following entry

a I , . . . , am

way.

whose

in

R0

depending

we and

in

er~tries of t h e p r o d u c t where

aij

a way that

m,

replace , c =

the o t h e r

hand

of

since

F

[aijXiYj

X i (resp. extending v i.

show

lift

Let

that has

Yi ) the

F

F

be

a

can b e

coefficients

c

that

c =

~aiv i 1

entry if

c q

by

c

c*

is even.

are quadratic

a basis with

to m a t r i c e s We

is a m i n i m a l

on

each

associated D

entry

i ai i

select

D*q_] D*q q= 0. F

such

we

matrix

to

that

we

where Then

f([aijXiYj)

is

the D*

resolution,

can

fix e l e m e n t s

c * = ~ i a i X-i clearly

expressions

the

of t h e t y p e

[~ijvivj

zero s i n c e

of

do this qin

We h a v e

f ( [ a i j X i Y j) = On

to

first

so for e a c h

Dq_] Dq

£ R O.

F

D q now h a v e

We

Observe

Yi

differential

q be the

Let

is

forms

of

R* .

component

Dq

image

homomorphism and

F*

of

of

D* q is odd,

Xi

complex ideal

in q n_~ in such * q-]"

if

[~ijXiYj

unique

the

sending

the

We shall

÷ F

To o b t a i n q

be

the q u a d r a t i c

be

M.

R-module. F

9 i)

of

each homogeneous

F

each

and

by all

(resp.

Xi ÷ R

R0 ÷ R

R-free

generated

Let

it is an e n t r y

of

197

Dq_]Dq =

0.

This

shows

that

D*q-] D*q = 0. established. Now

let

standard lic,

Y

that

The

existence

be a minimal

spectral

sequence

so it is a m i n i m a l Tor

R*

(M,k)

(~ij)

fi A,

of the

R*-free

argument

R*-free

so

[~ijXiYj

lifted

complex

resolution

the

total

resolution

= 0, w h i c h

of

R.

complex

of

M.

F*

means

is now

Then

by a

F*® Y

is a c y c -

Hence

= F * ® Y ® k = ( F * ® k ) ® ( Y ® k) R* R* R* k R* W

= ( F ® k ) ® ( Y ® k) R k R* This

proves

the

formula

in the

= TorR(M,k)®TorR(R,k). k

theorem.

RF~iARKS. ].

Any

R-free

all 2.

p

There from

complex

C

sufficiently exists

which small,

a minimal

is b o u n d e d c a n be

R*-algebra

fact

that

the h o m o m o r p h i s m

of L e v i n

[3],

i.e.

the

TorR(k,k) is s u r j e c t i v e . Rochandel

Then

[4],

the

R* ÷ R is a m i n i m a l 3.

The

integer

length than

of

m m.

or e q u a l

is a p r i n c i p a l R0-module estimate

to

m2

ideal

"best

the

also

of

(This

and

possible"

L. A v r a m o v

ideal

elements. ring

R.

is

This

large

A with

and

H.

of the a n g m e n t e d

See

in the d e f i n i t i o n

of all m × m - m a t r i c e s is

of

closure

case

of

÷ R

follows

in the

R* I

[3, T h e o r e m

2.5].

c a n be c h o s e n can be

comes

entries

Rahbar-

algebra

f r o m the

in R 0. )

from the

to be the

generated fact

is an R 0 - s u b m o d u l e

is c l e a r

by less that

of the That

following

* R

m = n.

Then

this

/(t] ..... t n ) 2 we h a v e

k[[X| ..... X m , Y ] ..... Ym] ] =

/( .... X i Y j .... )

i~i~m,

free

example:

kit] ..... tn] R = Choose

sense

÷ TorR(k,k)

resolution.

In that

R*

map

by a r e s u l t

acyclic

i.e. C = 0 for , P to an R -free complex.

resolution

the

induced

below,

lifted

] ~j~m.

R0

198

REFE RE NCE S [] ] [2 ] [3 ] [4]

T.H. Gulliksen, Massey Operations and the Poincar~ Series of C e r t a i n Local Rings. J. Algebra 22 (]972), 223-232. G. Levin, Poincar~ Series of Modules over Local Rings. Proc. Amer. Math. Soc. 72 (]978), 6-]0. G. Levin, Large homomorphisms of local rings. Math. Scand. 46 (]980), 209-215. H. Rahbar-Rochandel, Th6se, Caen ]979, appendice p.52, prop. 2]3.

(Note by the editor: For complementary results, c f. the mathematical introduction to these proceedings.)

Department of Mathematics University of Oslo Blindern - Oslo 3 (NORWAY)

THE RADICAL OF

~,(~S) ~ ~ , II

by Stephen Halperin

I.

INTRODUCTION

Let

S

be a simply connected

dimensional

in each degree.

with the Samelson product) Quille

topological

space whose rational homology

The rational homotopy Lie algebra is then also finite dimensional

[Q] showed that if no further restrictions

possible graded connected Lie algebras(finite By contrast,

severe restrictions

rational Lusternik-Schnirelmann

are imposed on category of

(equipped

in each degree.

are placed on

dimensional

is finite

~,(~S) ® ~

S

then all

in each degree)

~,(~S) @ ~

can occur.

if we assume that the

S , cat (S) , is finite.

(This is defined

O

in general zation

in [F-H] and coincides with the classical

S~

when

S

is a

L-S

category of the locali-

CW complex-[T].)

This article continues the study of the Lie structure of hypothesis

cat (S). finite) begun in IF-H-T] and [F-H-T-T].

~,(~S) @ ~

(with the

The techniques derive from

O

the earlier two papers,

and the fact that I have carried out this stage of the inves-

tigation alone is due only to the several thousand miles of ocean between Europe and Canada which made impractical

a collaborative

In [F-H-T] we showed that if graded ideals in series of

R

~,(~S) @ ~

(solv length)

Cato(S) _< m

If

if either

then the sum

is itself solvable, is

R , of all the solvable

and that the length of the derived

~ 4 + 21og2(m) . The ideal

homotopy Lie algebra. We also established Theorem.

effort.

R

is the radical of the

the

eat (S) < m then a graded idea] I c ~,(~S) ~ @ o dim ~,(~S) @ ~ is finite or if for all k

is solvable

if and only

2k-I I dim 12i < m . i=k Here it will be shown that the radical is in fact nilpotent

(and thus that every

solvable of

R

ideal is nilpotent) and the length of the lower central series (the nil length) 8+41og2m is < [ 2 ~ m ( m + 1 ) ] , if cat (S) < M . Explicitly we have --

Theorem

I. Suppose

O

S

simply connected,

each degree, and that

eato(S) ! m . Let

lowing conditions

are equivalent:

--

with finite dimensional I c ~,(~S) @ @

rational homology

in

be a graded ideal. The fol-

200

2k- I Z dim I. < [2m(m+1)]2 . i=k i Every finitely generated subalgebra of I

(i)

For all

(ii)

k ,

(iii) I

is solvable.

(iv)

is nilpotent.

I

is finite dimensional.

Under these conditions, moreover 8+41og2m nil length I < [ 2 ~ m ( m + I)] 2d < [2/~m(m+1)] where

d

is the length of the derived series of

Corollary. Let C > 0

I

be a graded ideal in

I .

~r,(f~S) ® ~ . Then either there is a constant

such that

P Z dim I i < C log 2 p , p > 2 , i=I -or there is a constant K > 0 such that (I)

P Z dim I. > K p ,

(2)

i=I

p > I .

l--

The ideal

I

is solvable if and only if it satisfies the first set of inequalities.

Proof. If

I

is solvable the first set of inequalities follow from Theorem I(i). If

I

is not solvable, then Theorem 1(ii) yields a finitely generated subalgebra,

infinite dimension, and the second set of inequalities results, Suppose now Denote by

RS

and

Theorem 2. If ideal in

S

RT

and

RT

(~I)(RT)

¢#(R S) .

ker ~# c R S

satisfies (~) and since

Im ~#

ker ~ # c R S

is an ideal in

J(~)

~1(j(~))

is an

also satisfies (I) it follows

is an ideal, for every homogeneous

is an ideal

~ ' ( J ( ~ ) ) . Thus

Im ~#

then

satisfies (I). Hence it is a solvable ideal; i.e.

[~'~#(Rs)] T ~#(RS)

RT

m

~#: ~,(~S) ~ ~ ~ ~,(~T) ® ~ .

satisfy the hypotheses of Theorem I, and if

while

Conversely, since

so does

is continuous, inducing

the radicals. Then we have

T

~,(~T) ® ~

Proof. Since that

~: S ~ T

I , of

in

c RS

Im ~# . This ideal satisfies (I). Hence and

~,(~T) ® Q . Since

(~I)(RT) c R S . ~ E ~,(~T) ~ 9:

J(e) c ~#(Rs) . This shows that

~#(R S)

satisfies (I) it is contained in

D

Corollary.

Let

F

~ E ~-~ B

be a Serre fibration for which

F

and

E

satisfy the

hypotheses of Theorem I. Then

(i~) -I (%> = R F Proof. Clearly, dim ker 7#

Im i# = ker ~

is finite, so

is an ideal, as is

ker ~ # c R F .

m

ker ~

. Moreover ([F-H])

201

The study of rational homotopy iterated integrals.

theory has, as one of its sources,

Either this way or via the standard methods,

compute the real homotopy Lie algebra forms on a smooth manifold,

Proposition

I. Let

S

satisfy the hypotheses

If

able ideals in

~,(~S) ® Ik.

Proof. Write

I~ ~ ~

{v }

proof of Theorem

of

and let

Ik. For

N

I

~

I and let

R c ~,(~S) ®

R @ 1~ is the sum of the solv-

be the sum of the solvable ideals in

x E I

I still works with

there is a constant

of Theorem

is an extension field then

L = ~,(~S) ® ~

q-basis

from the algebra of differential

S . This does not affect the radical:

be the radical.

Fix a

~,(~S) ® ~

Chen's work on

it is possible to

write

x = ~ x

instead of

such that for any

~

® v

x

L ® ]k.

C L . Now the

as coefficients;

in particular

k,

# {i E [k, 2k-1]IIi # O} < N . This will then apply to the ideals Theorem

1(ii),

Remark.

Part of Theorem

Lie algebras)by I

I

c R . In particular,

I

generated by the

each

I may be phrased

x

I

and

x C R ~ ~ .

I

of

locally nilpotent ~=~ I

~,(~S)

referred to

A X = exterior algebra The ground field id

~

m

nilpotent.

I. At points we shall the reader is

convention:

if

symmetric algebra

X

is a graded space then

(X even) .

throughout.

The main ingredient

S

(X °dd) ~

and by

and a review of the facts we require. Here I

shall simply recall that we use Sullivan's

Theorem 3. Let

I

the theory of minimal models and rational category:

[FHT; §3] for the notation

L

@ ~ :

solvable ~

The rest of the paper is devoted to the proof of Theorem need explicitly

in

(in the language of infinite dimensional

saying that for a graded ideal

locally finite ~

C R

x

in the proof of Theorem

I is

be as in Theorem I, and suppose

I c~,(~S)

@ ~

is a graded solv-

able ideal. Then for each k 2k I dim 12i+I ! [2m(m+ I)] 2 - 2m 3 . i=k In what follows, Theorem 3 will be first reduced about minimal models.

Two key lemmata are then proved

mata of [FHT; §5]), and are subsequently Theorem

If, moreover, cause ([F-H],[H])

of Theorem

~,(~S) @ ~

statement

(they are analogues of the lem-

used in § 4 to complete the proof. Finally,

I is deduced from Theorem 3 in § 5. The space

to satisfy the hypotheses

(in §2) to a technical

S

will be suppose d throughout

I and 2.

is finite dimensional,

then Theorem 3 is trivial be-

202

dim lod d ~ dim ~ o d ~ S )

® @ ! dim ~even(~S) @ ~ ! m

Thus in our proof of Theorem 3 we suppose that

~,(~S) @ ~

. is infinite dimensional.

2. THE REDUCTION STEP

In order to prove Theorem 3, it is clearly sufficient to establish Z dim I2i+i _< 2 m 2 ( m + 1 ) 2 - m 3 i=£ Fix

~ ~ ~'

degree of

[~,B]

with

3~ ~ 2%'. If

if

3% ~ 2%' .

~, B C ~ i=%I2i+I

(2.1)

are homogeneous then the

is even and lies in an interval of the form

[2r,4r-2] . It follows

from [F-H-T-T] that all such brackets span a space of dimension at most obvious i~Iductive procedure now gives a sequence homogeneous elements in

~i=%I2i+i

[Bi,B j] = 0, and

BI,B2,...,B N

m -I . An

of linearly independent

such that

i # j ,

(2.2)

m N ~ ~i=£dim I2i+i . We are thus reduced to proving that N < 2 m ( m + I) 2 - m 2. Moreover,

it follows from (2.2) that the sub Lie algebra, E , generated by the !

~i's consists of the span ~f the

B i s (in odd degrees) together with an evenly

graded space of dimension at most

m- I

Finally, for any homogeneous

spanned by elements of the form

~ C ~,(~S) @ ~ , and for any sequence

[Bi,Bi ] . i¥ ( 1 ~ i y ~ N )

we observe that [Bit[Sit_1[...[~i1,~]...] I n d e e d , by ( 2 . 2 ) we may s u p p o s e

= 0 ,

deg Bi

t ~ 4m. ~ ...

(2.4)

J deg ~i

t Yv = [ g i v [ ' ' ' [ B i 1 ' ~ ] ' ' ' ]

(2m < v < 4m)

a r e e v e n , and w h i c h l i e the

Yv

is zero,

Now l e t

and so

(AT,d)

i n an i n t e r v a I ¥v = 0 ,

have distinct of the form

v ~ 4m ,

product.

[r,2r-2]

which proves

b e t h e m i n i m a l model o f

is equipped with a scalar

" The s e q u e n c e o f e l e m e n t s 1 degrees at least m of which

Moreover,

S . if

~x

.

By [F-H-T-T] one o f

(2.4).

Then each p a i r E A2T

Tp+1, ~ (aS) ~ Q P is the quadratic part of

dx , x E T , then (2.5)

= ±<x;[~,S]> . Suppose the

~.

have been numbered so that

i

a = deg 131 < _ . . . . <__ deg BN = b ;
then

2~,+1 < a < b < 2~,' + I .

Divide

AT

by

T- ; then the elements of

r ] TLa+l'2aJ

203

become cocycles. elements

Since

b < 2a

x E T [a+1'2a]

those elements

x E T [2a+1'3a+1]

In this way we achieve spaces

L

~(~S)

by construction,

(i)

X p = O,

(ii)

X [a+1'2a]

= 0 ,

we can divide

minimal

model

the

~i

divide by

cocycles. (AX,d)

(2.5) a graded

contains

further by the

I < i < N . Finally,

which have now become

become via

P @ ~ . This Lie algebra

2%' < 3~)

<x,Bi>

a quotient

= Hom (xP+I;~)

Moreover,

(because

-satisfying

of

(AT,d) . The dual

sub Lie algebra,

(and hence

E)

L , of

by construction.

we have

pja. has a basis

(iii) For degree reasons

YI' .... YN

d

maps

satisfying

X [2a+1'3a+1]

dy i = 0

into

and



= ~ij "

A2x[a+I ,2a] ; moreover

this map

is injective. deg

Since in

X

that basis

E

to

[~N, BN] j 4~' + 2 < 6% + 2 j 3 a

is contained

X [2a+1'3a+1]

in

X J~a.

has as basis

to a basis

~1,...,~t

In particular

we have

it follows

In view of (iii),

elements

for

- I

wl,...,w t

that the dual

space

and (2.5) we may conclude

(t ! m - I )

which form a dual

Eeven .

L = E @ L>3 a ; i.e.

E

is complemented

by an ideal.

Now

put Y = X [a+1'2a] Thus

U

W = X [2a+1'3a+1]

is dual to the ideal

L>3 a , W

U = X k3a+2 is dual to

Eeven , Y

is dual to

Eod d , and

AX = AY ~ AW @ AU . It follows quadratic

part

from d2

(i),

(ii) and

(iii) above,

that the differential

d , and its

satisfy

d(Y) = d2(Y)

(2.6)

= 0 ,

(2.7)

d = d2: W ~ A2y . Let

32

and let

be the quadratic 8i, 8~ : U ~ U J

part of the differential

be dual to

d2u = ~i *Yi @ 8i(u) From

2 d2 = 0

we deduce

adB i , a d ~ . . J

in the quotient

+ lj ±w.j ® e!(u)j + I ® ~2 u .

(for the extension

of

8.1

to a derivation

however

u E U ;

(2.8) in

AU)

(2.9)

We shall use these formulae First,

(AU,d) ,

Then by (2.5) we get for

d2ei = ± 8 i ~ 2 .

3 - in § 4.

model

to complete

we establish

the proof of (2.3) - and hence of Theorem

our key lemmata.

204

3. THE KEY LEMMATA

We retain

the notation

of § 2. For

p ~ 0

put

S P = {~ = (o I , .... ON) IO i 6 ~ , o i _> 0 , I o i = p} . If

o 6 S

put

Iol = p , o! = H (o.!) i

P O

I

Y (Recall . . , o N)_

g Yi

=

Yi

is the ~asis

is written

Similarly A

If

(i)

we s e t

then

Y

and A

dual to the

o

= {4}

O

~ 6 Aq

There

satisfy d~'(~;o)

Bi.)

is written and f o r

~i ~ = (ml...gi...~q)

= 4 • If

3. I. Lemma. which

of

i+e

we write

If

1 < i < N

then

(ol,..,q i + I,.

~.(i~ + o ) .

1 < q < N

q = {~ = (~I , ...,~q) Imi 6 ~ ,

e 6 Aq

~i(~i)

and

I <

~I

<

"'"

< ~

q _< N } .

; it is an element

Aq_ I . Of course

of

I~I = q .

is a system of elements

~(c0;o) 6 AX (o @ Sm, ~ @ Aq , 0 <

q
~/(4;~) = yO , and q (_1)i-ly~ = Z A~(~i~o;~) , i=I i

(ii) q+1 I (-I )i_1~(~i~;~i +T) i=l

I~l = q > I, !o I = m

~ U~I = q+1

= 0,

and

> I, iTl = m - 1

.

>

Proof.

Extend

the projection

(Ax ® AZ,d)-~

(AX,d) -~ (AX/A mX,d)

(AX/A>mX, d) , Because

cat

(AX,d)

to a Sullivan model

< m

p:

there are elements

O

vT 6 AX(T 6 Sm+1) Then

dp v

o 6 S

~(i;o) p~(i;o) Thus

define

m

= vi+ ° - v!I + O

= 0, ~(j;i +o)

in particular

is aeyelie ker p

dvT = yT

= 0 , and so there are cocycles

pv$ = pv T . For

Then

such that

~(~;~)

= y

in

AX ® AZ

and

~(i;o)

and

d~(i;o)

such that

6 AX ® AZ

by

"

- ~(i;j +o)

= 0

yj~(i;o) -yi~(j;o)

it is a coboundary.

v'T

We now define

is a cocycle elements

= y in

= yi y

= yi~(4;o)

ker p . Since

~(~;o) ,

I~I = 2,

.

ker p

lol = m

in

as follows:

(a)

If

(b)

If

ok = 0 , k > j

d~(i,j;o)

is the biggest

is any element

in

ker p

for which

.

integer for which

ok # 0

set

= ~(i,k;j + $ k o) - ~ ( j , k ; i + ~ k o) .

It is then straightforward The same construction ~(~;o)

~(i,j;o)

= yi~(j;o)-yj~(i;o)

k > j

~(i,j;o)

then

6 ker p,

to check that the (applied

m 6 Aq , o 6 Sm

~(i,j;o)

inductively

satisfying

over

(i) and

satisfy q)

gives

(ii). Since

equations for

q > 2

Cato(AX,d)

(i) and

(ii).

elements i m

there

205

is a retraction

~: (AX @ AZ,d) ~ (AX,d) . Put

P(~;~) = ~ ( ~ ; o )

For the next lemma we need a little more notation. module.

Fix an isomorphic

suspension of algebra and Bigrade

Y AY'

copy,

with basis

Y', of

Y

with basis

sYi: deg sy i = deg Y i - 1

is a polynomial

N = AsY ® AY' @ M

.

Suppose

M

Y'i

and let

. Thus

AsY

is a free sY

AY-

be the

is an exterior

algebra. by putting

N q'p = AqsY @ APY ' @ M . The elements C S

f 6 N q'p

can be identified with the collections

f(~;a) C M ,

w 6 Aq,

, via P

I ~ sy~ A... A s y ~ ® ~ . (y')O @ f(~;o) ; ~,~ 1 q o. here (y,)~ = ~ (y[) 1 t NOW d e f i n e o p e r a t o r s ~1 a n d g2 i n N , h o m o g e n e o u s o f b i d e g r e e s f =

(I,-1) ~(y[)

as follows. = sy i

Let

and put

~

be the derivation

62 = ~ ® i d .

in

AsY ® AY'

g i v e n by

(3.2)

(1,0) ~(sy i)

and

= 0 ,

Then set

N

61(~ ~ ~ ~ m) = A short (AsY ~ A Y ' , ~ )

calculation is

~ sYi^~ i=1

® ~ ~ y:'m.~ 612 = 622 = ~1~2 + ~2G1 = 0 .

shows that

the classical

contractible

Moreover,

m o d e l a n d so (3.3)

H(N,~ 2) = 1 ® I ® M . On the other hand, because H(N,61) where

M

is a free

AY-module,

we have

= sYl A... A s y N @ I ® F ,

(3.4)

M = AY ® F . (This is essentially Lemma 5.6 of [FHT]). In particular Hq'P(N,~I) Finally,

= 0

we interpret

if

61

a simple calculation gives for

q < N . and

62

(3.5) in terms of the decomposition

f E N q'p

(~If) (~;~) =q~1(_/)1-1y~..f(~i~;~) and

i=l

q+l (~2f)(~;c~) = ~ (-l)l-lf(~i~;c0 i + o ) i=l 3.6. Lemma. Suppose f 6 N q'p (2 < q < N) q+l . (-l)a-lyi~ -f(aim;~) = 0 and i=l l for all

co C A q + l , a 6

Sp

l

m, ~ . There is then an element

co C A q + l , o 6 Sp_l . satisfies q+l . Z (-l)l-lf(;i~;~i+~) i=l

g 6 N q-1'p

such that

= 0

(3.2). Indeed,

206

q

q

i-I

X (-I) i=I for all

The lemma is stated

lied. Using g

the formulae

such that

in "component

X (-|)l-lg(~i~;~0 i +0) i=I

g 6 N q-1'p,

61g = f

because

Suppose

and

= 0 ,

-62f

= 0

and

hypothesis

(3.5)

q-I

> I

and we are required

to

we have by (3.5)

it is automatic is proved for

that

has bidegree

f = ~I g ,

~2g = 0 .

f 6 N q'p',

f = 61g1 ' gl 6 N q-1'p. ~2g I

that

p' < p , and that

Then

(q,p-I)

~i(~2gi)

=

our induction

such that

62g 2 = 0 .

(3.3)

g = gl + 6 1 g 3 . Then

Put

that

= 0 . Since

and

we apply

p = o

p = 0

that the lemma

g2 6 N q-1'p-I

~ig2 = 62g I Since

p . If

implies

~2(52gI)

yields

that it is how it will be app-

~I f = ~2 f = 0

62g = 0 .

on

q < N . Since

by induction

f 6 N q'p. As above

form" because

above we see that

We do this by induction

=

and

e ,~ .

Proof.

find

= f(m;o)

y~ -g(3im;c) z

to find

g3

with

61g = 61g I = f

~2g3 = g2 "

and

62g = ~2gi - 8 1 6 2 g 3 = 6 2 g I - 6 1 g 2 =0. D

4.

PROOF OF THEOREM

Recall

2.

the notation

at the end of § 2.

In particular,

e.: U ~ U 1

adS.. Extend the ~. to derivations in AU . Denote by F (AnU) i 1 p of the elements of the form 0. o ... o 0. ~ , ~ 6 Anu . Set 11 zp Fr(Anu) and note that,

= Anu,

that each

~

is dual to e~ :F

J Further,

r _< O ,

span

(4.1)

(4.2)

= 0 .

Recall J

the linear

in view of (2,4)

F4mn(AnU)

Since

is the dual of

a.

J ads. 3

is a linear combination its extension

(AnU) ~ Fp+2(AnU)

to

.

Anu

of vectors

of the form

[~i,si].

satisfies (4.3)

P

by definition 8 i :Fp(AnU)

~ Fp+I(AnU)

(4.4)

~ F 0 (An+Iu)

(4.5)

while by (2.9) 72: Fp(Anu) Next,

define graded

spaces

Ak, r c AX

by

207

= A kk+1X

@

AiY 0 AJw ® Fr_i_2n(AnU)

Z

i+j+n=k By (4.2) we have (4.6)

A~kx = Ak, 0 = Ak, I = ... ~ Ak, (4m+2) k = h ~k+1 X. From equations

(4.3),

(4.4), and (4.5) we deduce

d: Ak, r ~ Ak+1,r+ 2 . On the other hand, if

(4.7)

F~(AnU)

is a graded complement for

F (AnU)

and if we

put Bk, r = A
l Aiy O AJw 0 Fi_i_2n(Anu)• i+j+n=k

then AX = Ak, r @ Bk, r Moreover the linear maps

all YiA:

Yi ^Ak,r c Ak+1,r+l

and

~(~;~)

~B(m;o) 6 Bk,r+ I

(respectively

the

I~I = q)

(4.9)

of Lemma 3.1 can be so chosen that for

Iml = q +I . Then for a new choice (if necessif

I~] = q .

~(~;o) = ~A(~;o) + ~B(0~;o) with

~i'A(m;o) EAk,r+ I

- this is possible by (4.8). It is immediate that the

%(~;~))

On the other hand if From equation

if

P(~,o) ( Ak,r+ I

Proof of 4.10. Write (for and

satisfy

Y i A B k , r ~ Bk+l,r+ 1 .

q(2 _< q < N) , ~(~;c) 6 Ak, r

ary) we can arrange that

(4.8)

AX ~ AX

4.10. Lemma. Suppose the elements some

k, r.

~A(~;O)

satisfy equation (ii) of (3.1). 1001 = q +I

then (4.7) shows that

(i) of (3.1) and (4.9) we thus deduce for

d~(~;o) =q~l(_l)i_ly i=l

d~(~;o) 6 Ak+ I,r+2 "

]~I = q + 1

that

. ^~A(3i ~;°) z

and 0 = Since the

q+1 i-I Z (-I) y~. A~B(~iw;~) . i=I i ~B(m;o)

also satisfy equation

(ii) of (3.1) and since

AY-module we can apply Len~na 3.6. This yields elements that (for

~(~;o)

~ 6 Aq) ~B(m;o) =

q ( I~i-I Z - ) y~. A~(~i~;o) , i=I i

]~I = m

and q

Z (-I)i-I~(~i~;m i + o ) = 0 i=I Modify the ~(~;~) to ~'(~;o)

IO] = m - 1 by putting

.

AX

(t~1 = q - l )

is a free such

208

I

~'(m;~) =

It is straightforward that the conclusions We now revert

~(~;~)

if

I~l # q

~A(m;o)

if

le! = q

if

i~ I = q - 1

~(~;o)

- da(~;~)

to check

(given the properties

of the le=~a are satisfied, to, and complete

the proof

(2.3) and we do so by contradiction. by definition.

An iterated

we may suppose

that

~(m;o) ag long as

Write

Y(i;o)

equation

= f(i;o)

dY(i,j;o)

Since

AX

+T)

o

have just shown divides follows

implies

f ( m + I;o)

I~I = N , ~(w;o) CA],0,

shows that

(cf.

(4.6))

f(i;o)

then we may suppose C A SmX

and

now implies

g(i;o)

P(i,j;o) C A 2m+1 X. E A~m+1X.

The

that

o E Sm .

AX

algebra

AY , it follows

that there

such that

satisfy

,

• E Sm_ I ,

are satisfied

by the

f(i;o) ~ For any

i,j

we then get

~I = ~2 = "'" = °m

that

and hence

Yi

divides

h(o)

this element

=

I

and

o I.

for each

is zero,

=

0 , i > m

i . Thus

since it is also

.

Ym+1Ay] in

What we ^ ... A Y m

A!mX.

It

that g

= dY(m+1;o)

= dg(m+1;o)

C d(A>-m+Ix) c A>m+2x ,

contradiction.

We obtained that

~ )

, T E Sm_ I .

by putting

Ym+1 ^ Y a patent

,

~(i;o)

= ~(j;i+~)

= yj ^ h ( i + T )

and

3. We have to establish

of the lemma above

-yjW(i;o)

in

and hence the same equations

Now fix

~A' ~B

.

On the other hand the

Yi^h(j

with

= Yj ^f(i;°)

= Yi^h(o)

T(i;j +T)

for

.

+ g(i;o)

h(o) , a C S m ,

f(i;o)

derived

m

of Theorem

is a free module over the polynomial

are elements

.

we note that for

2 < N - ( m + 1 ) m ( 2 m + I)

= Yi AY(j;g)

Yi ^f(j;~)

q-1

Iw I = 2 ,

2 < N-k(k-I)(2m+1) if

Indeed,

application

C A zkX,

particular,

In

or

this contradiction

by supposing

N < 2 + (m+ 1 ) m ( 2 m + I) , and hence

(2.3),

N > 2 + ( m + 1 ) m ( 2 m + I) . It follows and Theorem

3, are proved,

m

209

5. PROOF OF THEOREM I,

dim ~,(~S) ® ~

Since for

finite we have necessarily

([F-H],]~H])

dim z,(~S) @ ~ ! 2m and since in this case the Lie algebra is trivially nilpotent may restrict

to the case that

dim ~,(~S) 0 ~

By the theorem of [F-H-T-T] bilities

for a graded ideal

some

either the

~ £ ~,(~S) ® ~

there are then exactly two mutually exclusive possi2k-I I is solvable, or for some k , Zi= k dim I2i

e'l

This shows that if

(ii) implies

for all

n . In particular,

suppose

I

It remains to prove that if I d

is nilpotent

[~i,B]

Then (i) is an immediate consequence of Thus

(iii) ~ (i). Moreover,

I

since finitely

(iv) ~ (ii).

is solvable with derived series of length

of nil length

d ,

< [2/~m(m +I)] 2d . We shall show by induction

= O,

B i 6 I,

~ 6 ~,(~S) ® ~ ,

(5.1)

q i [2/~m(m+1)] 2d . This is clear for

= ~,(~S) ® ~

and

d = 0.

Assume by induction that it holds for

F i+I = [I',F l] . Then

The induced representation, = ±O(B)e(~). . , a, B in

or for

that [BI[B2[...[~q,~]°..]

if

and

does (iv).

is solvable.

generated nilpOtent Lie algebras are finite dimensional,

then

a.l

I is not solvable then also (ii) fails. It also shows that 2k-I li= k dim I i ~ Ck, and so (i) fails. Thus either of (i) or

C ,

Theorem 2, and the theorem of [F-H-T-T].

on

subalgebra,

subalgebra.

(iii) as, clearly,

Conversely,

[F-H-T; Theorem 2.2] to obtain that for certain

generate an infinite dimensional

(ad~i)nB # 0

generate an infinite dimensional

for some constant

we

I: either

m . In the latter case we can apply ~1,...,~m 6 I

of nil length ~ 2m

is infinite.

FI/F I+I

homogeneous

(B i 6 I)

9 , of

Fi = 0

if

in

Fi/F i+I

I

elements of

can be represented

in

I'; set

F° =

i > [2/~m(m+1)] 2d-2 . satisfies

I . Thus an element ~,(flS) @ ~

8((~)e(~) = 0(81 ) o ... o 8(B2p)-V

by an element

[~l[~2[...[a2p,O]...] where

o

represents

v , a.1 6 1

The string of elements satify

and

deg 61 j

... ~ deg a2p .

Yi = [ai['''[a2p'°]''']

(i ~ 2p)

are all in

2deg yp > deg YI > "'" > deg yp . We have seen above that if

it satisfies

(i). Thus

(5.1) follows at once.

YI = 0

if

p ~ [2m(m+ I)] 2 . This holds

I

and

I

is• solvable.

in each

FI/F I+I and

210

REFERENCES

[F-H]

Y. Felix and S. Halperin, Rational LS category and its applications. Trans. AMS 273 (1982), 1-38.

[F-H-T]

Y. Felix, S. Halperin, and J.-C. Thomas, The homotopy Lie algebra for finite complexes. Publ. Math. I.H.E.S. 56 (1982), 387-410.

[F-H-T-T]

Y. Felix, S. Halperin, D. Tanrg, and J.-C. Thomas, The radical of ~,(~S) @ ~ , these proceedings.

[H]

S. Halperin, Finiteness in the minimal models of Sullivan. Trans. AMS 230 (1977), 173-199.

[Q]

D. Quillen, Rational homotopy theory. Ann. of Math. 90 (1969), 205-295.

[r]

G. Toomer, Lusternik-Schnirelmann category and the Milnor-Moore spectral sequence. Math. Z. 138 (1974), 123-143. S. Halperin Department of Mathematics University of Toronto Toronto, Canada M5S IAI

HIGH SKELETA OF CW COMPLEXES

by

S t e p h e n H a l p e r i n and G e r s o n Levin

i.

Introduction.

Roos and others

This paper follows the example of Avramov,

([A],

Felix,

[R]) who have begun a fruitful exchange of

t h e o r e m s b e t w e e n r a t i o n a l h o m o t o p y t h e o r y and the h o m o l o g y t h e o r y of c o m m u t a t i v e rings. Let R and S be local rings w i t h the same residue field Ik. h o m o m o r p h i s m R÷S,

inducing the i d e n t i t y on ~,

m o r p h i s m if the "fibre" TorR(s;k)

A local

is called a G o l o d homo-

has trivial M a s s e y o p e r a t i o n s and if

the induced map T o r R ( ~ , ~ ) ~ T o r S ( k , k )

is injective.

(If di~k

this latter c o n d i t i o n follows from the first -[A2].)

(Tor~(S;~))>l

The f o l l o w i n g

theorem was proved by the second named author in [L; T h e o r e m 13]. T h e o r e m A: ~.

Let R be a local ring w i t h m a x i m a l ideal m and residue field

There is then an integer n o such that for n~n o the map R ÷ R / m n is a

Golod homomorphism. For R regular this had already been proved by Golod w i t h no=2.

[G] in 1961,

In the c o m p l e t e l y d i f f e r e n t context of G e l f a n d - F u k s cohomo-

logy, J. V e y showed that the h o m o t o p y fibre of the i n c l u s i o n of the 2ns k e l e t o n of BU(n) was r a t i o n a l l y a wedge of spheres. later g e n e r a l i z e d by Ruchti

This result was

[Ru] who c o n s i d e r e d high skeleta of a finite

product of K(Q,ni)'s. Since the lexicon b e t w e e n local a l g e b r a and r a t i o n a l h o m o t o p y theory t r a n s l a t e s G o l o d h o m o m o r p h i s m s

to c o n t i n u o u s maps whose h o m o t o p y

fibre is r a t i o n a l l y a wedge of spheres,

and t r a n s l a t e s R + R / m n to the

i n c l u s i o n of the n - s k e l e t o n of a CW complex,

T h e o r e m A appears to sug-

gest a far r e a c h i n g g e n e r a l i z a t i o n of Vey-Ruchti:

n a m e l y that for any

1 - c o n n e c t e d CW c o m p l e x of finite type the i n c l u s i o n of a high s k e l e t o n should have as h o m o t o p y fibre a wedge of spheres

(up to rational homo-

topy type). This, however,

fails for the p r o d u c t n~2 Sn.

The point is that

there has to be some kind of finiteness a s s u m p t i o n on our space, corr e s p o n d i n g to the fact that local rings are noetherian. priate h y p o t h e s i s turns out to be semi-finiteness,

The appro-

w h i c h we define as

follows: Definition.

A simply c o n n e c t e d CW c o m p l e x R of finite type is semi

finite if there is a c o n t i n u o u s map E~R w i t h path c o n n e c t e d h o m o t o p y

212

fibre F, and such that for some M,N; H>M(E;Q) If HM(E;Q)~0, While the pair

: 0

and

H>N(F;Q)

and HN(F;~)~0 then

= 0.

(M,N) is a d i m e n s i o n pair for R.

(M,N) may v a r y with E, the d i f f e r e n c e M-N d e p e n d s o n l y

on R (given F'~E'+R c o n s i d e r E'×E). R The correct t r a n s l a t i o n of T h e o r e m A reads Theorem B:

Let R be a simply c o n n e c t e d CW complex of finite type w h i c h

is semi-finite w i t h d i m e n s i o n pair

(M,N).

If n>_max(M+l), 2N+3) then the

h o m o t o p y fibre of the inclusion Rn+R of the n skeleton is r a t i o n a l l y a wedge of spheres. Corollary:

If a 1 - c o n n e c t e d CW complex,

r a t i o n a l c o h o m o l o g y ring,

R, has f i n i t e l y g e n e r a t e d

then for all nhn ° (some n o ) the h o m o t o p y

fibre of Rn+R is r a t i o n a l l y a wedge of spheres. Proof:

We show R is semi-finite.

of H*(R;~).

They define

Let ~l,...,~r be the e v e n g e n e r a t o r s

~:R÷K=ZK(~;Iail ).

The fibre,

E, fibres over R

1

w i t h fibre ~K(9;l~il-l)

w h i c h has finite d i m e n s i o n a l cohomology.

The

1

E i l e n b e r g - M o o r e spectral sequence c o n v e r g e s to H*(E;Q) (H*(R;Q);Q).

Since H*(K;~)

f i n i t e l y g e n e r a t e d H*(K;9)

is a p o l y n o m i a l algebra, module,

from Tor H*(K;9)

and H*(R;~)

is a

this is finite dimensional.

This c o r o l l a r y may be regarded as the strict analogue of T h e o r e m A.

We are, however,

not limited to the spaces in C o r o l l a r y i, and

indeed we have T h e o r e m C:

Let R be a simply c o n n e c t e d CW c o m p l e x of finite type such

that for some mo, ~i(R)@~=0, Corollary.

i>m o.

Then R is semi-finite.

Under the h y p o t h e s e s of T h e o r e m C there is an n o such that

for nLn o the h o m o t o p y fibre of Rn+R is r a t i o n a l l y a wedge of spheres. Remark:

The heart of the proof of T h e o r e m B is T h e o r e m 2.1 in the next

section, w h i c h is its t r a n s l a t i o n into the h o m o t o p y t h e o r e y of cgda's. As is usual in this kind of exchange,

the basic idea of

[L] is still

the p r i n c i p a l element of the proof, but the d e t a i l e d t e c h n i q u e s n e e d e d are quite different. Remark:

r Suppose R = i~iK(Q;nl ) (the case c o n s i d e r e d by Ruchti).

Then

R satisfies the h y p o t h e s e s of both c o r o l l a r i e s and so the q u a l i t a t i v e V e y - R u c h t i result follows from either. N : ~(2ni-l)

In this special case we can take

and M=0 and so our r e q u i r e m e n t

is n~z(2ni-l)+3;

in fact in

t h i s l c a s e V e y - R u c h t i get a better bound for n. The authors w i s h to thank L. A v r a m o v and W. Singer for several helpful discussions.

213

2.

D i f f e r e n t i a l algebra and t h e m a i n theorem.

to [B-G],

[Ha],

[Su],

c o n n e c t i o n w i t h rational h o m o t o p y theory. rior algebra

The reader is r e f e r r e d

[Ta] for the theory of m i n i m a l models and its

(X °dd) ® symmetric algebra

Here we recall that AX=exte-

(X even) denotes the free com-

m u t a t i v e g r a d e d a l g e b r a over a graded vector space. ~, here and t h r o u g h o u t is assumed of c h a r a c t e r i s t i c

(Our g r o u n d field, zero.)

All g r a d e d

spaces are s u p p o s e d c o n c e n t r a t e d in d e g r e e s ~0, and ® denotes tensor p r o d u c t w i t h respect to ~

(as opposed,

e.g.,

to ®A ).

A c o m m u t a t i v e g r a d e d d i f f e r e n t i a l algebra ted if A°=k,

simply c o n n e c t e d if also AI=0.

q u i s m if H(¢)

is an isomorphism.

(A,d A)

is connec¢, is a

The e q u i v a l e n c e class of a cgda

(under the e q u i v a l e n c e r e l a t i o n g e n e r a t e d by quisms) h o m o t o p y type.

(cgda)

A cgda morphism,

is c a l l e d its

A cgda is said to be a w e d g e of spheres if it has the

h o m o t o p y type of a c o n n e c t e d cgda H w i t h d i f f e r e n t i a l zero and satisfying H+.H+=0. If A is a cgda,

a KS e x t e n s i o n of A is a cgda m o r p h i s m A~A®AX

where X admits a well o r d e r e d basis x

such that dx sA®A(X<~).

dgda m o r p h i s m A ~-~B extends to a c o m m u t a t i v e d i a g r a m

Any

(if H°(A)=H°(B)-~k)

~A®AX in w h i c h the b o t t o m arrow is a KS extension. S u l l i v a n model for ¢.

The q u i s m A@AK

It can be chosen so that dx ~ A + ® A ( X

(A+X), and in this case is called minimal.

<~

=~B is a )+I®(A+X) •

A S u l l i v a n model for ~+B

is called a S u l l i v a n m o d e l for B. If A is a u g m e n t e d then AX=k®A(A@AX) fibre of the KS extension.

becomes a cgda, c a l l e d the

If the e x t e n s i o n is a model for ¢:A+B then

the h o m o t o p y type of the fibre is d e t e r m i n e d by ¢ and is called the S u l l i v a n fibre of ¢. The m a i n theorem, 2.1

Theorem:

from w h i c h we get T h e o r e m B as a corollary,

reads

Suppose A is a c o n n e c t e d cgda a d m i t t i n g a KS e x t e n s i o n

A ~ A ® A X such that H>M(A®AX) (some M,N~0).

= 0

and

H>N(Ax)

Assume IcA is a d i f f e r e n t i a l

= 0

ideal such that A ~ n + i c I c A hn

for some n>_max(M+l,2N+3). Then the S u l l i v a n fibre of A+A/I is a wedge of spheres. 2.2 A.

Remark:

Let A ® A X A ~

Then @A(A@AXA)

be a S u l l i v a n m o d e l for the a u g m e n t a t i o n of

is a functor from g r a d e d d i f f e r e n t i a l A - m o d u l e s to

g r a d e d d i f f e r e n t i a l vector spaces,

and it sends c o h o m o l o g y i s o m o r p h i s m s

of m o d u l e s to c o h o m o l o g y isomorphisms. M®AXA,

As a g r a d e d space, M ® A ( A ® A X A) =

and we use this n o t a t i o n for simplicity.

Note that a special case occurs w h e n

¢:A+M is a cgda morphism,

so

214

that M is a differential

In this case M ® A X A has the homotopy

A-algebra.

type of the Sullivan

fibre of ~.

type of the Sullivan

fibre of A+A/I.

2.3

Lemma:

In particular

A / I ® A X A has the homotopy

Let r=n-N-l,

Then the inclusion >r ISA(A@AX A) + A-- ®A(A®AXA)

is zero in cohomology. Proof:

Let d be the d i f f e r e n t i a l

in t~ and put

F = (Ax)
A submodule

B = A®F.

of A~AX and a spectral

sequence

(filter by the degree of A) shows that the inclusion B~+A®AX

induces an isomorphism of cohomology. H(I'B)

For the same reason

= H(I®F)--C*H(I®AX)

= H(I.(A@AX))

(2.4)

and H(A~r.B)

: H(ALr@F) - ~ H ( A L r @ A X )

are isomorphisms. Now set W = B ~ n @ B n-l, where Then H(W)=H~n(B)=Hhn(AGAX)=0, tial A-module, I'BcWcA~r-B,

: H(A!r.(A®AX)),

~n-i is a complement because n>M.

and by 2.2, H(W®A(A@AXA))=0

because

Now consider

FJ=0

for ker d in B n-l.

Moreover, as well.

W is a differenNote that

(j>N) and r=n-N-l.

the commutative

diagram

(I.B)@A(A@A~A)

=

W~A(~AX A ) >r ~ -(A-- -B)®A(A®AX A) in which the horizontal arrows (2.5) and remark 2.2.

(2.5)

.(I@AX)®A(A®A~A) [J

= >(AAr~AX)®A(A®AXA ) induce c o h o m o l o g y isomorphisms

It follows

that the inclusion,

by (2.4),

j, is zero on

cohomology. On the other hand, because A@AX A is acyclic, ordered basis,

induction on the well

x , of X gives an i s o m o r p h i s m

(A®AX)®A(A~AXA)

= A®AXA~AX (A@AXA, d)@(A X,d)

as d i f f e r e n t i a l

A ® A X A - algebras

I (respectively

by A ~r)

([Ha]).

identifies

Multiplying

on both sides by

j with the map

incl.®id:(I®AXA,d)®(AX,d)C-+(Alr@AXA,d)®(AX,d). It follows that H ( i n c l ) : H ( I ® A X A) + H ( A ! r ® A X A) is zero. Proof of 2.1:

The commutative

diagram

= >r @AX- A A--

~ AQA~ A

proj >r ~AX--A i A/A--

215

and the acyclicity

of A~AXA,

imply that

H(proj):H+(A/I®AXA ) ~ H+(A/A!r®AXA ) is zero.

Thus there are cocycles

>r

in A-- /I®AX A which represent

a basis

of H+(A/I®AXA). Since n~2N+3, cocycles

2r=2n-2-2N!n+l,

is zero.

This

and the product of any two of these

shows that H + ( A / I ® A X A ) has zero multiplication,

and also shows that the cocycles we chose define a quism H ( A / I ~ A X A) =~A/I®AX A. O

3.

The second main theorem.

Theorem C. 3.1

A

KS complex

Theorem.

Let

Here we establish

(AX,d)

be a 1-connected KS complex

There is then a KS extension AX+AX@AY (i)

Y is finite dimensional

(ii)

in w h i c h dimX<~.

in which

and c o n c e n t r a t e d

in odd degrees.

The minimal model of AX®AY is finitely generated, generators

In particular, Proof:

the result which implies

is a KS extension of ~.

H(AY)

with

all of odd degree. and H(AX®AY)

are finite dimensional.

Suppose that we have a KS extension AX~AX®AY n with Yn finite

dimensional

and c o n c e n t r a t e d

even generators

this minimal model be denoted Extend the p r o j e c t i o n AU@AW-~U~k.

in odd degrees

and such that the first

of the minimal model of AX®AY n occur in degree n. AZ, and put U=Z
AU+U~k

(killing decomposables)

Because U is c o n c e n t r a t e d

in odd degrees,

minimal model of a finite wedge of odd spheres. concentrated

in odd degrees,

Now restricting cohomology

is mapped by d to a decomposable

<_n+l.

In AZ®Au(AU~A(w
in AU, which is then

z-w is a cocycle.

whose m i n i m a l model has for g e n e r a t i n g

ators is reduced by one.

Moreover,

AZ®A(W
Because

~ must be decomposable. Introduce

This gives a KS extension

This is finite dimensional,

An obvious

W is

z, of Z n

for some ~EAU®A(W
= AZ@A(w
an odd generator y with dy=z-~.

The first generator,

(n+l)-cocycle

Hence dz=d~

dz is odd, and U,W
is also a quism,

AU®AW is the

In particular,

and each W k has finite dimension.

in degrees

killed by the projection.

AZ~AZ®A(W
to a quism

the quism to AU®A(W
isomorphism

Z/(z)@W
Let

space

and the number of even gener-

if AZ--~AX@AY n is a quism then

~ (AX®AYn)®Az(AZ®A(W
and the cgda on the right has the form AX®AYn@A(W
the proof. D

4.

Proof of T h e o r e m B.

to cgda's.

Denote by ApL Sullivan's

functor

If R is as in Theorem B, with n-skeleton

from spaces

R n let

216

~':AZ'~ApL(R) be minimal models.

There

,

~ :AY-~ApL(Rn)

is then a h o m o t o p y commutative

ApL(R) + ~ A Z '

]

1

L0

ApL( Rn ) ~ Because injective

Hi(R;Q)

in

degree

with

an acyclic

plex

AZ,

the

same

KS c o m p l e x

a n d we c a n

is surjective. homotopic

~ Hi(Rn;Q) n,

~ ~

AY

is

an

is

true

isomorphism for

concentrated

extend

cgda d i a g r a m

~'

and

,'

in to

in

,'

degrees

Thus

degrees

~:AZ~ApL(R)

we c a n

Kn t o and

get

AZ'

a KS c o m -

,:AZ÷AY so

Note that ~ is still a quism and ApL(incl)oa

that

is

to ~ .

On the other hand put J=(Ay)>n@J n, where jn complements cycles


in (AY) n.

I = -1(j);

Then

the co-

H(AY/J)=H~n(AY)=Hin(Rn;~)=H*(Rn;~)=H(AY).

since ~ is injective (AZ) ~n+l

Because ~ is surjective,

in degrees

Put


c I c (AZ) in.

AZ/I=AY/J

and the commutative

diagram

AZ AY-identifies

the Sullivan

$~=ApL(incl)~, while by

fibres of ~ and the projection.

the Sullivan

[Ha;§20]

for the homotopy

the

,AZ/I

fibres of ApL(incl)

Sullivan

fibre of ApL(incl)

fibre of Rnt+R.

Since

and of ~ are identified, is a Sullivan model

To prove that this is rationally

wedge of spheres we have only to prove

([B-G])

a

that its Sullivan model

is a wedge of spheres. We are thus reduced to showing that the Sullivan is a wedge of spheres.

But the map E+R guaranteed

fibre of AZ~AZ/I

by the semi-finite-

ness of R gives a cgda m o r p h i s m A Z - I + A p L ( R ) ~ ApL(E)Form its Sullivan model A Z ® A X - ~ A p L ( E ) . Then H>M(Az@AX) Moreover,

again by

topy fibre,

[Ha;§20],

F, of E+R.

Since n>max

(AX,d)

It follows

(2N+3,M+I)

is a Sullivan model

= H>M(E;9)

= 0.

for the homo-

that H>N(Ax)=H>N(F;~)=0.

and since

(Az)An+IcIc(AZ) ~n we can apply

Theorem 2.1 with A=AZ to deduce T h e o r e m B. D

5.

Proof of Theorem C.

This is an immediate

consequence

of Theorem

3.1.

217

D e p a r t m e n t of M a t h e m a t i c s

D e p a r t m e n t of Mathematics,

U n i v e r s i t y of Toronto,

B r o o k l y n College,

Toronto,

City U n i v e r s i t y of New York,

Canada

M5S IAI

New York, N.Y.,

10453

U.S.A.

References [A]

L. Avramov,

Local algebra and rational homotopy,

A l g e b r i q u e et Algebre Locale, [A2]

L. Avramov,

[B-G]

A.K. B o u s f i e l d and V.K.A.M.

Golod homomorphisms,

Math.

144,

179

(1962), p. 479-482.

English Transl.

(1976).

Akad.

Soviet

(3) 1962 p. 745-748.

S. Halperin, Math.

ILl

On PL de Rham theory and

Soc. Memoirs No.

E.S. Golod, On the h o m o l o g y of some local rings, Dokl. Nauk SSSR,

[Ha]

(1984) p. 15-43.

these proceedings.

Gugenheim,

rational h o m o t o p y type, Amer. Math. [G]

in H o m o t o p i e

A s t e r i s q u e 113/114

Lectures on m i n i m a l models,

M ~ m o r i e s de la Soc.

de France 9/10, Paris 1984.

G. Levin,

Local rings and Gelod homomorphisms,

J. of Alg.,

37,

(1975) p. 266-289. JR]

J.E. Roos,

H o m o l o g y of loop spaces and of local rings,

the 18th Scand. p. 441-468, [Ru]

R. Ruchti, Math.,

[Su]

Conf. of Math.

Birkhauser,

(1980);

Progress

Proc. of

in Math ii,

1981.

On formal spaces and their loop space,

Ill. J. of

22 (1978) p. 96-107.

D. Sullivan, I.H.E.S.

Infinitesimal computations

47 (1977), p. 269-331.

in topology,

Publ. Math.

MATRIC MASSEY

PRODUCTS AND FORMAL MODULI by

O.A.

Introduction.

Laudal

It is now folklore

functor of an algebraic

geometric

that the hull of a d e f o r m a t i o n object,

in some way

is d e t e r m i n e d

by the a p p r o p r i a t e c o h o m o l o g y of the object and its "Massey products",

see

in Douadys

expos~

[M],

[May]. The

in [Car]

In ]975 I proved that,

first hints

in fact,

there

structure

induced by the o b s t r u c t i o n

hull,

[La] ].

see

This, actual

see f.ex.

results

k-algebra

A

and of an

in this

in two special

A-module

the general m a c h i n e r y

all the cases we have In §2 we prove

this

are c o n c e r n e d with the p r o b l e m of

c a l c u l a t i o n of these M a s s e y p r o d u c t s

In §] we recall

characterizing

[Pal ], [S & S ], for references.

and a forthcoming paper,

that of a

occurs

is a kind of M a s s e y p r o d u c t calculus

I n d e p e n d e n t l y many authors have p u b l i s h e d direction,

in this d i r e c t i o n

(]96]).

of

cases,

E.

[La] ] w h i c h

is common

for

in mind.

that the usual m a t r i c M a s s e y products,

properly

a d j u s t e d to our needs, for EXt;(E,E) d e t e r m i n e the formal moduli of the A-module E, i.e. the hull of the d e f o r m a t i o n functor of E. As a c o r o l l a r y we obtain the following

(2.]0)

Any complete by

local ring A

EXtA(k,k), i = ],2 n Ext 1 --÷ Ext 2.

result

is u n i q u e l y d e t e r m i n e d and the m a t r i c M a s s e y p r o d u c t s

219

§]

Formal Let

or an

moduli

X

be

and Massey

some

A-module

algebraic

E, and

A i = Ai(k,X;Ox )

k-algebra

A,

then

X

is an

By

[La] ], (4.2.4),

hull

of

object,

say a

the d e f o r m a t i o n

k-algebra

A

functor

÷ Sets

].

[Lal

Let

geometric

consider

Def x : !

see

products

the c o r r e s p o n d i n g

A i = Hi(k,A;A)

A-module

Def x

be

cohomology.

is the A n d r &

If

X

is a

cohomology,

and

if

E, A i = E x t ~ ( E , E ) . we k n o w

t h a t the

, is d e t e r m i n e d

formal

moduli

by a m o r p h i s m

of

X,

of c o m p l e t e

i.e.

the

local

k-

algebras o : T 2 = Symk(A2*)^ constructed In

fact,

the

using

(4.2.4)

only of

the

÷

T 1 = S~n~(AI*) A

"obstruction

[La] ] i m p l i e s

that

calculus" the

formal

of

A

moduli

H

has

form H = T 1 ~ k , T2

provided We

shall

A 1 and assume,

A2 in

a basis

A2*

Denote

dual

bases

Put

fj = o(yj),

of

T I

TI/(~). algebras

i

< ~

of

of

{x~ ..... y 2 ) A I

generated

resp.

k-vector

spaces.

{y] ..... yr}

of

that

A I* and

and

a basis

{y; ..... y~ }

the c o r r e s p o n d i n g

A 2.

fj | s

by the

~ : R ÷ S,

as

i = ],2.

j = ] ..... r. T h e n

for any

dimensions

follows, for

{x] ..... Xd} by

Now,

countable

what

dimkA Pick

has

by

(4.2.4)

of

[La]

the

_l T.2m

Moreover

H =

artinian

k-

is c o n t a i n e d

in

surjective

homomorphism

of local

such t h a t

mR.ker

Sor(H,R)

÷

DefE(R)

Mor(H,S)

÷

DefE(S).

~ = 0,

consider

ideal

(~)

the d i a g r a m

220

Suppose X

#: H + S

given a morphism

Defx(S), to a m a p

then ~

in the d i a g r a m

making

the

corresponding below,

resulting

we may

diagram

to the always

lifting

lift

the map

commutative

0

L ker

(i)

}I

S

s/m 2 The

obstruction

and

functoriality,

fact o(X

0 0 ~ ,~)

defined

for

lifting given

induces

6 A 2 ~ ker

to

the

a linear

Notice

R

is,

restriction

map

~ = A2(k,X;OxO

obstruction.

(2)

by

X

A 2. + k e r ker

by c o n s t r u c t i o n of ~,

~), w h i c h

that we h a v e

the

0 0 ~ i.e.

to

of

o,

A 2..

In

an e l e m e n t

is the u n i q u e l y

following

identity

o(X¢,~) = [ y~ ~ ¢{fj). k

Notice

also

that

Defx(S/m2) determined

the

corresponds by the

by an e l e m e n t

t¢

-- A I ~ m / m 2

t.l 6 m/m2_

then

the o t h e r

1

of

X

by the m a p

~]. M o r e o v e r

m a p on the

• A I* = m H / m 2 H

Defx(S/m2)

On

X

to the m a p

induced

t% thus

image

÷

cotangent

6 Al® m/m2

which

corresponds

to

hand,

having

fixed a b a s i s , al 6 A I.

Thus

there

is a one to one

correspondence

~] , ..., ~p n =

i] < i 2 < .,.<

of e l e m e n t s

(n] ..... n d) ip

is u n i q u e l y

level

the

isomorphism

t = d i=] x .~oti

~] (x.)l = t.l ' i = ] , .... d.

t# = zP=] alm ~]

an

under

X#I • If

that

Pick

~]

m/m2

find

sequences

Defx(S)

be the

d 6 N__

of

with

indices

i

{v] ..... Vp}

between

for

maps

#1

m 2, we --/m

and

A1 d J__nI = Z i = ] n i = N for w h i c h

n i # 0.

and

let

221

Consider

the

ideal

J n c_ k [u] , ...,u d]

generated

monomials

n] Put

R

= k[u]

~il

be the

n

generates {

Rn

let

.. Ud]/J n ' '

'"

image

..... Vp}

Now

by the

set of

t] t d{u] .o. u d 13i, t i > n i } .

of

Uil

S in

(resp.

S n) as

mn/m

. Fix

of

~1,...,~

6 A1

= R

n

R

n

/(u

nd ) .-- u d

(resp.

k-algebra, this

= and

let

S ). O b v i o u s l y and

induces

v1

vl,...,vp

a basis

basis.

and c o n s i d e r

the

corresponding

map

P ~]: H + S n l m ~

Definition

.

(1 :!).

Any

map

¢

making

the

following

diagram

n

commutative T2 o +

T1 p

R

n

~n

4~

H

~n_

S --

Sn/m is called

a defining

system

for the M a s s e y

product

<~1 ..... = ;n> = o(X ,~n ) 6 A 2 . P-Cn -%~en <xi]

we

~l = xl I ~,-ot

Suppose

shall

<x ;n>

for the M a s s e y

product

x i ;n>. P now

that

for some

fj = and

write

consider

any map

I£

N > 2

and

every

I=Na j ,n ~-n + higher

~n:

T1

+ Rn

n I] ~p = aj,~ ui] "''u" £ ker

such

j = ], ...,r

we h a v e

terms

that

~ n 0 ~ n = p0¢ n. T h e n

n,

%n(fj) _

we (3)

ip

find a j , ~ = y j ( < x * ;n>).

= k. A p p l y i n g

the

identity

(2)

222 It follows

that

fN 3

if we let

be the degree

N

(leading)

form of

f. , then 3 (4)

fN(x)

:

[nI=Nyj<x*;n> ,x n .

3 -Consider

the diagram: T2

(5)

where Let

~

¢N-] k[u] . . . . .

H

* k [U] ..... U d ]/m N CN-]

00¢N_l (xi) X~N_]

Ud ]/raN+ l

-= u i ( m o d _m), m = (u I ..... Ud).

£ D e f x ( k [ u ] ..... Ud]/m_N) c o r r e s p o n d

by a s s u m p t i o n to

TI

XCN_]

is a lifting

k[u] ..... U d ] / m 2 . N o t i c e

to

~N-]" N o t i c e

of the universal

also that

ker

lifting

~ -- m N / m N+]

=

that

of

•

X

k.(u_--n).

Inl=N An easy argument to

•

k[u I . . . .

(6)

Now

u

then shows ~.

dJ/m

N+]

o(X¢N_],~)

consider

T2

that the o b s t r u c t i o n

is given

for lifting

X~N_]

by:

:

In[:N[ <x*;n>®u n = J[ y~®(j [n_{=Ny j<x*;n>u n)

=

[

* N • yj®fj(u) . 3

the d i a g r a m

o

T~

*N

= k[u l , "'''Ud ]/(mN+2+m( N fNr) ) --- fl '''''

RN+] 1

~ ~N+ ]

(7)

H

___ ¢N. SN

¢N_l~

Since

SN

is

the o b s t r u c t i o n

k[u 1 .....

= k[Ul . . . . .

~N SN_ l = k [ u ] . . . . . Ud]/mN+l

for lifting

Ud]/(mN+l+(fl N .....

Ud]/mN

divided

X

fN) r- )

, we may

by t h e

ideal

lift

X~N_]

generated to

by

SN ,

CN-I therefore tative.

we may find maps

~N

and

CN

making

the d i a g r a m

commu-

223 for

Pick a m o n o m i a l

basis

with

and pick a m o n o m i a l

Inl

~ N-])

{un}nE~N -]

basis

m N / m N + 1 + ( f ] , . .., fN)r " Put

BN = BN_ ] U B N.

we have

in

a unique

(8)

relation

u----n=

(6)

by c o n s t r u c t i o n

implies

that

Write

ker

,;q+]

for

n6B N

For e v e r y

_n

n

all ker

u-xN =

with

Inl

( N

SN

_

o(X@N_I,~N)

for e v e r y

m

= O,

E BN

In--.~=N ~n,_m<x* ;n>

(9)

{un}

simply

~- ~n,m u_-m m_~BN -

Since

(take

SN_ ]

this

relation

(or BN if one

together

with

insists))

= 0

= (mN+]+(f~_ .....

f N ) ) / ( m'N + 2-+ m--( f N

m( f] ....

...,frN))

IN+1

of the

form

N+] N+2 N+] N N IN+ ] = _m /(m +m_ D m(f] .... ,fr )) n n We may assume that for n E ' u-- is {U--}nEB N +1 . BN+ ] , _

of the

form

Uk.U_-m

Pick a m o n o m i a l

For

every

(]0)

n

basis

with

for

for some In[

n _~-=

m

~ N+]

we h a v e

'

5_ l

mEBN+ ]

6 B N. Put

8n,m-

us

+ ]~

B~[+] = BN U B~+] .

a unique

relation

in

RN+ ]

(3' fN. n, 3 ] --

Let

(]i)

fN+]

then by d e f i n i t i o n

= ~N(fj)

of

= fN+

[

n

b.

] n~+1

u--

j.n-

o, the o b s t r u c t i o n

for l i f t i n g

X

to @N

RN+ ] ( ] 2)

is N+]

j

= [ y~ @fN j

3

+

~_ m6B~+]

(~*y ® bj,n u~n) j

--

224

Definition

(1.2).

Massey

The

these

(13)

[. bj, n yj ]

notations fN+1

=

is called

a defining

system

for the

E A 2,

put

~

m + ;m>u--

yj<x

E BN+ ] .

yj<x

n~m~

we have

for

we have:

[

] where

%N

products <x ;n> =

With

map

*

n ;n>u--

a~B~+ I

B' =

{nllnl:~}.

N

Consider

the d i a g r a m 9 T 1 CN+]+ RN+2

T 2

= k[u] , ...,Ud]/(mN+3+m.(f~/+1

,..., fN+]))r

!

(14)

%

¢~N+2

H ¢~+i ---

SN+ I = RN+ I

/( .N+] .N+] ) x] , ...,x r

SN

Since

obstruction therefore above Pick

we may

Put

find maps

to

~+]

¢N+1

and

for

BN+ ] = B N U BN+ ] . T h e n

{~}

SN+ ] . For relation

basis

every

~

in

SN+ ]

_u~ =

ideal

generated

, we may ¢N+1

by the

lift

making

X~N

to

SN+] ,

the d i a g r a m

the

with

~N+]

~EBN+ I

~ N+]

o(X N , XN+ I) = O,

following

identity:

[ fln,m <x*;n> nEB' -- --N+I is analoguous

I~I

ker

such

that

BN+ I

is a m o n o m i a l

we t h e r e f o r e

have

basis a

~fln,m -u~m ~EBN+ I

by c o n s t r u c t i o n

E BN+ I

which

XCN

by the

n {~--}~EBN+ I

(15)

(16)

divided

for lifting

a monomial

unique

Since

RN+ I

commutative.

B~+]. for

is

SN+ I

to (9).

= 0

this

implies

for e v e r y

225

Write,

again,

_N+] =

(f]

(mN+2+(f~+] fN+] -'''' r )~(mN+3+m( " N + ]_ _r]

~N+2' =

. fN+] ), , ~N+]

''

Pick

ker

'

r

/m%r]

a monomial

fN+])

,..,

basis

IN+2 = .m N + 2./ ( m.N + 3 +.m N + 2

for

n _u--

{~--}~6B +2 , where

form

is of the

B~+ 2 = B~+]U

BN+ 2

relation

~+2

in

m Uk* ~ -

form

For

n u-- =

(]7) of the

same

@ IN+ 2.

r

n

of the

,..,fN+]))r

for

every

we may some ~

assume m

that

6 BN+ ]

with

0 m(~]+] ''" ,fN+])) r

lal

and

~ N+2

for

n 6 BN+ ' 2 '

some

k.

we have

Put a unique

, m ~' fN+] ~-, 8 n,m _u-- + [. n, j ~£BN+ 2 3 -J

form as (]0).

Let -fN+2 fN+] 3 = CN+](fJ ) = J

(is) Again, ~+2

by d e f i n i t i o n

of

o,

the o b s t r u c t i o n

.

(]9)

for

lifting

X#N+2

to

fN+2

3

= j

Definition

(].3).

the Massey these

(20)

Clearly k ) 0

n ~ c u-. ~ ( B ~ + 2 3,a --

is o(X#N+ ] , ,

With

+

The

products

notations,

~j+2 =

this

#N+] <x;

we have

the

cj ,n--ua)

j

is called

n> = Zjcj,~

Y j*®

([

yj

_

a defining 6 A 2, for

system n

for

£ BN+ 2.

following

m + [ <x. l>u ! + [ yj<x * ;m>u-[ yj<x* ;m>u--n ~B~ ~B~+~ a~B~+2

process

we obtain

map

Y~ " fN+]+ ~ 3 ~6~+2

m a y be c o n t i n u e d

a diagram

indefinitely.

For

every

226

T2

~

T1

(21)

~N~k

RN+k+]

+

+~'

H

~N%k

N+k+]

SN+k ¢~N+k SN+k-]

a monomial with

basis

J~J

( N+k

(22)

inducing

the

{u~ -- J~6BN+ k

for

such

SN+k'

there is a unique n m U~ u_-~~n,m -~ E B N + k -- _

relation

that

in

for e v e r y

n

SN+ k

,

identity

(23)

~

~n,m<X*;n>

= 0 ,

m E BN+ k.

n E B ~ + k -- _ And

there

IN+k+ ] such

is a c o r r e s p o n d i n g

of

that

ker in

~' N+k+]

RN+k+ ] n u_-- =

(24)

where

we have

=

put

basis

{~}m£B~+k+

] for

the c o m p o n e n t

(_N+k ,fN+k. N+k N+k f] .... r )/~(f ..... fr ) • IN+k+ ]

we have

for every

~

with

, n 8' 0n,m -u- + [ n , j

~_

I~I

( N+k+]

M~+k 3

BN+k+ 1 = BN+k U BN+k+ ] .

Moreover,

(2s)

fN+k+1

= ~N+k(fj)

= fN+k +

~

3

nEB'

3 The

obstruction

(26)

for l i f t i n g

X~N+k

to

~j~

--

RN+k+ ]

N+k+]

n u--.

--

is

) = ~ y~ . fN+k+] j

o(X0N+k,~+k+l

3 = ~ y~ . ~j+ k J

Definition

(].4).

the Massey In particular,

The

map

products we

find

~N+k

+ [ ,

is called

a defining

<x ;n> = Zj Wj,n yj for every

* We,_n~_) n ([j YJ"

£EBN+k+ ]

k,

E A2

for

system ~

for

E BN+k+ ] •

227

fN+k+] 3

(27)

Notice

that by (4.2.4)

(28)

H

of [La] ] we have

lim

=

k+1 ~ ~ yj<x*;n>u n I=o n6B~+ 1

=

SN+k

therefore (29)

H = k[[u~

. . . . .

Ud]]/(~ ~

~d )

. . . . .

where

(30)

fj = lira f~!+k. k÷= 3

Formally we may therefore write c0

* > n ~j = ~ ~ ~]v'<x- ,_±-n._u-" i=o _6BN+ n ' 1

(3|)

§2

Massey products

for

ExtA.(E,E)

In this paragraph we shall shall

let

X,

in §1, be some

let

A

be any

A-module

concerned with the deformation

k-algebra

and we

E. We shall thus be

functor of

E

as an

A-module

DefE: ! ÷ Sets defined as follows, { S ~kA DefE(S ) = ~ A As is well known,

÷

End(Es)i %

ES ~

÷

End(E)

E S ~S k = E

the corresponding

is

cohomology

S-flat?

/ iso.

is

A i = Ext~(E,E) The deformation [La| ], parallels global

theory

for modules,

the corresponding

theory and a relative

of [La] ] holds.

as hinted at on page ]50 of theory

theory,

There

and the main theorem

There are no surprises,

leave the details to the reader.

for algebras.

is a

(4.2.4)

and we shall therefore

228

Pick

any

free

resolution

the a s s o c i a t e d HomA(L.,L.).

single

L. of

complex

By d e f i n i t i o n HomAP(L.,L)

Let

d i : L i + Li-]

be

dP: is d e f i n e d

as an

A-module,

and

consider

of the d o u b l e

HomA(L.,L.)

complex

we h a v e =

H H o m ( L m , L m _ p) m>o

the d i f f e r e n t i a l

Hom~(L.,L.)

of

L.,

then

÷ HomP+I(L.,L.)

by

dP({=~}i~o) Clearly

E

HomA(L.,L.)

= di0=~_ I _ (-I)P J 1 od z - p

is a g r a d e d

differential

associative

A-

0

algebra,

Lemma

multiplication

(2.1).

There

being

is a n a t u r a l

i EXtA(E,E)

Consider

the c o m p o s i t i o n

any s u r j e c t i v e

of

Hom

(L.,L.).

isomorphism

i . = H (HOmA(L.,L.)),

morphism

i ) 0.

~: R ÷ S

in

1 , such

that

m °ker ~ = 0. --R Assume

there

{L.,di}, This

exists

i.e.

means

a lifting

of the

that

free

there

{L.o k S, di(S) }

resolution

exists

L.

of

a commutative

0

where

÷-- Ho(L.

+-- L o ® k S

÷--

for e v e r y

shall

see that

resolution E = Ho(L. )

of to

of the

form

÷

L] ~ k S

d2(S) ~

L2®k S

d3(S) ÷

d] ÷

L]

d2 ÷

L2

d3 ÷

i, the c o m p o s i t i o n di+ ] (S)

We

LO

complex

E.

diagram

d] (S) 0 "~-- H o ( L . ® k S )

of the

any

o di(S) such

Ho(L.~kS) S.

=

lifting

= ES

, and

0. is,

in fact,

that

ES

an

A ~k S - f r e e

is a l i f t i n g

of

229 Both contentions may assume

are obviously

they hold

corresponding

for

true

S

statements

for

di(s) }

conversally to

S

A ~

of

S-free

obstruction

For every

are

is a c o r r e s p o n d i n g

S. By a s s u m p t i o n

we

the

{L.~kS,

of

E

lifting

we have

di(S) ) and

ES

of

is,

{L.,di} itself,

an

E S. di(S) }, and let us compute

{L.~kS,

an o b s t r u c t i o n

i, pick a lifting

L i ~ k S + Li-] ®k S, to

to prove

Given a lifting

E S = Ho(L.~kS)

{L.~kS,

for lifting

clearly,

there

lifting

of

lifting

problem.

to

a lifting

resolution

Pick one such

is then,

{L.,di}

that any such

determines

so by induction

R, we are through.

it is easy to see that

{L.~kS,

S = k,

S. If we then are able

But first we have an existence to

for

di(S) }

R. This o b s t r u c t i o n

for lifting

di(R): '

R. This

to

the

E

to

S

R.

Li® k R + Li_ ] k®k R

is obviously

possible,

of

di(S):

since all

Li

A- free.

Since

di(S)0di_] (S) --: 0

maximal Li_2~k R

ideal

mR

of

is induced

and

since

I = ker

R, the c o m p o s i t i o n

~

is killed

by the

d~(R) 0d~_] (R) : Li~kR

+

by a unique map

Oi: L.l ÷ Li-2 ~k I The

family

{Oi}i) 0

defines

an element

O E Hom2(L.,L.) One checks defining

that

d20 = O, so that

It is easily d~(R)'s

Moreover,

O is a 2 - c o c y c l e

of

HomA(L.,L.),

an element O(Es,~)

the

~k I

if

seen that lifting

6 Ext~(E,E) o(E S, ~) the

o(E S, ~) = O,

£ Homl(L.,L.)

~k I.

is independent

of the choice

di(S) 'sthere exists

~k I such that d ~ = - 0

.

an e l e m e n t Put

of

230

d, (R) = d'.(R)+~. , l 1 1 then one

finds di(R)

and Now

0 di_ ] (R) = 0

{L.mkR , di(R) } let

is a lifting

{L.mkR;di(R) }

R, then

there

be any

is an exact

of

{L.~kS , di(S) }

lifting

sequence

of

a long

exact

Hn(L.@kR)

÷

H n (L" ®kS)

÷ H n _ ] ( L . ® k I)

÷

....

÷

H] (L. ~kS)

÷

÷

H

from w h i c h

o

(L.~

I)

it follows

÷

E ~k I

o

(L.®

R)

+

÷ Ho(L.®k s )

0

that

Hn(L.~kR) 0

di(S)} ~ 0

sequence

Hn(L.®kI)

H

to

of c o m p l e x e s

÷

÷

R.

{L.®kS , di(S) I

0 ÷ {L.-kl, di~1 I} ~ {L.®kR, di(R)} ÷ {L.%S, inducing

to

*

= 0

Ho(L.~

for R)

n ) ], ~

ES

+

is a l i f t i n g

of

and 0

is exact. Therefore

Ho(L.~R)

Moreover,

given

= E

two

liftings

The

family

element

{ni}i) O

÷ Li-] is a

liftings induce

R.

1 = ],2 E~

and

of E~

of

maps

~k I.

]-cocycle

of

Hom'(L.,L.)

defining

~ 6 Ext~(E,E).

In this w a y we o b t a i n {liftings making space

to two

di(R)]-di(R) 2

ni: Li

to

{L.~kR , di(R) l} ,

{L.~kS , di(S) }, c o r r e s p o n d i n g E S , the d i f f e r e n c e s

ER

the

a surjective

of E s to R}

set of

liftings

( t o r s o r ) over

We h a v e

established

the

map

x Ext~(E.E)

÷ {liftings

of

R

Es

Ext~(E,E). following,

to

of E S to R}

a principal

homogenous

an

231

Propositi0n

(2.2).

Let

E S £ DefE(S)

{L.~kS,

di(s) }

of

defined

obstruction O(Es,~)

L.

to

correspond

S. T h e n

E Ext~(E,E)

there

to the

lifting

is a u n i q u e l y

~k I o

given

in terms

defined to

above,

such

R

if

that

O(Es,~)

is a p r i n c i p a l

T h u s we h a v e entirely

O(Es,~)

= 0

in t e r m s

of the c o m p l e x

products,

apply

<x

turn out to be

the d i f f e r e n t i a l Pick of

a basis Ext

for

xi

and

(E,E)*

n>

for

in §] mn/m~

let

an

£ = the

graded

Denote

for

HomA(L.,L.) iff

~ I

m a y be

ES

lifted

calculus its

by

for

ES

to

Ext~(E,E). Def E

given

liftings.

of §]

In fact, "ordinary"

and the

compute <x ,n>

the of

§]

Ext~(E,E)*

Ext I

and

X i £ Hom~(L.,L.) Yj

Massey

products

of

HomA(L.,L.).

{x ..... x~} of

over

of

N+k"

k-algebra of

liftings

(torsor)

and

n 6 B'

j = ] ..... r,

(n] ..... nd) k-algebras

with S

that we h a v e

and a b a s i s

and

{y] ..... y r }

{y ..... y r }

the

Ext 2. be a c o c y c l e

6 Hom~(L.,L.)

representing

be a cocycle

d I~I = Z i = ] n i = N

and

R

. Fix

in

Rn

the

the b a s i s following

and c o n s i d e r {~]---~p}

as of

slightly

identities v I = Uil,

ui = 0 insisted

set of

yj.

. Recall

(])

the

space

some g e n e r a l i z e d

i = ] .... ,d,

confusing

on.

= 0

L.

--

dual b a s e s

representing Pick

of

the c o n s t r u c t i o n s

{x] ..... Xd}

corresponding Let

then

homogeneous

"

will

O

at h a n d a nice o b s t r u c t i o n

this we shall

Massey

2-cocycle

R.

Moreover,

Using

of the

upon because

1 = ] ..... p.

if

i ~

it m a k e s

{i I ..... ip} the n o t a t i o n s

more

streamlined

later

232

We shall pick a monomial

basis

for the

k-vectorspace

S

n

of the

form in] md {u] -oou d i 04mi
m%_n}

as {urn}mEan

With this done, element Let

let

a],...,~

E Ext~(E,E)

P

#] : H ÷ S /m2

be the corresponding

be the induced

Assume

is given a defining

there

product


E DefE(S ~)

Then and

the

Z~=] ~i ~ ~i E Ext~ ~ --ms/m2--n"

DefE(Sn/m~)

E~n

and consider

E%] E~n

of

deformation

map of

system

and let

E#I E

E.

~n: H ÷ S n

(see (] .])), corresponding

for the Massey to a lifting

E~].

is represented

by a lifting

{L.,Sn/m~;di(Sn/m~)

is represented

by a lifting

{L.~Sn;di(Sn) } _ _

} of

L.

of

{L. ~Snlm~;d i (Snlm~) }. The family of

A o k Sn-linear

di(Sn): is uniquely family of

Li ®k Sn

determined A-linear

maps

÷

Li-]

®k Sn

by the restriction

maps

~i,m: Li ÷ Li_ ] ,

m E Bn

defined by di(Sn)iLi~] --

=

[_ =i,m ~ _~mEB ---

With this notation,

we may assume

di(Sn/m2n) ILie] --

where

--

--

n

=

~ I=]

ai

o u~I '~i

21 = (0 . . . . i,O ..... O) E B . 11

--

to

L i ® ], thus by the

233

According

to (]) we may also w r i t e d i ( S n / m 2 ) I Li~]

= ~ ~i

--

For e v e r y

m

£ Bn

~ Vl" ,_E 1

the

family

{~i,m}i

is a c o c h a i n

am 6 H o m I ( L . , L . ) such

that

~--~I is a c o c y c l e • ~1,o

1 = ] .... ,p , and Since

=

representing

d

{~m}m6~n

satisfies

the

7

~

for all

following 0 ~

ml+m2=m _ml

m2

cohomology

class

~i

i ~ 0

we

that

the

family

, i.e.

the

find

identities

= 0

for all

m 6 -

_n

miEB n Moreover

the o b s t r u c t i o n

obstruction is e a s i l y

o(E n,~ seen

)

for for

lifting lifting

to be r e p r e s e n t e d

m~

Proposition ~i

(2.3).

of

R

{Li~ S , di(S

)}

(& priori),

to

R

,

cocycle

6 Hom2(L.,L.). A

Given

a sequence

of

then

a defining

system

corresponds

Hom~(L.,L.),

*

to

n

£ Ext~(E,E),

<~],...,~p;~>

E#n

by the,

~ ~ 0 ~ m l+m2= n ml m2 --i

such

~

~

p

cohomology

to a family

t h ~ for e v e r y

0 a

= 0

= d.

for

classes

for the M a s s e y {am}m6~n

product

of ] - c o c h a i n s

~ £ Bn

m. EB --1 n and ~i

such '

1

that =

~.

], ....

Conversally,

any

i > 0

and

~

represents

p.

such

family

give

{~m}m~ ~

rise

to a

n defining

system

'

i ) 0"

i '

di(Sn) 0di_ ] (S n) = 0

the

for the M a s s e y

product

<~],--.,~p;~>°

234 Moreover,

given

<~] , ...,ap;n>

such a defining is represented

[ a 0 ml+m2= m m I

system,

by the

the Massey

product

2-cocycle

6 Hom2(L.,L.). A

a

m 2

miEB n

Proof.

This

is just the observation

that a lifting

E

of

E

Cn corresponds thus

to a lifting

tO families

**

{L.~kSn;di(Sn)

{~i,m}m£~n_ _

di(Sn) { Li~]

relation

proves

=

[_ m6B

di(Sn)

a. ~ u~m . l,m --

defined

translates

by ** defines

{L.~Sn,-d.l(Sn/m2)_n }' thus also a lifting such

E n

corresponds

to a map

into *. C o n v e r s a l l y

E~n

¢ : H ~ S

of

E~] . Finally

, i.e.

any

to a defining Q.E.D.

Remark

(2.4).

defining

In the light of (2.3)

system

either

the map

Remark

(2.5)

•

for the Massey #n Let

depends

(n.ll,ni2 .... ,n.lp). Given there

depending

be given

then the Massey upon

let the notion



{am}

n = (n] . .• . . nd)

only

m = (m] ..... mp)

we shall

product

or the family

i ~ {i I ..... ip},

defined,

~], .... ~p

to

on the situation. ni

=

<~] ..... ~p:n>,

and the

0

if

p-uple

a] .... ,ap 6 Ext~(E,E)

is no confusion

refer

such that

product

of a

and any

p-uple

in writing

<~1 , ..., ~p;m> . Suppose system

*

a lifting

system.

for

},

n

di(Sn) 0di_] (S n) = 0

that

¢]

{L.~kSn/m~;di(Sn/m~)

such that

--

The

I of

p = 1 for

and

<~;n>

n = (n), is a family

a]=~

6 ExtI(E,E),

{am}o(m
of

then a defining ]-cochains

235

am E H o m ~ ( L . , L . ) such

~i,~ = d i ,

that,

property

that

i > 0

for e v e r y

and

system

represents

0 ~ m < n-l, -

If a d e f i n i n g

~]

exists,

~

-

~

~1+~2=~

a, w i t h

o ~ ~1

the

= 0. ~2

then

m1+m2= n

ml

m2

o~m. ~m-] In p a r t i c u l a r defined

and

for

n =

(2)

the M a s s e y

is r e p r e s e n t e d

by the

product

<~,(2)>

is a l w a y s

2-cocycle

~0~.

These

6 ExtAI(E,E)

then

the

are

the

"Bocksteins" If

p = 2

and

n = (],]),

{~m}m6{(O,0),(],0),(O,]) represents for

~ , and

<~]'~2; ( ] '] ) >

~(0,J) ° ~(],0)" Now,

having

Massey

computing defining Assume,

formal

in §]

Then

some by §1

is r e p r e s e n t e d

cohomological we

of

the

[

a.

n

system

by

+

a(] ,0)° ~(0,])

is the

symmetrized

expression

for the

shall

procede

for

cup-product. (defined)

as in §],

for the

power-series

ideal

of

fj -- o(y j)

T1

6 T1 =

as

+ higher

terms

j = ] ..... r

N > 2. (3),

~j,n

=

yj<x

w

;n>

where,

by a s s u m p t i o n

<xi],xi2, .... x i ;~> is (uniquely) d e f i n e d . P u~ and c o n s i d e r the d i a g r a m Put ~j = ~i~i= N a j . n _ %N- ]

~(],0)

E.

formal

x~

{di)i>0'

family

~2 ' is a d e f i n i n g

a set of g e n e r a t o r s

moduli

that

=

represents

may be w r i t t e n

fj

~(0,0)

<~],~2;(] ,] )>

step by step

as

where

,

k [ [ x ] ..... x d]]

for

which

a purely

the

)

~(0,])

Thus

products

~]'~2

induces

I~I < N,

defining

and

systems

corresponds

for

all

therefore

Massey

§]

products

to a f a m i l y

<~*;n>

(7).

=

The map

w <x ; n >

236

(2)

{a m } m E ~ N _ ]

of d.

1

]-cochains

of

HomA(L.,L.)

, and

~ is a cocycle e. --I N__. Moreover, for every

(3)

[

~

such

that

representing m

for every

i > 0,

~i,o

x . , e = (0,.~_~_~,0,.. 1 --I i

O)

6 BN_ ]

0 ~

= 0

~iEBN _ ] Let

di(SN-]):

defined

Li~SN-]

+ Li-] ®k SN-]

be the

A ® SN_]-linear

map

by m

d i ( S N _ ] )I L i ® ] = ]

(3)

Then

~.

~BN-]

implies

universal

[

that

deformation

u-- •

~ ' ~ --

{L.o k SN_] ; di(SN_]) I of

L.

to

S 2 defined

is a lifting

of the

by the map

#]: H ÷ k[u] ..... Ud]/m2. Recall

that

$]

if one wishes,

corresponds to

S2

to the d e f o r m a t i o n

defined

by the

d [ X~® ~, 6 E x t ~ ( E , E ) ~ i=l 1 l By c o n s t r u c t i o n E~N_]

L.,

or of

E

element

m/m 2 = DefE(S2).

{L.~ k SN_ ] ; di(SN_] ) }

induces

the d e f o r m a t i o n

E DefE(SN_ I ).

Sticking

to the

a6B~

=

the

2-cycle

notations

{m£~dllml=~ }

Y(n)

of §] , and

the M a s s e y

=

~

(8)

and

(9)

Zn (B~ ~n,m Y(n)

translates

=

into

noticing

product

that

<x*;n>

for every

is r e p r e s e n t e d

by

o

ml+m2=n ml

§1

of

the

is a coboundary.

m2

following.

For

every

_m E B N

,

237

Now,

pick

for every

m £ B --

a

]-cochain

N

m

such

(HomA(L.,L.)

that (4)

d am = --

and consider

[

~n,m Y(~)

n6B' -N

-- --

the family

(5) {~m}m

Let,

for every

6 BN

i ) O, di(SN):

Li~ S N + Li_1 ~ S N

di(SN) I Li~ ] = Z~£BN~i,m~_ _ _u~m.

Then

be defined

(4) translates

by:

into

di(S N) 0di_ ] (S N) = 0. Consequently

{L.~ k SN; di(SN) }

{L.SkSN_] ; di(SN_]) } E%N~ ~ DefE(S N)

of

to

SN

is a lifting

, and induces

E~N_] . E N' again,

of

therefore

corresponds

a lifting

to a map

:

4~N

H + S N which we now fix. According

to (].2)

products

<x ;n>

induces,

a family

defining

system

By definition,

is a defining

for

~

see

(5), we shall

(].2),

refer

~

these

for the Massey is induced

to any such

Massey

by,

family

and

as a

J <x ;n>, ~ 6 BN+ ]•

products

see §] (]]),

this obstruction

system

6 B~+] . Since

for the Massey

of the obstruction, By (2.2)

~}I

products

are given

in terms

o(E~N , ~N+]). '

is defined

by the

2-cocycle

O =

{Oi}

where

0 i = d i ( ~ + 1)0di_1 (~+~), dl(RN+]): dl ( RN+ ]

Li® RN+ ] ÷ Li_]® RN+ ]

any

lifting

of

di(SN).

such that

di(~+,)J then

being

streight

forward

Li® ] =

calculation,

~_ = i , m ~(B N using

~

§]

(]0),

shows

that

Pick

238

Oi =

,7

(

Y,

Y

n6B'N+ ] ImJ ~N+] ml+n_12=m

r J=]

+

Remember

[ (

~] Iml

that

Comparing

~N+I

di(S N) 0di_ ] /S N) = 0. (].2)

(2.6).

and §] (]]), we have proved the following

Given a defining

Massey products by the

3 iY+m--2=m~3-~' j " "i,m~ ai-] 'm2 ) f~'~

m

this with

Proposition

8'm,n'~i,mOai_],m2 ) u n ]

system

<x ;n>, ~ £ ~ + ]

{a }mE~N+]

, <x ;n>

for the

is represented

2-cocycle ~[

Y(n) =

y

#'

a

lmJ <~+1 m1+m2=mm- m,n

o~

mI

m2

miEB n By §1 (16) we know that for every ~m = Y , ~n,m Y(n) -n6BN+ ] is a coboundary. and consider

~N+I

2-cochain

6 Hom~(L.,L.) such that

am ( H ~ m ~ ( L . , L . )

d

m = ~m '

the family {am }m£~N+ ] •

(6) Just

Pick one

the

m 6 BN+ ]

as '

above, for

(6)

the

is

Massey

There are relations procedure

seen

to

correspond

products

§1, (17),

<x

;n>,

(18),

to ~

a

defining

system,

i

E BN+ 2 .

(19), and we may copy the

above.

We end up with the following, Proposition

(2%7).

Given a defining

Massey products the

*

,

system

{am}m6~N+k_] W

<x ;n>, n 6 BN+ k , <x In>,

2-cocycle Y(n}

=

~

~

Iml
%,n ~ ° ~ -- -- ml m2

m i 6 BN+k_ ]

for the7

is represented

by

239

Moreover,

the p o l y n o m i a l s

k fN+k = [ ~ 3 i=o n(B' --

induces

§]

(22)

pick a c o c h a i n

and a

--

m

=

n

--

then the

as

Theorem

[

such

that

Hom~(L.,L.)

6

if we

for e v e r y

with

~n,m Y(n) -- ---

'

_£B

N+k

family

is a d i f i n i n g

§2

(23),

m

d a

We may,

j = ] ..... r

N+I

identities

m 6 BN+ k

yj<x*;n>u ~

refering

system

to §]

for the M a s s e y

(28),

(29),

(30),

products

<x ;n>, n

£ ~+k+]

sum up the c o n t e n t

of t h i s

follows

(2.8).

Given

is d e t e r m i n e d

an

A-module

E,

the

by the blassey p r o d u c t s

H = k[[x I .....

Xd]]l(f I .....

formal of

moduli

ExtA(E,E).

H

of

E

In fact

fr )

where . = ~ [ yj<x*;n> f3 1=2 £ ( B ~

Cqrollary field

(2.9). k

Any

complete

is d e t e r m i n e d

by

xS.

local

k-algebra

Ext~(k,k),

A

with

i = ],2

and

residue its

Massey-products.

Proof.

Obviously

A

is the

formal

moduli

of

k

as an

A-module. Q.E.D.

"

240

BIBLIOGRAPHY [Car]

Cartan H. Seminaire ]960-6]. Paris, ]962.

Institut Henri Poincar6,

[La] ]

Laudal,

O.A., Formal Moduli of Algebraic Structures, Lecture Notes in Mathematics No 754, Springer-Verlag ]979.

[La2 ]

Laudal,

O.A., Groups and Monoids and their Algebras. Preprint Series, Inst. of Math., University of Oslo, No 12 (]982).

[M]

Massey,

W.S., Some Higher Order Cohomology Operations, Symposium International de Topologia Algebraica, p.p. ]45-]54, La Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City ]958.

[May]

May, J.P., Matric Massey Products, (]969), p.p. 533-568.

Journal

of Algebra

]2

Palamodov, V.P., Cohomology of analytic algebras (russian), Trudi Moskovskogo Matematitseskogo ob~Qsi¢~ (Vol 44) ]982, pp. 3-6]. [S&S ]

Schlessinger, M. & Stasheff, J., Deformation theory and rational homotopy type & The Tangent Lie Algebra a Commutative Algebra, Manuscripts, ]982.

of

Department of Mathematics University of Oslo Blindern,OSLO 3 (NORWAY)

A METHOD FOR CONSTRUCTING BAD NOETHERIAN LOCAL RINGS

Christer Lech

We wish to introduce some new ideas for obtaining results involving socalled bad Noetherian local rings. Taken together, these ideas form an alternative to the well-known method by Rotthaus,

first applied in [7]-

We shall display them by indicating a proof of Theorem I below.

THEOREM I. A complete Noetherian local ring S is the completion of a Noetherian local domain if and only if the following conditions hold. (i)

The prime ring of S is a domain that acts on S without torsion;

(ii)

Unless equal to (0), the maximal ideal of S does not belong to (0) as an associated prime ideal.

Before going further, let us state, without proof, a more comprehensive theorem, which,

in essence, comprises not only Theorem I but also the

main results in the articles [3] by Brodmann and Rotthaus and [5] by Larfeldt and the author. The same basic principles can be used for the proof -- not, however,

in the same simple guise.

THEOREM 2. Let S be a complete Noetherian local ring and ~ a n

ideal of S.

Then, in order that ~ be generated by a prime ideal in some Noetherian local ring having S as completion,

it is necessary and sufficient that

the following conditions hold. (i)

For W = {x~ S; ~ : x = ~ } ,

the ideal ~S W of S W is generated by a prime

ideal in a flatly embedded subring of SW; (ii)

Unless equal to ~,

the maximal ideal of S does not belong to

as an associated prime ideal.

242

Theorem ] rounds off a line of research beginning with Akizuki [1], Sect. 3, and having Brodmann-Rotthaus [2], Prop. (15), as its most recent exponent. The author has been strongly impressed and spurred by the last-mentioned result. He has also had the benefit of stimulating discussions with Christel Rotthaus and Ralf FrSberg.

For the proof of Theorem ] we shall need certain notions concerning subrings R of a local ring (S,m). In reality these notions will merely concern the corresponding local subrings R R ~ ~ . We shall say that the injection R - - ~ S rs#O

is flat if S is flat as a R-module,

for r ~ R -{O1,

torsionfree if

s ~ S - {01, unramified i f ~ S ( R ~ ) ~

rational etc if the field extension ~ ( S ) / k ( R R ~ )

residually

is rational etc. Here

k( ) indicates the residue field of a local ring. Similarly A will indicate the completion of a local ring and -- the algebraic closure of a field. Following Well in [8], we shall say that a field extension K/k is re~lar

if K and k are linearly disjoint over k.

In the sequel, let S be the complete Noetherian local ring of the theorem and ~ its maximal ideal. Let further ~ be the minimal finite subset of Spec(S) whose union consists of all zerodivisors in S.

The theorem has the form of an equivalence between two statements about S. Let us introduce the following third statement as a connecting link:

There exists a ring injection R - - > S

which is

(I) flat, (2) unramified, (5) residually regular, (4) torsionfree.

This statement is implied in an obvious way by the statement that S is the completion of a Noetherian local domain. Moreover,

it implies,

243

through (4) and (2), the validity of the conditions (i) -(ii) theorem. When discussing the reverse implications,

of the

we may strengthen

condition (ii) into

(iii) ~

does not belong to (0) as an associated prime ideal,

as the possibility ~ = (0) causes no difficulties. theorem,

it suffices to prove the two implications

(i)-(iii) (*)

Thus, to prove the

-~

~>

(*);

'S is the completion

of a Noetherian local domain'.

The second implication is of minor interest from our present methodological point of view: k(S) is countable,

it can be treated by rather general means, and when

it can be got round entirely by strengthening (~) so

that rationality takes the place of regularity.

In the proof that follows

some of the details will be omitted.

The desired Noetherian local domain is obtained by a ring construction embodied in the concept of straightness defined below.

DEFINITIONS.

A morphism A ~

called telescopic

B of commutative rings with l-elements

is

if there exists a well-ordered generating set {xiliE I 1

for B over A such that, for each j ( I, A [ I x i l i ~ j } ]

is free as a module

over A [ { x i l i < j}]. A morphism A --~B of local rings is called straight if it is unramified and can be presented as a composition of a telescopic morphism and a subsequent localization.

Since the given injection R - - > $

can be assumed to be local and since

every straight local morphism is obviously flat, we get the desired result by combining the following two propositions.

244

PROPOSITION. Let k m ~ K

be a separabel field extension, and let R m ~ R 1 ,

R - - ~ R 2 be two unramified flat local morphisms, both inducing k ~ K residual field extension. Then there is an isomorphism

~R

as

2 respecting

the ~ - a l g e b r a structure and the given identity of the residue fields.

PROPOSITION. For any field extension K/k and any local ring R with residue field k there exists a straight local morphism R ~ R k ~K

I with

as residual morphism. For any such morphism R - - ~ R ~ the statements

indicated below hold true. R Noetherian

~

R I Noetherian;

K/k regular

~

~R I prime (V~ (Spec(R)).

The first proposition can be obtained as a consequence of [6], Thm. 82, which ensures the existence of a q - a l g e b r a h o m o m o r p h i s m ~ I --~R~ respecting K. It is not hard to see that any such homomcrphism is indeed an isomorphism. The second proposition is in substance contained in [4] apart from the very last assertion, which can be made evident by the device of extending local domains into valuation rings. (The regularity hypothesis might be unnecessarily strong, but separability, at least, is needed.)

Let us now turn to the more fundamental first implication. Each of the properties (I) -(4) expresses a condition on R. Assuming (i) and (iii) to hold, we shall show how to construct a subring of S that satisfies them all. The first three can be summarized as follows: All S-linear maps S n - - > S and S n - - ~

given by matrices over R and ~(RR~?~) resp.

must have kernels generated by elements in R n. Thus R must exhibit a sort of completeness in its relation to S, namely by providing

~ultimate'

solutions for all linear equations of certain types. Clearly S, as a subring of itself, meets these demands. However,

there is a simple way

245

for obtaining a countable

subring with the same property.

It consists in

forming the union of an increasing sequence of countable subrings so chosen that the linear equations arising from one ring have appropriate solutions in the next. The existence

of such sequences is clear from the

fact that each submodule of sn is finitely generated ( n = 1 , 2 , 3 , . . . ) . The described procedure for complying with the demands of (1) -(3)

forms

the basis of our construction.

Concerning the property (4), which means that R ~ $ begin by making a few observations. of S, by (i). Secondly,

First,

simple adjunctions

: (0) ($ E~),

let us

(4) holds for the prime ring of the form R - - > R [ s ]

will

preserve the validity of (4) (in the natural implicative sense) if the element s either represents a transcendental

over R in each S/~ ( ~ ( ~ )

satisfies a relation of the form a s = b with a,b (R,

a/O;

or

let us refer

to these two types of adjunctions as transcendental and fractional resp. Finally,

the property of a ring extension to preserve the validity of (4)

is ~transitive' with respect to arbitrary well-ordered

towers, not only

finite ones.

Put together,

these facts allow us to conclude that (4) holds for any

subring of S that can be obtained from the prime ring by a possibly transfinite

succession of transcendental and fractional adjunctions.

It suffices to show that our basic procedure can be performed within that framework. Thus it is enough to prove the following assertion: For any countable subring R of S enjoying property (4) and any linear equation over R or k ( R R ~ )

as considered above,

it is possible to

incorporate a S-linearly complete system of solutions by means of simple adjunctions of the two permitted types.

What resources in transcendentals assumptions

do we have? It follows from the

(notably (iii)) that S contains a complete discrete valua-

tion ring which maps injectively into each of the rings S/~ ( ~ ) .

246

Hence S contains indeterminates

an uncountable

in each S/@.

ically independent rings

Suppose

we can maintain

a countable

subset

of sn determined

of a generating

permitted

adjunctions.

arbitrary

(minimal)

Remembering vectors

of R,

by the equation

existence

say with a n ~ 0 . alx I + ... + a

generating

if

set.

Let E be the submodule

x = 0. We must n n

system in accordance

of transcend~ntals,

O,-al) , (O,an,O .....

show the

from R by

with Nakayama' s lemma.

and observing

-a2) , ....

(0 . . . . .

adjunctions.

that the n-1

O,an,-an_l)

system whose all n-1

the remaining n:th components

first comp-

In view of the

can then be adjoined

adjunctions.

for equations

of modification

are larger.

can be used as a modifying vector. exclusively

(uncountable)

system for E which can be reached

A similar argument applies possibilities

by omitting,

subring of S for which (4) holds and

onents can be adjoined by transcendental

by fractional

for given countable

the independence

belong to E, we can find a generating

defining equation,

but,

Such a system can be obtained by modifying an

the abundance

(an,O .....

sense,

of the original

then that R is a countable

that al,...,a n are elements

independent

It is true that these elements are algebra-

only in the absolute

of constants,

necessary,

set of elements representing

transcendental.

over k ( R R ~ ) .

But here the

In fact, any element

This leads to adjunctions

of ~

that are

n

247

REFERENCES

[I] Akizuki, Y., Einige Bemerkungen ~ber primate Integrit~tsbereiche mit Teilerkettensatz.

Proc. Phys.-Math. Soc. Japan 17, 327 -536 (1935).

[2] Brodmann , ~., Rotthaus , C., Local domains with bad sets of formal prime divisors. J. of Algebra 75, 386 -394 (1982). [3] Brodmann, M., Rotthaus, C., A peculiar unmixed domain. Proc. Amer. Math. Soc. 87, 5 9 6 - 6 0 0

(1983).

[4] Grothendieck, A., El@ments de g@om@trie alg@brique, Chap. O, ~ 10. 3 . Inst. Hautes Etudes Sci. Publ. Math. N°SS (1961). [5] Larfeldt, T., Lech, C., Analytic ramifications and flat couples of local rings. Acta Math. 146, 201 -208 (1981). [6] Mat sumura , H., Commutative algebra, 2nd ed. Benjamin/Cummings, Reading, Mass. 1980. [7] Rotthaus, C., Nicht ausgezeichnete, Math. Z. 152, 107-125

universell japanische Ringe.

(1977).

[8] Well, A., Foundations of algebraic geometry. Amer. Math. Soc., New York 1946.

Department of Mathematics University of Stockholm Box 6701 S-113 85 STOCKHOLM (SWEDEN)

YET ANOTHER PROOF OF A RESULT BY 0 G 0 ~

Crister Lech

As a further illustration

of the ideas in [2] we shall give a summary

proof of the following theorem of Ogoma ([3])-

THEOREM (0goma). There exists a non-catenary,

normal, Noetherian,

local

domain of dimension 3.

The proof of Ogoma has been simplified by Heitmann ([I]). Both Ogoma and Heitmann apply a method of Rotthaus (cf [2]). Though different respect,

in this

our proof will have certain basic features in common with theirs.

Proof. We shall use the following notation: k is a countable field,

S = k[[X,ZI,Z2,Z3]]/(ZIZ2,ZIZ3);

X,Zl,Z2,Z 3 are the uatural

images of X,ZI,Z2,Z 5 in S;

M_= (X,Zl,Z2,Z3) , P_= (Zl,Z2,Z3) , P1 = ( X ' Z l ) '

P2 = ( x ' z 2 ' z 3 )

(: Spec(S).

A local subring R of S will exhibit the truth of the assertion if it satisfies the conditions

(1)

k[x]

(2)

R-->S

(3)

-Pi = ( R f l ~ i ) s

(4)

R0Z=(O).

In f a c t ,

~

listed below.

R; is flat and unramified;

( i = ~,2)~

i t follows from ( 1 ) ,

domain having

S

as completion.

for i= 1,2, Pi = R ~ P i '

(2) and (4) that R is a Noetherian local In particular, R has dimension 3. Putting,

we have --i P =pi S by (3). We conclude that ht(P2) = I

as the primary decomposition xR = x S D R

= P I ~ P2 ~ R

=PI~P2

must be irredun-

dant, and further that coht(P2)= coht(P2)= I. Hence R is non-catenary.

249

(4),

The only singular prime ideals in S are M and ~. In view of therefore

true for every ~

Spec(R) - I R ~ M )

it is

that ~S has a regular minimal

prime ideal. Thus every prime ideal in R is regular,

except the maximal

one. As depth(R)= depth(S)= 2, R is normal by the Serre criterion.

To show the existence

of a ring R satisfying

the conditions,

make slight changes in the proof of the implication in [2]. The construction

we need only

~(i) -(iii)

~

should start with k[x] rather than the prime

ring of S; the r~le of ~ should be taken over by {~}; generators and ~2 should be incorporated by transcendental multiple

adjunctions,

as each generator can be modified by a our proof.

The method of Rotthaus has the advantage R/~ has a very simple structure rings are indeed essentially the examples

To attain this property, that ~ = ( R ~ ) S of non-zero

of providing rings R such that

for all ~

Spec(R) -{(0)}.

finitely generated

These factor

over a field. As a

of Ogoma and Heitmann are pseudo-geometric. we could sharpen condition

for those ~

Spec(S)

ideals in S generated

that this stronger condition ~S prime for ~ S p e o ( R )

of ~I

in R. (The latter task can be performed

of x2.) This finishes

consequence,

(~)'

(3) by demanding

that appear as minimal prime ideals

by elements

in R. It is not hard to see

could also be satisfied.

Then we would have

- {(0)}. If k is chosen to be of Qharacteristic

O,

this gives the result.

REFERENCES

[1] Heitmann,

R., A non-catenary,

normal,

local domain. Rocky Mountain J.

of Math. 12, 145 -148 (1982). [2] Lech, C., A method for constructing

bad Noetherian

local rings. These

Proceedings. [3] Ogoma, T., Non-catenary Math.

6, 147 -163

pseudo-geometric

normal rings. Japan. J. of

(1980).

Department of Mathematics, University of Stockholm, Box 6701, S-113 85 STOCKHOLM (SWEDEN)

MODELE MINIMAL RELATIF DES FEUILLETAGES

pag~ Da~

LEHI{ANN

I. I n t ~ o d u ~ o n . Soit

V

d'un feuilletage

une vari~t~ connexe paracompacte F

(codimension

Ib : ~b(F) l'inclusion naturelle de la sous-alggbre pour

F

(c'est-~-dire

vecteurs

X

qui vgrifient

tangent aux feuilles de

Un point de base et

~DR(V)

x°

Cm

(dimension

n)

munie

diff~rentielle

des formes

~

basiques

q).

On notera

~

~DR(V)

ix~O = 0 F)

et

LXW = 0

une fois choisi dans

d'une augmentation,

Ib

pour tout champ de

dans l'alg~bre de de Rham. V

permettant

admet un "modUle minimal

de munir

~(F)

relatif"

(~b(F) 0 *,],D,¢) rendant commutatif les graduges

le diagrarmne

suivant de morphismes

de

~R-alg~bres diff~rentiel-

:

{2DR(F)

+

~F

(-~,d)

JF

gDR(V)

+

%

~b(F)

(~b(F)@~ 'D)

251

o~

(~,d)

d~signe une

~-alg~bre minimale au sens de Sullivan

~5] ,

~(F)

@

d~signe le produit tensoriel gradu~ en tant qu'alggbre gradu~e, mais avec une diffgrentielle l'id~al de de base

D

"tordue"

~b(F) @ ~

Xo,

~

(~ u c ~ ) ,

D(| @ u) - I @ d ~

engendr~ par l'id~al maximal de

est un morphisme de

~b(F)-alg~bres

induisant un isomorphisme en cohomologie, relle dans que

V

de la feuille

(~b(F) @ ~ , D )

finie ~

V

~b(F)

(~b(F) 0 ~,D)

relatif au point graduges

d@signe l'inclusion natu-

Rappelons

~b(F)-isomorphisme

pros une fois

D'autre part,

x . o

appartient

diff~rentielles

JF : F + V

contenant

est bien d~finie ~

~b(F)-homotopie

morphisme,

F

u

(ef. S. Halperin pros,

et

~

[5])

bien d~-

fix~ dans sa classe d'iso-

~tant connexe, on a l e

Lemme I. (i)

@~(F)

(ii) (iii) (iv)

Ib

est cohomologiquement

connexe,

est injectif en cohomologie de dimension n'a pas de g~ngrateur en degr~

on peut choisir

(~b(F) 0 ~ , D )

degrg (v)

~(F)

@ ~

O,

dans sa classe de

de fa~on que, pour tout gl~ment tienne ~ l'id~al

I,

u

de

~,

~b(F)-isomorphisme

D(lOu) - 1 @ d ~ u

appar-

engendr~ par les formes basiques de

> O,

la classe de

~b(F)-isomorphie

point de base

de

(~b(F) @ ~ , D )

ne d~pend pas du

x . o

Ce lemme sera dgmontr~ au § suivant.

Dans le cas o3 le feuilletage

F

triviale de base une vari~t~

W

f~ : ~DR(W) ÷ ~DR(V),

au moins si

H~(F,~)

et si

et

H~(F,~)

ou

est eompacte par exemple) mologie, de sorte que

-

H~(W,~) -

(~,d)

est une fibration

de dimension

~F :

q,

~I(W)

f : V + W

l'inclusion

Ib

localement

devient

op~re de fa~on nilpotente sur

est de dimension finie en chaque degr~ (si ~

÷ ~DR (F)

induit un isomorphisme en coho-

est le modgle minimal rgel de la fibre

dule de la fibre est ~gal ~ la fibre du module" Dans le cas ggn~ral d'un feuilletage,

(cf. S. Halperin ~F

V

F : "le mo-

~5]).

n'a en g~n~ral plus aucune rai-

son d'induire un isomorphisme en cohomologie, ne serait ce que.parce que les feuilles F

n'ont plus ngcessairement

routes le m~me type d'homotopie.

On se propose de d~montrer le :

252

Th~or~me : Supposons : (i)

V

est compacte, orientable, de dimension

la cohomologie

(ii)

H~(F)_ de l'alg~bre

Poincarg pour la dimension chaque espace H~(~)

(iii)

H~(F)

q

n,

~b(F)

v@rifie la dualit~ de

@gale g la codimension de

F,

et

est de dimension finie,

est de dimension cohomologique finie

(Hi(~) = 0

pour

i

I

suffisamment grand). Alors : H~(~)

(i)

vgrifie la dualit~ de Poincar~ pour la dimension

des feuilles de

p = n-q

F,

Ii existe une application naturelle injective

(ii)

¢I o8

H~(V,~)

faisceau

: H~(7)

÷

o H (V,~b)

d6signe la cohomologie de ~<

V

~ coefficients dans le

des germes de fonctions basiques.

2. Rappels su~. la s ~ i t e spectrale d'un ~euilletage (Vaisman [ 1 ~ ) . Soit

Q

un suppl~mentaire dans

des vecteurs tangents aux feuilles de en d~finissant ArQ ~ @ AS(rF) ~,

r,s s ~DR (V) = ~r,

on munit

~DR(V)

T(F)

d'une bigraduation ~ C

du fibr~

et de la filtration par des id~aux diff~rentiels :

La suite spectrale

~r,s

F,

du sous-fibr~ vectoriel

comme l'espace des sections

Fr~DR(V) =

(O ~ r ~ q,

T(V)

0 g s ~ p),

@ ~i,~ . i~r

(E~'S,$ i)

converge vers

de la diffgrentielle ext~rieure ~r,s

d d2~

od ~

associ~e a son support dans le rectangle H~(V,~), dDR

et

E~,s = ~r,s.

La restriction

admet alors 3 composantes :

~r,s+I ~r+l,s ~r+2,s-I

253

et la relation

implique en particulier

(dDR) 2 = 0

I

(do)2

:

= O,

dld ° + dod I

= 0 ,

(dl)2 + dod 2 + d2d ° La diff~rentielle

~

de

E

o

coincide avec o

=

O,

d ,

de sorte que

o

r,s = Hs.~r,~ El ( ,do)r ~b

Notant

le faisceau des germes de

faisceau des germes de section

0

÷

f~ ----+

C~ d o

~r,o

ArQ ~ @ As(T(F)) ~,

f~r,l ~br

constitue une r~solution fine de est paracompacte,

de

r-formes basiques et

d

o

(les faisceaux

~~r,s

sont fins d~s que

et le lermme de Poincar~ dans chaque feuille de

de Rham g~nfiralis~ permet donc d'affirmer que

ElO,S De la relation diff~rentielle

~1 ]

~i ]

(dl)

sum

2

d 1 sur

Er,o 1

: le th~or~me de

est encore figal ~

HS(v,2 ).

3. D ~ m o ~ t ~ o n

~(F)

o = HS(V,~b)

E~ ~ = H(~

= ~(F)

E •2 ,o = HE(F )

=

+ dod 2 + d2d ° = O,

de la suite spectrale, d'o~ =

Erl~ s

V

F, param@tr~

on a :

I f Er,o

21

le

la suite

par l'espace des feuilles, implique l'exactitude de la suite)

En particulier,

~r,s

,do) ,

on d~duit clue d I

induit une

qui coincide avec la diff~rentielle

E~ '~ = H ~ ( H ( ~ , d o ) , ~ l ] ) .

En particulier,

de sorte que

est la cohomologie des formes basiques°

du lemme I.

La suite exactes des termes de bas degr~ dans la suite spectrale pr~c~dente fournit -

un isomorphisme

:

H~(F) ~ H°(V,R),

254

- une suite exacte :

0 -+ H (F)

Ib

* HI(V,]R) ÷ E2o '1

d'oO la partie (i) du lemme, puisque tie (iii) est un corollaire de Notons

~(F)

la

V

÷

H2b (F)

H2(V,IR)

~+

est connexe et la partie (ii). La par-

[(i) + (ii~ o

~-sous-alggbre diff~rentielle gradu~ de

~(F)

d~finie par : ~b(F) = m ~|b(F) = suppl~mentaire de ~b(F) = f~i b(F)

Notons

pour

B I (f~b(F))

relatif de

Ib - a,

partient ~

~(F)

~b(F~

soit

0 ~.

f~(F) ,

i >. 2.

a : ~b(F) + ~ ( F )

en cohomologie puisque

dans

l'inclusion naturelle, qui induit un isomorphisme

est cohomologiquement connexe. Dans le module minimal

~b(F) @ ~ , Puisque

a

il est clair

que

D(l@u) - I @ d ~ u

ap-

:induit un isomorphisme en cohomologie, on en

d~duit la construction d'un module minimal relatif de

Ib

en faisant la somme

amalgam~e

a

f~b(F)

' ~b (F)

d'oN la partie (iv) du lemme. Puisque suppl~mentaire

Ib est injectif en cohomologie de dimension I, il existe un ^l ~R(V) de BI~DR(V) dans ~DIR(V) tel que I b ( ~ ( F ) ) C ~ R ( V ) .

Si l'on d~finit donc ~DR(V) = IR i' inclusion naturelle et

Ib

et

^i i ~DR(V) = ~DR(V)

a' : ~DR(V)

se factorise ~ travers

Le module minimal de

~+

~b :

pour

~DR(V)

i >. 2,

induit un isomorphisme en cohomologie,

{%(F) ÷ ~DR(V)

~b

.... ~D~(V)

°7-

\

~b(F)

de faqon que

a'.[ b = Ib.a.

255 ne faisant intervenir aucun point de base dans du module minimal

%(F)

@ ~

~

ne d~pendant pas du point de base

÷

~DR(V)

x

V, de

on en d~duit une construction 1b

tel que

a'.P = ~.(a 0 ]~),

d'o~ la partie (v). O ~

4. D ~ m o ~ t g o n

du ~or~me.

Gottlieb a d~montr~ (E3J)

que si, dens un espace fibrg, la base et

l'espace total v~rifient tous deux la dualit~ de Poincar~ en cohomologie avec dimensions respectives

q

et

n = p+q,

et si la fibre a une dimension cohomolo-

gique finie, alors cette fibre v~rifie aussi la dualitg de Poincar~ en cohomologie pour la dimension

p.

La conclusion (i) du th~or~me est une transcription alg~bri-

que de ce r~sultat, qui g~n~ralise un th~or~me de F~lix-Halperin cas oO la pseudo-homotopie (Thomas (~b(F)

~9])

de la "base" est de dimension

[2] (relatif au

I) : elle se d~montre

par rgcurrence sur la dimension de la pseudo-homotopie

en l'occurrenee),

d'o~ la conclusion

de la "base"

(i) du th~or~me.

La filtration

o ?) d~finit une suite spectrale 0 ~ s ~ p)

~r,s

convergeant vers

El-r,s = ~(F) 0

Notons

H~(V,~)

cohomologie

~u]

et

# 0 e HP(~)

De la commutativit~

o7

tandis que

r,s

~i

HS(~),

u 6 zP(~)

~r

de support inclus dens le rectangle

suites spectrales multiplicatives On a alors :

= %

:

(et

v e ~(F) et

[v~

~r,s ~

~

induit un homomorphisme

+ Er,s i

E2-r,s = H~(F) O H S ( 7 )

si

H~(F) = 0)

des ~l~ments induisant des classes de # 0 e H~(F).

~ , o 0 ~l~°'P

.....

HP(v,a~(F))

~'P Hn(v,~)

de

"

du diagramme

f~(F) e HP(~])

(0 ~ r ~ q,

,

Hn(v,~)

256

et du fait que

Ev]. Eu] # 0

dans

E~'P,

~°'P(10

on d~duit

[u~) # 0 e HP(v,~)

I

Puisque

H~(~)

et puisque llapplication

v~rifie la dualit~ de Poincar~ avec H~(~)

H~(V,

÷

)

induite par

[u] # 0 e H P ( ~ )

,0~ ~l

est un homomor-

phisme d'alg~bres, on en dgduit la conclusion (ii) du th~or~me.

]° )

Supposons v~rifi~es simultan~ment les 2 conditions suivantes (c'est,

par exemple, le cas si si

F

F est une fibration de Seifert ggn~ralis@e,c'est-g-dire

a toutes ses feuilles compactes et est localement stable E6]) : (o)

V

est compacte orientable,

(i)

F

est minimalisable,

(ii)

dim ~o,p = 1 -2

D'apr~s Kamber et Tondeur E6], de Poincarg ,

Soit

X

une

~(F)

p-forme sur

V

volume associg ~ une m~trique riemannienne sur doX = O,

X

d~finit

EX]o ~ E? 'p,

[xJ! = O e E o,p 2 •

[X]l e E~ 'p, portionnel ~

~X]I.

L'inclusion

homomorphisme d'alg~bres .~ compacte, puisque iF(X)

"~

D'apr~s

v~rifie automatiquement la dualitg induisant sur chaque feuille V

E~,

rendant

X

d~finit donc

~ ' P ( | ~ [u~)

est pro-

induit, pour toute feuille

~

3F : H~(V'~ ) ÷ H~(F'~)

et

~-

jF(I_X]o

le

minimale. Puisque

diX = O ,

Par consequent, JF : F ~ V

F

F

F

un

) ~ 0

si F est ~o,p JF o ~1 = ~F '

est une forme volume. De la relation

on

d~duit donc la

Pro~os~on 1.Si

V

~F : H ~ ( ~ )

est compacte orientable, si ~

H~(F'~)

F

est minimalisable et si

est injectif pour toute feuille

F

dim E~ 'p = 1,

compacte.

2 °) De faGon plus triviale encore, mais souvent v~rifi~e en pratique, on a la

Propos~ion 2.i

Si

~

v~rifie la dualit~ de Poincar~ en dimension

p,

s'il est possible

257

de choisir

u 6 zP(~)

de telle fa~on que compacte

F

dans

[u] # O e H P ( ~ )

~(] @ u)

et

~ : ~b(F) 8 4 ~

induise une forme volume

~F(U)

÷ ~DR(V)

pour une feuille

particuli~re, alors : ~F : H ~ ( ~ )

÷

H (F,~)

est injectif pour cette feuille compacte.

3 °) Lorsque les hypotheses du th~orgme sont v~rifi~es, l'application

n'est pas n~cessairement surjective, conm~e le prouve l'exemple des droites de pente irrationnelle

~

sur le tore, chaque fois que

II est ais~ de voir que, pour tout nombre est le module minimal de

S !, Si

que les constantes tandis que la forme ferm~e dx-~ dy

~

~(F)

~

~st un nombre de Liouville :

rationnel ou non, ~ =

est irrationnel,

~(F)

F. Si

H~(r,~)

~

(Al(x),dx=O)

ne contlent

ne contient que les formes

d~finissant

E1Kacimi a alors d~montr~ que

~,

~-proportionnelles

est un nombre de Liouville,

a une dimension infinie ~I].

4 =) On pourrait chercher ~ interpreter

~

comme le module de la feuille

g~n~rique (rev~tement eo~mlun ~ toutes les feuilles) lorsque celle-ei existe (cf. Haefliger [4~) : il n'en est rien, puisque pour les droites de pente irrationnelle sur ie tore, la feuille ggn~rique est

~, alors que

~

est le module de

S I.

5 = ) Les r~sultats exposes sont ~galement valables si, au lieu de prendre toutes les formes basiques du feuilletage, on considgre seulement celles appartenant ~ une certaine sous-alg~bre diff~rentielle gradu~e de associ~e g certains types de Le module ~

~(F),

g~n~ralement

F-structures transverses.

d~pend alors de la structure en question. De plus, ~ l'~tude

du module minimal relatif, se greffent des questions d'irrationnalit~ de morphismes entre

~-alg~bres diff~rentielles gradu~es admettant des

Q-structures donn~es,

qui mesurent, en quelque sorte, en quoi ces feuilletages different de fibrations d'oO une th~orie de "l'homotopie irrationnelle" (cf. ~7]).

258

REFERENCES

[|]

EL KACIMI-ALAOUI A.

Cohomologie f e u i l l e t ~ e - Ex~nples de c a l c u l s ,

-

Th~se de 3~me cycle, Universit~ de Lille I, 26 juin ]980. E2]

[3]

FELIX Y. - HALPERIN S. -

GOTTLIEB D.

-

L.S.-category, (Transactions of American Mathematical Society,

]983).

Poinca~£ d u a l i t y and f i b r a t i o ~ , (Proceedings of the American Mathematical Society 76.1.79).

~4~

HAEFLIGER A.

-

Groupoldes d'holonomie ~t c l a ~ s i f i a ~ , (A para~tre dans les comptes rendus des journ~es de Toulouse 1982 sur la g~omgtrie transverse. Ast~risque).

E5]

HALPERIN S.

-

E6]

KAMBER F. et TONDEUR P. - Foliations and megric~, (Differential Geometry ) Birkhauser - |983).

E~

LEHMANN D.

-

L e c t u r ~ on minimal mode£s, Pub. I.R.M.A. Lille I, Vol. 3, fasc. 4, 1977.

S t r u ~ w t ~ de M a u r ~ - C a ~ n ~ Fo-structur~, I - f e u i l l e t a g e s de Ma~er-Cartan ( P r e p r i n t ) . II - espaces classifiangs,

Ast@risque,

E8]

RI~@fLER D. et SULLIVAN D. - Currewgs, flows and diffeomorphisms, (Topology 14 - 1975).

E9~

THOMAS

~0~

J.C.

-

Communication priv~e.

VAISMAN I .

-

Vari~t~S riemanniennes f e u i l l ~ e s ,

116, 1984, 134-148.

(Czechosl. Math. Jal 21 - 197|).

E.R.A au C.N.R.S. 07 590 UNIVERSITE DES SCIENCES ET TECHNIQUES DE LILLE U.E.R. DE MATHEMATIQUES PURES ET APPLIQUEES 59655 - VILLENEUVE D'ASCQ CEDEX (France)

LUSTERNIK - SCHNIRELMANN

CATEGORY

Jean-Michel

The introduction important Thomas,

results,

of L.- S. category

LEMAIRE

F@lix's and Halperin's

fundamental

but some seasoning

and also Jean-Claude

Lie algebra of a finite complex.

to the leading theme of this conference, The following

INTRODUCTION

(Nice]

due to Yves FElix and Steven Halperin,

local algebra as well.

material,

AN

into rational homotopy theory has lead to

on the structure of the rational homotopy

According

:

such results

are of interest in

notes are meant to serve as an introduction

paper [FH] : they therefore contain

of geometry which m a y make reading

They also provide an opportunity

to thank Jan-Erik Roos for his kind invitation

1983, which was a most pleasant and profitable

Definition A ~ X

fines

oat

X

is categorical

Definition

be a topological

rical covering

exists,

1.3 Examples

:

oat

space with base point

X

admits a finite covering

of

X

one sets

X = O

cat S n ~ I

* E X . A subset X , homotopio

h : X

h(A) ~ *

X , the L. - S. category of

there exists a covering

one.

if there exists a continuous map

such that

1.2. If

Symposium

AKO ELEMENTARY PROPERTIES.

1.1. Let

to the identity,

no original

[FH] easier.

to give this set of lectures in the Nordic Summer School and Research

§ I. - BASIC D E F I N I T I O N S

to

by

by categorical subsets,

X , to be the least number

n + I

categorical

subsets.

cat X ~

iff X for all

is contractible. n ~ I

.

n

one de-

such that

If no finite catego-

260

This definition originates

in the work (1934) of the two eponymous authors,

proved that any smooth function on a compact manifold of category least

n + I

admits at

critical points. Actually their definition required the categorical

sets to be closed, topic,

n

who

but only contractible

in

but the homotopy need not extend to

X

(i.e.

A ~

-X

must be null-homo-

X ). Later Fox (1941) modified the de-

finition by requiring the sets to be open, and proved that category so defined is a homotopy invariant,

while it is not if one insists that the sets be contractible

in

themselves.

The definition

adopted here is due to George Whitehead

to be equivalent to Fox's for

One can reformulate space of the

(1954) and can be shown

cw-oomplexes.

this definition

as follows

T nI X ~ X n+ I

: let

be the sub-

(n + 1)-fold product which consists of those sequences

(Xo, xl, ..., Xn)

such that

x.i = *

for some

i . Clearly the following definition

is equivalent to (1.2).

Definition

(1.4] . The category

fold diagonal

A : X

• X n+1

eat X

of a space

factors through

X

A

OY

of basE~-point

be the loop space on

preserving maps

Theorem

[1.5) (Ew]

I°f class

s n .

[SX,

). if

iff the

(n + 1)-

up to homotopy

: let

SX

be the suspension

Y . Recall that the set of homotopy classes

Y] ~

oat

s n

Tn "~I X

G. Whitehead made the following crucial observation X , and

is

X n+l

~

of

X

n TI X

Ix,

OY]

is a group. Then

x s n , the group

~8X, Y]

This result shows that category is some kind of "homotopical

: is nilpotent

nilpotency".

shall see that the rational category of a l-connected space actually is the

We

261

"homotopical

nilpotency"

some elementary

of its Sullivan m i n i m a l model.

topological

the homotopy type of

Before we briefly collect

facts. From now on we assume that spaces in sight have

cw-complexes

of finite type, with a base point. Details can

be found in [W],

J Lemma

1.B : If

ticular,

in

X

cat

is a homotopy

Lemma

I.? : Let

then

uo U u I U

Proof

: Since

H*(X n+1,

pairs).

is a homotopy retract

R

be a ring,

. Let

cat

Now

f

exists

; R)

. If

cat X s n ,

.

u° x u I x

, the cross-product ~

~

(X n+1,

T~)

o" . x u n

be the inclusion

lies

(of

Proof

h

x u I x ... with

and

jh ~ A , where

j * k* = 0

x Un) : Tn1 X ~

j

-X

is

the

in the long exact sequence of the

.

Let

f : X

be a map,

-Y

and

Cf - Y U f

C×

be the m a p p i n g

cone

. One has cat

(Of)

~cat

X +

: By (1.6) one may assume that

Cf = Y U CX

and both

Y

and

Then one easily sees that if A o, A I, ... , A n , CX

J ~oplication

1.9 :

Proof : CP(n) fore

m

Uu n

A* k* = h * j * k *

T~]

Lemma 1 . 8 .

then

cat X ~ cat Y . In par-

uo, u l, o.., u n 6 ~ ( X

k : (xn+1

IU

X s n , there

(X n + l ,

of

Y , then

Then

inclusion.

pair

and

u i E H (X, * ; R)

uoUu But i f

invarianto

..o U u n = 0

T nI X ; R]

of

cat ~ ( n )

~ n

CX

I

f

.

is a closed cofibration.

have the h o m o t o p y

Ao' ~I'

... , A u

is a categorical

Then

extension property

is a categorical

covering

of

YU

CX

in

Cf

covering of .

. Y ,

•

cat OP(n] I n .

is the mapping

cone of the Hopf map

by (I .8) and induction

the Chern class of the canonical

on

line bundle,

S 2n-I

n o Now if one has

.OP(n-

c E H2(O~(n]

cn ~ 0

and

I)

, there-

; Z)

cat ~(n)

is >- n

262

by ( I , ? )

.

"

One may p r o v e is

along

n . Incidentally,

cat X -< n

if

X

cat X < n + I

t h e same l i n e s

that

the category

of a product

of

n

spheres

from (1.5) on we have seen several good reasons to set

can be covered by

n + I

categorical subsets,

instead of

which was the original convention.

Lemma (1.10). Let homotopic. Then

F

i

cat

E

P~ B

be a fibration sequence such that

B . Let

H : B x I

, 8

p-1(A]

with

~o = idE

that

F has the h .e.p. in

that

k~id

E and

pP = H(p x I )

k(F)

Observe that if

i

. Hence

is categorical in

be a homotopy with

HI(A ) = * . By the homotopy lifting property of

and

is null-

E s cat B .

Proof : Plainly it suffices to prove that is categorical in

i

K,~I(p-I(A))

is null-homotopic,

= *

E : let

and

by exactness of the homotopy exact sequence.

A

and - E

one may assume

k :E

~ E be

k.~ I ~ id E .

• ~.(B)

TT.(p) : ~.(E)

if

~ : E x I

= F . W.n.l.g.

E , and thus is categorical in

= * E E . Then

H ° = id B

p , there exists

~I(p-I(A))

E

such

E

is injective

The converse is not true in general,

but it is true for rational spaces : this is [FH]'s first theorem, the mapping theorem, which we now discuss.

§ 2. - THE MAPPING THEOREM

We begin with recollecting some rational homotopy theory. Let

is

S

be a l-connected space. The ~urewicz homomorphism

a Lie algebra

map, w h e r e t h e

Lie

structure

is

given I

bracket, and on

H.(~S)

by

Milnor and Moore asserts that

(2.1)

h.

: "rr.(~S)

®Q

[a, b] = ab - (h

on

~.(OS)

Samelson

. A fundamental result of

induces an isomorphism of

~ ~ PH.(C~S ; Q)

by t h e

I

I ) lal Iblba

Lie algebras

263

where

P

stands for the (Lie subalgebra

We can choose a retraction

r i : H i(08

of

h.l

; G]

Hi(~s

r ." ~

i :

; ~i(f~)

--K(~i(~ ] ®Q,

: ~S

The latter induces an isomorphism

(2.2)

, for each

elements.

~ ' z (OS) ®

which can be viewed as a c l a s s i n

ri

of) primitive

on

~ ~

~. ® Q

, o r as a map

i) . Thus the product map

m

K(L(~

®Q)

® O, i)

i=I where t h e p r o d u c t i s g i v e n t h e weak t o p o l o g y , homotopy groups and t h e r e f o r e

is a rational

l - c o n n e c t e d space has t h e r a t i o n a l

i n d u c e s an isomorphism on r a t i o n a l

e q u i v a l e n c e . Thus any l o o p space on a

homotopy t y p e o f a p r o d u c t o f E i l e n b e r g - M a c Lane

spaces. We can now prove the mapping Yves F@lix during

Theorem

Proof Q-vector of

p

TT.(p)

p : E

*

-- B

, hence

is a surjective

Then

-~ F

This w i ! i

cat(E)

s cat(8)

follow from the existence

i -- * . To construct

sequence

for

s , we observe

groups

fibre

~.(j)

i ~

so that

#.(j)

maps

U.

± : F

±js

: ~.(~8)

s

~ E of and

. ft.(F]

v e c t o r spaces, by t h e e x a c t n e s s o f t h e homotopy s e -

quence. W r i t e

%(m)

spaces,

are

of a section

p : indeed

that

rational

.

to prove that the homotopy

in the fibration

map o f r a t i o n a l

see [FL].

space is a space whose homotopy

By Io10, it suffices

j : ~8

simple proof was found by

be a map between simply connected

is injective.

is null-homotopic.

: the following

; for a generalization,

: Recall that a rational spaces.

the fibre ij -

the conference

2.3. Let

such that

theorem

:, U. e Ker ~ . ( j ) isomorphically

on to

~.(F)

. Then,

by 2.2,

264

~8-- 0 i~I and

the restriction of

j

is the required section

K(Ui, i) x 0

K(Ker ~ i ( j ) ,

i)

i~I to

0 K(U i, i) i~I •

s

is a homotopy equivalence, whose inverse

We shall see in the next section that L. - S. category "localizes" well, that is, if

X

is the localisation of the homotopy type of

0

cat(Xo] s cat(X)

. Setting

X

at all primes, then

Cato(X ) , the rational category of

X , to be

cat

..(Xo] ,

we can reformulate the mapping theorem as follows :

I (1.11') : Let

p : E

• B

be a map such that

spaces are l-connected. Then

Cato(E ) s Cato(B)

is injective and the

~.(p) ® Q .

We conclude this section with another result of [FH], which we derive from the mapping theorem. The following concept, due to D. Gottlieb, was brought to the attention of rational homotopy theorists by H. Baues : Definition (2.4). The Gottlieb group as follows :

~ : Si

- X

Gi(X ]

represents an element in (id, ~) : X v Si

extends to

Gi(X )

GL(X)

~i(X)

defined

i f the map

- X

X x Si .

We leave to the reader to check that indeed

is the subgroup of

is the image of

at the base point and

~ (X) 0

~i(ev]

Gi(X )

, where

actually is a subgroup of ev : So(X )

is the group of self-maps of

- X X

~i(X] :

is the evaluation

homotopic to the

identity.

Theorem (2.5) ([FH] Thin I I I ) . n

.

Let

X

be a l-connected space of finite category

Then

Ca)

V i ,

G2i(X] ® Q - 0

co

(b)

E

dim 0 G2i+1(X) ® Q < n .

i-1 Tn other words, the groups

Gi(X )

are torsion except at most

n

of them which

265 occur in odd dimensions

Proof of 2.5.

: moreover the sum of their ranks is at most

Let us prove (b) first. Let

ft. : S

2r.+I !

n .

~ X , i = I ..... s

1

represent linearly independant elements in

easy

Godd(X ) ® Q . An

induction on

s

shows that

" (~1 .....

~

: (

" ~ s2ri +I

)v

s 2r.+1 V S 1 i=1

:

I-IS s2ri+ I I ] i¢=~I

extends to the product

to

~s ]

=X

Indeed, assume that

g

(~I'

....

~J)

can be extended

J

s2rj +I

X . Thena further extension to

~I~ S 2ri+1

is

3 given by the diagram S 2ri+1 ) v S 2rj+1

~j v id

XvS

2r.+I 3

(id, ~;) d_ X g /

I s2ri+l

/

F

~.xid 3

/

"

X x S 2rj+Iz"

Localizing we obtain a map of rational spaces s

s2ri+1 0

0

'~ich is injective on homotopy groups. By 1.9, the category of a product of (rational] spheres is

s , therefore

s ~ cat(Xo) = Cato(X ) ~ n

s

by the mapping

theorem. A proof of Ca] along the same lines is a little more involved : we need dames's reduced product construction. Let duced product (Z)~

i=o

zi/

be a connected,

pointed cw-complex

; the re-

is the quotient space

((~I'

"

Concatenation gives *~

Z

"'~j-1'

(Z)~

*' ~

'

3+I .... zi] ~ (z I ..... zj_1, zj ..... zi] )

the structure of a topological monoid with unit

( ) . dames's theorem asserts that the canonical map

multiplicative homotopy equivalence

(Z)~

Z

~ - ~ SZ . Now if

- 0 SZ ~ : mr

extends to a ,X

266

represents an element in : ( m r )~

Gr(X ) , one easily sees that

extends to

~ X , Localizing we obtain a map

Sr

sr+l

~

o

If

~

O

r - 2s , ~ S 2s+i ~ K(Q, 2s)

0

and

~

is injective on homotopy groups iff o

r e p r e s e n t s a non-zero element i n

G2s(X ] ® ~ = G2s[Xo)

is a polynomial algebra on one generator of degree cat[K(O,

2s]) = ~

G2s(X ) ® 0

2s

. But

H (K(O, 2s)

;

and therefore

by lemma 1.7. Thus the existence of a non-zero element in

would contradict

the mapping theorem.

Theorem 2.5 is a key ingredient

I

in the proof of [FH] Thm. IV, which says that

the rational homotopy groups of a finite complex are either zero for large enough degrees or grow exponentially.

§ 3, - SPACES OF CATEGQR,Y,,,I AND THE HOMOTOPY SUSPENSION DIAGRAM By definition

1.4 , a space has category

~ I

if

the diagonal can be factored

*

on

IX, Y]

through the wedge up to homotopy A

X

~XxX

h''-

"~Xv The map

h

defines a natural composition

X

h ,, X v X

X

law

(f'g]-

by

f * g ~ (f,g) o h

Y

which admits the trivial map as unit.

One says that theorem

h

is a co-H.spaoe structure on

(I.5) says that

this is why

~ (X)

[SX, Y]

is abelian for

n

Suspensions structure map

is abelian if

X

X . Observe that Whitehead's is a co-H.space

: incidentally

n ~ 2 "

are canonical

examples of co-H.spaces,

; but there are examples of co-H.spaces

with the "pinching" map as which do not have the homotopy

267

type of suspensions

: the simplest

example i s

one may show~ u s i n g t h e Hopf i n v a r i a n t , ture

that

(which extends the standard one on

only if

~ ~ 0(6)

S 3 U ~ e ? , where this

S 3)

~ E ~ 6 [38) "

= Z/12Z :

space a d m i t s a c o - H . s p a c e s t r u c -

iff

~ ~ 0(2)

. This is a torsion phenomenon of course,

, but is a suspension and over the rationals

things are much simpler :

Theqrem

(3.1)

(I. 8ernstein).

Every simply connected space of category

I has

the rational homotopy type of a wedge of spheres.

Tnis result will follow as an easy exercice from the characterization

of rational

category on the Sullivan model that we will discuss in the last section. We now give a proof which avoids models because it leads to interesting

side comments. We need

the

B o t t - S a m e l s o n theorem ( 3 . 2 ) .

The P o n t r y a g i n a l g e b r a

H.(O SX ; k)

, k

a field,

i s i s o m o r p h i c t o t h e t e n s o r a l g e b r a g e n e r a t e d by t h e graded v e c t o r space ~ ( X ; k)

.

We can now prove (3.1) for suspensions (2.1) and the Bott-Samelson

theorem that the Lie algebra

t o t h e f r e e L i e a l g e b r a g e n e r a t e d by ~.(X

: it follows from the Milnor-Moore

~.(X

; Q)

~(~

SX) ® Q

; we choose a b a s i s

theorem

is isomorphic

(x)

of

; Q) , and representatives

x

The family of maps

(x~)

: S

~OSX

.

defines a map

l×L x:

VS

~SX

lx t+1 whose adjoint groups

: V S

---~SX

induces an isomorphism

: since both spaces are simply connected,

valence.

on rational homotopy

this is a rational homotopy

•

We now introduce the "homology suspension diagram"

equi-

268

OX * DM

,

~SOX

X

(3.3) f

OX * ~X

VoH ~ X v X

f

c

j -XxX

in which is the evaluation map A

is t h e diagonal

j

is the inclusion

H

is the mopf map

V = (~ v ~) o ~

~(t, X) - k(t)

H(k, t, ~) = (t, X.p)

where

~ : SOX

~ SOX

v SOX

is the pinch map.

Theorem (3.4). The diagram (3.3) is homotopy commutative, the rows are fibration sequences and the right-hand square is homotopy-cartesian.

By homotopy-cartesian,

we mean that if one replaces

A

or

j

(or both) by a

fibration, the pullback square is homotopy equivalent to the given square.

S k e t c h of proof of 3 . 4

(a) the right hand square is homotopy commutative A~(t, m) = (m(t), m(t)) jr(t, m) =

(w(2t), ~o(0))

t S~

1 I

(m(1), w(2t - I)) t > 5

The required homotopy is a "simplicial approximation" of the diagonal in

I x I

(b) the right-hand square is homotopy catesian : a standard way to replace by a fibration is to consider the evaluation map at the ends

269

x [°'10

. × x x

Composition with the inclusion of constant paths ¢

is a homotopy equivalence.

those paths in contractible,

X

Now the pullback is

g : X

it is not hard to construct a homotopy

* OX

A , and

E-X

and

E+X

are

equivalence

~ ,, E-X UOX E+X

(c] it remains to show that the homotopy fibre of DX

is

E-X U ~ X E+X , that is, the set of

which start or end at the base point. Since

S~ X

the join

- X [0'I]

j

has the homotopy type of

: one may consider the fibre square

E-X x OX U OX

x OX

DX

x E+X

= E-X x E+X

(~(1), ~(o)) XvX

in which

E-X x E+X

r

is contractible,

OX

* F~X

J

•

and construct

• E - X x OX U OX

We leave the details to the reader (see [Si]).

a

XxX

weak equivalence

x E+X ,

•

Now, in the homotopy cartesian square

SDX

X

(~.s) X v X ~'\,,,-

the existence of

h

J

• X x X

is equivalent to the existence of a homotopy section

~

of

Thus : I Proposition suspension,

3.6. A space has category •

~ I

iff it is a homotopy retract of a

270

The proof of (3.1] is achieved if we observe that a retract of a free Lie algebra is free - in fact, any subalgebra

(3.7) RemarkS

: we call diagram

of a free Lie algebra is free.

(3.3) the homology suspension diagram because the

Serre exact homology sequence for the fibration G. Whitehead's

OX ~ P~

exact sequence for the homology suspension

= ~(0

On the other hand, if we apply the functor

(3.3),

diagram

L ~ ~(×)

setting

- S OX

•X

is

(see [W]).

.) ® Q t o t h e whole

we have

=~(SOX) ~ L{U-L)

where

UL

denotes the augmentation

ideal of the enveloping algebra of

is the free Lie algebra functor from vector spaces to Lie algebras,

Finally,

L , and

L

and

the map

2(~) ~(sn×)

.~(×)

:

is surjective,

because

O~

has a section

~IX : FiX

O8(OX)

. We therefore get

a diagram with exact rows

L(U'L ® U'E)

~ E(O-£)

O

=

O

- L(~'C ®~J"l~) - - - " L

II

l

Lt L

in which the right-hand square is a pullback. Lie algebra over

Proposition

Q

3.9.

occurs as

Let

L

~(×)

for some

'-'

J-L

l

x L

~

O

-g

Since by Quillen's theorem any graded X , we can conclude

be a graded connected

J : LEL

L

Lie algebra over

Q , and

.LxL

be the canonical map, represented

by the unit matrix.

Ker j

are free with minimal generating vector

and

j-1(a(L))

spaces isomorphic to

of UL ® ~

L~L and

UL

respectively,

Then the Lie subalgebras

u

271

Let us conclude this section with observing that the analogous statement holds for

discrete

groups

- and can be d e r i v e d

from Gruschka% theorem

: as a c o n s e q u e n c e ,

the fundamental group of any co-H,spaoe (connected) is free,

§ 4 . - .,THE GENERALIZED SUSPENSION DIAGRA¥~ THE GANEA FILTRATION,

AND L . - S .

CATEGORY

FOR DG ALGEBRAS. We wish to generalize the homotopy cartesian square (3.5), to g e t

X(n)

n

X

I~ n TI X r

I~ j

To achieve this, we may replace either back

xn+1

&

or

j

by a fibration and take the pull-

X(n) , whose homotopy type is then well-defined, We will then have, by general

homotopy theoretic nonsense :

I (4.1)

cat

X ~ n

iff

X

is a homotopy retract of

Of course, we must try to describe Again we may first replace

&

X(n]

X(n)

to make (4.1) of any significance.

by the evaluation map at integral points

e : X[O'n]

r Xn+1

. (X[O],

X

~(1) . . . . .

X(n))

which is a fibration. Then

X(n)

Note that each piece

.

n U i=e

{X ( X [ O ' n ]

{X I X(i) ~ ~}

} X(i)

= *}

is contractible : thus

equipped with a standard categorical covering,

X(n)

comes

272

Exercise :

cat[X(cat

X)) = c a t X .

Moreover the intersection more-than-formal

space, say

o f two p i e c e s has t h e h o ~ o t o p y t y p e o f

a n a l o g y between t h e s t r u c t u r e

here

a deformation retract. Indeed associative

H.space

OX

and t h a t

(n + 1}

of a projective

affine spaces given by

the intersection of two affine charts admits X(n)

is the

n-th

SI

as

projective space of the homotopy

, and on t h e o t h e r hand one checks t h a t

GP(n)

~ K(Z,2)(n),

0 K [Z, 2) ~ S 1

Another approach to finn

X(n)

C P (n) , together with its covering by

homogenous coordinates :

with

of

~£ . There i s a

X(n)

through a construction

Pi : Ei

~ B , i = 1,2

, due £o

inspired

W. G i l b e r t ,

is

to convert

by t h e Whitney sum o f v e c t o r

be two fibrations with fibers

be the projection of @he mapping cylinder of

Pi

on

~1 x ~ 2

into

a libra-

bundles. Let

F i , and let

~i : Zi

~8

B . Let

Pl ~ P2 : Zl x E2 UElXE 2 E 1 x Z2 = E I ~ E 2

be the restriction of

j

- B x B

' and

Pl @ P2 : EI#~E2

-B

be defined by the pullback square

EI

iE2 Eli Pl @ P2

Pl &~ P2

&

B

~ BxB

one may check the following

Proposition 4,2 (a]

P l ~ P2

and

Pl ~ P2

are Hurewicz fibrations with fibre

F 1 ~ F2

(the

join of the two fibres] (b)

Let

~ ~ E+X

~X

i s a homotopy e q u i v a l e n c e

be t h e p a t h space f i b r a t i o n

with fibre

, Then t h e r e

273

E+~!E+X

~

~

>#< E+X

such that the above triangle commutes (c] If lence

p : E

...=. B

E U F CF

up to homotopy

is a fibration with fibre

, ~ E ~ E+B

PIE ~ P , F I C F

= *

F , there is a homotopy

equiva-

such that the triangle

E U F CF

where

"n'>)&(n+1)~xn+1

~

" E ~ E+B

, c~mutes

up to homotopy.

=l

From this we readily deduce the

Theorem 4 . 3 .

In the diagram

nx
,. X(n] ~ E+X~
F2X * ( n + l )

n ., T I X ~

the rows are fibration Moreover,

sequences,

n+1)

~X(o)

is a cofibre sequence for all

n ~ I .

co-fibre" process, ~ X

X(n + I) sion of case with

due to T. Gangs, has been defined

is the m a p p i n g ~

n

square is hometopy cartesian.

the sequence

~x * ( ~ + I )

: X (n)

X

,, ,.,, n+1 j,,-~r~(n+1]

and the right-hand

- X(n+O •

This theorem shows that the sequence of spaces

n

~,(n+l).,

X (n)

is generated

whose inductive step is the following ; convert

~

n

into s fibration

cone of the inclusion of the fibre and

which maps the cone on the fibre to the base-point. X(O) ~ *

by a "fibre-

to get the sequence

X(n)

~

@n+1

n

: assume

; then is the exten-

Initialize

the pro-

274

Observe that if n

: X(n)

~ X

is

X

is

k-connected,

(k(n + 2) - 1)-connected. Tnerefore

on homotopy groups in degrees limit of the spaces

k ~ I , then the fibre

X(n)

~

n

~ X ~(n+1)

of

induces an isomorphism

s k(n + 2) - 2 . This shows that the homotopy direct

has the homotopy type of

filtration of the hemotopy type of

X

X . The sequence

X(o)

is e

called the 8an6a filtration.

Also the above inductive "fibration-cofibration" process localizes all right at any set of primes

~ , so that

[X(n)]~ ~ X£(o)

From (4.1] we deduce t h a t cat(x~) In p a r t i c u l a r

~ c~t(×)

.

we s h a l l d e f i n e c a t e ( X ) , the r a t i o n a l

cat(Xo) : it is the lower bound of cat(X')

when

spaces with the same rational homotopy type as

X'

L . - S . c a t e g o r y of

X , t o be

runs through the class of

X .

It turns out that the Gan@a filtration has a nice interpretation in terms of the Sullivan minimal model. Recall that the Sullivan minimal model space

X

d

V maps

isomorphic to V

into

Hom(~.(X), Q]

A~2V : the ideal

and we may consider the DG-algebra homotopy type which we denote by Now

of a l-connected

is a DG-algebra, freely generated as a graded commutative algebra by a

vector space rential

~

~/~+I

. Thus

~

-~ (AV, d) , and the diffe-

A>-Pv ~ ( ~ ) P

is a differential ideal,

. This algebra represents a rational

X[n] , whose minimal model is a model of

~/-~+I.

:

Theorem 4.4 ([FH] VIII and IX)

(a] The Canna space X[n]

X(n)

has t h e r a t i o n a l

and a wedge of spheres

~n(X) c

.X[n]

~(X)

homotopy t y p e o f t h e wedge of

, such that the composition

v ~n(X) - - ~ X ( n )

c

,X(n

+ I)

is null-h~notopic (rationally]. (b) eatoX ~ n

iff

~

is a retract of the minimal model of

~/~+I

|

275

To prove 4.4, F@lix and Halperin first translate the pullback square

x(n)

-x t°'n]

Xc

into

J

. ~

-

, Xn + l

to obtain a (neither free nor minimal) model of

DG-algebras

show that this model is quasi-isomorphic

X(n]

. Next they

~/~+1

to the direct product of

and a

trivial algebra. This trivial algebra is huge in general

X(1) = S e X

while

~/~

is the trivial

the reduced homology of of

Hem(V, Q) = s~(X)

~I(X) in

rational homotopy type of

: for instance if

algebra

is isomorphic

~.(SOX) X[I]

O ®V

CP(~)

(n) ~ CP (n]

is when

, while clearly

discussion

~(X) = ~ ]

CP(~]

of the rational

h~otopy

~ n

(4,4)(b]

i.e.

X~

has the ~(X)

(over the integers

, for all

type

of

This p r o v i d e s an i n t e r n a l

is

K(Q, 2r) : in !)

n .

X(n)

, see It]

.

means that a space has

iff its model is a homotopy retract of a DG-algebra

~ n : compare with (1.5)

of nil-

!

definition

of the L.-S, category of a DG-algebra.

It is an open question whether the model of any space of category quasi-isomorphic

X(1)

, that is, when

I n ] ~ CP (n] £

We conclude with pointing out that Theorem rational category

to a supplement

= s ~=---~) . The only case when

we already mentioned the hemotopy equivalence

Fo~ e f u r t h e ~

potency

, d = 0 , VoV = O ; t h u s

(as vector spaces)

an abelian algebra on a single generator of odd degree, particular,

n = I , one has

to a DG-algebra of nilpotency

and has been checked by F@lix and Thomas for

n

actually is

n : this is obvious for

n = I ,

n - 2 .

REFERENCES, [FH]

Yves FELIX and Stephen HALPERIN, R a t i o n a l L . - S . c a t e g o r y and i t s tions,

T r a n s , A,M.S, 2?3 (1982) p p , 1-38,

applica-

276

EFL]

Yves FELIX and Jean-Michel LEMAIRE, On the mapping theorem for L,-S, category,

[Ga]

Topology, 24, 1985, 41-43.

Tudor GANEA, L . - S . c a t e g o r y and s t r o n g c a t e g o r y ,

t11o J . Math.

11 (1957)

p p . 331-348.

[Gi]

W. GILBERT, Some examples f o r weak c a t e g o r y and c o n i l p o t e n c y ,

Ill.

J . Math.

12 (1958) p p . 421-432.

EJ]

Ioan JAMES, On c a t e g o r y i n t h e sense of L . - S . ,

Topology 17 (I£78)

p p . 331-

348.

ELS]

J e a n - M i c h e l LEMAIRE and F r e n q o i s SIGRIST, Bur l e s i n v a r i a n t s

d'homotopie

rationnelle li@s & la L.-S. cet~gorie, Comm. Math. Helvo 56 (1981),

EW]

103-122.

George ~tHITE~EAD, Elements of Homotopy Theory, Graduate Texts in Math. Springer-Vet lag.

[L]

Jean-Michel LEMAIRE,

8ur le type d'homotopie rationnelle des espaces de

Gan@a, in Homotopie Alg@brique et Alg@bre Locale, Ast@risque n ° 113-t14 (1984),

pp 238-247.

Jean-Michel LEMAIRE Laboratoire de Meth~matiques U.A. eu C.N.R.S. n ° 168 Universit@ de Nice Parc Valrose F - 06034 NICE-CEOEX

S~RIES DE BASS DES MODULES DE SYZYC~IE par

Tousles n~th~riens,

anneaux consid~r~s sont des anneaux commutatifs, locaux,

de m~me corps r~siduel

Soient socie ~

M

Jack LESCOT

(R,m)

k.

un anneau local et

M

un R-module de type fini. On as-

deux s~ries formelles :

la sgrie de Poincar~ : P~(t) =

Z

bi(M)t i,

o5

bi(M) = dim k Tor~(M,k),

la s~rie de Bass : IM(t) =

Consid~rons

~ '~i(M)ti , i~>O

P" :

jective minimale de

M

"''--~ Pn+l --~ Pn -~...--~ Po --~ 0 et soit, pour

n i~me module de syzygie de

Dour

n > O,

~i(syzn(M))

une r~solution pro-

syzn(M) = Im(Pn --+ Pn-1 )

le

bi(syzn(M)) = bi+n(M)

pour

M. II est clair que

i > O. On montre ici comment les IR(t)

i M oil D.(M) = dim k EXtR(k,~).

sont d~termin~s par

M (on note

I~(t)) :

Th~or~me A.-- On a :

l~yzn(M)(t) = (bn_1(M)+...+tn-lbo(M))IR(t)-tn-ll~(t)+(1+t)tn-IIFn(M)I(t),

o__~ IFn(M) I(t) gradu~

d~signe la s~rie de Hilbert d'un certain espace vectoriel

F (M), associ~ g n Soit

E

M.

une enveloDpe injective de

le dual de Matlis de

k

sur

R

et soit

M v = HomR(M,E)

M. Ii existe un produit homologique associ~ ~

M :

R R v ,k) -~ Tor.( R R v ,k), Tor.(M,k) ®R T or~(M

et

F (M) est un sous-espace vectoriel de Tore(MY,k), d~fini ~ l'aide de ce n produit. La situation est simple Dour le module M si le produit est nul.

Dans ee eas

IFn(M) l(t) = I~(t). Ainsi Dour

M = k :

278

Th~or~me B.- Soit n>O

(R,m)

un anneau local non r~gulier alors pour tout

:

l~yzn(k)(t)

= (bn_1(k) + tbn_2(k) +...+ tn-lbo(k))IR(t)

+ t n P~(t).

On utilise ces r~sultats pour montrer que la dimension syzyg~tique introduite par Roos dans

y(R)

[11] est infinie pour la plupart des anneaux qui ne

sont pas de Gorenstein. Afin de mesurer la complexit@ de l'anneau th~or~me A, on pose la question suivante Existe-t-il un entier type fini, on ait :

On montre qu'il e n e s t

o(R)

W(M)

le sous-espace

de

classes d'anneaux.

:

de

Tor$(RV,k)

W(M) = Im s~

o~

associ~ ~

M

engendr~ par les valeurs du s : R--~ R/J

est la pro-

Ce r~sultat est utilis~ pour donner une nouvelle d~monstra-

tion d'une caract~risation Szpiro

M

Vp > o(R) ?

on d~finit le produit homologique

M = R/J, on a

jection canonique.

tel que pour tout module

bien ainsi pour quelques

Dans la premiere section,

produit. Lorsque

vis ~ vis de la formule du

Fo(R)(M ) = Fp(M)

Voici le plan de cet article

et on ~tudie

R

:

des anneaux de Gorenstein,

due g Peskine et

[10]. Dans la deuxi~me section,

on d~montre les th~or~mes A et B, et dans la

troisi~me section, on ~tudie l'existence de D'autres propri~t~s et applications

o(R)

pour quelques

du produit homologique

cas. associ~

un module se trouvent dans [9], papier auquel nous ferons r~f~rence pour quelques d~tails. O. NOTATIONS ET RESULTATS PRELIMINAIRES Soient

(R,m)

O.1. Soit A =

@

p>0 TAl(t) =

A

V

un anneau local,

k

un k-espace vectoriel,

un k-espace vectoriel

son corps r~siduel. on note

p ~ IA Itp

p~O

p

IVI

gradu~. Si pour chaque

la s~rie de Hilbert de

A.

sa dimension. p,

IApl < =

Soit on note

279 Soit

f : M--~ N

un homomorphisme de R-modules, on note

f~ (resp. fp)

l'homomorphisme induit en homologie :

f. : TorR(M,k)--~ T o r R ( N , k ) ( r e s p .

Soit

M

fp : TorR(M,k)--~ P Tor pR(N,k)).

un R-module de type fini. Les modules de syzygie de

M

sont d~-

finis ~ un isomorphisme pros g partir d'une rgsolution projective minimale par

syz°(M) = M e t ,

pour

n > O, par

P.

syzn(M) = Im(Pn--~ Pn_1).

0.2. Duals de Matlis On choisit pour l'anneau siduel Si

k. Si

M

f : M--~ N

R

une enveloppe injective

est un R-module, soit est un

R

M v = HomR(M,E)

homomorphisme, soit

induit entre les duals de Matlis, ainsi si

E

le dual de Matlis de M.

fv : Nv __~ M v

a C Nv

de son corps r~-

l'homomorphisme

fV(a) = aof

(composition

des applications)• Nous ferons un usage constant de l'isomorphisme canonique E ~ R v = HomR(R,E ) dans

E

identifiant les ~l~ments de

E

~ des applications de

R

et vice versa.

La formule de dualit~ de ([3], chap. VI,5.3) montre : Pour tout R-module

M

et pour tout

p 6 ~, il existe des isomorphismes

fonctoriels : TorR(MV,k) ~

P

Comme

kv ~ k

Ext~(k,M) v.

il en r~sulte imm~diatement : V

Pour tout R-module de type fini En particulier

P~ (t) = l~(t).

Rv IR(t ) = P R (t) =P~(t). On notera aussi que P~(t) =l~(t)°

Le probl~me du calcul des s~ries de Bass est ainsi ramen~ g u n

calcul

de s~ries de Poincar~ (pour des modules qui ne sont pas n@cessairement de type fini). Rappelons enfin que de l'anneau

R

TorR(RV,k)

peut ~tre d~finie par

n'est jamais nul et que la profondeur [l] :

Prof R = inf {i I EXtR(k,R) # O} = inf {i i TorR(RV,k) # O}. I

I. LE PRODUIT HOMOLOGIQUE ASSOCIE A UN MODULE I•I. Soient ~valuation, e ( a ® b )

M

un R-module et

8 : M @ R M v --~ E ~ R v, l'homomorphisme

= b(a). On associe ~

e

un ~roduit homologique :

280

R R V ,k)--~ Tor~(RV,k) Tor.(M,k) O R Tor.(M

par composition du produit ext@rieur :

R Tor,(M,k) %

Tore(MY,k) -o Tor.~ R'M ®RMV,k)

et de l'homomorphisme induit en homolo~ie par

0 :

R ® R MV,k) -~ Tor~(RV,k), 8. : Tor.(M

([3], chapitre XI). On note

< ,>

l'application R-bilin@aire correspondante

et on l'appelle le produit homologique associ~ ~

M.

On d~signe par : R~RV ,k ~ le sous-espace vectoriel gradu@ de Tor.~ j image de R M v ,K) ~ T o rR. ( M , k ) % T or.( par le produit associ~ g M, Wp(M) sa composante W(M)

de degr~

p, (Wp(M) = 0

si

D
le sous-esDace vectoriel de

W~

TorR(M,k)~Torm(MV,k), p ~ q On a donc

(W (M) = 0 P,q

Wn(M ) =

si

~ Wp,q(M) p+q=n

On v@rifie facilement que

TorR+q(RV,k)p D
ou si

(la somme n'est pas directe).

R R v ,k ~ W(R) = Tor.( )

W(E) = Tor$(RV,k). Notons aussi que

image de

q
W(M~N)

et de m~me

= W(M) + W(N).

Les outils essentiels pour travailler avec le produit homologique associ~ ~

M

sont fournis par la proposition suivante :

1.2. Proposition.- I) Soit Six

C Tor~(M,k) 2) Soit

et

f : M --* N

y E Tor~(NV,k)

0 --* M 1 i

M-~S M2 --* 0

on a

un homomorphisme de R-modules. <x,f~(y)> = .

une suite exacte de R-modules et consi-

d@rons les homomorphismes de connexion associ@s ~ cette suite et g s a duale :

6

R

: TorR+1 (M2,k) --~ Tor.(M 1 ,k)

6' : TorR+l(My,k) --* TorR(M2,k ).

Si

x E Tor +1(M2,k)

et

R v y E Torq+l(Ml,k)

on a

<x,6'(y)> = (-I) p < ~ ( x ) , y > .

281 Preuve : Soit X une R-alg~bre diff~rentielle gradu~e, r~solution minimale de

k

sur

R [6]. On choisit

structure multiplicative de (M®RX)®R(MV®RX) soient

(Ti)i E J X

--+ R V ® R X

a =

sur

R. La

permet de d~finir un produit

et

b =

~ b.T. b. E M v, ~ l'~l~ment a ® b j ~ J j j' J

~ bj(ai)TiT j. On note i,j ~ jxj

n~aire correspondante. Si

X

de la fa~on suivante :

E aiTi, a i E M i~J

on fait correspondre

une base homo$~ne de

a et b

< ,>

l'application R-bili-

sont des cycles, il en est de m~me de

et pour les classes d'homologies correspondantes dans Tor~(M,k), Tore(MY,k) R v Tor.(R ,k) on a : c~ = < c~(a),c~(b) > . D a n s la d~monstration qui

et

suit, pour tout homomorphisme phisme induit :

l) Soient b =

~ b.T. jCJ j j

fV(b) =

M®RX

f : M--~ N

on note encore par

f

l'homomor-

--~ N ® R X .

a =

E a.T, un cycle de M ® R X qui relgve x et iCJ I l un cycle de N v ~ R X qui rel~ve y. On a f(a) = ~ f(ai)T i et iEJ

~ (bjof)Tj. On obtient j CJ

=

E = (i,j)Ej×j(bj(f(ai)))TiTj ,

ce quid~montre le point I) en prenant les classes d'homologie. 2) Pour tout R-module plexe

M®RX.

Soit

classe d'homologie que

s(~) = a. Alors

classe d'homologie

a

M

d~signons par

un cycle de

x, et soit d(%)

M2®RX

~

q+1

tel que

est dans

Im(i)

~(x). De mgme, soit

repr~sente la classe d'homologie

de degr~

b

et

d(h)

p+l

M®RX

qui repr~sente la de de~r~

i-|(d(~))

h

p+!

de degr~

un ~l~ment de

est dans

tel

repr~sente la

M]v ® R X

un cycle de

y, et soit

iV(h) = b. Alors

= <~,d(h)>

la diff~rentielle du com-

un ~l~ment de

qui reprgsente la classe d'homolo~ie degr~

d

Im s v

et

Mv ® R X

q+| de

(sV~l(d(h))

~'(y). On v~rifie facilement que

et que

= . Comme

d(<~,h>) = - (-l)P<~,d(h)>, les cycles

<~,d(h)>

et

(-I) p

sont homologues. Ce qui ~tablit le point 2) en prenant les classes d'homolo~ie.

].3. Corollaire.- Soit

M

I) Pour tout

(p,q) E Z × ~

2) Pour tout

J EZ

on a

un R-module de type fini. on a

Wp,q(syzr(M)) c Wp+r,q_r(M).

Wo(svzr(M)) ~Wo(M) j ° j

et si

Preuve : I) II suffit d'~tablir le r~sultat dans le cas g~n~ral en d~coulant puisque

Wj(syzr(M)) =0.

j< r

syz1(syzr(M)) = syzr+l(M). Soit

r=1, le r~sultat P.

une r~solu-

282 tion projective minimale de O--~ syz1(M) --~ P

M. Consid~rons la suite exacte :

--~ M---+ O

et soient, comme dans la proposition prEc~dente,

o l e s homomorphismes de c o n n e x i o n 6'

R

: Tor.+l((syzl(M))V,k)

--~ Tor~(MV,k). S o i e n t 8

y E Tor~((syzl(M))V,k). Comme x' E Tor~+l(M,k), et alors blit que

<x,y>

est dans

2) Comme

R R ! ~ = T o r . + l ( M , k ) --~ T o r . ( s y z ( M ) , k ) , x E TorR(syzl(M),k) P

est surjective, on a

x = 6(x')

et

avec

<x,y> = = (-I) p <x',6'(y)>. Ce qui ~taWp+l,q_l(M).

Wj(syzr(M)) =

diatement de I).

E

W

(svzr(M)), l'assertion 2) r~sulte immE-

P+q=J P'q

Pour un module monog~ne, l'espace

1.4. Th~or~me.- Soient tion canonique. Alors on a Preuve : Soit

M

J

W(M)

admet une interpretation simple :

un ideal de

R

et

s : R-~

R/J

la projec-

W(R/J) = Im s~.

un R-module annul~ par

J.

L'homomorphisme Evaluation du debut du paragraphe admet la factorisaV

tion :

factorise

M®RMV

(R/j)v

s ~,,,~ Rv. Par c o n s e q u e n t ,

le produit

associE ~

M

en : Tor~(M,k)R ~ T o r ~ ( M V , k ) . . . Tor~((R/j)v k) ........

On en d~duit teur canonique

T

W(M) c Im s~ de

R/J

en particulier

v s~~ Tor,(R R v ,k).

W(R/J) c Im s v

d~finit une classe d'homologie

Le gEn~ra-

e~(T)

dans

Tor~(R/J,k). On v~rifie facilement que l'application R

: Tor~((R/J) I' application

v

R

v

, k ) - ~ Tor~(R ,k)

s~. v On en d~duit l'inclusion inverse

coincide avec Im s v c W(R/J).

Le corollaire suivant est une consequence immediate du th~or~me pr~c~dent et de sa d~monstration.

1.5. Corollaire.- Soient

J

tion canonique. Pour tout R-module

un ideal de M

R,

annul~ par

W(M) c W(R/J) = Im s~. Donnons maintenant quelques exemples :

s : R--~ R/J J

on a

la surjee-

283 1.6.- Soit

s : R-~

k

la projection

canonique de

R

sur son corps

r6siduel. Si R n'est pas un anneau r~gulier, on a d~montr~ dans [7] (ou [8]) v s~ 6tait nul. Autrement dit dans ce cas W(k) = O, et par consequent pour

que

tousles

modules de syzygie de

1.7.- Soit

M

k

on a aussi

d. Alors l e d igme module de syzygie de (corollaire

En particulier

de type fini v6rifie

1.8.- Soient

corollaire

1.3).

M

W(M) = Tor.(R R v ~k)

N

N

un R-module et

par

co-syzr(N)

1.3, on d~montre que

tails). Supposons que

N

est un R-module

libre et

1.3). Donc n6cessairement

sur un anneau local r~gulier tout R-module M

une resolution injective minimale de de co-syzygie de

= O (corollaire

un R-module de type fini, de dimension projective finie

W(R) = W(syzd(M)) c W(M), W(M) = Tor~(RV,k).

W(syzr(k))

et par consequent

W(M) # O.

j. = O__~ jo__, j1 __~...__~ j n _ ~ . . . N. D6finissons

pour

r >O,

les modules

= Im(J r-1 __~ jr). De mani~re similaire au

W(co-syzr(N))

c W(N)

(voir [9] pour des dE-

soit un R-module de dimension injective finie r(de type fini)o N est une somme directe de modules injec-

Le r igme module de co-syzygie de tifs tous isomorphes ~ et donc

E

[I]. Par cons6quent

W(E) = W(co-syzr(N)) c W(N)

W(N) = W(E) = Tor~(RV,k).

Ce dernier point permet de donner une nouvelle demonstration d'un r6sultat de Peskine et Szpiro ([10], th~or~me 5.5).

1.9. Th6orgme.- Pour qu'un anneau local noeth~rien il faut et il suffit qu'il existe un id6al R/J

J de R

R

soit de Gorenstein,

tel que le R-module monog~ne

so it de dimension in jective ' finie. Preuve : Rappelons qu'un anneau local noeth~rien

il est de dimension injective

est dit de Gorenstein si

finie [I]. La condition est clairement n~cessaire.

Pour la r~ciproque, notons

s : R --~ R/J

W(R/J) = Im s,v (th~or~me

D'autre part, puisque R/J est un R-module de R v W(R/J) = Tor,(R ,k) (I .8). Par consequent

1.4)

dimension injective finie, s, : Tor ((R/j)V,k) que l'anneau

R

--~ Tor (RV,k)

est un homomorphisme

surjectif.

IIen

rEsulte

est de dimension injective finie, donc qu' il est de Gorenstein.

II. UNE FILTRATION SUR 2.1. D~finition.croissante

la surjection canonique. On sait que

(Fn(M))n6~

Tore(MY,k) Soit sur

M

un R-module,

Tor~(MV,k)

on d~finit une filtration d~-

de la mani~re suivante

•

284

R

Fo(M) = Tor (MY,k) Vx E TorR(M,k),

et pour

j < p,

R v F p (M) = {y ly E Tor.(M ,k)

'

et

<x,y> = 0}. II est clair que les

espaces vectoriels gradu~s de caractgris~e par :

p>O

F (M) sont des sous P et leur intersection F (M) est

TorR(MV,k)

Foo(M) = {y l y E Tor,(M ,k) et

2.2. Th~or~me.- Soient

M

Vx C Tor (M,k), <x,y>}.

un R-module de type fini et

nombres de Betti. Alors la s~rie de Bass de

syzn(M)

(bp(M))pE~

est donn~e pour

ses

n > 0

par la formule : l~yzn(M)(t)=(bn_l(M)+tbn_2(M)+...+tn-lbo(M))IR(t)-t Preuve : Soit pour tout

p C~

P.

n-I M n-I IR(t)+(l+t)t IFn(M)l(t).

une resolution projective minimale de

les suites exactes : O - ~

syzP+1(M) --~ P

M. Consid~rons

--~ syzP(M) --~ O

P et les homomorphismes de connexion associ~s ~ ces suites et aux suites duales :

6

: Tor~+|(syzP(M),k) --~ Tor~(syzp+l(M),k),

6' : Tor~+]((syzP+](M))V,k) --~ Tor~((syzP(M))V,k).

En it~rant ces homomorphismes, on obtient pour

8n

n > O :

R n : Tor +n(M,k) --~ Tor~+(n_l)(syzl(M),k ) --~...--~ Tor~(syz (M),k)

v R v ,k). ~,n : Tor~+n((svzn(M))V,k ) --~ Tor~+(n_ I)((syzn-1 (M)),k)--~. ..--~ Tor~(M

Posons en outre

6°

Tore(MY,k). Notons que morphisme

et

~n

8 'o

les identit~s sur

Tor~(M,k)

sur les composantes de degr~ sup~rieur ou ggal ~

x C Tor~(M,k)

et

et

est un homomorphisme surjectif de degr~ -n (iso-

y E Tor$((syzn(M))V,k)

de la proposition 1.2, on obtient :

n). Soient

des gl~ments homog~nes, ~ partir

<x,~'n(y)> = ± <~n(x),y>.

La d~monstration du th~or~me se fait en plusieurs ~tapes : Point 1.- On a Soient Si

z

x E Tor~(M,k)

Par suite

F (M) = Im(8'nl. n

un ~l~ment homog~ne de avec

j < n, on a

Tor~((syzn(M))V,k)

et

y = 6'n(z).

<x,y> = <x,6'n(z)> = ± <6n(x),z> = O.

Im(6 'n) c Fn(M ).

Pour obtenir l'inclusion inverse, nous avons besoin du

285 Lemme.- Soit O -~ N] --~ R n s N --~ O une presentation minimale de R v v N. S i y E Tor,(N ,k), on a s,(y) ~ O si et seulement s i i l existe x E TorR(N,k) o

tel que

<x,y> # O.

Admettons le len~ne pour l'instant. Soit

R v y E Tor (M ,k)

un ~l~ment

homog~ne qui n'est Das dans Im ~,n. On peut trouver un entier r, tr ,r+| tel que y E I m ~ et y ~ Im ~ . En particulier, il existe z E TorR((syzr(M))V,k), ~l~ment homo~gne, tel que pas dans l'imaF~e de

O
y = ~'r(z), et qui n'est

R r+l (M))V,k) --~ TorR( (syzr (M))v ,k), 6' :Tor.+]((syz

homomorphisme de connexion associ~ ~ la suite exacte : O --~ (syzr(M)) v

sV. pV --~ (syz r+1 (M)) v --~ O. Cela signifie aussi que r sV(z) # O. Doric d'apr~s le lemme, il existe u C TorRo(syzr(M),k) tel que

# O. Soit <x,y> = <x,~

vr

x E TorR(M,k)

Preuve du lemme : Soit x E ToroR(N,k)

tel que

~r(x) = u

(z)> = ± # O. Par consequent

tel que

n'est pas dans

F (M). n

y E TorR(NV,k). Supposons qu'il existe

<x,y> # O. Comme

surjective, on peut trouver

alors y

So : T°rR(Rn'k)o --~ T°rR(N'k)o

x' E TorR(Rn,k)

tel que

est

s,(x') = x. On a

v * y)" # O. R~ci<x',s,(y)> = <s.(x'),y> = <x,y> # O, (1.2), et donc sV( v proquement supposons s.(y) = O. Soient (el) , |
Rn

et

ji : R --~ Re i --~ R n. Les ~l~ments

e.

d~finissent des classes d'ho-

i

moloF~ie dans

TorR(Rn,k)

et on v~rifie que les applications

: TorR((Rn)V,k) --~ TorR(RV,k) sV(y) # O, on peut trouver on a

i

tel que

et

j,

sont identiques. Puisque

"i'V(sV(y)) # O. Posons j, •

x=s.(c~(ei))

x E ToroR(N,k) et <x,y> = <s.(c~(ei)),y> = # O, (1.2).

Point 2.- Pour chaque de degr~

r

l'homomorphisme

G 'n

induit un isomorphisme

-n ~,n : Fr(syzn(M))/Fr+1(syzn(M)) --~ Fr+n(M ) /Fr+|+n(M).

Comme l'application Tor~((syzn(M))V,k)

an

est dans

est surjective, un ~l~ment homog~ne Fr(syzn(M))

x E Tor~(M,k), j < n+r, on a :

y

de

si et seulement si pour tout

<~n(x),y> = O

ou de fagon ~quivalente

<x,6'n(y)> = O. On a donc l'~quivalence : y E Fr(syzn(M)) ~=~ ~,n(y) EFr+n(M). D'autre part, le point precedent montre que pour tout

r :

~'n(Fr(syzn(M))) = Fr+n(M) o On en d~duit le r~sultat.

Nous sommes maintenant en mesure de d~montrer le th~or~me.

286

A partir de la lonF~ue suite exacte d'homologie associ~e ~ la suite exacte

0 -~ (syzn-l(M)) v --~ pV -~ (svzn(M)) v O, et en tenant compte du n-I ~ --~ I, on peut ~crire :

point

IR yzn(M)(t) = bn_ I(M)IR(t) - IR yzn-l(M)(t) + (l+t) rF l(syz n-l(M))] (t). Ce qui gtablit la formule pour

n = I. On peut aussi gcrire :

n-1

IRsyzn(M) (t) = bn_l, (M)l~(t)+tlsYZK ~

n-1

(M) (t)-(1+t) IFo(sYz

.

n-1

(M))/Fl(sYz

(M))[(t) •

En utilisant le point 2, on obtient : n-1 (M~t)-(l+t)tn-IjFn_l(M)/Fn(M)l(t).

IRYZn(M)(t) = bn_ I(M) IR(t) + tlRYZ

Ce qui permet d'obtenir la formule par r~currence.

Remarque I.- On peut d~montrer la formule du th~or~me 2.2. en utilisant les r~sultats de compl~mentaire

Foxby

[4]

sans toutefois obtenir une description de terme

(|+t)tn-lIF (M)[(t), ce qui est le but recherch~ ici. n

Remarque 2.- On peut d~finir sur la filtration

Tor~(M,k)

une filtration similaire

F. . Cette nouvelle filtration intervient dans une formule ex-

plicite de la s~rie de Poincar~ des modules de co-syzygie d'un module M [9].

2.3. Exemple.- Soit finie

d,

@ < d+t-n

TorR(MV,k)

F.

se d~crit simplement :

(Voir [9] proposition 1.2.9 pour la d~monstration).

3

IF (M) l(t) = a +...+ a n

pM(t)

un R-module de type fini, de dimension injective

IM(t) = a o + . . . + a d td. La filtration

F (M) = n j Ainsi

M

o

t d-n d-n

et comme d 'autre part

IR(t).IM(t -I) [4], on obtient des formules explicites pour la s~rie

de Bass des modules de syzygie de

M.

Ii est clair que la filtration

F.

est triviale (Vn Fn(M)=TorR(MV,k))

si et seulement si le produit homologiques associ~ ~ dit si

M

est nul, autrement

W(M) = O. Dans ce cas, la formule du th~or~me 2.2 se simplifie et on

obtient : 2.4. Th~orgme.- Soit s~rie de Bass de

syzn(M)

M

un R-module de type fini. Si

est donn~e pour

n > 0

W(M) = O

par la formule :

la

287

l~yzn(M)(t) = Lorsque

R

(bn_1(M)+tbn_2(M)

n'est pas un anneau r ~ u l i e r ,

le th~or~me B de l'introduction Dans

+...+ tn-lbo(M))!R(t) + tnl~(t).

puisque

W(k) = O (1.6) et on obtient

l~(t) = P~(t).

([11], chapitre 7), J.E. Roos introduit la dimension syzyg~tique

d'un anneau local

R :

y(R) = inf {t I tout t i~me module de syzygie de type fini est projectivement ~quivalent ~ un (t+I) i~me module de syzygie}, ou n'existe Das de tel

t. (On dit que

si on Deut trouver

pet

q

anneau de Gorenstein,

on a

des autres cas, on a

y(R) = =,

2.5. Proposition.~d(R) # 1

on a

Preuve

M

tels que

et

N

y(R) = ~

s'il

sont ~rojectivement

M@R p ~ N@Rq).

Si l'anneau

~quivalents R

est un

y(R) = dim R. Nous allons voir que dans la plupart

Soit

([l]], probl~me 3.8, p. 249).

R

un anneau local de profondeur

d. Si

y(R) = ~.

: Si

y(R) < ~

et

q > p ~ y(R), alors on v~rifie facilement que

tout pleme module de syzy~ie de type fini est projectivement ~quivalent ~ un i~me module syzygie. Soit M = syzr(k), r > d on va montrer que M n'est

q

pas projectivement

~quivalent ~ un r+2 igme module de syzy~ie.

n'est pas r~ulier,

le th~orgme B permet d'~crire

Comme l'anneau R

:

IRM~RP(t) /IR(t) = (P+br_l(k)+tbr_2(k) +...+ tr-lbo(k))+trp~(t)~ /iR(t)

C'est une s~rie formelle g coefficients non nul de

IR(t )

seuls les Soit

r-d

est

l~yz (N)@ Donc si

Rq(t )

/ IR(t)

a au moins ses

Wj(syzr(M))

M = O

on ait en fait

n-l-d

on doit avoir

est

I,

~.

premiers

coefficients

dans

~.

n < r+1. Par consequent on a

y(R) = =.

III. UNE CONSTANTE ASSOCIEE A L'ANNEAU Soit

P~(t)

de la s~rie ci-dessus sont dans

le th~or~me 2.2 on v~rifie que

M~R p = syzn(N)@R q

n~cessairement

~. Comme le premier terme

~d(R)t d, et que le terme constant de

premiers coefficients

n > d, en utilisant n

dans

R

un R-module de type fini, on a vu (corollaire si

j < r. On peut esp~rer que si

W(syzr(M))

de la faGon suivante

:

r

1.3) que

est ehoisi assez grand

= O. On est conduit ~ d~finir une constante

o(R)

288

3.1. D~finition.-

Soit

(R,m)

un anneau local.

o(R) = inf {r I pour tout R-module S'il n'existe En utilisant de comparer tions,

pas de tel entier,

M de type fini, W(syzr(M)) on pose

W(syzr(M))

de

M

et de

= 0, et, Fr(M)

aussi ~tre d~finie

par

du th~or~me

syzr(M))

= F(M),

= 0}.

a(R) = =.

le point 2 de la demonstration

les filtrations

Posons

2.2 (qui permet

on constate

sont ~quivalentes.

que les asserDonc o(R) peut

:

o(R) = inf {r I pour tout R-module

M de type fini, Fr(M ) = F ( M ) } .

Ainsi

de l'anneau

o(R)

mule du th~or~me Question Avant pri~t~s

mesure

la complexit~

2.2. Comme

W(R) ~ 0

on a toujours

: Pour un anneau local

de donner des exemples

simples

de

R

(R,m),

vis ~ vis de la foro(R) ~

I.

a-t-on toujours

o(R) < ~ ?

o~ il en est bien ainsi, notons

deux pro-

o(R)

a) o(R) > prof R = d. En effet, jective b) Soit

il existe des modules d. On a alors

x

un ~l~ment

(R) < o(R/xR) On obtient

M

W(syzd(M))

de l'id~al

de type fini,

de dimension

pro-

= W(R) # O.

m,

x

non diviseur

de z~ro alors

+ 1

ce r~sultat

des modules

en comparant

de syzygie

sur

les modules

de syzygie

R/xR (pour les details

sur

R

voir [9],

1.3.7).

3.2. Proposition.s : R ~-~ R/a

la projection

l'application Preuve (thEor~me a. Donc si

V

s.

canonique.

: L'application

1.4). Tout W(R/a)

l'application

(R,m)

un anneau local de socle Alors

o(R) = |

!

; notons

si et seulement

si

est nulle. s~

rleme-module

= O

o(R) = I. R~ciDroquement injectif.

Soit

on a soit

f : R/~--~ R n,

Par consequent

R/a

est nulle si et seulement de syzygie

M

avec

W(M) = 0 (corollaire xl,...,x n

si

W(R/a)

= 0

r > 0, est annulE par

1.5), et par suite

un syst~me

g~nErateur

de

m

;

f(y) = (yxl,...,yx n)

est un homomorphisme

est un premier module

de syzygie.

Si

289 o(R) = I, on a

W(R/a)

II existe Soit

lier.

des anneaux

(R,m)

en outre que

= O.

R

R

n'est pas de la forme

l.ll). Consid~rant

I~R/m2(t) R --

la suite exacte

on en d~duit que la condition IR(t)

= (Im21-1m/m21t+t2).p~(t)._ condition

le produit

d, alors Preuve

Tor~(RV,k).

o~

:

est satisfaite

par B~gvad

est un anneau r~gu-

)

R v --~ ( m 2 ) V - - ~ O ,

si et seulement

d'anneaux

v~rifiant

avec

sa s~rie de Bass qui est calcul~e

(R,m)

On suppose

si

cette der-

[2].

des anneaux artiniens

Soit

2

([2] ou [8] exemple

sv

O--~ (R/m2) v

o(R) > 1 : il suffit

la condition

dans

pr~c~dente

[7] ou [8].

un anneau local de Gorenstein

de dimen-

o(R) = d+|.

: En effet la seule composante

non nulle de

On en d~duit que pour tout R-module

la profondeur

(B,b)

fibr~ de deux anneaux v~rifiant

3.3. Proposition.sion

B/b 3

L'existence

a ~t~ d~montr~e

On notera qu'il existe

m_3 = O, de socle

= (Im/m21-t).P~(t)

s~ = O

nitre

et de consid~rer

o(R) = I :

un anneau local tel que

Dans ces conditions

de prendre

tels que

de

R

est

d

3.4. proposition.-

M,

on a n~cessairement

Soit

(R,m)

Tor$(RV,k) ~

Fd+I(M)

est

= F (M). Comme

o(R) = d+].

un anneau de Golod,

n = Im/m21. Alors

o(R) ~ n+2. (Pour la d~finition Preuve on montre

: En utilisant

Soient

phisme de R-modules. O < p < l+Im/m21. Soient de

M

(R,m) Pour que

([9],

[5] ou [6]).

de Ghione et Gulliksen

([5] th~or~me I)

f, = O

f : M --~ N

un homomor-

il faut et il suffit que

de type fini et

tenu du point

Fn+2(M ) = F=o(M)

l'homomorphisme

O --~ (syzr(M))V

un anneau de Golod et

fp = O

pour

1.3.|O).

un R-module

M . Compte

pour ~tablir que r > n+2,

l'argument

voir

le

Lemme.-

minimale

des anneaux de Golod,

P.

une r~solution

I de la d~monstration

projective

du th~or~me

2.2,

il suffit de montrer que pour tout

de connexion

associ~

s r'v ~ pV --* (s y z r+! (M) )v --* 0 r

g la suite exacte est surjectif,

ou ce qui re-

290

vient au m@me que

s: 'v = O. Le corollaire 1.3 montre en particulier que

Wo,q(syzr(M)) = =0 si

q < r. Par consequent, pour tout

x E Tor~(syzr(M),k)

et tout

y E Tor~((syzr(M))V,k), q < r, on a < x , y > = O. En utilisant le r,v s = 0 q q < r donc finalement que s: 'v = O en utilisant le lemme precedent.

lemme de la d~monstration du th~or~me 2, on peut conclure que pour

Bibliographic [I] H. BASS.- On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28. [2] R. B~GVAD.- Gorenstein rin~s with transcendental Poincar@ series, Math. Seand., 53, 1983, 5-15. [3] H. CARTAN, S. EILENBER~.- Homological Al~ebra, Princeton Univ. Press, Princeton, N.J., 1956. [4] H.B. FOXBY.-

Isomorphisms between complexes with applications to the

homological theory of modules, Math. Scand., 40 (1977), 5-19. [5] F. GHIONE, T.H. GULLIKSEN.- Some reduction formulas for the Poincar@ series of modules, Atti. Accad. naz. Lincei

LVIII Ser., Rend., CI.

Sci. fis. mat. natur., 58 (1975), 82-91. [6] T.H. GULLIKSEN, G.L. LEVlN.- Homology of local rings, Queen's papers in pure and applied Mathematics, n ° 20, Queen's Univ., Kingston, Ontario, (1969). [7] J. LESCOT.- La s@rie de Bass d'un produit fibr@ d'anneaux locaux, Comptes-rendus, Acad. Sci., Paris, 293, S~rie A (1981), 569-571. [8] J. LESCOT.- La s@rie de Bass d'un produit fibr@ d'anneaux locaux, S@minaire d'Alg~bre P. Dubreil

et M.P. Malliavin (1982), Lecture

Notes in Mathematics 1029, 218-239, Springer 1983. [9] J. LESCOT.- Produit homolo~ique associ~ g u n

module et applications,

Pr@publication n ° 14 (1983), D~D. de Math. et de M~canique, Univ. de CAEN. [lO] C. PESKINE, L. SZPIRO.- Dimension projective finie et cohomologie locale, Inst. Hautes Etudes Sci. Publ. Math., Paris, n ° 42 (1973), 47-119. [11] J.E. ROOS.- Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos, Commutative Algebra, Durham 1981, Ed. by R. Sharp, London Math. Soc., Lecture Notes, Vol. 72, 1982, 179-203.

Dgpartement de Msthgmatiques, Informatique et M~canique Universit~ de CAEN 14032 CAEN CEDEX - FRANCE

ON THE SUBALGEBRA GENERATED BY THE ONE-DIMENSIONAL ELEMENTS IN THE YONEDA EXT-ALGEBRA by Clas L~JFWALL INTRODUCTION. Let k be a field and R a ring with a ring epimorphism R --> k. Then k is a module over

R

and

E = EXtR(k,k)

is a graded algebra under the Yoneda

product. We study the structure of this algebra in two situations. In the first ease

R

is an augmented algebra over

second case

R = (R,m)

k

(in general non-commutative) and in the

is a local (commutative noetherian) ring with

R/m = k .

We are mainly interested in the second case, but within this theory it is natural to consider certain algebras of the first type. For example the Yoneda algebra itself is an augmented algebra and the homology theory for about the structure of complex for

E

E

gives information

E . Another example is the homology algebra of the Koszul

R . The Yoneda algebra for local rings has been studied by Levin [9]

and SjSdin [15], [18] end Roos [13] . Sj6din determines the structure of

E

when

the local ring is a complete intersection or a Golod ring. In [13] Roos gives an example of a local ring for which

E

is not finitely generated (this answers

negatively a conjecture by Levin [9] ). In this example the subalgebra of

E

generated by the one-dimensional elements plays an important r$1e. The main goal for our work will be to "compute" this subalgebra in the two eases described above and examine to what extent it is a good approximation of the whole Yoneda algebra.

Stmumar~

In part I of the paper we consider rings of the first type, i.e. augmented algebras over a field

k . If

we prove that the subalgebra Yoneda algebra EI

E = EXtR(k,k)

R A

belongs to a certain class of such algebras, generated by the one-dlmensional elements in the

is equal to the free non-commutative algebra on

divided by the two-dimensional relations ker(E I ~ E I

-~

E2). This theorem

and its corollaries may essentially be found in Pride/ [12]. He studies algebras satisfying

A = E

and we prove that this condition is equivalent to the "FrSberg

formula" (see [5] ) being true. When the cube of the augmentation ideal of zero, we prove that where

T(V)

E ~ A 8 T(V)

R

as left A-modules and as right T(V)-modules,

is the free non-commutative algebra on the graded vector space

We also give a formula for %he Hilbert series the Poincar6 series of

R

is

HE

of

V .

E (which is the same as

and is defined by the formal power series

~n>0dimk(En)zn) in terms of

H A . As an application we study the homology algebra

of the Koszul complex for a local ring with imbedding dimension three. Our methods in part I applies to get the rationality of the Poincar~ series of a class of local rings, indeed let where

p

R = k[tl,...,tn]/monomials of degree two + (tl,...,tn)P

is any number > 2

then the Polncar6 series of

R

is rational.

292

In the second part of the paper we study local commutative rings (R,m). We give an equivalent condition for a local ring~nomomorphism ¢: (R,m) --* (S,n) Indeed, if

¢

to be a Golod homomorphlsm in the sense of Levin [8] .

is surjective the condition is as follows:

There is an exact sequence of Hopf algebras k --~ T(V) where

T(V)

--* EXts(k,k ) --~ EXtR(k,k)

--~ k

is the free non-commutative algebra on

V = ~>2Vi

and

2

Vi+ I = Ext~(S,k)

for

i~I .

We prove that for some rings there is a differential which is a free

R

is two-homogeneous or

are fulfilled~i.e. then

E

is a

R = GrR

such that m

= 0

"semi-tensor-product"

~7]

HomR(U,k) = A . This is the

and if both these conditions

(with respect to the m-filtration) and m h = 0, (see Smith

non-commutative algebra. If furthermore (see SjSdin

U,

R-algebra with divided powers~containing the Koszul complex

and contained in a minimal resolution case when

R-algebra

) we prove that

PR

A

~9]

) of

A

and a free

is nilpotent as Hopf-algebra

is rational.

Part I. Non-commutative algebras

Notations and basic facts

I.

k

is always a commutative field.

2.

A graded vector space

IVil = dimkV i < ~ power series

for all

V = ~.>oVi

is locally finite if

i>0 . The Hilbert series of

Hv(z) = [i>01Vilz i

V

is the formal

293

3.

if

V, W

are graded vector spaces then (V m W) n = ~ i + j=n (Vi ~ Wj)

space with

vector space with

and

is a graded vector

Homk(V,W)

Hom~(V,W)n = ~ i H O m k ( V i , W i _ n )

is a graded vector space with h.

V m W

V~=n H°mk(Vn'k)

is a graded

, especially -

We use strict sign convention. This means that when we in a defining

formula replace

a,b

by

b-a , we must multiply by (-I) deg(a)'deg(b)

For more details about this principle we refer to Gunnar SjSdin If

a

and

h

are bigraded elements with bigrade

we use the sign

(-1)sls2+tlt2

when

formula. The graded commutator for 5.

a ~ b Let

and

UCX

[a,a] = a 2

~

and

or

V

and

{x 6 X;

6.

0

and

b

V c X* , f(x) = 0

V0

(sl,t 1)

is defined by

y

and

s1+t I

(s2,t 2) ,

U0

as

odd or even. {fE X*: f(U) = 0}

then,

(imp) 0 = ker¢ ~ .

and

X

then, (U~V) 0

=

U0 + V0

and

U 00 = U ,

f C V} .

There is a natural map V~

.

ab - (-1)SlS2+tlt2 ba ,

means the set

for all

~6]

are interchanged in a

according to

X--~

are suhspaces of

(U + V) 0 = U 0 ~ V 0

where if

[a,b]

he a linear map

(ker@) 0 = im~~ U

a

be graded vector spaces. Define

=(X/U) R. Let

If

V ~-- HOmk(V,k)

V~

--,

(v ~ v)*

defined by

f~g(x~) = (-1)stf(x)g(y)

for f E(v*) s

and

g ~ (V*) t

294

This map is a monomorphlsm, indeed suppose and

gl,...,g n

0 + [1
are linearly independent, and

n~1

--~ 0

is minimal with this

property. gl

linearly dependent of

g2,...,g n

<=,~>

im g~ £ im g2 + "'" + im gn

kerg1~kerg 2~

... ~ kerg n

(cf. no.5). So, if We suppose that

gl

g2~...sg n , then there is an element gi(y) = 0 fl = 0 7.

A graded al~ehra

R

g1(y) @ 0

f1(x) = 0

but

for all

x , i.e.

n .

is an associative ring with unit which is also and

I ~R 0

and

is augmented if there is a map

c: R

--~ k

of graded algebras such that

I ). R

is connected if

of

R

8.

V 8n

e

R 0 = k . If

as algebra is

I/I 2

IRil < ~

V ~ ... 8 V V

for all

i .

is locally finite, the Hilbert series

(n factors). T(V)

, n

Xl,...,x n

is the free is a bigraded

is called the "tensor" degree and

is called the "pure" degree and is written

dt(a)

are elements of

x I 8 ... ~ x n ~ and the empty string

T(V) ~ = HOmk(T(V),k)

ideal and is written

(the tensor algebra). T(V)

vector space, T(V) = -~n,s>0 II (V~n) s ~

instead of

Ri+ j .

H~Ig(x,y) = ~i>_0Hii/ii+1(y)x i

is short for

a E T(V) . If

Ri'R~C

is called the a u ~ e n t a t i o n

is locally finite if

non-commutative algebra on

for

such that

R = ~.>0Ri

I(R) (or just

s

y~V

but this contradicts the minimality of

e(1) = I . The kernel of

R

is linearly independent of

for i = 2,...,n . It follows that

a graded vector space R

(where gi is the dual of the map gi: V ÷ k )

T(V) ( )

and

dp(a)

we write

respectively (Xl,...,x n)

is equal to

I .

is also a bigraded vector space and it is an algebra

with the following definition (cf. no.6):

295

Let

f,gET(V) ~

be homogeneous

elements of degree

(n,s)

and

(m,t)

respectively..Then f'g(x ~ y) = (-1)n'm + s-t f(x).g(y) where

x ~(vSn) s

T(V)

and

y 6 (vSm) t .

is also a Hopf-algebra

to be primitive.

If

V

(see

~(M) = T(M)

T(M)

as bigraded

= (X 2 ..... Xk)

and

x'y :

) by requiring the elements of

is locally finite, the dual of

Hopf-algebra and we denote it is possible to define

~I]

modules.

Let

If

R

(Y,dy)

of

B.

Y:

(Yl . . . . .

Yl ) '

8 (xi+ 1 ..... x k)

~(M)

,

dp(x)

even.

inductively

(using the formula

is any ring, we define the Yoneda 10roduct (A,B,C

in the following way(see

PB

'

(i). (j) Y ) •

ExtR(B,C)

and

Xk)

(XI,X'y) + (ml)dp(Yl)'~P(X)(y1,x'y)

This defines the structure of

9.

= (x~ . . . . .

y = (Y2 ..... y£) . Then

= (x1,~.x(n-1))

(x+y) (n) = Li+j=n x

is also a

T(V W) . This is even a Hopf-F-algebra and it -.over a commutative ring j (ef. ~ar~ ~j ): for any graded module M ~

Ax = ~0
T(V)

V

Let

~

are left

O

ExtR(A,B)

~ ExtR(A,C)

[15] ): Choose projective resolutions

fEZHom(PA,B)

are complexes, t h e n

Hom(X,Y)

R-modules)

and

gEZHom(PB,C).

has a d i f f e r e n t i a l

(If d

PA

of

(X,d X)

A

and

d e f i n e d by:

dr(x) = dy(f(x)) - (-1)deg(f)f(dx(X))). There is a map e o ~ = f

where

~ZHom(PA,PB) e: P B - - ~

j i.e.

d B o ~ = (-I) deg(f) f o d A , such that

B . We now define

[g~ o If]

as

[g o ~] •

This product is independent of all choices and gives,for any left R-module ExtR(A,A)

a structure of a graded associative ring.

A ,

296 If

R

is a graded augmented algebra, ExtR(k,k)

is a bigraded algebra.

The definition of the product is unchanged. If resolution of

k , then

in the definition of The subalgebra of

10.

Let

R

so the sign

generated by

Extl(k,k)

is written

(B,d) is a graded free resolution of

B -- R 8k T(1) = R-T(1) , ,xn). =. x1-(x2,. . d

over

is R-linear and for

s if

I . R

defined

(x I ..... Xn)~T(1)

,x..x.+ .,xn) i l I'''

~''"

is &cyclic,

defined by r(I

and

s(1.(x I ..... Xn) = 0

ExtR(k,k) = HHomR(B,k) = H(T(I)~,d ~)

spaces. We will now examine the Yoneda product. Suppose and

k

has bidegree (-1,0) . (B,d)

s(r'(x I .... ,xn) = (r,x I ..... x n) k-linear. We have

d

,Xn) + ~1
.

since there is a homotopy operator

dt(f) = n

(-I) deg(f)

follows the principle given in no.h.

is a bigraded R-module and

s

is a graded free

be a graded augmented algebra with augmentation ideal

d(x I ' . B

has bidegree (-1,0)

EXtR(k,k)

The Bar-resolution as follows:

~

d

(P,d)

and

as bigraded vector

fE T(I)n~

where

dp(f) = j .

d~f(x) = (-1)n+If(d(x)) = (-1)n+1~1
Suppose

d~f =

0 , we define a "lifting"

~

by:

~(x I ..... Xn. k) = (-1)n'k * j.dp(x I ..... Xk) f(xk.1,...,Xn~(Xl .... ,Xk) and

~

R-linear. ~

shows that

is homogeneous of degree

df = (-1)nfd . Let

representing elements

(-n,-~)

f~znSHon~(B,k)

If] ~ExtRS(k,k )

and

and

and a simple computation g~zktHomR(B,k)

[ 4 ~Extkt(k,k)

. Then,

[g]'[f] : [g o f] ~EXtR+k's+t(k,k) and g o f(x I ..... Xn+k) : (-1)nk*stg(x I ..... xk)f(xk. I .... Xn+k) , One easily proves that

d*(f'g)

= d*f'g + (-1)dt(f)f'd~¢g

297 which

shows that

H ( T ( I ~ , d ~)

is an algebra,

and it follows

from no.8

and no.9 that

E:~R(k,~) = H ( T ( I I ~ , ~ )

[~] ).

as algebras (see also Ca~an 11.

Suppose

R

graded locally

Suppose graded

is a graded locally finite vector

spaces

Yn = R 8 Xn

and

further there complex

or if

for each

d: R s ~ Xnt

Y = nI ~'> 0"Y

d

making

(Y,d)

to a

-~

"-~

Yn+l,j

j

(e.g.

-~

if

Ynj"

X . = 0 nj

Rs+l ~ Xn-l,t-1

Yn-l,j

-~

for almost

all

--~

) " Then for each

"'" n

for each

j

j

= ~n(-1)nIHnj(Y)l

gives the formula,

"lemma"

following Let

X n , n>O_ ~ are

t

differential

HR(Z).HX(-1,z ) = l-LH(y)(-1,z) This

and

and

is a homogeneous

~n(-1)nlYnjl which

algebra

such that, "'"

is finite

finite

R

is due to J6rgen

proved in Lemaire

be a connected

a minimal Tor~j(k,k)

resolution = 0

for

Backelin. ~]

~
It has several

locally

finite algebra.

a resolution

of the type

. Hence the lemma applies

HR(Z).PR(-I,z)

applications,

= I .

The construction

R 8 Tor~(k,k)

X . = 0 no d:

We have

for

j @ n

R s @ Kn

Y . = R. ~ K nj 0-n n

Rs+ I

. Put

HH(y)(-I,z)

and

Y

where

( PR is the same as the Hilbert X

nn

= K

n

and

Kn_ I .

. = Y' n3 n,j-n

= HH(y,)(-z,z)

and hence we get the formula HR(Z)'HK(-Z)

of

and we get

of Suppose now,

e.g. the

:

augmented

yields

•

= HH(y,~(-z,z)

. Then,

TorR(k,k)

series )

298

We are going to use this formula several times in the paper. For instance if

(R,m)

is a local ring with imbedding dimension

r

and

R = GrR

with

respect to the m-filtration, we may apply the formula above to the Koszul complex of in

[5].

12.

Let

R . In this case

R

Suppose

and we get the formula mentioned

be a graded augmented algebra with augmentation ideal

I 0 = Jacobson radical of

is an R-module with each and

HK(-Z) = (l-z) r

IM = M , then

Proof: Suppose

(i.e.

R0

is local), If

M = ~>0~

a finitely generated R0-module (or

I 0 = 0)

M = 0 .

M.l = 0

Ro-module (or

Mn

R0

I = ~.>OIi

for

i
i 0 = O) implies

and

Mn finitely generated

Mn = 0 .

Two-homogeneous algebras

Let

R

be an augmented graded algebra over

ideal with degree

Definition.

R

vector spaces

If

R

k , and let

I

be the augmentation

>_0 .

is two-homogeneous if there is a decomposition as graded I = V ~

12

such that

V2~I 3 =

i.s two-homogeneous then as a k-space

0

R = k ~

(I 2 is short for

V ~

I-I).

V21~ 13

(and conversely).

Proposition I. I V

(R = k ~ " ~ n %

with

(;~

If

R

V I I V2 ~ with as above

%

is two-homogeneous then the subalgebra generated by I3) is isomorphic to G V @ V

and

%

is two-homogeneous.

T ( V ) / ~ where, as a k-space,

c.ELi.> " 3v @i . Conversely

T(V)/~

.

299

Proof.

Consider the morphism

is in the kernel, then

x2

T(V)

--~ R . If

~i>2xi , x~E~ V ~

must be in the kernel since

Hence the kernel has the stated property. Conversely two-homogeneous if indeed let I , V

I

in

=

~]~

~$

with

. If

~ ~ q2~ ~

y = z = x . Hence

I ,

R = k ~

V 8 V

--* V 2

where

~

(V~

V~

V ~

13

Let

and let

I , V y ~ V2

C ~ - > 3 V~i the images of and

y$~

z E Is and

~ = 0 .

be the multiplleation map

im(¢*)~V ~

(I2) 0

V# .

(V @ V)K" by the map given in Notations no.6).

H(T(I~,d ~) . Let

I . We have

with

and

Z

be two-homogeneous with augmentation

R

From no.t0 in Notations we have that

algebra is equal to

V~

and

is certainly

then,

is embedded in

~

T(V)/~

y - z ~ ~ , which implies that

is the two-sided ideal generated by

Proof.

I 8 I

V2 ~

T(V)

, then there is

Theorem 1.1 (cf. [12,Theorem 2.5] ) Ideal

g~ C V 8 V

be the augmentation ideal of

T(V)/02

such that

~

V 2 ~ I s = {0} .

m

EXtR(k,k)

as a bigraded

be the multiplication map

Ext~(k,k) = ker(m@) = V ~t (we identify in the sequel

as a subspaee of

I~). Hence,

(Ext~(k,k)) 2 = V~ @ VM/im(m*)~V* 8 V ~ =

V~8

V~/im(@~)~V*8

V*

and generally (Ext~(k,k)) n = (V~)Sn/im(dW)~(V~) 8n Since

dW(V ~) = 0

we have 6~ C. im(d ~) ~ (V~) 8n . We will now prove the other

inclusion, i~(dW)r~(V~ @n = { f ( ( V ~ 8n : ~ g ~ < i < n _ 2 ( V

8i ~ V 2 8 vS(n-i'2)) * and f = @ o d}

this set is contained in, ~0
g

such that,

fl @ " * ®

f = g o (I 8i ~ } 8 ]@(n-i-2))}

fn = fE M i , then there is a

300

g(v1~'" .~v.@v.+.v. l z / i+2_Sv.+~...SVn~ I ~ Let

x

be a fixed element in

= f(v18...Sv n) = + f1(vl)'...'fn(V n) .

V 2. Then

x = v,v'

g(v18...~v.SxSvl i+3~ . "~Vn) =. c(x)f1(vl)f2(v2)" . . c(x) depends only on

x . It follows that

c

for some

v,v'~ V

and

"''fi(vi)fi+3(vi+3 )° "''fn(Vn ) " is a linear form, and hence

g ~ ( V * ) ~i ~ (V~)~ ~ (V~) ~(n-i-2) . Hence, i m ( d # ) ~ ( V ~ 8n C~o
CO~Qllar~ 1.1... ( c f .

[12, p. 42] )

graded vector space and

8 (im(¢~)~VW8

Let

R = T(V)/~

V W) 8 (V*) 8(n-i-2) C

V = ~i>oVi where

be a l o c a l l y

~[ = g~ ~> .2 _

Then

R

is bigraded and

EXtR(k,k)

finite c V8 i

and

'

1

~

l

is trigraded, where we let the first

degree be the homological degree, the second arising from the tensor degree of R

and the third arising from the pure degree of [Ext~(k,k~

where

¢

Proof.

= ~,qExt~'n'q(k,k)

is the multiplication map

Since

R

V ~ V

R . We have

= T(V~)/(in~ ~) -~

VL .

is connected in the first degree, it follows from the

Bar-resolution that Ext~'n(k,k) and since

R

=

(V~)~n/im(d~)~ (V~) @n

is two-homogeneous and

V

is locally finite

im(d*)~(V*) ~n = (im(%*)) .

Corollary 1.2

(cf. ~ 2 , Proposition 9. 5

k-algebra with augmentation ideal Let

Let

and suppose

R

be a graded augmented

I/I 2

is locally finite.

A = [Ext~(k,k)] . Then, [Ext~(k,k~

where

I

)

%: I/I 2 @ I/I ~

-~

: T(I/I2)/(ker(%))

I2/I ~

is induced by multiplication.

,

301

Proof.

A

is a graded augmented algebra. _Put V = Ext~(k,k) = (Iii z) .

A = k ~i>ivi -we get that

A

, so

A

is two-homogeneons. From the proof of Theorem 1.1 ~since I/I 2 is locally finite;

is a quotient of

T((iiI2)@)/(im @ ~ ) ~

be the map induced by multiplication

in

A . Then

Let

@: V @ V

--* V 2

im @~ = ker @ . Hence,

im @ ~ = (im @~)0 = (ker ¢)00 = ker ¢ (of. Notations no. 5).

Remark.The corollary holds also for a local commutative noetherian ring

R

with maximal ideal

I , indeed Jan-Erik Roos has proved in [13] that in this case,

Corollary

k[x1~...Xn]

1.3

Let

algebra on the variables

denote the free graded strict commutative

Xl,...,x n

with certain degrees

deg x i ~ 0 , i = 1,...~n

Let

fi = [j<_kbijk~j~

, bijk~k

, i = 1,...,r

be homogeneous elements. Put R = k[x I ..... x j / ( f l ..... fr ) " Then A = [~t~(k,k)]

= k<~ 1 .....

~n>I(¢1

. . . . . %)

as bigraded algebras, T.l has bldegree (1,deg Xl) , and ¢i

([Tj,Tk]

= ~j!kcijk[Tj'Tk]

' cijk~

k , i = 1 .....

s

is the graded commutator defined in Notations no.4) and

i = 1,...~s

(Cljk)jk

is a basis of the solutions of the linear equation system lj~kbi~kXjk = 0 , i = I ..... r

(hence Also

s = n(n+1)12 - r ). [Ext~(klk)]

= R

where

R

is bigraded giving

xi

bigrade

(1,deg xi).

302

Proof.

Let

Themap

V

V

--*

be a graded k-space with basis {Xl,...,Xn, R

defined by

X.

--~

x.

i

Let

~: V @ V

--~

R = T(V)/(ker ¢) F 1 .....

where

Fr

R

be the restriction

[Xi,X3]

= X.SX.

l

-

D

of this map to

satisfying the conditions

from Corollary

Following Priddy

algebra

R

(-1)degxi'degx~

. ker @

X,SX. j

im 4~~ =

R .

i,5

is generated by = 1,...,n

l

(ker ¢)0

in the corollary.

is generated by

The last statement

follows

Koszul algebras

[12] we call an algebra of the form

a "homogeneous

pre-Koszul

is a "homogeneous

only if the "FrSberg formula" a complex

R

T(V)/(~ 2 )

al~,ebra". A homogeneous

Koszul algebra"

In this section we will prove that

construct

--~

1.2 . Q.E.D.

Homogeneous

c V 8 V

T(V)

V ~ V . Then

A ~ T(V~)/(im ~ )

F i = [j
elements

~2

extends to a map

1

and from Theorem 1.1 and

deg X i = deg xi}.

if

[Ext~(k,k~

is a homogeneous

PR(x,y)'HR(-x,y)

= I

with

pre-Koszul = EX%R(k,k)

.

Koszul algebra if and

holds. To do this we first

(U,d); which is the foundation of the material of this

paper. The complex coincides with the "Koszul resolution"

defined in Priddy

[12]

when it is acyelic. From now on 1/12

R

is a two-homogeneous

local l~ finite.

I = V~

12

algebra with augmentation

and

V2g]l 3 = [O} . Let

as usual be restriction

of multiplication.

spaces

Kn = 0

Kn

as follows:

is defined for Then

K

for

ideal

~: V 8 V

I -~

n
and

K I = V . Suppose

in: K n

--~

18i

Hence for

n

~81 n

V 8 V @ Kn_ 1

n>2 , K

= n

n-2 ~ V ~i 8 ker @ ® V 8(n-i-2) i=O

........ ,

Kr

V 8 Kn_ I .

is defined by Kn+ I = ker(V @ K

V2

Define graded locally finite vector

r<_n and there is an inclusion map

n+1

and

V 2 @ Kn_1)

303

U

= R @ K

n

d: U n

--~

is a graded free R-module. There is a homogeneous map

n

Un_1

with

d 2 = 0 , indeed

d(R @ K0) = 0

I@i d: R @ Kn where

m: R @ V

--~ R

n>1

m@1 n

R @ V ~ Kn_1

~

is induced by multiplication

It is easy to see that

and for

d2 = 0

R @ Kn_1 in

R .

and it is formally proved by the following

commutative diagram I@i R®K

n

R @ V @ Kn_ I

n

I@I@in-i

m@1 ~

R @ Kn_ I

i

I 1@in_1 m@I~1

~v~v~K_2

, R~v~_

18¢@Ii

im@1

R @ V 2 @ Kn_ 2 d 2 = (m81)(1@in_ I )(m@1)(1@i n)

U = .~>0Un

~ U

(-1,0) . The complex

Proposition

(U,d)

R @ Kn_ 2 Kn ~

= 0 .

is a bigraded R-module and

d: U

--* U

is of degree

has the following two properties

PI.

~: Un/IU n

P2.

Z U ~ 12U + BU n n n

1.2

m@1 ~

and from the definition of

(1@¢81)(1818in_1)(1@in) Put

2

K = ~>0~ n

--~

and

IUn_i/I2Un_ I for

is mono for

n~1

n>1 . --

[Ext IR(k ,k )]

are isomorphic as bigrade@

vector spaces. Proof.

K

is a s~hspaoe of

V ~n

~a

K~

~

(V~)~n/(K) ° . It fonows

from no.5 in Notations that (Kn)0 = Hence

~(V*) @i @ (ker ¢)0 @ (V~@(n-i-2) K 9~= T(V~)/~im ~ )

from Theorem 1.1 .

= ~(V*)@i @ im ~* @ ( V ~ 8(n-i-2)

as bigraded vector spaces. The statement now follows

304

Proposition

1.3.

Suppose

R

is two-homogeneous,

=kllvliv Consider the following

Suppose

U = R ~k X

(b)

~XCV~X

(c)

ZU

(U,d)

C

12U

and

identity map,~This

n

+ BU

X

(U,d)

of R-modules.

is a graded vector

for

n

(U',d')

n>1 --

s~isfy

f: (U,d)

--*

homomorphism

are graded algebras algebras~

where

for a complex

space and

X0 = k ,

,

homomorphism

unique

properties

(a)

n

lil 3

such that

.

these

conditions.

(U',d')

such that

is an isomorphism. (R ~ X,d)

Then there

and

fo: R

Moreover

(R ~ X',d')

--* if

is a R

X

is the and

X'

are differential

i.e. d(u°v)

(and the same formula

= du.v + (-I) deg(u)

for

u-dv

d' ), then the homomorphism

f: U

--~

U'

is an

algebra homomorphism.

Proof.

Suppose

f. : 1

we get a commutative

X. 1

--~

X: 1

is defined

--'~ X

Because

--~

n

V ~ Xn_ I

X~ n

--~

--*

V ~ X' n-1

an isomorphism.

If

(using the fact that

fn-1

(a) and (b)

--*

Xn_ 2

[ 1~fn-2 V

~ X' n-2

f : X n n

--~

we extend the maps

f (or the "five-lemma")

is an algebra homomorphism,

d': X' n

is an algebra homomorphism.

of

V2

d'

and hence

By linearity

U'. The uniqueness

--*

i 1~fn-1

of (c) the rows are exact,

defined by the diagram. f: U

, From

d

--*

d' 0

i~n

diagram d

0

for

--~ Q.E.D.

V ~ X' n-1

X' n

is uniquely

{f.} i

to a map

shows that

f

is

then it is easy to see

is injective)

that also

f

n

305

Theorem 1 . 2 V

locally

Suppose

finite

and

R R

is

two-homogeneous,

generated

by

V

R = k jj_ V j_[ V 2 jJ_ 13 ,

as

an algebra~

then

the

following

equivalent:

are

(i) R

is a homogeneous Koszul algebra,

(ii) Ext1(k,k) (iii) R

generates

is a homogeneous pre-Koszul algebra and

(iv) R

ExtR'P'q(k,k)

= 0

for

n ~ p,

is a homogeneous pre-Koszul algebra and all matric Massey products of

EXtR(k,k)

(v) R

are zero,

is a homogeneous pre-Koszul algebra and satisfies the

"FrSberg formula"

(vi)

EXtR(k,k),

the complex

(vii) R

PR(X,y)'HR(-x,y)

(U,d)

= I ,

defined above is acyelic,

is a homogeneous pre-Koszul algebra,

R = T(V)/(~ 2)

and

R

has distibutive associated lattices in the sense of Backelin [3]

§2. Proof.

We first prove that (ii) implies that

R

is a homogeneous

pre-Koszul algebra. We know from Proposition

1.1 that

(~ = ~ 2

C If.

of

j~ ~ 3

with

U I --~ U 0 , i.e.

we get T(V)

Henee

Z = ker(R 8 V

EXtR(k'k) ~- K2 = ~ 2 --~ R ~ V

g~/V~

~2

" Now

" Hence

and

~3

-~

~>_3

R) .

+ ~V (i)

we refer to Priddy [3]

Let

Z

From Proposition

Z/VZ ~- ~ 2

--~ g~V = and

Z/VZ (ii)

-~

~

-~ ~ 2

--*

Z

be the kernel 1.2 and (ii)

" The natural map

--~ 0

" It follows that

is generated

are equivalent by definition and from Corollary 1.1

it follows that (ii) and (iii) are equivalent.

Backelin

V ~i

where

induces sm exact sequence 0

by

@~2 C,V 8 V

R = T(V)/~

For the proof of (i) <=> (iv)

[12] and for the proof of (vi) <=> (vii) we refer to

306

(ii)--> (vi): Suppose that

Hi(U) = 0

for

i~p-1 . Suppose

We want to prove that d: h + 1

[Ext~(k,k)] i = Ext~(k,k) n
Hn(U) = 0 . Put

and

for

H.(U) = 0 1

Zn = ker(U n

--~

from PI). From Proposition 1.2 and since

i
--~ Un_1) . We have

Z ~ n

IU

~

.

is an isomorphism. Now Nakayama's lemma (see Notations no. 12) may

he applied. Indeed, since the "tensor" grade gives --~

(which also follows

n

n+1<_p we get

(Zn/iZn)le -- ~~x ~.n+1 [Extl(k,k)]n+1 -- K + I R ( k , k ) --

n

for

ZnlIZ n

is mono because of PI page 13 and the fact that

d: U

We elaim

"-~ Zn . We must show that this map is epi. But, [: Kn+ I : Un+I/IUn+ I

Hence

i~p.

Z n

R = T ( V ) / ( ~ 2) R

with

g~2CV

8 V

(see above)~

a structure of a graded connected algebra and

is a homomorphism of graded modules.

(vi) => (v): The complex

U = R ~ K

satisfies the conditions given in

Notations at the end of no. 11. Since

K

has one additional grading, we get

the following formula HK(X,y)HR(-x,y) = HHu(x,-x,y) Since

(U,d)

is acyclic (v) will follow from (vi) if we can prove that dU~iU

HK = PR " But hence

implies that

We will prove that

has differential zero and

follows that

n~1

H..(U) = 0 mj

holds. Moreover HK(x,y )

It follows that

(Ext (k,k)) i = Ext~(k,k)

Hi(U) = 0 for

of

if

x n+1 - term

PR(x,y)HR(-X,y )

K i = Ext~(k,k) and

for

j=0,1

so formula (I) shows that the xn+1 - term

and

for

i<_n .

(Ext~(k,k)) n+1 = ~xL = ~n+1 R (k,k) . From the proof of

(ii) => (vi) we have that

Hence

Hom(U,k)

Ext~(k,k) = K ~ . n

(v) => (ii): Suppose that

The

(1)

PR(x,y)

for

i<_n-1 and from P2 page 13 it i>O . Hence of

HK(X,y)HR(-x,y)

for

i+j=n+1

is zero.

is also zero since we assume that (v) i<_n by assumption and Proposition 1.2.

are equal up to dimension

Kn+ I = Ext~+1(k,k)

H..(U) = 0 10

and hence

n

of the variabel

(Ext~(k,k)) n+1 = Ex~n+1(k,k)

x .

,

307

Remark

The proof originates

essentially

(v) <=> (vi) in the ungraded commutative

Suppose

R

is a graded augmented

13 = 0 . Then trivially sections

are valid.

V = I/l 2 , so

R

In fact

(U,d)

bigraded vector space

K

of

k

in both degrees.

0 is exact for all

--~

n>0

Kn+ I

it follows that

Cn+ I

has tensor degree

Put

n>_0

=~>O~Zn.--n

and

~ then

Put d --~

V 8 Kn

is locally finite.

We will n o w define a

Un = R 8 K

d --~

that yields

is b i g r a d e d

n

and

(C n = 0 for n<1).

Since

V 2 8 Kn d: V 8 Kn+ I

--~

. There is a map p such that d d Kn+ I - - - V2 Kn_ I

V 2 ~ Kn

. Hence

Choose a k - l i n e a r homogeneous

Y = U 8 T(C)

~

map

is homogeneous

. Then

p

Cn

~:C n

--~

0 V2

8

Yn = (U 8 T(C)) n =

is a b i g r a d e d

Yn-1

as follows:

d'

d'(18v18...SVk) for

i=1,...,k

free R-module.

is R-linear for

and X~Un,

n>0

= a(v1)Sv28.,.Sv k

are homogeneous

Kn_ 1

also in the tensor degree.

p

--~

d

V 2 8 Kn_ I

d'(xSv1~...Sv k) = dxSv18...Sv k

1

and

n+1

Un0 8 Cnl 8 ... 8 C n

d': Yn

where v . E C

V

V 8 V

has a differential,

Cn+ I = ~n(U)

11 V 8 K n ~

1

Define

n

p ~ = id . Observe that

C = ~ L cn

--

,

and

R .

is the cokernel of

o is exact for

g~2 ~

that

section.

U 8 T(C)

I

and

Un = Kn

such that

over

tensor degree n ~ the degree zero]

is h o m o g e n e o u

Cn

Suppose furthermore

such that

ideal

so the results of the previous

R = T ( V ) / ( ~ 2 + V 83) Where

as in the preceding C

a minimal resolution

algebra with augmentation

is two-homogeneous,

is bigraded.

We may construct

We give

R

case.

(I(R),,,),,, 3 = O.

case

The

from Christer Lech. He proved

elements,

d'

is clearly

308

homogeneous of degree (-1,0,0) and

Propos~tiqn I.~

Proof.

(Y,d')

a(v)~V2y

for

Ho(U) = k

also

v~ C .

is acyclic.

We have an exact sequence of complexes, 0

--* U

where the differential on Suppose

(d') 2 = 0 , since

nhl

and

--~ Y

--~ Y 8 C

Y 8 C

Hi(Y) = 0

is

d' 8 1 . Since

1
for

--~ 0

. We claim that

HO(Y) = k

H (Y) = 0 .

The following sequence is exact

Hn+I(Y 8 C) Since C has degree

~

Hn(U)

(the "first" one)

---* Hn(Y)

>2

Z n ( Y ) ~ C n = {0} . Hence

Cn+ I

is a part of

Hn+I(Y ~ C) Hn(Y)

Hn(Y)

Hn+I(Y @ C)

--~ Hn(U)

--+ Hn(Y ~ C)

to

Hn(Y 8 C)

Hn(Y ~ C) = (H(Y) 8 C)

--

But

--~

~

= C n

--* Hn(Y 8 C)

n

is the zero map.

and the restriction of the map

Cn+ I

is

a

which is epi. Hence

is mono and therefore

Hn(Y)

must be zero. Q.E.D.

(The technique of the proof is due to Gerson Levin.)

Corollary ! . h

The graded vectorspace En'n+P = ~ Z

K ni=n

Also

E n'n+p = 0

for

p>n/2 . C

E n'r = Ext~'r(k,k) 8 C

nO

nI

is obtained from

XHc(x,y) + 1 = HK(x,y)(1 Proof.

8 ... ~ C np K

is given by

for

p>_O .

by the following formula,

- Hv(Y)X + Hv2(y)x2)

The definition of the tensor degree of

K

and

C

.

gives the first

formula. To prove the second formula we use (I) on page 16 . We have Cn+ 1 = Hn,2(U)

where the sum of the indices of

~(U)

is the tensor degree.

Hence, HK(x'y)HR(-X'Y) = ~n>1 xn+2Hc --

(y) n+1

+

1 = xHc(x,y) + I . Q.E.D.

309

We could also give a formula for the Poincar@ series of will define one additional degree of series of Let space

R

and thereby obtain a Poincar@

in four variables.

A = K* = ~Ext~(k,k~ E

ExtR(k,k)

is filtered by

trigraded vectorspace

and

E = ExtR(k,k)

E ~ EIE D (EI)2E ~

(E1)rE/(EI)r+IE

The coefficient before ~+t=q

= [E I]

....

. The trlgraded vector

Let for

xnyPzqu r

in

PR(x,y,z,u)

must consider tlhe left A-module structure on is a minimal resolution of

= HOmk(V,k) and

g C HomR(Yn,k)

k , so

• Lift

g

HA

E . E = HomR(Y,k)

to a map

go: Yn

gl £ HomR(Yn+I,YI)

g£Homk(Knl

8 Cn2 ~ ... ~ Cnk , k) . It is enough to define

and extend it linearly to d'

into

such that

8 C nI

~ C

~ ... 8 C

n2

and

nk

d'gl = (-1)ngod'

Yn+1 " There are two factors of

R 8 K C

~ ... ~ C n2

The differential of the last one is in

, namely nk

~ C

ni+I

~

... ~ C

n2

nk

V2Y

•

and for this reason we

find a map

n1+I

be the

is the dimension of

Er+1'r+1"sEn-r-l"p-r-1"t

We are going to compute this series in terms of

K

Qr

.

Er'r's~n-r'p-r't/~+t=q

mapped under

r>O

~where x,y,z are the variables ) for the homological degree, PR (x'y'z'u) = ~r>O H (x,y,z)u r , ) t h e tensor degree and the pure -- Qr L degree respectively.

Definition.

(Y,d')

R , but first we

, hence n

dg1(Cn1+l @ Cn2 ~ ... ~ Cnk) ~ V 2 and hence g1(Cn1+1 8 Cn2 8 ... 8 Cnk) E VY I and therefore f o gl(Cn1+1 ~ Cn2 @ ... ~ Cnk) = 0 .

. Let

f(HomR(Y1,k)

--* YO " We must . Suppose gl

on

Yn+1 8 k

Yn+1 ~ k that are

=

310

An element in

K

8 C ni+I

xfK I

and

Y(Knl

8 ... 8 C n2

may be written as

8 Cn2 8 ... 8 C

. Define

(_1)n + deg(x)deg(y) then

~'gl = (-1)ngod'

g1(x 8 y)

E = A 8 T(C ~)

is a subalgebra of

E

and

is defined by the following: Let x@(K

8 T(C))

Theorem 1.3.

E = A 8 T(C ~)

I

and

R

x 8 y~K

8 T(C)

g(Homk((T(C))n,k)

such that

•

y £ (T(C)) n

g(y)'x .

be a graded augmented algebra with augmentation ideal

13 = 0 . Suppose

V = I/l 2

is locally finite. Let

be the map induced by multiplication PR(X,y,z,u)

.

as right T(C~')-modules

T(C@)'A): Let

then m [(x 8 y) = (-I) n'm + deg(x)deg(y)

Let

f(x)g(y)

as left A-module. We can also prove

(however nothing is said about the product

and

as

and

Thus we have proved that T(C ~)

where

g(y)-x(K I ,

f 0 g1(x 8 y) = (-I) n + deg(x)deg(y)

that

x 8 y

nk

: XHA(Xyu,z)/(1

in

@: V 8 V

--~ V 2

R . Then,

+ x - HA(xy,z)(1

- Hv(Z)Xy + Hv2(Z)X2y2))

where A = [Ext~(k,k)]

and

where

Proof.

PB(x,y,z,u)

= T(V~)/(im¢ ~)

is defined On page 19.

It follows from above that n,n+p Qr

~

I~Z nl.=n-r

K

r

~C

nl

8 ... ® C

np

Hence, HQr(X,y,z)

= %~0(xy)rHKr(Z)(Hc(xY,z))PyP

= (xy) r HKr(Z)/(I

- YHc(XY,Z)).

According to Corollary 1.4 , XyHc(~Zy,z) + I = HK(xY,z)(I and if we use the fact that

H K = HA

- Hv(Z)Xy + Hv2(Z)X2y2)

we obtain the formula for

PR(X,y,z,u)

.

311 Remark.

If we put

Poincar@ series for Application. Let

(R,m)

z = 0 R0

and

y = I

and

u = I

in

(2)

we get the

obtained in [101 .

(This application is due to Gerson Levin.)

be a commutative local ring with

Im/m21 = 3 , which is not an

artinian complete intersection. Wiebe [20] has proved that in this case (HI(K))3 = 0 , where

K

augmented algebra with

is the Koszul complex. Hence

H(K)

is a graded

(I(H(K))) 3 = 0 , and the results above may be applied.

From Avramov [2] we get

PR(X) = (1 + x ) 3 P H ( K ) ( x , t , x , 1 ) Hence the problem of rationality for where

PR

.

is reduced to that of

HA(X,X)

A = ~[Ext!(K)(k'k)] ~ J

Theorem 1.h.

Let

(R,m)

be a local ring with imbedding dimension

which is not an artinian complete inter~tion. Then

(3)

H(K) = T(V)/(g~ 2 + V @3)

Suppose

S = T(V)/(~2)

where

~2 ~V

Let ~ V

K

3 ,

be the Koszul complex.

and

V = I(H(K))/(I(H(K))) 2 .

is a homogeneous Koszul algebra then,

(~)

PR(X) = x(1 * x)3~I . Xms(-X,X) - ~H(K)(-x,~. Proof.

By Theorem 1.1

A = [Ext~(K)(k,k) ] = [Ext~(k,k)]

, and by Theorem 1.2

HA(X,y)Hs(-X,y) = I . Hence by Theorem 1.3

PH(K)(X,I,x,1) and

(4)

Remark. three if

follows from

= xl(t + X)Hs(-~,~) - F~(K)(-x,x ~

(3) •

Levin [9] has proved the rationality of (HI(K))2 = 0 . His formula is the same as

"S = H(K)"). A ~ a m o v

has proved the rationality of

and three relations", and again the formula for

PR

PR (h) PR

in imbedding dimension (in this case for "three generators

is given by

(h) .

312 A Poincar$ series.

We end Part I with a theorem, that is a generalization of the method above to compute the Poincar@ series for satisfy the condition

Theorem 1.5. (&~2 C V 8 V

V

to algebras that not necessarily

13 = 0 .

Suppose and

H(K)

S = T(V)/(~ 2)

is a homogeneous Koszul algebra

is locally finite). Let

p~3

and

R = T ( V ) / ( ~ 2 + V~ ) then PR(-x,y,z,u)

= x p-2 .

(Hs(Xyu,z))-I/(xP-2

I + (Hs(XY,Z))-IHR(XY,Z))

and particularly

PR(-X,l,z,1)

Proof.

Since

as above

R

= x P - 2 / ( ( x p-2 _ 1)Hs(X,Z ) + HR(X,Z))

.

is two-homogeneous we may construct the complex

[see pp. 12-13). Put

Cn+ I = Hn(U)

(Cn = 0

for

(U,d)

n<1)._ By Theorem 1.2

there is an exact sequence 0 for

--~ K.l

n>0

and

d -'* V 8 Ki_ 1

i = n+p-2 . Hence

--

for

d d --~ ... --~ V p-I @ Kn_ 1 C

has tensor degree

--~ Cn

--~ 0

n+p-2 . Exactly as

n

p = 3

we get a minimal resolution of the form

EXtR(k,k) = A ~ T(C ~)

as left A-modules where

U @ T(C)

and

A = [Ext~(k,k)] = Exts(k,k)

As in the proof of Theorem 1.3 we get PR(x,y,z,u) = We have

On+ I = Hn,p_1(U)

HA(XyU,Z)/(1 - yp-2Hc(XY,z))

where

n+p-1

is the tensor degree, hence by (I)

we get HA(X'Z)HR(-X'Z) = ~n~l xn+p-I(-I)P-IHCn+I (z)

+

I •

Hence xP-2Hc(-X,Z) = I - HA(-X,Z)HR(X,Z) We now put this into the formula above for

PR

.

and use the fact that

HA(-X,Z) = (Hs(x,z)) -I , which follows from Theorem 1.2 o

313 1..5.

Corollary

Let for

p>3

R = k[x 1 ..... where

deg(x i) = 0

x.x. i j

for

and

1
6~

x n] / Ol + (x 1 . . . . .

Xn)P

is generated by elements of the form

then,

pR(-X) = x P - 2 / ( ( x p-2 _ 1)Hs(x ) + HR(X)) S = ~[x 1 .....

where

Proof.

S

xn]l,~

.

is a homogeneous

Koszul algebra according to FrSberg

Part 2. Commutative

[5]

local ringg:

Notations.

(R,m) k = R/m

is always a local noetherian

ring with maximal ideal

U

U = __~'>0 Ui

is a graded and each

(skew) commutative

Ui

differential

ZU , reduced homology

HU , divided polynomial

variables to kill cycles~free All R-al~ebras --~

R/m

extension,

are considered

such as reduced cycles

algebra, the process of adjoining

etc. we refer to [6] ).

.

(-I) deg(x) M

and

as augmented with the natural augmentation

We use the following sign notation

If

algebra over

is a free finitely generated R-module

U 0 = R (for more details and other related definitions

R

m .

is the residue-field.

An B-algebra R ,

commutative

and

we write N

: ~ = (-I) deg(x) + 1 .x , instead of

(-I) x .

are submodules

of

L , we write

M/N

instead of

M/M~N

.

314

Serre. R Talgebras.

Definition.

An. R-algebra

if the following

Proposition Then

=>

2.1

Let

X

Suppose

is an

n~1

and

if

deg(S)

since

of the definition get

dXn_ I ~

Hence

The Koszul

Proof. killing

2.1.

Corollary

etc.

in Serre

and

[I~

).

s g ZX , s ~ BX + m2X

element

dXn_ I - ~n s E m2X

from

dx n ~

that

m2X

we have

.

in

X' (S (i) = 0

, hence

for

is not a unit

xn

x n E mX . As a consequence

ZX c mX

, hence

, since

x' { mX'

3 in appendix

{ m2X ' , where

we get

Xn_ I g mX

, finally we get

(Proposition

s E mX

deg(Xn_1)

and we

> deg(Xn)

. Q.E.D.

to chapter

IV in Serre

[14]

).

is an S-R-algebra.

an S-R-algebra.

that represents

2.2.

IV

is a homogeneous

. It follows

is trivially

cycles

R-algebra)

S-R-algebra.

of an S-R-algebra

complex

R

(Serre

u~ U 0

d(x 0 + xiS + ... + XnS(n))

. Hence

dXn_ 2 E m 2X

Corollary

(i.e.

m2X

S-R-algebra

S-R-algebra

is odd). We get

s ~ BX + m2X

or

to chapter

be an

x' = x 0 + xIS + ... + Xn S(n) i>I

u~mU

(c), appendix

X' = X<S: dS = s>

Proof.

is called an

property holds du~m2U

(cf. condition

U

a basis

The Koszul

for

There is a free extension

U = K<...S .... : dS. = s.> I

I

i

) such that

m/m 2

U

complex

is obtained by

.

of the Koszul U

complex

is an S-R-algebra

and

7U C m2U + BU .

Proof. Ui

Put

UI = K

and suppose

is an S-R-algebra

represent

a basis

and

~.uic 0

Ui

m2U i + BU I

for z.ui/(m2U i + BU i) . l

is constructed for

j
for . Let

i>I

such that

S1"" "" "Sn

~ 0 .

315

Define

U i+I

as

U i+I

ui<s1 ,...,Sn: dS~ = s 0.> . It is clear that

is an

S-R-algebra (Proposition 2.1) and z.ui+I~: m2U i+I + BU i+I j Finally, define

U

as the union of all

Proposition 2.2.

for

j
Ui .

Let

U

be an S-R-algebra. It is possible to imbed

in a minimal resolution

Y

of

(i.e.

Y = U ~R F , F

d(u~f)

= duff

Proof. ~(yi) and

is a graded free R-module, F 0 = R , and

y0 = U

for

F i. = 0 0

~
for

and suppose and

j>i

i>0

and

yi

yi = U ~R Fi ' where

is graded free, F i 0=

Fi

R

= (du).y

- ~.~y

for

u~ U

and

y ~ yi

is considered as a U-module in a natural way). Choose a free R-module such that

Fi+ I

~

•

N

"

.yi

F i + 1 ~ k ~ Zi Y1/(mZ'Y~l + Ba Choose a map

a:Fi+1

~

I Define

d

j~

on

Z'Yla

i

Fi+i/mFi+ I yi+1 = yi

) .

__~ ~.yil such that the following diagram commutes

Fi+1

Put

is constructed such that

and

d(u.y)

(yi

k , which is a free differential U-module

- ~.df).

Put

= 0

U

~

ziYi/(mZi Yi + BiYi) •

U ~ Fi+ I = U ~ F i+I , where

U 8 Fi+ I

(5)

F i+I = F i

I]

Fi+ I .

by the "derivation" formula,

d ( u ~ f) = d u ~ f - Z - ~ ( f )

Then

d2 = 0

and for

ueU

and

y = u' 8 f ~ y i + 1

d(u-y) = d(uu' ~ f) = d(uu') ~ f - ~ ' . ~ ( f ) = (du)-y

- ~-dy

.

we get

= (du)'y - ~.(du' ~ f) + ~'~''~(f) =

316

B.Y i+I = im(~) + B.Y i . From (5) we get

We have

1

i

im(~) + B.Y I + mZ.Y i = ~.yi . i

i

i

Hence by Nakayamas lemma ~.yi = im(e) + B.Y I = B.Y i+I 1

Hence

Hi(Y i+I) = 0

Hence

yi+1

i

and also

follows that

an

S-R-algebra).

Y

for

Z Y C mY . We have

j
X~Zr(Yi)

for

j
Finally put

ZUCmU

ZYICmYI

yi .

Y = ~i>O U

is acyclic (so far we have not used the fact that

We will now prove that

and

~i.(yi+1) = ~j(yi) = 0 J

satisfies the induction hypothesis.

It

ZYJCmY J

1

is

and suppose . Suppose

i>I

r>i

and

,

~ u r ~ F 011Ur_ ~ ~ F IL[... ]IUr_ i ~ F i • We write and

x = [0
a k = (akl ..... akn k)

where

is an

b k = (bkl ..... bknk )

nk-tuple of elements in

% ~ bk = 11<1<% %1 ~ bkl We have

d(a i) ~ b i = 0 , hence

If

we have

r=i

Suppose now for all

and

ak~mUr_ k

k , so

d(ap)~ m2Ur_ p ak~mUr_ k

ai~ R

for

d(a i) = 0

for all

k

Ur_ k

and hence

k>_p+1 (p+1<_i). We have

such that

since

and

aiEmUr_ i

a ( a i b i ) ~ B ( Y i-I) , hence by (5)

ap~ mUr_p

Fk

"

d(~o~k<_ p a k ~ bk) E m 2 Y I . Hence and hence

is a basis for

if

ai~m

r>i . .

dbk~ zyi-lcmyi-I

d(ap) ~ b p ~ m2Y i , hence

p
0
x ~ m Y ~. Q.E.D.

It is probably well-known that the Ext-algebra is generated by the one-dimensional

elements if and only if any minimal resolution of

k

is an

S-R-algebra, but anyway it follows from a lemma by Roos [13] , which is a more general result.

317

Lemma 2.1.

Let

(R,m) be a local ring, Then [Ext~(k,k)]

is the

kernel of the natural map: EXtR(k,k )

--~

ExtR(R/m2,k )

Proof.

See

Let

be a minimal resolution

Y

[13, Cor I, page 291].

of

k . The dual of the map above is the

natural map H(Y/m2y) By the lemma

[Ext~(k,k)]

--~

H(Y/mY) = Y/mY .

= EXtR(k,k)

if and only if this map is zero in

degrees greater than zero, which is the same as saying that

Y

is an

S-R-algebra.

Corollar~ 2.3. U

satisfying

[Ext~(k,k)]

ZUgm2U

+ BU

= EXtR(k,k)

if and only if any S-R-algebra

is a minimal resolution.

\and Coro~.lar~- 2 . 2 J Proof.

The "if-part" clearly follows from above~ Suppose

S-R-algebra

satisfying

resolution

Y

of the form

lowest index such that ~ ( F i ) c mZU . Take now an S-R-algebra

ZUCm2U

Y = U 8 F . Suppose

and therefore we conclude

Proposition 2.3. Y , which

~: F i

y ~ F i , y ~ m F i . We have

Let

U

is an

2.2 gives a minimal

F @ 0

F i + 0 . We msO~ choose

generated by its one-dimensional

resolution

+ BU . Proposition

U

and --~

i ZU

is the such that

d y e m2y , hence

from above that

Y

EXtR(k,k)

is not is not

elements.

be any R-algebra such that there is a minimal

is a free differential

U-module

(cf. Proposition 2.2) ,

Y = U 8 F , F 0 = R , thsn, (a)

U 8 k

is a subalgebra of

Y 8 k

and

F 8 k

Y 8 k , F 8 k

is a quotient Hopf algebra of

is the algebra cokernel of

U 8 k

--~

Y 8 k.

318

(b)

If

HHom(1,¢): HHOmR(U,U) --~ HomR(U,k)

the augmentation)~ k

(c)

If

dFCm2y

Y ~ k , U ~ k

(d)

U ~ k

U

--~

Y ~ k

--~

F ~ k

--~ k

is an S-R-algebra then, F ~ k

is a quotient Hopf algebra of

algebra eokernel

is

we have an exact sequence of Hopf algebras:

--~

and

i s epi (¢: U --~ k

F ~ k

If both the conditions

is a subalgebra of

Y ~ k

and

U ~ k

is the

--~ Y ~ k .

in (b) and (c) are valid, then we have a split

exact sequence of Hopf algebras:

k i.e.

--~ H o ~ ( F , k )

--~ Hom~(Y,k)

Ho~(Y,k) = H o ~ ( F , k ) ( ~ U o ~ ( U , k )

(a) , (b) and (c) may also be formulated

algebra

HOmR(Y,k)

k

in terms of the dual Hopf

.

(a). The algebra structure on

complexes

-~

( s e m i - t e n s o r p r o d u c t , see Smith [19] ).

Remark.

Proof.

--~ HomR(U,k)

@: Y ~ Y

--*

this kind by induction.

Y

Y ~ k

is determined by a map of

lifting the identity on

Suppose

@

k . We define a map of

is defined up to degree

n , such that

¢(uly I ~ u2y 2) = (-~)Ylu2 u ~ % ¢ ( y I ~ y2) where

u.~l U

(F ~ F)n+ I

and

yiE Y

for

i=1,2

. Define

¢

on

(Y ~ Y)n+1 ~ first on

and then in general by ¢(ulf I ~ u2f 2) = (-1)flu2 UlU2@(f I ~ f2 )

where

u.~ U i

and

f.~ F I

for

i=1,2

this defines a lifting

@: Y ~ Y--~

a subalgebra of

and that

hence

F @ k

Proving Hom(F,k)

F ~ k

Y ~ k

. It is straight-forward

Y . It is clear from this that

~'>I

Ui ~ F ~ k

is the algebra cokernel of the map is a coalgebra quotient of

is a subalgebra of

to check that

Hom(Y,k)

.

Y ~ k

is an ideal in U ~ k

--*

U ~ k Y ~ k ,

Y ~ k .

is the same as proving

is

319

Let

fEHom(F,k)

. Define

up to dimension for

u( U

zero on

and

n,

f ~ ZHom(Y,Y)

such that

f

y ( Y . Define

f

inductively. Suppose

is zero on on

Yn+1

it

is easy to see that

It

follows

is

zero on this ideal if

Hom(F,k) (b).

that

f

this

f

and

is a subalgebra of

Hom(~j>__l U 8 Fj ,k)

f~ZHom(Y,Y)

u(U

in some way, as

and

a lifting

f'

in itself.

such that

%(F ~9 F ) C F

prove that

Ui ~ F

--~ Hom(U,k)

may be chosen to be

f

maps

U

into

+ mY . Since

Yi~mUi

~ Fr_ i

for

Hence

f'

. o

is an algebra map, i.e.

on

zero

f6Hom(U,k)

U , and it is a there is a lifting

U .

and

d(F ~ F ) ~ m 2 ( Y

+ mY . But dye m2y

Y = [1<_i<~r Yi dYE m2y • Then

f

Hom(Y,k) . Hom(Y,k)

U

d y ~ m 2 y => y E F

Y~'>I

of

is, and thus we have proved that

We first prove that it is possible to define the map

such that

.

v~F

fEZHom(Y,Y)

1 ~ . > 1 Ui 8 F 8 k

right ideal since, by assumption, to every

and

Fn+ I

is a two-sided ideal. It is a left ideal since a lifting

of a map which is zero on

if

}(uy) = (-1)Uf-uf(y)

first on

for

defines

maps t h e i d e a l

We must prove that

(c).

and

is defined

U , and generally by f(u 8 v ) = ( - 1 u f . u f ( v )

Again,

U

f

then where

dFCm2y y£mY

--~ Y ,

it is enough to

so it is enough to show that . Suppose

y~ ~ U i 8 Fr_ i

y 1 ~ mU I ~ Fr_ I , since i<s-1

8 Y)

~: Y ~ Y

U

for

1
is an S-R-algebra. Suppose

then,

d([s
@(F 8 F ) C F

and also that since

ysE mU s 8 F

+ mY

U @ k

eoalgebra of

. Hence

it follows that

F 8 k

y ~ mY . is a subalgebra of

is the algebra cokernel of the map

__II~>I u @ F. 8 k _

r-s

is an ideal. The fact that

U 8 k

F 8 k

Y 8 k

--~ Y 8 k ,

is a quotient

J

Y 8 k

will follow from Theorem 2.1 below, which says that

320

Hom(U,k) (d).

is equal to the subalgebra generated by

Hom(Ui,k ) in

Hom(Y,k)

.

This follows from (b) and (c). Q.E.D. On the structure of Ext(k~k).

We want to apply the proposition to the case when all Massey products of HU

are zero. To do this we start with the following proposition~ whose proof has

been obtained in collaboration with Gunnar SjSdin. Propos%tion 2.4. HU

Let

U

be an R-algebra.

are zero. Then there is a differential on

locally finite free R-module, such that Here

T(F)

Suppose all Massey products of U @R T(F) , where

U @R T(F)

F

is a

is a differential~ R-algebra.

is the "shuffle" algebra defined in Notations no. 8 , and

T(F)

is graded by means of its pure degree.

Proof.

Since products of reduced cycles in

annihilated by

m = Z0 U . Let for every

Fi+ I @ k = Hi U . Choose a map Fi+ I @ k

li: Fi+1

i~I --*

U Fi

are boundaries, HU

be a free R-module such that

ZiU , that lifts the isomorphism

--~ Hi U . We are going to extend this map to a map

and by means of this define a differential on

is

l: T(F)

--~ U

U @R T(F) , such that

U ~R T(F)

becomes a differential R-algebra (it is well-known that it is possible to define on

~

U ®R T(F)

becomes a complex). We will define

T(F) inductively with respect to the tensor degree of

(1)

~

is R-llnear, zero on

as (ii)

T(F)

(iii)

k and zero on the decomposable elements of

F-algebra.

dl(x I ..... Xk) = ~1
i.e.

~

d(x I ..... Xk) = ~1
for

u~U

and

and

d

such that~

d o X = (~ @ X) o A i.e.

(iv)

such that

f~T(F).

T(F)

321

It is easy to see that (ii) - (iv) implies that of

U ~ T(F)

d2 = 0

is proved in the same way as Proposition

is defined on

T(F)

for elements

(1) - (iv) holds. Then tensor degree

I ~ I o A

n , indeed let

of tensor degree

and the acyelicity 1.3. Suppose now


such that

is zero on the decomposable

x,y

be elements

in

1

T(F)

elements of

of positive degree

then, (~ ~ ~) o A(x-y)

and for

= (I ~ ~)(ax.ay)

x = (x I .... ,xk)

= 0 + (~ ~ ~)(x ~ y + (-1)Xyy

of even positive

(pure) degree,

(I ~ I) 0 A(x (n)) = (I ~ I)(Ax) (n)

~_

xI

xll

. . . . .

....

..

n = 2 , but X(x).Mx)

x

.

.

.

.

(x p+,~

•

.

(I g l)(x ~ x) , which appears for

is of even degree implies that

l(x)

is of odd degree and hence

(l ~ l) o k(x)( = the set of k-fold Massey products

defined by

l(xl),...,l(Xk)

Hence, for every

, indeed a defining system is

x([T(F)) n = the elements

{Aij}

of tensor degree

{l(xi,...,xj)}.

n , there is a

such that (ii) holds. By Lemma 2.3 below we may choose a basis of the

free R-module elements

n

(T(F)) n

that, as a subset,

of tensor degree

to define (T(F))

.

(Ax) (n) =

= 0 .

We have

l(x)

where

p

the only term, that might not be zero is

~ x) = o

1

on a basis.

n . Since

contains

(I ~ l) o A

a basis for the decomposable is R-linear,

it is enough

It follows that it is possible to define

1

on

such that (i) and (ii) holds.

Lemma 2.2. an al6ebra

Suppose then,

1

is zero on the decomposable

(U ~ T(F),d)

is a differential

elements of

algebra where

T(F) d

as

is given

by (iii) and (iv) above.

Proof.

We must prove that

d(x.y)

= (~)-y

+ (-1)Xx.(c~)

for

x,y~U

It is easy to see that it is enough to prove the formula for prove it by induction on

dt(x'y).

~ ~(r)

.

x,yE T(F)

. We

322 ^

Let =

x = (x I ..... Xk) ' Y = (Yl ..... yn ) , x = (x I ..... Xk_ I)

and

(Yl . . . . . y n _ l ) . We get

d ( x . y ) = d ( x . ~ ~ Yn + (-1)xkY ~.y ~ xk) = d ( x . ~ ) ~ Yn + + (-1)XkYd(x'y)

+ (-1)X[(x'dy)

8 xk + ~(x'y)

=

(dx'y)

8 Yn + ( - t ) x k Y + X k ( x ' d y )

~ Yn + ( - 1 ) x k Y ( d x ' y )

~ Xk +

8 Xk~ = ( d x ) ' y + ( - 1 ) X x ' ( d y )

The first equality follows from the definition of the product in

. T(F) , for

the second equality we use (iii) and for the third we use (i) and the induction hypothesis. For the fourth equality we use (iii) twice indeed,

(dx.~) ~ Yn + (-1)xkY(d~'Y) 8 ~ = ((dx 8 X k ) ' y ) 8 Yn + %(x)'y + ( - 1 ) X k Y ( d £ ' y ) = (d~ 8 xk)-y + ~(x).y = (dx)'y the expression

Lemma 2 . 3 .

Proof. Let

(-1)x[...]

T(F)

x'y = x'y

B

and

,

F-algebra.

T ( F ) 8 k = @(F 8 k )

~i = Si " Let

8 xk =

is treated in the same way.

is a free

{S.}I be a basis for

=

is

a free

T(F) 8 k

F-algebra (see e.g. Andr@ [I] ).

as F-algebra. Choose

be the set of all monomials

x(~) = ~(n) , ~

By Nakayama'slemma, we get that

x!nl I" (n) 11 "'''XlpP

is a basis for B

{xi) , such that

T(F) 8 k

is a basis for

T(F)

•

Since

as vector space. as R-module.

This ends the proof of Proposition 2.4.

Corollary 2.h.

¢

Let

¢: (R,m)

-~

(S,n)

be a local ringhomomorphism, then

is a Golod homomorphism in the sense of Levin [8] if and only if there is

an exact sequence of Hopf algebras k

--~

T(V)

--~

Exts(k,k)

where the right map is induced by

~

-~ and

EXtR(k,k)

V

--* k

is a graded vector space such that

323 V I = (n/mS~

and

Vi+ I = (Tor[(k,S)) R

(for the "if-part" it is supposed that by

n/mS

and

for

Tor~(k,S)

ihl are annihilated

n ).

Remark.

The corollary may be applied to trivial extensions. In this case

there is also a homomorphism

@: S

--~ R

such that

@ o ¢ = id , hence the

sequence splits to the right (i.e. it is a semi-tensor product).

Proof. of

k

Suppose

over

Hi(X ~)

@: R

--~ S

R , X # = X 8R S for

is Golod. Let and

F

X

be a minimal resolution

a free S-module such that

Fi+1 8S k

i>0 . We know from Levin [8] that all Massey products of

= H.(mX ~)

--

i

are zero, which enables us to apply Proposition 2.4 and also to choose (and

d ) such that

(X ~ 8 T(F),d)

1

becomes a minimal algebra resolution.

Since the lifting property holds for

X ~ , we may apply Proposition 2.3 (b)

and get the desired exact sequence, where the coalgebra structure on the natural one. That also the algebra structure of

T(V)

is

T(V) is the natural one

follows from an explicit formula for the lifting of an element in Homs(T(F),k)

(this is formally the same as the one used in Part I page 2O).

Conversely given the exact sequence we get a formula for the Poincar@ series of

S

which shows that

Theorem 2.1. of the form HOmR(U,k)

Suppose

@

is Golod (see Levin [8] ).

R

is a local ring with a minimal resolution of

Y = U 8 F , where

U

is an S-R-algebra and

is equal to the subalgebra of HOmR(Y,k)

dFgm2y

generated by

k

. Then HOmR(Y1,k)

i.e. ~o~(~,k) = [~(k,k)] Proof.

It is possible to give a direct proof by proving that the image of

the multiplication map

~o~(U 1 ,k) ~ Ho~(U_t,k) is

HOmR(Un,k) . Firstly

HOmR(Un,k)

dFCm2Y

--~ ~o~(Yn,k)

implies that the image is contained in

, and secondly one proves that the dual of the map

,

324

Hon~R(Ul,k) ~ Hor%R(Un_l,k) ---~ HomR(Un,k) is a monomorphism,

which follows from the fact that

U

is an S-R-algebra.

It is however also possible to use Lemma 2.1. We have to prove that the algebra cokernel of the natural map

%: H(Y/m2Y)

U ~ k . From the proof of (c) in Proposition dy ~ m2y Hence

im(¢)

is equal to

=>

Theorem 2.2.

Suppose

a graded vector space of

EXtR(k,k)

Proof. ~UCm2U

U

and Proposition

2.3 (c) gives that

is a local ring and

V = ~.>iVi

, such that

[Extl(k,k)]

8 T(V)

and also as left

[Ext~(k~k)]

-modules

m~ = 0

all Massey products

2.h gives an algebra resolution

the differential

d

(U ~ T(F),d)

BUCmU of

of

is a sub-Hopf-algebra

and as right T(V)-modules.

U

with the property HU

of the form

ZUCm

2

U + BU

are trivially U ~R T(F)

the map

, may be chosen such that

is a minimal algebra resolution.

and if we put

The last isomorphism

and since

U ~ T(F)

EXtR(k,k)

structure of

m 4 = 0 , then there is

T(V)

=

+ BU . Since

as algebras,

U 8 k

and

is an S-R-algebra,

Hence

2.3 we have

Apply Corollary 2.2 to get an S-R-algebra

Proposition

is exactly

~ .

(R,m)

EXtR(k,k ) as coalgebras

Y/mY

y ( F + mY .

F 8 k

is the algebra cokernel of

--~

V = (F 8 k) ~ = Hom(U,k)

zero.

. Since

I , defining

im(1)~m2U.

Thus

we get an isomorphism

~ T(V) = [ E ~ ( k , k ) ]

as coalgebras

8 T(V)

e

follows from Theorem 2. I. The assertion about the algebra

EXtR(k,k)

follows from explicit lifting formulas as on the pages

19 and 20. It is however also possible to apply the theorems h.h and 4.7 in Milnor-Moore

[I I] :

325

At first we consider the Hopf algebra map

A = [Ext~(k,k)]

--

~xtR(k,k)

= B

It follows from the proof of Proposition 2.3 that Theorem 4.h in [11] now gives that

B = A ~ T(V)

@

T(V) = k ~A B . as left A-modules.

Secondly we consider the coalgebra map

B = E~tR(k,k) (arising from the algebra map that

T(V) = k ~ A

right

--~ U

[~t~(k,k)]

= A

--* Y ). It follows from Proposition 2.3 (a)

B . Theorem h.7 in [11] gives that

B = A 8 T(V)

as

T(V)-modules.

Remark. rings with

Compare this theorem with the result obtained for non-commutative m3 = 0

Definition.

Let

in Part I page 20.

(R,m)

be a local ring, then

ring with respect to the filtration

Theorem 2.3.

Suppose

(R,m)

GrR

is the graded associated

F R = mPR . P

is a local ring with

m 3 = 0 . Then

A = [Ext~(k~k)] = [EXt~rR(k,k) ] and there is a graded vector space

V , such that

of

and

EXtR(k,k)

and of

EXtGrR(k,k)

ExtR(k,k) as coalgebras

and as l e f t

PR(x)

Remark.

Proof.

The fact that

We construct

GrR-algebra, where

T(V)

= E X t G r R ( k , k ) = A ~ T(V)

A - m o d u l e s and as r i g h t

= POrR<X) = x H A ( x ) / < t

PR(X) = PGrR(X)

(U,d)

T(V)-modules. Moreover

+ x - HA(X)~(-x)).

was first proved by Levin.

as in Corollary 2.2. We have

d1: U ÷ s-IU

(GrU,Grdl) is a

is a map of filtered modules and Fo(S-Iu) = FD+I(U)=

= mP+Iu. It is easily seen to be an S-GrR-algebra. Since have

is a sub-Hopf-algebra

Z G r U ~ m 2 G r U + BGrU , and hence

ZUcm2U

HU = HGrU = m2U/BU .

+ BU

we also

326

It trivially holds that

U ~R k = GrU ~GrR k

HomR(U,k )

=

HOmGrR(GrU,k)

and •

Hence everything follows from Theorem 2.1 and 2.2. The formula for

PR follows

from Theorem 1.3. Q.E.D.

Next we are going to look for sufficient conditions for a ring to satisfy the premises of Theorem 2.1.

Definition.

A local ring

a minimal resolution S-R-algebra and

Lemma 2.4. U

F

(Y,d)

(R,m)

is said to have "property E" if there is

of

of the form

k

Y = U ~ F , where

is a free graded R-module with

A local ring

(R,m)

dFcm2Y

U

is an

.

has property E if there is an S-R-algebra

satisfying du~ m~U

=>

u~m2U

+ BU

or

u~U 0 ,

(6)

duffm3U

=>

u~m2U

+ BU

or

uEU 0

(7)

if furthermore

then = [ExtlrR(k,k)]

Proof. have

The condition (6) implies that

.

Z U ~ m2U + BU , in particular we

B0U = m . We use the same notations as in the proof of Proposition 2.2.

Suppose

zYiC m2y I + BY l

may be chosen such that

and

d F l ~ m 2 Y I-1

for

i<j , then

d: F. 0

--~ yj-1

dF.C m2y j-1 . We must prove that also 3

~YJC m2Y j + BY j Suppose

x

~zrYj

' x = ~o~<.<_j

ak ~ b k

We know from Proposition 2.2 that say

a~ = du . Then

hypothesis. Suppose

where

a k E Ur_ k

x f mY J . If

x - d(u ~ b j ) ~ Z Y J - I ~ m 2 Y J r>j

and

ak~ m2U

for

r=j

we have

+ BY j

k>p+1

and

b k ~ Fk aj~m

for

0
= B0U ,

by the induction

. Since

dbk~ m2Y j

we get

$27

dap 8 b p E m ~ U to

8 Fp . Hence

x' = [0<_k<_$ ~

8 b~

a p E m 2 U + BU

with

a~Em2U

and it follows that for

x

is homologous

k~p . At last we get

x ~ m2y ~ + BY ~ . Suppose (7) is true. Put

R' = GrR

and

(U',d') = (GrU,Grdl). U'

is an

S-R-algebra satisfying ZU'E

and since

~2U'

+

BU'

is homogeneous we also have

d'

d ' u ' E ~ 3 U ' =>

u E ~ 2 U ' + ZU' .

Combining these facts we get that (6) is true also for

EExt~rR(k,k) ] = Ho~,(U',k)= ~x~:~le.

Ho~(U,k)

~t is not always tr~e that

U' . Hence

: [Ext~(k,k)]

[~x~(~,~]

. Q.E.D.

: [~Xt~r~(~,~]

,

as the following example shows:

R = k[x,y]l(x 2 + y3,~) is a complete intersection and

~>~R(,<,~<~ : [~<,<,k~] ( o f . S~Sdin [15]

, [%2 ,T2] )

: ~<<~,~>/<~

). On the o t h e r hand GrR = k [ x , y ] / ( x 2 , x y , y

~)

and by Theorem I. I

r~Xt~r~(~,~ Theorem 2.h.

Let

(R,m)

:

~<~,~>i(~.

be a local ring. R

has property E if any of the

following conditions is satisfied:

(i)

R :

kilt I~. . . . . .

t lIln ~ il ( f

are linear forms in (ii)

m~ = 0

(iii)

R

..... fr) {t~t k]

+

and

.

is a complete intersection.

where

f. 1

for

g~ C(t1,...,tn )3.

i=1,...,r

328

(iv)

R

is "trivially"

Golod, i.e. it is possible to choose cycles in

the Koszul complex

K

representing

a basis for

H(K)

, such that

all products of these cycles are zero.

(v)

Im/m21 ~ 2

R = R/~

or

where

(R,~)

i s r e g u l a r , ~ C ~2

and

is generated by two elements.

Proof. by

(i). R

is two-homogeneous

(tl,...,tn)

(V

is concentrated

R = k ~ We construct Suppose

V ~

V

equal to the k-space generated

in degree zero),

V2 ~

13 .

an S-R-algebra as in Corollary 2.2 and we use the same notation.

Ui = Xi ~

for some

with

Xi

V ~ X i 11 V 2 ~ X i ~

and that the differential

13 ~ X i is homogeneous

of degree

+I

with

j.e. d
follows that

vv

•

U i+I

du(m3U

--

s1~...,s n

- -

~

may be chosen in

has the same properties

has these properties and

.

where

X

= 13 8 X , then

as

Ui

is the union of all u(mZU

+ ZU , but

V ~ Xi

and it

and hence also

U = R 8 X

X i . Suppose now

ZUCm2U

+ BU . Hence

u6 U (7)

of Lemma 2.h is true and we are done. (ii).

This ease follows trivially

from Lemma 2.h and we have also proved

it in the proof of Theorem 2.2. (iii). for

HI(K)

Let

K

be the Koszul complex and

, where

sl,...~s r

represent

sl,...,s n

represent a basis

a basis for

Z~(~)/(m2K + BK) and

si( m2K

for

i=r+1,...,n

K<S1'''''Sr where

dS i = s i

for

, then a minimal resolution of

is given by

: dSi = si> ~R R<Sr+1'''''Sn >

i=r+1,...~n

. It follows from Proposition

K<SI,...,S r : dS i = si> is an S-R-algebra.

k

2.1 that

329

(iv).

Let

X

be any R-algebra that has "trivially"

equal to zero, i.e. there is a set

{s.} 1

~X , such that

i

s.-s. = 0 l j

for all

all Massey products

of cycles representing

and

j . Let

a basis for

s E {s.} . We claim: l

Y = X<S: dS = s> has also trivially

all Massey products

zero. Indeed, suppose

y = x 0 + xiS + x2S(2) + ... + XnS(n) is

a homogeneous

to

X'n = ~risi

cycle

in

where

Y . We g e t

riE R , so

dx

y

= 0

n

x

, hence

is homologous

n

is homologous

to

x0 + xlS + "'" + x~-Is(n-1) + x's(n)n Since d(x~S (n)) = 0 , the same argument may be repeated and we get eventually that

y

is homologous

to

x~ + x~S + ... + x'S (n) n where ~Y

x ~ ( [Rs i

for all

in this manner,

j . If we choose cycles representing

it follows that the product of any two of these is zero.

Thus the claim above is proved. a complex

U

zero. Since

that

has property E.

(v).

If

R

ZUcm2U

+ BU

U

Golod, we may construct

has trivially

all Massey

it follows as in the proof of Theorem 2.2

This follows from (iii) and (iv) (for a proof, see [6] ). Q.E.D.

Remark.

The condition

(i) of the theorem is equivalent to

and two-homogeneous.

Indeed if

and two-homogeneous.

Conversely suppose

then

is trivially

as in Corollary 2.2 such that

products R

a basis for

R

is a quotient of

R

k ~t I ..... t n ~

By definition of a two-homogeneous tl,...=t n

satisfies R

(i) then clearly

R

complete R

is complete

is complete and two-homogeneous, , since

R

is an algebra over

k

@

algebra we may choose the variables

in such a way thai the relations between them are of the given form.

S30

It is possible to prove (i) of the theorem by using the complex (R 8 K, d) defined on page 12-13 for two-homogeneous algebras. In the commutative ease this complex has additional structure. K

is defined there as a suhspace of

T(V)

on page 5, it is fairly easy to

and, using the definition of

see that

K

T(V)

is a sub-Hopf-F-algebra of

T(V) (where the elements of

V

have

degree one). This also follows from [16, Theorem 2, page 17] , since the dual of the map

K

--~ T(V)

this map is equal to

is the natural map

W(f)

T(V ~)

--~ T(Ve)/(im@q)

for a map of graded Lie-algebras where

and W(')

is

the universal enveloping algebra functor. Moreover it is an almost direct consequence of the definitions on page 5, that the differential on

R 8 K

is compatible with the algebra structure and the structure of divided powers, Proposition 1.3 page lh tells us that this d{fferential algebra structure is unique and also that the differential algebra defined in the proof of (i) is isomorphic to

CorollalV

(R 8 K, d).

2.5.

Suppose

by linear forms in

R = k 6 1 ..... tn~ / ~ + ~

{ti-tj}

and

~C(tl,...,tn)S

where

g~ is generated

and one of the following

two conditions is fulfilled:

( a)

(t I ..... tn)4

(b)

k~1,...,tn]/( ~ k

such that

~_. is a homogeneous Koszul algebra and there is

(t I .... , t n ) 2 ~

Then there is a graded vector space

C ( t I ..... tn )k+1

V = __/I~>IVi

and a split exact sequence

of Hopf algebras

k

--~

T(V)

--~

EX~tR(k,k )

--~

KExt~(k,k)] ....

i.e.

EXtR(k,k)

is a semi-tensor product of

a

T(V)

and

-'*

k

I

" -rExt~(k'k)7" n

331

Proof.

Suppose (a) is fulfilled. We want to apply Proposition 2.3 (d)

so we must prove that the complex defined above has the lifting property. But this is easy: Let

f @HOmR(Un,k)

= HOmk(Kn,k). Put

f(x I ~ ... ~ Xn+ r) = (-l)nrf(xr+1 ~ ... ~ Xn+r)(X 1 ~ ... and extend

f

to

R 8 Kn+ r

Xr )

by linearity. We get the split exact sequence

and the argument for the fact that the Hopf algebra structure on

T(V)

is

the natural one is the same as in the proof of Theorem 2.2. Suppose (b) is fulfilled.

Let

be the minimal resolution of HU = H(R 8R' U') cycles in

k

be the ring

over

kIEt I ..... t n ~ / ~

(tl,...,tn)kU

[Ext~(k,k~

. Let

U'

R' given by Theorem 1.2. Then

has all Massey products zero(trlvially),

is a Golod homomorphism fact that

R'

representing a basis for

since we may choose R'

HU . Hence

R

(see [8] ) and we may apply Corollary 2.4 and use the =[ExtR,(k,k)]= HomR(U,k)

, which follows from

Theorem 1.1 and 1.2. Q.E.D.

EExt~(k,k~

as enveloping algebra of a graded Lie-algebra.

It has been proved by Andr@ [I] and SjSdin EI6~ , that = universal enveloping algebra of a graded Lie-algebra

EXtR(k,k) = WL = L = ~.>iLi

where

i

L I = Ext~(k,k). Let ~

be the graded Lie-algebra generated by

that

.

W~

= [Ext~(k,k)]

L I . We claim

The following lemma is due to Gunnar SjSdin. Le~mna 2.4. WL

Suppose

generated by

Proof.

~

~

~

L

are graded Lie-algebras,

is equal to

is imbedded in

WL

(PBW) (see e.g. SjSdin [16] ). Let

W~

.

by the theorem of Poincar@-Birkhoff-Witt A

be the subalgebra of

. There is an epimorphism of algebras diagram commutative

then the subalgebra of

W~

--~ A

WL

generated by

which makes the following

332

--~

W~ The map

W~

--* A

--~ A

--~ WL

WL

--~ A is a monomorphism, because of PBW. Hence also

is a monomorphism.

Q.E.D.

We know, by PBW, that the Hilbert series of

has the form

(I + x)60(I + xS)62..../(I - x2)61(I - x~)63-... where

6i = dimk~i+ I .

Definition.

The natural numbers

Definition.

(cf. Sj6din [17~ )

nilpotent,

i.e.

6i = 0

are called the subdeviations

[Ext~(k,k)3

= W~

for sufficiently large

tde fined above l Since ~ Y ' i s generated by

Theorem 2.5.

{6i}i>0

Let (R,m)

~I

' W~

~

is

i .

is nilpotent if

be a local ring and

6 0 = dimk(m/m2)

is nilpotent if

R

of

K

6.1 = 0

i .

for some

its Koszul complex. Then

, 61 = dimk(ZIK/(m2K + BK))

and

82 = dimk~Z2K/(m2K + BK)) Proof.

We will use Lemma 2.1. Consider the minimal algebra resolution

obtained by killing cycles (see [6~ ). Let H(Y/mZY)

--~ Y/mY

By Lemma 2.1, the algebra cokernel of

¢

Y

be the natural map

•

~

is equal to the dual of

~Ext~(k,k)~

We are going to compute the dimension of the ideal generated by im(¢) in degrees ~3 • Choose a basis {[i,...,~nI}

is a basis for

{[1,...,Se }

for

ZIK/(m~K + BK)

ZIK/BK and

such that

Sn1+1,...,selE mZK . Put

333

E = K<Si: dS i = s.l , i = 1,...,si> . Choose a basis for

{~I'''''~E2}

Z2E/(m2E + BE)

and

for

Z2E/BE

such that

is a basis

{~1'''''Un 2- }

Un2+l , .... uc2E m2E . Put

F = E . We have of

F. = Y. J

for

j~3 • Since

is an S-R-algebra, there are no elements

K

K/m~K in im(@). Suppose y = ~riS i + x

is an element of degree two in

r.~R i

with Y

for

such that

i=I,...,~ I

and

x~K

d y E m 2Y • Then

~risiE m2K + BK which implies that ~1~i!nlrisiE m2K + BK and hence

ri6 m

for

dx6 m2K

i=l,...,n I , it follows that

We have proved that im(@) is generated by z = [riU i + [xiS i + x 0

with

riE R

SnI+I,...,ScI for

hence

x{m~

.

in degree two. Suppose

i=1,...,E 2

and

xi~ K for i=0,..,E I.

We get [riui6 m~E + BE and as above this implies that

d(~xiS i~ hence

dxi(~ m2K , hence

im((~) is generated by

+

r.~ m i

for

i=1,...n 2 . It fellows that

Xo)~ m2E

x i @ mK

and finally also

Unz+1,... ,UE

Xo~ mK . We have proved that

in degree three. Thus we have proved that 2

a basis for the algebra cokernel of of degree _<3 S 1,...,Snl

in the variables

and

(~ in degree <--3 , is given by the polynomials

T 1,...,Tg0 (the Koszul variables)

U 1,...,Un2 . Hence

6 o = s o , 61 = n I

It remains to prove that Z2E/(m2E + BE) ~- Z2K/(mZK + BK) .

and

,

62 = n 2 .

334

Suppose y = [r.S. + x 1

is an element in element

x'

with

1

Z2E . 7T~en riE m

in

xE K

for all

i , so

y

is homologous to an

Z2K . Suppose x'E m2E + BE , x' + d([xiS i + X O ) ( m S E

.

Then

dxi~mSK

for all

Remark 1.Let Let

yn

Y

i =>

x.¢mK1_

for all

i

-->

x'~mZE

+ BK

=>

x'6 m2K + BK

be the minimal algebra resolution obtained by killing cycles.

be the subalgebra of

Y

generated by variables of degree
6'n = dimk~nYn/(mSyn + BYn) are invariants of

R , since

yn

to see, using Lemma 2.1, that

6

are invariant complexes (see [ 4

)" It is easy

< 6' . We have proved equality for n -- n

n<2

and

it is possible to use the same technique as in the proof above to prove equality for

n~5

but the equality is probably not true in general.

Remark 2.

If the complex

U

obtained from Corollary 2.2 satisfies

HomR(U,k) = ~Ext~(k,k)] we get the subdeviations as

ai = diem ZiU~/(mSU I + BU I) with notations as in the corollary. This is the case if,for instance,some of (i) - (v)

of Theorem 2.4 is fulfilled.

Christer Lech has derived an equivalent cond~.tion to

62 = 0

when

R = k litI ..... tm]] /(fl ..... fr ) where

f. I

i=l,...,r

are linear forms in

result by means of Theorem 2.5.

{t.t.} . We are able to prove his i J

335

Theorem 2.6.

Let R = k lit I ..... t ~

where

f. m

i=1~...,r

independent.

are linear forms in

{t:t.} z j

and

f1'''''fr

are linearly

Then 62 = 0

Proof.

/(fl ..... fr )

<=>

{t~fj}

linearly independent.

We may apply Notations 'no. 11 to the bigraded Koszul complex for

GrR

and get

(1 + x)nI~..R(-X) = H H ( K ) ( X , - X ) where

hi) = dimkHi)(K) , here

We have Let

h11 = r , h12 = 0

c

i

= 1 - h11x2 + ( - h 2 t

is the Koszul degree and

and

h21 =

j

+ h12)x3

is the ring degree.

62 ,the last equality by Theorem 2.5.

be the dimension of the linear space spanned by

{t.'f.}. From above i j

we get -

62

((n~2) =

-

C) + n(

-

n+1 (

2

)

-

r)

- (~)n

(~)

+

=

c

-

mr

"

Thus 62 = 0

Corollary 2.6.

c = nr

<=>

{t:f.} IJ

linearly independent.

Let = k

where

<=>

I .....

/(fl

.....

fr

+

g ~ C ( t 1 , . . . , t n )3 . Then nr - 62 = the dimension of the k-space spanned by

Proof.

{t.'f.} . i j

Let R' = k litI ..... tn]~ /(fl ..... fr ) "

By Theorem 1.1

Theorem 217.

[EXtR1(k,k)] = [Extl,(k~k)~

Let

(R,m)

+ ...

.

be a local ring with

m4 = 0

is nilpotent and one of the following conditions is valid

such that

336

(i)

R

(ii) Then

PR

Let

Massey products

R ~ GrR .

U

be given by Corollary 2.2. Since

=

Hu(X)I(I

has all

-

Hs~u(X))

(sHU)i+ I = Hi U . From Theorem 2.1 and 2.4 we have HomR(U,k)

The condition

= [Ext~(k,k)]

[Ext~(k,k)]

is nilpotent

is a finite extension of

says that

HHu(x)

Suppose

is rational if

Ui

U

implies

Hu(X)

is rational and that

is a finite free extension

(a)

R

has characteristic

p>0

(b)

it is possible to extend "derivations"

j

K

and

,

(b) is satisfied.

is homogeneous

to

U .

We use the same

as in the proof of (i) of Theorem 2.h. Suppose

such that

of

[6'] which

is valid:

R ~ GrR . We will prove that

notations

.

K . Now we apply a theorem by Gulliksen

one of the following conditions

on

m 4 = 0 , HU

zero. Hence PR(X)

U

,

is rational.

Proof.

where

p>O

has characteristic

in the ring degree,

j

is a derivation

i.e.

j(x i) c x i-deg(j) Let

S

be a variable of degree

it is enough to show that But we may choose

s

in

j(s)

i+I . In order to extend is a boundary

V ~ X i , hence

and this is a boundary since

Z.U i c J

and hence we may choose the extension the ring degree

m2U i + BU i of

(since this construction

formula on page 41). Q.E.D.

j(s)

j

j

to

Ul<S: dS = s> ,

(see [6, Lemma 1.3.2 page 16~). is a cycle in for

V @ X i-deg(j)

~
j(s)~ dX

such that it is homogeneous

is unique, ~

in

must satisfy the lifting

337

Corollary 2.7.

Let R = k

where

fi

be a local ring with

1,...,t

gi

and

/(fl,...,fr,gl,...,gs) +

R = GrR , i.e.

tl, ..., t n

are homogeneous elements of degree three. Suppose

linearly independent. Then Proof.

mh = 0

are homogeneous elements of degree two (in the variables

degree I~ and is

(R,m)

PR

of

{tif j}

is rational.

By Corollary 2.6 we have

and from Theorem 2.7 we get that

ti

62 = 0 . Hence [Ext~(k,k)] PR

is nilpotent,

is rational.

References [I]

M. ANDRE, Hopf algebras with divided powers, J. Algebra 18 (1971), 19-50.

[2]

L.L. AVRAMOV, On the Hopf algebra of a local ring, Izvestija Akad. Nauk SSSR set. mat. 38 (1974), 253-277.

[3] [4]

J. BACKELIN, A distributiveness property of augmented algebras, and sQme related homolo~ic~l results, part of thesis, Stockholm 1982 (defended January 21, 19~3). H. CARTAN, Homologie et cohomologie d'une alg~bre gradu@e, S@minaire Henri Cartan, 11e ann6e 58/59, expos6 15.

[5]

R. FROBERG, Determination of a class of Poincar6 series, Math.Scand. 37 (1975), 29-39.

[6]

T.H. GULLIKSEN, G. LEVIN, Homology of local rings, Queen~ Paper No. 20, Queen's University, Kingston, Ontario, (1969).

[6'] T.H. GULLIKSEN, On the Hilbert series of the homology of differential graded algebras, [~

Math. Scand., 46 (1980), 15-22.

G. LEMAIEE, Alg~bres connexes et Homologie des Espaces de Lacets, Lecture Notes in Mathematics, No. h22, Springer Verlag, Berlin, 1974.

[8]

G. LEVIN, Local rings and Golod homomorphisms, J. Algebra 37 (1975), 266-289.

[9]

G. LEVIN, Two conjectures in the homology of local rings, J Algebra 30 (1974), 56-7h.

338

Do]

C. LOFWALL, The Poincar~ series for a class of local rings, Preprint series, University of Stockholm, No. 8, 1975.

[Ii]

J.W. MILNOR, J.C. MOORE, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211-26h.

[12]

S. PRIDDY, Koszul resolutions, Trans. A.M.S. 152 (1970), 39-60.

[13]

J-E. ROOS, Relations between the Poincar@-Betti series of loop spaces and of local rings, Lecture Notes in Math., 740, Springer-Verlag, Berlin, 1979.

[18

J-P. SERRE

Alg~bre locale multiplicit@s, Lecture notes in Mathematics,

No. 11, 3 ed., Springer Verlag, Berlin, 1975.

[~~]

G. SJODIN

A set of generators for ExtR(k,k),

Math. Scand., 38 (1976), 1-12.

[16]

G. SJODIN. Hopf algebras and derivations, J. Algebra, 64 (1980), 218-229.

[~7]

G. SJODIN. A characterization of local complete intersections in terms of the Ext-algebra, J. Algebra, 64 (1980), 214-217.

D8]

G. SJODIN. The Ext-algebra of a Golod ring , Journal of Pure and Applied Algebra, 38, 1985, 337-351.

L. SMITH, Split extensions of Hopf algebras and semi-tensor products, Math. Scand. 26 (1970), 17-41. H. WIEBE, Uber homologische invarianten lokaler ringe, Math. Ann. 179 (1969), 257-274. Department of Mathematics University of Stockholm Box 6701

s-113 85 STOCrmO~ (SWEDEN)

THE

GENERAL

EXTENSION

OF A L O C A L

RING

AND M I X E D

MULTIPLICITIES.

D.REES.

The

present

Stockholm School was

paper

during

in A u g u s t

able

1983.

to e x t e n d

extension

is an e x t e n d e d

a Research

appears

Symposium

During

the

considerably as t h e o r e m

version

preparation

the main

4.6

of a l e c t u r e

associated

with

of

at Summer

the m a n u s c r i p t , t h e

result

of

will

also

below.lt

given

the N o r d i c

that

author

lecture,and

be d e s c r i b e d

this

in

this

introduction. The

proof

existence earlier set-up

of

this

of e l e m e n t s

part

of

the

in w h i c h

r e s u l t r e q u i r e ~ the of i d e a l s

paper

We

start

with

ideal

m,residue

another

constructed (also the

also

in terms

of

in the

In the

(k = l,...,n.)j term

for

set

of

elements

of

of g e n e r a l

I is a set

of

paper,we

the

elements

the

is

then

m [ X I , X 2 . . . ].

similarly

defined

result

elementary

the

ideals

of

results

notion

of an

! = ( I i , . . . , I s)

an a r b i t r a r y

I I,.. .,Is. NOw w r i t e

~ X(N.+k)u(j,k)j

I .An i n d e p e n d e n t the

of

the

ideals

I i , . . . , I s.

=

basis

u(j,k)

Nj=nl+'''+ni~-l"

this

Q.This

definition

yl,...,y s chosen,and

definition

of Q g , a n d

set

of g e n e r a l

j = i ..... s

over

set

l,...,s

form

T(yj)

of Qg

standard

of

j

of

of the

(Xl,...,Xs)

First,choose

is

XI,X2,...

fact.

introduce of

Q we

This

general

of

choice

ideal

the

this

of a set

ideals

ideal QN

elements

the

the

prime

of r i n g s

from

Q has

d. W i t h

reasons).Qg

XI,...,XN.A

stem

that

extension.

of e l e m e n t s

of

consequence

the

of a

of e l e m e n t s

T is an a u t o m o r p h i s m

One

at limit

the

be d e f i n e d .

of i n d e t e r m i n a t e s

is n o e t h e r i a n , a n d

in two s t a g e s .

xj where

direct

mostly

yj a standard

general set

can

dimension

typographical

Q[XI,X2...]

Qg

section

set

its

of

general".The

indicating

Krull

of i n d e t e r m i n a t e s

that

for each

the

elements

and

proof

to a d e s e c r i p t i o n

(Q,m,k,d),this

as the

of g e n e r a l

is done

general"

termed

and

"sufficiently

devoted

k (=Q/m) Qg

ring

section

first

are

to Q a c o u n t a b l e

set

implies first

set

of Q. T h i s

ring

the

of a f i n i t e

independent

We now

local

be d e s c r i b e d

A.Grothendieck proved

ring

field

by X ( 1 ) , X ( 2 ) , . . .

1ocalisation

Qg can

a local

by a d j o i n i n g

denoted

which

therefore

such"sufficiently

maximal

associate

is

definition

let

is i n d e p e n d e n t also

is the f o l l o w i n g . Let

P range

over

the

set

of

of

the o r d e r

X(!)

denote

prime

ideals

340

of

Qg

minimal

endent

of

concerned we

the

set

of

ideals

X l , . . . , x s and

se

depends

a sub-set

independent

of

a set

of

of

Xl,...,x

good,this

meaning

of on

an

choice

Then

of

a restriction

if

a sub-set

P be

X(~).

choice

with

place

ideals

over

the

this

the set

now

good

consider

prime

ideals

ideal.Then t of one is

the of

p is

).Next,we

of

the

main

we

consider

on

the

field

of

v to

of

v and

v

to

Q,the

denote

it

of

Q definied

ideals

J of

is

to

be

of

m-valuations.We

condition

over

k. N o t e to

is

then say

Now

by

a restriction

is on

set

of

of !

is

independent P.We

p = Q~P,where

of

require

P ranges

Q.Further,if

p is

htp

=

that

union. Finally

Q in

p,and

the

local

over such

t,then

their

paper,we of in

of

L(v).If

the by

consider an

the

ring

the

a prime

p contains

there

another

m-valuation lecture

group

the

that

term

v(J)= set

an

v on

is

only

(Qg)p/p(Qg)p

ingredient

on

Q.The

referred

additive

p being

p # m,we

same

to

to

functions

the

definition

of

J is

now

v is

good

Q.We >0

on

consider if

it

v an the

restricted J is

on limit

of

v(J)

v(J),where

integers,

p the

lifting

function

independent.We

m-valuation

term

of

above. First

group

~.We

term

the

~$~(v(x)).If

of

linearly

+htL(v)

to

set be

restricted two

classes

satisfies

the

the

degree

left-hand

right-hand

side

side in

of

Q.Finally,an

come

to

the

of

final

the

main

~

(ll,...,Id)

and

a finitely

M

= d-I

transcendence

function. Let be

follows

that

!,it

a set

is

a general

condition

be

case. First

= dim(Qg).

of

value

ideal

we

proof

the

that

the

prime

let

is

( X l , . . . , x s)

(Xl,...,Xs).Suppose

concept

finite.The

t(v) t(v)

and

!

indep-

will

that

where

equal

We

a particular

over

given

value

apply

m-primary,are

Q is

F of Q / p , w h e r e p is a p r i m e i d e a l of P restrictions that it is > 0 on Q / p and

further

will

m-primary,v(J)

the

v,with

m-valuation.We

good

over

the

that

fractions

lift

that

Qg(this

ideals

ideal

of

is

from

the

in

of

on !.

s-t.

a valuation

m/p. We n o w ideal

prime

section

differs

only

elements

dim(Qg/P) prime

minimal

dimension

theorem. This

here

subject

of

Qg

a good

third

ideals say

place

I. a n d is m i n i m a l J prime ideals P meeting

the

In the

of

p = P~Q

that

set

of

~.We

ideals

regular

given

the

of

general

parameters s

set

set of

htP+ We

the

=

if of

all

e(!

be

an of

; M)

the

the

an

c a s e s . If

we

residue

above

section,which

generated

parameters

of

is is

need

we

devoted

to

the

to d e f i n e s~of

set

of

;Mg

)

of

L(v) htL(v)

define

= e ( x l , .... x d

v at

if

Since

independent Qg.Then

good,then

K

is

proper

independent Q-module.

field

equation

v is

m-valuation

theorem. First

X l , . . . , x d is

of

v

most is

a

=0.

statement

a multiplicity ideals

I is

with

s=d,

independent,it

general

elements

of

341

w h e r e Mg = M ~QQg. Then e(l,M)_ is i n d e p e n d e n t It is an a d d i t i v e

function

on the c a t e g o r y

It is also a " m u l t i - l i n e a r " f u n c t i o n that,if

I = ( Ii, ...,lj_ 1 . lj .....

I"

is also

on the v a r i a b l e s

I d) and I'

_

both i n d e p e n d e n t

e(_l" ;M) = e(l function

plicity,reducing

we must

that J is any ideal

introduce

of T e i s s i e r ' s

set

of Q such

is to e x p r e s s

l'(v)

e(l,J

that

ll,...,Id_l,J

;M) in terms

the

I. c o n t a i n e d

is

over

sets

so

also

of

good

multilinear

it

was

this

In what ideal

m,residue

by the n o t a t i o n

field

sense

denote

m[X I ..... XN].It

was

IT

such

a

defined

above.

of a local

rin$.

will

by QN the

denote

in

dimension

for

implies

its

function

If

all

Q/L(v),where uniqued(I",v)

Ii,...,Id_

above

the

formula,d(I_",v) defined

1 are

sum are

all

proper,

lecture.

local d,this

ring with maximal being

is clear

set of i n d e t e m i n a t e s

summarised

that

Q ~

stated

Q1 ~

write X(J) in place of X . ) . T h e n l of Q [ X I , . . . , X N] at the prime ideal Q2 ~

is the same

them by m g ( m N ) . T h e

to k ( X I , X 2 , . . . )

over Q(as

occasionally

localisation

of the rings Q N , O r , w h a t

isomorphic

the

be a n o e t h e r i a G

Q [ X I , X 2 .... ] at m [ X i , X 2 .... ].The maximal we will

formula

in with

k and Krull

of

d(I",v).The

dealt

Q will

this and

Id(v)_l)

occurring

be a c o u n t a b l e

in the i n t r o d u c t i o n , w e

union

the

Q.In

values

(Q,m,k,d).

Let XI,X2,...

we will

coefficients

which

follows

of

the

Extension

on

integer

I_" = ( I 1 . . . . .

valuations

case

I. The General

m-valuations

existence

in

)v(J)

~d(l"(v),v)e(l'(V)Qv;M

II

ideals

determines

the

;M) =

non-negative

= dim(Q/L(v)).The

m-primary,then and

all

taking

independent d(v)

in

3

l.+p/p of Q / p , w h e r e 3 Qv for Qp and My for_ Mp. Then our

is

function

ness,and

let I be

to be the set of ideals

over l - l ' ( v ) . F i n a l l y , w r i t e

sum being a

is indep-

of m - v a l u a t i o n s . First

to be the set of ideals

d e f i n e l"(v)

e(l,J is

multi-

( l l , . . . , I d _ I) c o n t a i n i n g

--

expression

mixed

some notation. Let v be a good m - v a l u a t i o n , a n d

we define

p = L(v).Next,we lj ranges

,then

Id)

; M) + e(l_' ;M).

is a g e n e r a l i s a t i o n

that I is an i n d e p e n d e n t

objective

as a b o v e . T h e n

= (Ii,.. .lj -I' I', 3' ...,I d) are

to it if I i , . . . , I d are all m - p r i m a r y . .

Now s u p p o s e ideals,and

Q-modules.

I i , . . . , I d in the sense

and _I ''= (Ii,... ,lj -i ,Ill'j,..., set of ideals and

a general

endent.Our

of ( X l , . . . , X d ) .

generated

sets of ideals

This m u l t i p l i c i t y

d-I

of the c h o i c e

of f i n i t e l y

residue

.... We will

thing,the ideal

den~

by Qg the

localisation

of

of Q g ( Q N ) is m Q g ( m Q N) and

fields

of Qg'QN

and k ( X I , X 2 , . . . , X N) and will

are r e s p e c t i v e l y

be d e n o t e d

by

342

kg and

k N respectively.

dimension QN

is clearly

a consequence

noetherian

of a more

p20).However LEMMA

Finally

we give

and,in

general

an ad hoc

l.I

If I is a f i n i t e l y

Let

U l , . . . , u K be a basis

~(I

m

we

can

find

= mQg,it

g if I'

that

Qg'QN

N such

follows

fact,the

both

result

same

is

true

of Qg. This

of A . G r o t h e n d i e c k ( [ 2 ] , P r o p

is

10.3.1.

+ m n)

ideal

of Q g , t h e n

= I

that

of

I and

let x be an element

Ul,...,Ur,X

a11

all

tile ideals

I+m

/'~(I+m n). l g to QN.Ilence,as

belong n

have

g

a basis

of

in Q

.Therefore,

= I ~QN,

since

QN

n)

O(l+m

g

=

~ QN

¢

=

(l'+mn)QN

I'

~

1.2.

Since Qg of Qg

Qg is n o e t h e r i a n .

the maximal

ideal

of Qg

is n o e t h e r i a n . H e n c e , i f

contained

in l.Let

I' be the I ~

is f i n i t e l y

I is any

ideal

IQg~Qg

of Q -

ideal

generated,the

completion

of Q g , I Q g

a finite

generated

g

by

~o

=l'Qg~Qg~

I = I'

We c o n t i n u e the

proof

First

is f i n i t e l y with

has

this

,

#-~ (I +m n) l

hence

cases

I

is n o e t h e r i a n ,

THEOREM

into

Krull

g

that

x~

and

have

proof.

generated

t

Then

we note

d.

basis

basis.Then = I' ~

I

g

generated.

a number

is

left

to the

suppose

that

h:

of e l e m e n t a r y

results

concerning

Qg~n

many

of

ring

Q

reader.

Q--->Q'

is a h o m o m o r p h i s m

a local

a local

ring Q ' . T h e n we have a natural e x t e n s i o n h of h m a p p i n g g Q' o b t a i n e d by m a p p i n g the i n d e t e r m i n a t e s X I , X 2 , . . . into t h e m s e l v e s . g F u r t h e r , i f I is the kernel of h , t h e n Ig=IQg is tile kernel of hg. In

Qg into

particular,taking

Q'

= Q/l,we

obtain

(Q/l)g As a c o n s e q u e n c e , Next,suppose isomorphism pg,bOth

being

the

the

Note used

in the

Qg and

last

and

the

g

.

to be d i m ( Q / l ) , d i m l ideal

of Q.Then

localisation

as Q - a l g e b r a s . T h i s the

fact

that

consisting ideal

= INQg=

least

I ChQN

to

paragraph,we isomorphisms

M be a f i n i t e l y

have

are

have

will

of QN

that

isomorphism a natural

ideal

htp.

I be an ideal

if N is

I ~Q

this

prime

htpg-Let

= Io).

large Then,for

(QN)g

with

Qg in

is true

will

be

be d e n o t e d

isomorphisms

generated

that

identified

N for which

I and

the natural also

implies

the

is n o e t h e r i a n .

of QN(SO we

integer of

Qg

= diml. g there is a natural

of Qg at

of e l e m e n t s

(IN)g(here

of d e f i n i t i o n

in a d d i t i o n

(Qg)N.BOth

Next,let

of

a basis

way.).The index

that

(Qp)g

by I N the

enough,I

natural

termed

use

is d e f i n e d

p is a prime

considered

I has

enough. D e n o t e N large

that

between

Now we make of Q g . T h e n

if diml

an i s o m o r p h i s m

Qg/l

by i(I).

between

isomorphism

Qg and

(QN)g

between

as Q - a l g e b r a s .

Q-module. Then

we

define

M N and

343

M

a s , r e s p e c t i v e l y , M ~ Q Q N and M ~ Q Q g . W e can g as f u n c t o r s f r o m the c a t e g o r y of f i n i t e l y respectively,the

generated simply

categories

Qg-modules.These

reflects

the fact

of

finitely

functors that

M - - - > M N and M - - - > M

generated

generated

are

QN and

consider

both

Qg are

Q-modules

QN-mOdules

faithful

and

faithfully

g

to, and

finitely

exact.This

flat

extensions

of Q. Nowsuppose L ( ( Q / m ) N) Hence by

both

Hilbert

the

ideal

that

M is of

have

value

functions

passage

finite l,the

f r o m M to M N or M

a consequence,it

and Qg. We

simply

m-primary

ideal

We give THEOREM

1.3.

associated We

.To be p r e c i s e , i f

g

of M

recall

of M we

see

P'.We

can

that

of M

take

in X I , X 2 , . . . are

). g preserved

are

I is an m - p r i m a r y

that

result

is P.By

g

,and

that

u to be of

the

form

coefficients

define

P is

contains

the

of

the c o n t e n t

by the

for all By

the

of

prime

and

prime

the

P ranses

associated

over

the

of M if M c o n t a i n s

u to be a s u i t a b l e

an

element

c(u)

w h o s e a n n i h i l a t o r is g u(X) d e n o t e s a p o l y n o m i a l

of

of

these

sub-module c(f)

f(X).Then

of

of M

f to be

a classical

in the a n n i h i l a t o r c(f).This P'

prime,is

= Pg,where of

the

P' of

that

i(1)

u.

P'

P = P ' ~ Q.

individual

the a n n i h i l a t o r

associated that

implies

of

w i t h M.
i(P)JN

P o f I. associated

the m o d u l e

with

Qg/I.Now,if

I we m e a n

IN = If-~QN,

the

prime

ideals

I = INQg. H e n c e

the

result

= O.

annihilators

such

of g e n e r a t o r s

be one

the c o n t e n t

therefore

ideal

of Qg

a set

r

therefore

the

has f(X)

u to be the

and

some

hence,being

primes

of

= c(f)rc(fu)

of Q and of

of M

Q.Let

is c o n t a i n e d

P is a p r i m e

ideals

over

of u(X)

and

I is an ideal

with

g

,where

taking

coefficients

c(u)

intersection of u(X)

associated

associated

Q-module,then

u(X),where

that,for

c(f) r+l

by e l e m e n t s

them. H e n c e

C o r o l l a r y . If

section.

in M . F u r t h e r , P '

in X I , X 2 , . . .

states

annihilator

coefficients

P

u is an e l e m e n t

by the c o e f f i c i e n t s

is g e n e r a t e d

ideals

are QN some

of M.

c(f)r+ic(u)

P'

this

senerated

prime

if,for

L(Q/I) = e ( l ; Q ) .

of

P is an a s s o c i a t e d

of Q g e n e r a t e d

the

Q is C o h e n - M a c a u l a y

is an a s s o c i a t e d prime ideal of M if P is an g g ideal of M . C o n v e r s e l y , s u p p o s e that P' is an a s s o c i a t e d

(see N o r t h c o t t [ 4 ] )

Hence

;M ) g g if Q is C o h e n - M a c a u l a y , s o

by d e ! e m e n t s last

the

= e(l

that

P

with

generators.We

ideal

g ideals

polynomials

generated

the

annihilator

prime

ideal

are

that

u whose

prime

fact

I generated

prime

associated

the

of

e(l;M)

follows

I f M is a f i n i t e l y

element

which

use

a proof

prim e i d e a l s

one

L ( ( Q / m ) ) and g L(M)=L(MN)=L(M

that

and,consequently,multiplicities

e(IN;MN)=

Now

implies

of Q , t h e n

As

Bur

length.Then,since

above

Qg/I

344

is

isomorphic

2. G e n e r a l

(QN/IN)g

to

elements

DEFINITIONS.

of

Let

and

the

of

Qg

set

constructed

We

term

general such

elements

that

the

) be

elements

THEOREM

2.1.

- -

of

the

elements

follows. Let

of

For

Nj

automorphism

T of

of

each

mean

ideals a set

j choose

the

theorem.

of Q . T h e n , b y

I if t h e r e

exists

z. = T ( x . ) J J

an

of

a set

= nl+n2+...+nj_l.Then

we

an

a

of

Xl,...,x s

senerators

u(j,k)

take

(j= 1 ..,s) ,• independent set

automorphism

(j=l, .... s ) f o r m

elements

T of

of

Qgover

a standard

set

Q of

I.

If X l , . . . , x s ;x'! ' ' ' ' ' x's set

a set

of I we

x. = ~ X ( N + k ) u ( j k) J ~., J ' e l e m e n t s x I, • . . ,x s _of _ Qg

of

of

elements

elements

general

are

as

__°f Ij.

a set

from

s

of g e n e r a l

(k = l,...,n.)3

follows

ideals.

I = (Ii,...,l --

standard

result

~ = ( I i , . . . , I s)

Qg

over

Q such

are of

independent

sets

of Q , t h e n

there

assume

that

sets

of

general

exists

an

that

Without

loss

of

T ( x . ) = x~ J ] generallty,we can

standard

sets

of

genera]

elements ~j

xj

two

ideals

of ! , t h a t

both

is,we

can

~=~,X(Nj+k)u(j'k)

of

elements

take

(j=l ..... s)

~ j

xj' where

generators

of

Further,we

can

for

each

to X ( M j + k )

complete unit we

and

1,...,s

(aj(k,i))

of

now

basis

must

have

adding it

two

sets

of

two

automorphisms

of X ( 1 ) , X ( 2 ) , . . . , w h i c h

for

of

Nj=n1+...+n[j_l,Mj=ml+...+mj_l.

(u(j , i), .... u ( j , n j ~ is a m i n i m a l for e a c h j.

j = 1,...,s

fact,descrlbe

a minimal

Hence,by

and

(j=l ..... s) are

and

fixes

basis

of

Ij

T 2 T I . T 1 is X(r)

k = l,...,njand

if r > M s+l' otherwise

arbitrarily.Now T~ has to s a t i s f y ~j Z T2(~X(M.+k)u(j,k))= ~k(M.+ilw(j,i)

We w i l l , i n

=

that

that

X(Nj+k)

j

j = l,...,s

assume

by a p e r m u t a t i o n

is c o m p l e t e d

form

--=.~X(Mj+k>w(j'k)

n.< m. J-- J c o n s t r u c t T as a p r o d u c t

We n o w

sends

I.j for

j,so

determined

for

:

( u ( j , l ) .... , u ( j , n j ) ) , ( w ( j , l ) , . . . , w ( j , m j ) )

k = 1 ..... of an

suitable

to an m i~x m

Q,for define

T 2-

.We

can

u(j,k)

=

nj.Since

write

j (k,i)w(j,i)

the

elements

u(j,k)(k=l

l.,it follows that,for each j,the J nj. xn.j s u b - m a t r i x with determinant rows matrix

.....

nj)

nlxm. matrix J J a u n i t of Q.

of

an m . x m . i d e n t i t y m a t r i x , w e can J J Aj = ( a j ( k , i ) ) w h i c h has d e t e r m i n a n t

each j.With this extension -I T2 ,by f i r s t d e f i n i n g

of

the

definition

of

a

a.(k,i) J

345

~ T2-1(X(Mj+i) for

j = i .... ,s,i

This

can

now

= 1 ..... mj

be e x t e n d e d

and

to an

T2-1( ~X(Mj+i)w(j

T 2-

(X(r))=X(r)

automorphism

,i)) =

if

r>Ms+ 1 • satisfies

which

~aj(k,i)X(Mj+k)w(i,j)

= as

°

=l~'#j(k,i)X(Mj+k)

~X(M.+k)u(j,k) 3

required. In

the

following

corollaries,we

will

denote

the

ideal

( X l , . . ° , x s)

by X ( ~ ) . Corollary on

the

i) To w i t h i n

set

Corollary of

I and ii)

x I ......

Corollary

not

The

iii) with

ideals

is

of

P ranges

X(1)

follows

we w i l l

now

an i n d e p e n d e n t of _I is

is c l e a r l y prime

ideal

2.2.

Let

(Xl,...,Xs)= P be

i

I and

ideals

a sub-set of over

ring

of

X(i)

p is

not

on

only

the

choice

all

ii)

least prime

only

ring

prime

prime

must

pg. H e n c e

of Qg. T h i s follows

at m o s t

case.We

property that

s and

further

elements

the

dimension

term

a prime

if

independent set

over

of

set

of

ideals

general

X(!),and

prime

elements

suppose

that

of Q , a n d of ~. p=Pp-~Q is

ideal

of Q c o n t a i n e d

in P,

ideals of

dim(Q

ideal

P

to

o

Qg m i n i m a l

over

X(1)

whose

(Qg)p/p(Qg)p, s,whose

then

maximal

R is ideal

a oregular is g e n e r a t e d

of X l , . . . , x

P is

Qg

of

Q is p, rin~

ideal

is e n o u g h

ideal

an

of d i m e n s i o n

images

= d. S i n c e

of

_l=(ll,''',Is)

general

Then

prime

the

a chain

one ideal

of

= dimQ.

minimal

If R d e n o t e s

minimal

height

independent

with

It

have

minimal

of Q.

X(!),then

situation

ideals

set

X(1)

only

i)Clearly,any

dim(Qg)

of

parameters

an

over

= d,

the

bY the

a set

X l , . . . ,x s .It

intersection

.local

that

independent

of

+ dimP

ideals

a particular

of

ideal

the

with

a set

at

prime

ideals

choice

in

be

a]l

the

a good

idea]

Q/p

P is

iv

if

of

prime

concerned

an

of

prime

dim

iii

be

say

prime

ii

be

depends

on I__

set

equality

a good

a minimal

since

on

set

of m i n i m @ l only

I = ( I I ' ' ' ' ' I s ) be

let

the

d e s c r i b e . We w i l ]

a local

Let is

only

either

set

we will

minimal

d-s,with

P of

the

htP

LEMMA

a Q-a1$ebra,Qg/X(1)

of X l , . . . , X s .

depends

over

depends

independent

ideals

at m o s t

or

P~Q

In w h a t

Xl'''''Xs

as

choice

Q rNX(!)

ideal

If

the

which

the

xs

associated set

isomorphism

on

of

in R. s -Q contained in P is c o n t a i n e d

show

that

a good

dim(Qg/pg)

prime

~ P I c" "'" ~ P d contained

/pg)

is

at

ideal

=d°It of

in

is at m o s t

height

s,there

p. d exists

=m with P =P.Then P is a g s o in P and so m e e t s Q in p. H e n e it

least

d.

346

We will

prove

automorphism that ients

iii)

and

of Qg over

iv)

Q,we

t o g e t h e r . By

may

assume

applying

that

X(!)

a suitable

is

standard,and

hence

x. is a l i n e a r form in the i n d e t e r m i n a t e s X I , X 2 , . . . w h o s e c o e f f i c l g e n e r a t e l i , t h e i n d e t e r m i n a t e s e n t e r i n g in d i f f e r e n t x i b e i n g

distinct.

P = PNQg. H e n c e

Choose

htPN=htP

b e l o n g to Q N . T h e n , d e n o t i n g P ~ Q N by PN' s = s and d i m P N = d i m P = d - s , i . e . P N is good. F u r t h e r ,

we can

Q by Q/p

and

write

N so

replace R N for

,replace Qg

that

the

Xl,...,x

assume

1ocalisation

by QN

to o b t a i n

that

Q is a d o m a i n . F i n a l l y

of QN at P N . T h e n

iii),iv),with

the

we will

R =(RN)g.Hence

assumption

we

that

can

Q is a

domain•

ring

We

can

of

fractions

construct

R N in

Q . T h e n , i f F denotes the F [ X I , . . . , X N] latter

in X I , . . . , X N over

there and

prime

over

is

the

F,and

one

ideal

of

fractions

prime

s and

is c l e a r l y

the

of height

so

regular

generated

by

the

we

over

take

the

elements

of

localisation

of

(Xl,...,Xs).But

linear

s. S i n c e

a localisation

is

the

ideal

by e l e m e n t s

of Qg m i n i m a l

stage

of n o n - z e r o

independent

) is g e n e r a t e d

R N is

first

set

of Q,R N is

over

s linearly

so is

the

to the

minimal

s p r i m e ideal

of height

stages. For

respect

ideal

proved. F u r t h e r

ideal

maximal

by

(Xl,...,x

is only

iii)

field

at a p r i m e

is g e n e r a t e d

minimal

two

of QN with

Of

forms

any

prime

QN,it

X(!)

idea~of

follows

meeting

the

Xl,...,x s Qg

that

Q in zero,

of F [ X I , . . . , X N ] at a

of h e i g h t images

s. F u r t h e r

its

of x I, • ..,x s. This

proves

iv).

THEOREM and

2.3.Let

let X(1)

I = (Ii,...,I ( X l , . . . , x s) be

~(or

the

ideal

over

X(1),and

I i , . . . , I t are

they let

iii) iv)

that

in p but

prime

ideal

prime

only

prime

ring

R = (Qg)p/p(Qg)p

dimension

s-t,

whose

As form,which

where,as

. . . . .

in the we will

ideal

minimal

local

e~ (xt+ l

prime

maximal

X(!)

of Q

elements

of

of Qg m i n i m a l

so n u m b e r e d

over

pgiS

that

minimal

meeting

over

Q i___n_np,

(Xl,...,xt)contained

is a r e g u l a r

ideal

ideals

ideal

t,and

over

minimal

of

are n o t . T h e n

of Q of height

ideal

set

of g e n e r a l

I i , . . . , I s are

It+l'''''Is

is the

the

set

P be a good

p =P~Q.Suppose

contained

(x I .... ,xt), P is the only pg

) be an i n d e p e n d e n t an i n d e p e n d e n t

generate).Let

i) p is a good

ii)

s

local

is g e n e r a t e d

by

ring the

in P,

of images

xs),

proof make

of

the

lemma

we will

take

X l , . . . , x s in s t a n d a r d

explicit,and write ~j xj = ~~=uI( j , k ) X ( N j + k )

earlier,Nj=nl+...+nj_l.Let

Ns+I=N

j= and Nt+ l

=

N

v

1 ..... s.

347

Write the

same

I

!

QN,PN

Q'

= Q/p,and

meanings

"N-

as in the

and

Q'

in the

Q~/P~

is the

in P,and

N-s+t.

Q'

Further

two

follows

the

residue

transcendence

I

QN/PN,Of

the

htp+

dimQ' that

a set

it have

dimension

we have

proved

dimension

of

We

IsQ N in

Q~.

Further,x*

ideals.

ideal

set

define

minimal

1emma

over

first

of

(yl,...,ys),where all

contained

I t + l , . . . , I s are

in the

lemma,are

not

linearly

by Y t + l , . . . , y s and

t(E'/F')

and

the

of E'

second

over

F'

so is

> d-t.These

two

i.e.,htp

inequalities

ideal.Further hence

pg must

to the

local

rings

obtain

~ d i m ( Q NI/ P ~ ) + N

the

ring

imply

pg c o n t a i n s all

minimal

be minimal

~

t

= d-s+N equality

and

X l , . . . , x t which

prime

over

ideals

over

( X l , . . . , x t) and

p~,

Q~ = Qg/(X I ..... x t) and write

images

of ideals

is a minimal a minimal the

t+l''''' ideals

x* s _I*

is

= dimP

=

of Qg'QN'

prime prime

images a

ideal

I*)s

htP*

=

Q~

modulo

of Q~,

for

( X l , . . . , x t)

and

has

of x* .. x* which, t+l'" ' s' I* of I IQN,...,

t+l''''' of a

(I~+ l, ....

= d-s,and

ideal

I*

sub-set

have

implies

sets

that

meets

of g e n e r a t o r s

P is

QN'

(Xl,...,Xs).If

the

in PN' the

only .Let

s

set

is

s-t,so

t+ parameters

of

an

in

independent

that

contains

PN'

must

properly

contains

it,and

it must

intersection

contain

of PI with

at

in QN'

P*

prime

and

least

Q properly

.Finally

ideal

is

set a

of

good

dimension

one

the

contains

ideals p.

over ideal

Q contains

PN,in

so has of

prime

of PI with

so is either

we c o n s i d e r

minimal

P,. be a n o t h e r

intersection

QN'

that

of Q'

fractions

fractions

over

since

= s+N-s+t

with

implies

of

Ii,...,I t are

degree

i n e q u a l i t y . We

prime

of

ideal

is g e n e r a t e d

to P'PN

intersection be P,or

field

extension

fields

of Q * . . H e n c e we can apply the lemma. S t a t e m e n t iv) is g iii) follows if we o b s e r v e that the minimal prime ideals

( X l , . . . , x s) which

the

have

of Q'/P' is i s o m o r p h i c to k(X 1 ...,X N) N N ' ' N over k,the residue field of Q'.We now

of

of

( X l , . . . , x t) all

ii).The

the prime

zero,and

P"

elements

Finally,dimP*

immediate,and of

degree

is clearly

general

the

field

denote

"*".Then

d-t.P*

that

all

parameters,and

introduce

are

prime

defined.We

generated

be the

non-zero,and,as

j d-t..Hence

in turn

Q$,so

QN,PN

i).

QN,/(Xl,...,xt). by a d j o i n i n g

is

dimQ'+N-s+t

= dimQ/p

of

we

E'

that

dimension

p is a good

is part

Now

F',E'

N ~ htP N + t(E'/F')

dimQ'+t(E'/F')= whence

a finitely

transcendence

applications,the !

and

F.It

s-t.Henee,the

so has

hence

T of Qg. Let

P/pg

as a l r e a d y

of x.z in F ' [ X l , . . . , X s ] . B u t , a s in p , y l , . . . , y t are

over

height

of

P" is a m i n i m a l

in p , y t + l , . . . , y s are

independent

and

ideal

being

ideal.Let

r e s p e c t i v e l y . Then

image hence

contained

make

prime

lemma,N

zero

F'[Xl,...,XN]/P",where

has

the

is a l o c a l i s a t i o n

N

PN' meets

of Q'

Yi

for

similarly. Now O'/P'

and

P'

of Qg, p,its

which

case


this

It+l ''''Isand

PI

so

348

To ent

conclude

sets

ideals

of

elements

to j o i n t

necessarily

tile r e l a t i o n s h i p

reductions.

distinct.

We

Let now

I

of

be

R(j)

term

a set each

will

denote

of

elements

j and

there

the

set

of

integers

a set

a set

of

of d

some !R

we

that

mean

r.>0,j

(rl,...,rj-1,...rd).Then

y l , . . . , y d of Q a j o i n t exists

independ-

introduce

R = ( r l , . . . , r d) be a set of d i n t e g e r s . T h e n , b y r r2 rd I 1 !I 2 ...I d r _<0,1j r= Q. If R is as a b o v e , w i t h the r e s t r i c t i o n

then

reduction

positive

of !

integers

we

if y j ~

Ij

R = ( r l , . . . , r d)

that IR= _

Our

consider

Let

where,if

such

general

section,we

(Ii,...,Id),not

notation.

for

this

object

is

R(I}

+

Yl !

to p r o v e

that

an

~R(2) Y2 !

independent

X l , . . . , x d of

I is --

a joint

reduction

We

some

further

definitions

need

Let

T = (tl,...,td)

Then,if

be

R=(rl,...,rd)

is

+ +...

a set

a set

of

of

of

R(d) Yd ~

set

of

general

elements

I . --g which

now

follow.

independent

integers,we

indeterminates will

write

over

T R for

Q.

the

product

negative

exponents

(u 1 ....

,u d)

sub-ring in

r1 rd t 1 ...t d , being allowed.We will

by

of

U.We

Q[T,U]

_I R , r l , . . . , r

now define

d being

allowed

ring is graded by s e t s .th J d e g r e e of R ( 1 ) . LEMMA

2.4.

Let

x.

- -

integer being

of

be

that,if

multigraded

of

all

to

be

finite

element

into

Now divide but

we

the

note

passage

of

replace set

then

to R ( I g )

implies

Mn

of

that,if

be S i . H e n c e This

with

those

the

unR(!)

which

do

Q by Qg

prime

is

replaces element

is a n n i h i l a t e d

to

be

the a(R)

negative.This

being

termed

the

of

we

R(j)

= x j!

independent of

not

of

these

x.tjJ of denote

those

Qg of of

which

contain

and !

ideals

S.I c o n s i s t s

that,if

or j

a non-zero-divisor

sub-sets,S 1 consisting

consisting

R(!)

and

that

U l U 2 . . . u d = u is

associated

two

uj

~a(R)TR,with

(rl,...,rd),r

R ! Qg fh X j Q g

ideals

ring sums

by

I . . T h e n t h e r e e x i s t s an J > o R = (rl, ... , r d ) w i t h r J r.j ,r.1 for i # j

unrestricted,such

Since

-1

tj

positive,zero,

d integers

a general

J

r °j s u c h

the

consisting

denote

R(!),the

n. We

divide

contain

with

uR(l we

(Pl)g,...,(Ps)g

Ig is not

this

prime set

and

S2

ljtjR(1).

pl,...,Ps,When by

of

l.t j j R(1) --

by ! g = ( l l Q g , . . . , I d Q g ) . W e

associated

set

contained

) into

g consider and in

this any

will

still

two

classes,

R(1)_ , t h e n set

will

ideal

the now

in S 2.

the m o d u l e n (unR(l_g):Xjtj )/unR(l_g) by

by M

(lj tj )N for

some

integer

N depending

on

n. But

349

M

is a f i n i t e l y n d e g r e e R is z e r o

) - m o d u l e and h e n c e any e l e m e n t of M of g m if the j d e g r e e r. is s u f f i c i e n t l y large. 3 N o w let B d e n o t e the i d e a l of R(I ) c o n s i s t i n g of all f i n i t e sums R --g R ~.a(R)T w i t h a(R) c o n t a i n e d in x.Q ~ I . T h e n B has a f i n i t e b a s i s

consisting we

can

and

of

find

hence

generated

elements

an

R(I

th

of

integer

the

form

q such

that

B = x j t j R ( I g ) : U ~.

Now

~b.T w i t h b. in Q .It f o l l o w s that J O q g the e l e m e n t s b.u all b e l o n g to R(I ) R 3 --g s u p p o s e that zT b e l o n g s to B. T h e n

uq.zT R = x.t.W where

W is

Then,by that

a homogeneous

the

first

r.-q-i

part

rj

of

the

is s u f f i c i e n t l y

03

>r~,W ~uqR(l ) and J --g

3

element

3

of R(Ig)_ w h o s e

proof,W

will

large.Hence

hence

j

th

belong

we

can

degree

to u R ( l

find

is

g such

r?

r.-q-l.3

) providing that,if

a

zT R b e l o n g s

o

to x j t j R ( l _ g ) . T h e r e f o r e , i f

r.>r.. 3

3

X j Q g fN(Ig) R = xj(l_g) R(j)

THEOREM

2.5.

If

X l , . . . , x d is

I = (ll,...,Id),then

observation disposes

prove

this

that,if

of

the

( 0 : I i . . . I d)

case

d=0

= 0,which

suppose

that

satisfying d'

= dimQ

this

result

I i . . . I d is

the

and

hence result

condition

this

restriction

for

J =

I_7 (O:(I

.

Id,)q).Then

ring

Q'/J,and

Consider

the

that has

most

d. T h e n all

can

nilpotent,when

the

is

for

we

show Q of

I .

is

restriction

non-zero-divisors.

all

local

then

d'<

I

ideals

I~ ..... I~,

result

is

implies

d. Let

=(I I,

.Id

)q)

" d ),where either

zero,which

true. H e n c e that,if

Q

without

J=(Q:(I I

ideals

is

rings

,where

holds

of

the

that

a11

that

product

the

Ii,...,I d contains

the

their

with

trivial.This

the

dimension q so

of

--g

I i , . . . , I d, it

elements

commence

ideals

for

that

choose

Q'.This

we

implies

may

assume

r l , . . . , r d are

all

large,then iR --g ~

and

impose the

ideals

can

ti~e set

ll...Id,is

result

proved

of

rings

we

that

sufficiently

of

we w i l l

that

true

fact

been

I~ = l . + J / J . T h e n d i m Q ' < d i m Q and 3 J each contain a non-zero-divlsor,or

result

in each

of

on d . W e

X l , . . . , x d are

sets

local

of s e n e r a ]

reduction

induction

that

and

is at

a joint

by

set

nilpotent,the

we

implies

a non-zero-divisor,and For

independent

X l , . . . , x d is

--

We will

an

obtain

the

result

for

Xl!g

R(1)+

"''Xd,!g

Q,ll,...Id,

if we

R(d')+

increase

J rl,...,r d each

by q. We By

now

taking

come

to Q N . N O w

take

let

xj

be

the

set

of

general

apply

our

to

the

inductive

X l , . . . , x d in s t a n d a r d Q'

of

elements

large

of

the

can

xj of

in Q',SOg the

that

set !'

assumption,and

hence

proof. Suppose

find

= QN/XlQN,I' j = IjQN+XlQN/XlQ

image

inductive

sufficiently

part form,we

N such

N for

that

that

d>0.

x I belongs

j = 2, ... , d , a n d

x 2 , . . . , x d' f o r m

an

independent

(I~ .... ,I~)

of Q ' . T h e n

obtain,for

rl,...,r d

we

can

350

and

this

can

be

and,hence,by

R I ~ --g

XlQg+X2!g

I R _g =

(XlQg ~_

R(d) +...+Xdlg

written

]emma

2.4,at

the

I -g which

R(2)

completes

the

R

l ~R ) (+ X22 !)g + o _

expense

= x

fig

of

R(1)

R(d)

...+Xd~g

increasing

rl,...,r d further,

R(d)

+...+XdIg

proof.

3.m-valuations. We ring

commence

this

(Q,m,k,d).We

of Q / p , w h e r e P restrictions on

section

begin

F

p is

a prime

its

ii)it suppose

that

p,we

define

v(I)

I is

m-primary,and

implicitly the

ideal

to

value

takes I is

be

m-valuations

of

v on

Q.We

the

place

on

the

on

field the

a local

of

fractions

following

additive

>0

on

Q.Then,if

is

value always

apply

set

the

of

minimum

will

is

ideal

p # m,this

v(I)

group

values

an

the

excluded).We

function

obvious

defining

a valuation

v: i)

Now

by

with

of

Qlp,and

of

>0

v on

term

ideals

of

on

I is the

finite(the

the

group

contained

ideal

case

that

integers,

mlp.

not

in

I + p / p . If

p=m

m-valuation

Q.Note

of

to

will v,or

v(I)

be to

has

the

properties a

V ( l l l 2)

= V(ll)+V(l

b

if

12,then

c

if

I ~ p,then

of

all

I1 ~

2)

v(ll)! v(l 2) The

ideal

be

termied

LEMMA

p consisting the

3.l.Let

limit

ideal

elements

of

V l , . . . , v N be

v(I)

v and

a set

= v ( I + m r)

in

Q for

written

of

for

which

r large.

v(x) = ~ , w i l l

L(v).

N distinct

m-valuations

on

Q.Then,

N

if

~a.v.

--

for

I

all

m-primary

Let

r be

the

L(VN).Then,we replacing

Q by on

Now

ideals

l,al,a2,...,a of

let

any

assume

of

non-zero

large,vi(cQ+mr)

which

it

follows

N are prime

on

element

aivi(c)

of for

= 0

immediately

zero. in

the

suppose

V l , . . . , v N are F of

= v.(c)

a11

ideals

r. F i r s t

that

fractions

~ from

distinct

induction

may

field

c be

sufficiently

by

Q/p,we the

= 0

number

proceed

valuations

(I)

ii

set~(Vl),...,

that

r=l.Then,by

derived

from

Q. Q.Then,by i =

c)

above,if

l,...,N,and for

all

r is

hence c # 0 in

Q.

that

N Z aivi(x) __

= 0

for

a11

x#

0 in

F.

i

Then,by (Note

the

theorem

that,since

the

of

independence

value

group

of

of

valuations,al=...=aN=

each

valuation

group

O. is

Z,any

two

351

valuations Now suppose

among

that

thatL(vj)# but

Vl,...,v N are

suppose

that

Vl,...,v N

p if

contained

are

j>N'.Now

in

so

let

L(v.) 3

for

these

for

all

cb

numbered

all

a,b

are

(n+l) in

the

any

way

Then n = v.(cQ+m ) i

any

non-zero

before,that the

elements

v on

are

Qg

to

of now

and

purpose

we

appendix

>N'.

of

their

all

1emma to

The

of

Vo,VN which

given

is

due

First

we

introduce

some

of

fractions

F and

residue

field

of v is v d e g r e e t(K

If

an v

elements

algebraically

that

let

fractions

Q'

of

inequality

that

dimQ'>O,it is

We

turn

the

now

v is

form

the

Q',so to

the

p

n+l.Hence

hence

of

have

we

ca,

have

Q/p,implying

thus

reduced

inductive

to

Q'QN

to

in

let

E be

E,~O of

a more

and

k. We

m -valuations g For this

general

form

Samuel[12],vol.ll,

(Q,m,k,d)

a finitely

on

r-l,and

hypothesis.

respectively.

Zariski

as

r to

between

Q and write

to

the

an

m-valuation of

g ~",a

being

finite,

Vg(f)

to

v

to

local

be

a

local

generated

>0

on

m. T h e n

t(v/Q)

.N,where and

Min

the

v(a

on

for

t(E/F).We

~

d+t(E'/F)~

s is can

bounded

now

Q with

~ runs

take

the

the

over

have

E'

be

are

v

apply

the

the

obtain

dimQ'+s above

that

First,let monomials

by

d+t(E/F)-l,proving

Q is we

f be in

belonging

a unique

K

v on

now

and

L(v) = 0.Then

a

of

in

s = t(v/Q).

Q,Qg. Suppose

follows.

now

centre

Q,Q'

rings

images

Q [ Z l , . ..,z s I p . L e t

t(E'/F)~

coefficients

),We

the

their

of

we

as

ring

that

that

rings

Qg

O

pair

that

finite,and

v

be

p and

such s v o v e r k. Let p be

be

follows

t(v/Q)

extension

in that

> t(v/Q)

in

d+t(E/F) Since

by

relationship

on

extension

Zl,...,z

independent

Q[Zl'''''Zs]'and of

n such

/k).

d+t(E/F)-I

field

find

d = dimQ,then

Choose

dimension

l
I< i < N' ---

n o t a t i o n . Let

a valuation

K

transcendence 3.2

v be

can

follows,due

field F.Let

we

Q not

replaced

our

the

2.

of

of

fractions

by

consider

lemma

proof

L ( v i ) = p if

L(vi),and

j > N'

in

z e r o . We

follows

restrictions the

ideals

= 0

field

are

going

Q not

with

extension

LEMMA

the

the

require

Abhyankar.

domain

x in

al,...,aN,

statement We

j

the

~la c Q + m )-n ~ a . v . ( c Q + m n+l) 0, t~l i vi( m "~=i i i = d e s c r i b e d . C l e a r l y we c a n r e p l a c e c by

= aivi(x) for

among

that

v . < c Q * m n) = n v . ( m ) J 3 equations r e m a i n t r u e if n is

~a.v.(c) L~ i I c choosen

where

independent.)

p minimal

c be any e l e m e n t

v.(c) i

and

linearly

r > l.Choose

a

local

define an

domain

the

general

element

of

XI,X2,...

,the

to

we

extension

Q.Then to

Qg

and

Qg

sum

define

i t s e l f . If

of

352

L(v)

# 0,then

LEMMA

3.3.

we

Let

use

Q be

the

a local

such

that

L(v)

= 0 and

such

that

L(w)

= 0.Then

Suppose contains

an

greater and

can

the

this

~sa

).Then

power

of

the

implies

that

w

an

replacing

Q.

m-valuation

on

extension

least

of

hence

Q such

X l is

an

of

not

v

equal

to

Q

to Qg

the

algebraic

> v

over

to,and

O

v to

Q by

a polynomial

ideal

v

QN

(f).Multiplying

g

such

the

QN+I

hence,

of

exists

maximal in

that

QN+l. Replacing

there

wl(f)

coefficients

such

extension

wN

that

that of

integer

general

extension

with

Q/L(v)

v be

be

w(f)

the

generator

f(Xl)

let

the

.B w i t h

N = 0 and in

with

.

w N is

general

that

a polynomial

and

= 0.Let

w = v

coefficients

a suitable

But

not

domain

t(v/Q)

f =

w(a

assume

X 1 with

find

Min

definition

g f a l s e . Let N be

is

element

than

W N + 1 is

QN,We in

this

same

of

f by

Ov,We

that

v

can

(f(Xl))=0.

g f i e l d of

residue

O

,and V

hence an

over

k,modulo

THEOREM N such We P is

3.4. that

Let w

find

= P~QN'

the

field

images

N in

maximal

ideal

m g -valuation

of

O

on

Qg. T h. e n.

s t a g e s . Let

elements may

extension

of

assume

w

,and

fractions

of

QN

let

Q'

be

extension

same

that

of

of

w N on

contains

hence

that

w is

K

w

is

the

we

of

can

find

replacing Next,we

over of

not

Q

can

.Let

on

Q'

N'

by

fractions

integer w

such

N>N'

to QN" that ,where

such

that

in O

these

.Hence

g of

an

QN,/PN,

find

contained k

.

w N of

g QN[Zl,...,Zs]

a valuation

field

t. h e.r e . e. x i.s t .s

restriction

elements

of

w localisation

the

the

Then,by

L(w)=0.

basis

t ( W N / Q ' ) = 0 and

general

of

P=L(w).Then

QN''

that

a transcendence

Z l , . . . ,z s and

as

an

seneral

two

by

of

be

the

,we

form

WN.Then

w

is

generated

PN'

of

the

m -valuation. g

whose w elements be

at w

the is

Q' and g

centre

the

this

is

the

Qg.

4.Mixe d Multiplicities. DEFINITION.Let d ideals

of

Q be

a

Q subject

independent

set

m g -primary.(This ideals).Let

M

multiplicity

of

local to

the

general

condition be

to

and

elements states

a finitely

e(~;M)

ring,

is

definition till,i.e.,if inductively

clearly

of

I is

an

Q-module.

( I i , . . . , I d)

be

X l , . . . , x d is X(I)

=

( X l , . . . , x d)

independent Then

a set

we

of

an

set

define

i_~s

of the

be

independent

e(xl,...,Xd;M) d=O,then

l,then

that

senerated

I =

that,if

of

e(xl,...,Xd;M" This

let

condition

e(~;M

of

that g

we

) g the

choice

shall

) = L(M),while

use if

of

xl,...,Xd.The

is

that

due

d >0

we

define

to W r i g h t it

by

e(x~..... x~; M~)=e(-

x 2 , . . . , X d- ; M g / X i M g z - - e ( x

-

2 .... X- d ; ( 0 : X l ) M

) g

353

where and

x 2 ..... x d are (O:xl) M

are

the

both

images

of

considered

x 2 ..... x d in Q = Q g / X l % as Q - m o d u l e s . l t

and

is c l e a r

Mg/XlMg

that

e(~;M)

g s an a d d i t i v e and

that

an

we c a n

independent

take

in

for

of

lj=

X l , . . . , x d in

definition

the

the

choose

set

(~2 .... , ~ d ) , w h e r e we

on

is n o n - n e g a t i v e ( s e e

observe is

function

category

~

general

right-hand

side

to r e d u c e

the

in

is

a l-dimensional

local

two

results

elements

of

the

we

can

Q-modules,

theorem to

6).We

QN a n d ( x 2 ' ' ' "

-

set

of

write

also

,~d)

ideals

N of QN = Q N / X l Q N . T h i s

form. T h e n

is

clearest

the

inductive

two

terms

if

e(l;M)

= e(l;M')-e(~;M")

to m u l t i p l i c i t i e s

the

definition

define

x 1 belongs

= M N / X l M N , M '' = ( O : X l ) M N , t h e

referring

convenient.This

can

generated

form

enables where

that

IjQN+XlQN/XlQ standard

= (12' .... Id )'M'

us

finitely

example,Nothcott[5],p308 N such

e(!;M) where

of

proof terms

of of

b e c a u s e , if

Hilbert

x is

ring, then

which

hold

t hat

a)

e( I I I 2 ; M )

b)

if

functions

large

case,and

ring

to

the

n. We

which

of of

we

d=l,

more

an ideal

Ig

recall

and

use

I of

hence

without

will

the

QN.This case

is o f t e n

element

is a r e d u c t i o n

for

in

the

results

a general

XQg

= L ( I n M / I n+iM)

over

certain

on

in

we

proof the

next

lemma = e(il;M)+e(12;M)

11,12

have

e(ll;M) LEMMA

4.1.

a)~et

independent

sets

~"

is

an

I = ( I i , . . . , I d)

and

let !"

independent

I'

integral

= (ll,...,Id_l,l

closure,then

~)

b__~e

= (ll,...,Id_l,ldl~).Then

set,and e(!";M ) = e(!;M)+e(~';M

b)Let

same

= e(12;M)

of

d ideals,and

the

I = ( l l , . . . , l d)

_I* = ( l l * , . . . , I d * ) , w h e r e

lj*

be

is

an

the

)

independent

integral

set

closure

of of

ideals,and

let

l..Then3

e(!*;M ) = e(!;M) In b o t h case,we equal if

must to

and

I.

cases have and

in

only

if it

is m - p r i m a r y

,the

simply

a) For

b),we the

for

necessarily same

can

reduce

tile proof

ht(XlQg+...+Xd_iQg) the

case

d=l

is m - p r i m a r y . first

= d-I

a single Since

statement

of

the a)

to the and

ideal

d=l,ln

the

first

dim(Qg/XlQg+...+Xd_iQg) is

product

case

an of

follows,while

independent two

set

m-primary

the

second

ideals is

above.

time,using Xd_l(~)

we

integral

replace symmetry

the of

ideals e(~;M)

I i , . . . , I d by in

I 1 * ,.. . ,Id*

I 1 .... , I d , N O t e

XlQg+...+Xd_iQg,then

Id*Qg+Xd_l(~)/Xd_l(~)

the

closure

of

that

can

integral

closure,so

we

that,if

b)

above..

at

a

is not

IdQg+Xd_l(!)/Xd_l(~),but apply

one

we w r i t e

has

the

354

We some

now

come

to

preliminary

consisting that

of

the

d-I

ideals

(I1,...,Id_l,J)

m-primary elements

of ! ~ a n d

concern

is

is w i t h

of

is

ideal).Let

Xl,...,Xd_l,y

main

theorem

explanation.

We

Q,and

also

let

y be

we

an

a general

J

range

of

over

example,J

of

general

will

set

independent

element

set

multiplicity

paper,which

independent

let

1 be

independent

the

the

an

m-primary(for

Xl,...,Xd_

an

of

fix

!

=

all

ideals

could set

of

J such

require

(Ii,...,Id_

be

I)

such

any

general

that

elements

of ~ , J . O u r

function

e(!,J;M ) = e(xl,...,Xd_l,Y;M) considered Q-modules ideal

as

an

additive

and

as

a finction

(x I .... ,Xd_ I) In

general

of

function of

on

J.As

av(!;M)

is

all

m-valuations

theorem

set

of

the

coefficients

a

v. L e m m a

(l;M),and v -We n o w

of

ali

note We

to

states

ring

the

4.2.

where

the

negative

play of

There

is

integer

We

start

p342,Theorem

the

a

an v all

over equal the

18),which

parameters

to

we

some

being

over

have

all

the

first

term

does

is

l-dimensional

to

Northcott([3],Theorem

the

case

the

for

save

we

the

determines

over p = Q

Qp

a good

and

Mp

Q(P),M(P)

the

ideal.

ring

J(P)

of

set

P,and

prime

the

and

JQg+P/P lemma

for a finite

function

write

e(I,J;M)

on

not

save

uniquely

also

idea]

zero

=

all

(Qg)p

the

Q(P).These

following

proves

e(!,J;M).

for

all

Q,and

of

for

the

partition

can

write

it

the

av(!;M)

a finite

formula

to

form

is

a non-

number

of

v.

multiplicities([5], Xl,...,Xd_lly in

the

of

the

form

~e(J(P);Q(P))e(IQp;Mp) prime

ideals

involve

1-dimensional

the

by by

av(!;M)v(J) m-valuations

apply

is

role.The

expression

e(J(P);Q(P)) In

X(!)

equation

P range

also

2.3,p

denote

Xl,...,Xd_l,Y.We

a

generated

by

on ! , M , d e f i n e d

equation

X(I).We

theorem

for

good

an

I for

n o t a t i o n . Let

/PM and g g temporary

e(!,J;M)

Q(P)

given

above

of

associativity

the

ring

the

also

formula

exists

with

M

only

of

The

we

depending for

notation,denote

will

set

that

zero

further

of

Q(P)-module

sum

is

ideals

of

( M g ) p . We

existence

LEMMA

some prime

convenience

will

integer

that

a consequence

Qg/P,the

notations

implies

that

(l;M),and c a n be u s e d to d e f i n e v -d e r i v e m a n y of its p r o p e r t i e s .

module

finitely

denote

a

minimal

that,as

the

Q,which

introduce

good

will,for

and

3.1

of

will

Gav(!;M)v(J)

a non-negative v of

category

Qg.

terms,the

e(!,J;M) where

the

earlier,we

local

P minimal now

over

consider

X(1).Note

the

first

domain. Further

= e(yQ(P);Q(P)).

have

6Cwith

M.We

a formula

the

for

observation

e(yQ(P) ;Q(P)) that

due

term.

355

lengths

and

multiplicities

notation,this with

the

this

ring

runs

integral

of

these

of

w. T h e n

has is K

as

closure

finitely

w

of

many

a discrete is

are

equal

f o l l o w s . Let Q(P) maximal

over

its

of

ideals,and

of

k

K

g

of

ideal all We

is

we

P.The

can

:k

is

an

r(Q(P),w) that

integer

it

is

is

N such

that

w(y)

= v(J),where

determine

choosen

the

same

e(J(P);Q(P))

on

formula

is

whose

limit

Qg

factor

about

which

we

extension

is

the

be

a general the

now

each

field

integer. Now

w

v,we

=

at

residue

]w(y).

to

v is

that

g

a ramification

that

y is

the

m-valuation

a positive

that

w may

w

an

localisation

Northeott's

= ~r(Q(P),w)[K

determines

assume

follows

different

w

factor

note

choose

we

then

valuations

need

can

and

the

our

associated

fractions(recall

its

w

,and

domains).In

valuations

de~te

w

Each

the

field

ring).Let

extension

e(yQ(P);Q(P))

l-dimensional

range

in

valuation

a finite

in

w

general

element

restriction have

turn

an

to w ( y ) . of

of

of

w

wN

JQN.It to

Q.Since

expression

~d(!,v)v(J) V

where

d(l,v)

finite

set

recall

that

is of

a non-negative

v with

limit

different

integer

ideal

P meet

Q

p.

in

equal

Again

to

zero

referring

different

for to

all

save

theorem

p. H e n c e , w e

a

2.3,we

have

derived

the

moment,not

a formula e(!,J;M) where

av(l;M)_

=

~.~a (l;M)v(J) v v --

= d(l,v)e(IQ_ _ p;Mp).

Tile e x p r e s s i o n

we

have

derived

for

a

(I;M) V

particularly We

will

convenient,and

first

v a partition I'(v)

to

is,p).We on

be

LEMMA

of the

will

v.l"(v)

consider the set

also

our

the set

of

I into

sub-sets

Suppose

we

to

theorem

write

now

number

whose

maximal

apply

the

pQp

the

l"(v)

2.3,we

ideal,namely

see and

ideal

have

the

that

is

following

factor

ring

to

depends

and

two

factors.

with

I"(v).We

each

define

to

lemma.

= e(!' (v) ;M(v ~ 1 so

P has

that

only

Qp/pQp

bythe

is

images

e(!Qp;Mp)

Xl,...,Xd_l.Then

= e ( x I ..... X d _ l ; M p ) = e ( x t ~ l

turn

the

associate

It+l,...,Id_l.Then

xlQp+...+xtQ

formula I of

= e(l'(V)Qg;(M we

now I'(v)

ll,...,Id_ of

generated

associativity

e(~Qp;Mp)

ideals

consists

further,the

Xl,...,x t Xt+l,...,Xd_

simplify

I. c o n t a i n e d in the l i m i t i d e a l of v ( t h a t J Q v , M v f o r Q p , M p to i n d i c a t e their dependence

e(IQp;Mp)

11,...,1 t and

to

e(IQp;Mp).We two

is,for

--

is

ideals

= I-I'(v).We

of

this

factor

4.3.

Now

objective

consists

if

refer

we

minimal

a regular

of

the

back

prime

local

ring

Xt+l,...,Xd_l.We

with we

one

!'(v)

now

partition

obtain

.... X d _ ~ ; Q p ~ p Q p ) e ( x

I .... xt ; ( M p ) p Q P

) ) = e ( l ' ( v ) ; M v) Pg

the

other

only

on

factor

the

set

d(!,v).Our l"(v)

and

object not

on

is the

to

showthat

whole

set

~.

356

To

this

lj+p/p.Note have

end we will

belongs

anm

use ! " ( v )

to d e n o t e

t+l .... ,d-l.

ideals

of

(Q/p)g

ideals

contains

of

and

the

set

of

now

calculate

of Q/p

minimal the

form

ideals

p.We

of

consists

the

set

of

ideals

the

P'/pg,where

P'

of height

use

lj+p/p,where

of

the

set

is a prime

d-l.Hence

lemmas

4.2

and

4.3

ideals

of good These

ideal

P'

meet

lj

e(!"(v),J+p/p;M/pM).

(Xt+l,...,Xd_l)+pg/pg.

( X l , . . . , X d _ I) w h i c h

can now

e(!"(v),J+p/p;M/pM)

ideals

= Q/L(v).Consider

over

(Xt+l,..o,Xd_l)+Pg prime

containing

Q/p

set ! " ( v ) . W e

l l , . . . , I d _ I so that ~ " ( v )

for j=

minimal

ring

I b e l o n g s to l ' ( v ) , t h i s ideal will be zero. Henc£ we J r e d u c e d the set ~ to the set ~ " ( v ) . W e now c h a n g e

to the o r i g i n a l

Again,renumber

prime

the

that,if

essentially

notation

consider

lj+p/p

prime

are

the

of Qg w h i c h

ranges

over

Q in a p r i m e

the good

idea]

p'

to c a l c u l a t e

as ~d(~"(v'),v')e(l(v');Mv,/PMv,)V'(J)

where

v'

ranges

l(v')

is

the set

suppose hence the

that

we

the

vector

Next as

that

the

sum

I"(v)

that

take

I(v)

ideals. d(!,v)

d(!"(v),v).This

:k ],taken w g r e s t r i c t i o n to Q/p

then

we

that

have

that and

see

we

that

replacing

is e m p t y , a n d

of

those

d(I,v)

of

fractions the

of Q/p.

above,

valuations

w on

is p r e c i s e l y

the

depends

on

only

same the

set

d(I,v)

O.Then

of

we d e f i n e

such

ideals

Q by Q / p , w e to this

d i f f e r e n t l y . First

that of Q/p

can

apply

case. If we

I an i n d e p e n d e n t summarise

set

this

d(!,v)

L(v)=p

as above.

#O.Now

containing the

of d-i

dim(Q/p)-I

definition

follow

suppose

this

ideals

of

procedure ( I i , . . . , I d _ I)

in the s t a t e m e n t

of

the

lemma.

l"(v)

l"(v)

contains

e(!,J;M)= is the

set

dimp-I

and

ideals~and

above

on

d(l"(v),v)e(l'(v),Mv)V(J) I +p/p of Q/p w h i c h are n o n - z e r o 3 is an i n d e p e n d e n t set of i d e a l s

tile field

preliminary

a lemma,which

in the

m-valuations.

this

E

of i d e a l s

v is a v a l u a t i o n

conclude

definition

L(v)=

set

L(v)=O

that,with

4.4.

occurring

that

approach

= d(I"(v),v).We

LEMMA

of Q / p , a n d

proved

p,and

particular,

it.

v is an m - v a l u a t i o n

case

where

To

we

redefined

l(v)

is d e f i n e d , b y

is v. This

hence

L(v') ~

the d i m e n s i o n

field

over

the

following

that

is s i m p l y the

that

lj ~ L ( v ' ) . I n

whose

Then,by

of Q , d ( ! , v )

that

we note

e ( l ( v ) ; M v / P M v)

be an i n d e p e n d e n t

in

First

term

Q is a d o m a i n

let

= v.

of Q such

such

= M v / P M v over

suggests

suppose

lj+p/p

Mp/pMp

the

and

Now

v'

factor

as we have This

ideals

~r(Q(P),w)[K

= Qg/P

as d(!,v)

the m - v a l u a t i o n s

of

space

consider

Q(P)

over

indicates

formula

of

fractions

discussion,we

belong

that

of~Q/p.

require

one

the v a l u a t i o n s

to a r e s t r i c t e d

.

class

further v of

357

DEFINITION.Let K v of

t(v)

v over

k.

denote

Then

we

t(v)

We

have

equivalent We

already to

now

the

+

that

restricted

to

be

good.

LEMMA

If

in

lemma

We

recall

restrictions condition

that

of

that

that

=

t(v)

the

v is

degree a good

< dim(L(v)).Hence

that

L(v)

is

valuations

has

w

v in

on

Qg

dimension

the

and

in

1emma

v)

lemma

such

I and

this

good

occurring

4.4,d(I"(v),v)e(I'(v),M

the

of

residue

m-valuation

field

if

d-1.

valuations

mg-Valuations Qg/P

that

statements

prove

4.5.

transcendence

say

ht(L(v))

seen two

the

will

is

4.4

that

4.4

can

non-zero,v

are

obtained

P = L(w)

further

definition

the

be

is

good.

as

the

satisfies

residue

the

field

K

'

of

w

is

generated

w

is

to

of

a finite

QN"

by

It

w N and

Now

let

let

F',E'

follows

p=

be a

that

t(E'/F')

R'

and,since

3.4

field

fields

of

the of

O

v

of

Q',R.Then

the

at

dimension of

dimO v +I(E'/F') 1 kR

is

to

the

+n-t-d+1

residue

degree

t is

now

the

collect

state

our

field

N over height

of

4.6.Le___~t ~

Let v be

a good

of

seen

Q',and R

[R] at the c e n t r e of v generated extension of

domain,it can

is

also

apply

= dimR'

+ t ( k R,

/Kv)

+ t ( k R,

/Kv)

of

of

Q/L(v)

of

the

form

J is

any

ideal

K

I

R CR'~

0 and ~ wN transcendence

has

v

p,which the

=(ll,...,Id_

ideals

I contained

proves

lemmas

I)

such in

he

an

that

L(v),and

I +L(v)/L(v)

- -

ideals,and

M in

is the

any

the

hence

kR,

degree

has

d-t-I

result.

and

parts

of

theorem

independent

ht(L(v))=t. let

where

l"(v) I

set Let

be

does

the not

of

ideals

l'(v) set

be of

belong

of

the

Q.

set

ideals to

l'(v).

3

of

Q

such

finitely

that

generated

l,J

is

an

independent

Q-module,then

we

can

set

of

write

form e(~,J;M)

~,d(I"(v),v)e(!'(V)Qv~Mv)V(J)

where

v ranges

over

all

good

to

taken

zero

if

either

as

the

above

3

be

of

and

theorem.

m-valuation

of

e(~,J;M)

already

Q'.Further

= R'.But

k. H e n c e

together

main

THEOREM

Then,if

kN

~

k and We

2.3

over

E = QN/PN

have

of

ring,we

,

transcendence over

O

local

local

let

P

w N of

extension

algebraic

extension

a finitely

a regular

general

we

generated

1ocalisation O

the

w N is

that

equality

i.e. where

is

Q/p. Further

v l-dimensional

a is

for

is

w

K N of

a finitely

be

contains

Q'

d-t

k

that

fractions

of

R'

v l-dimensional.Since

dimension

write

1ocalisation

it

we

theorem

= N+t-d+l,where

a

and

.We n o w c h o o s e N so g consider the r e s t r i c t i o n

residue

localisation

is

QN

of

the

l-dimensional.Let

w. T h e n

of

= Q~PN,and

the

w

extension

from

that

Q ~P

R is

O

elements

hence

that

is

algebraic

is

t(v) = d i m ( L ( v ) ) - l .

m-valuations the

number

on of

Q,and ideals

e(I'(V)Qv;M in

I'(v)

is

v)

is

not

to

358

equal

to h t ( L ( v ) )

ideals

or

v a l u e s , d e f i n e d for contain

d v-I

followin$ Q/L(v) that

if l ' ( v ) Q v is not

of Qv. F u r t h e r , d ( l " , v ) all

containing

I K.

independent

ideals(dv=

additiona]

ideals,and .

.

.

the is

greater

proofs to be

of

terms

in the

lemma

4.2.The

To p r o v e

can,without

is

a domain.

true

of

obtain

to

only

4.2

of Q / L ( v )

d(l",v)_

two i d e a l s

of

ideals

has

set

of

integer which

the

of i d e a l s

of Q / L ( v )

of Q / L ( v ) , s o

of

such that

of

Now l e t

J

of

~,KIK2,J.We

d(!,Kl,V)+d(!,K2,v)

proof

of

to 4 . 5 . T h e has

this

in

of

be

any of

that

m-primary

use

e(~,K1K2,J;M)

of

proof

v I in p l a c e

the

e ( ! ' ( v ) , M v) of

occurring

in

is

the

that

~,Ki,J

d ideals,and

the

= e(~,J,K1K2;M

last

one.

of v. S e c o n d l y , 0 and

symmetry

in

the n u m b e r

L(Vl)=

ideal,so

Q containing

now make

to w h e n

reducing

further

write

generality,assume

ideals

as

is c o n t a i n e d

the e x p r e s s i o n

requiring

fix v , a n d

theorem

statement

the e f f e c t

occurring

statement

,we first

set

e

set

ideals

.

the

to those

loss

independent

of

to be zero

sum

this

we

part

lemmas

taken

ideals

function

KI,K 2 are

of

independent,then d(!,Ki,K2,v)=

The

I" of

set

non-nesative

I is an i n d e p e n d e n t

is an i n d e p e n d e n t

l

I,KIK 2 is also

sets

dim(Q/L(v)).The

property.lf

dv-2

(i =1,2)

an i n d e p e n d e n t

is a f u n c t i o n , t a k i n $

of

the

that

Q

is

an

same

is

function

)

e(~,J,KI;M)+e(~,J,K2;M) e(I,KI,J;M)+e(!,K2,J;M) Now

suppose

(i = 1,2)

in the

M = Q.The

fact

ideals

are

we

replace

can

we

expand

form

that

linearly

the

given

the

terms

in

the

functions

independent

e above

e(!,KIK2,J;M)

theorem v(J)

according

that

further

to lemma

3.1

implies

of V l ( J ) . S i n c e is the

take

of m - p r i m a r y

field

we

that

L(Vl)=O,it

of Q and

that

e((~,Ki)'(Vl);Q) = e((!,KiK~)'(Vl);Q) = l.Hence

that

are

of f r a c t i o n s we

are

left

the e q u a t i o n d(!,KIK2,Vl)= Note We

theorem in the

that

conclude

this

is also paper

to two s p e c i a l

first

m-primary,and First

d(l,v)

d(!,Kl,Vl)+d(!,K2,Vl

symmetric

by a p p l y i n g

situations

and

in the set the

of

formula

re-derive

)-

ideals

given

some

I.

in the

results

last

already

literature.

The

also

Qv

set

e(~,Ki,J;M)

assuming

with

follows

aboveand

on the

by the c o e f f i c i e n t

and

situation hence

we c o n s i d e r assumed

general

that

we

necessarily

consider form

the m u l t i p l i c i t y

an

is w h e n

independent

function

to be m - p r i m a r y . l f , X l , . . . , X d _ 1 , y

elements

of l , J , t h e n , b y

theorem

l l , . . . , I d _ I are set

e(~,J;M)

of

where

ideals

form

of Q.

J is

is an i n d e p e n d e n t

2.5,they

all

a joint

set

of

reduction

359

of

~,J.But

of

the

the

it

was

ideal

set

of

ideals

multiplicity mixed

~,J

function

multiplicity

theorem

4.6.Let

and

v(J)

and

contains

this

true

in

t of

only

as

that

is

the

is

of

this

sense

formula ideals if

reduces

the

being

require

J

J = xQ

Now

hence

on

consider

must an

Qv

is

Teissier's

given

in

L(v).If

these

are

ht(L(v))=t,

the

a primary

good

m-primary,

prime

such

factor

reduces

v is

all

a minimal

of

the

coefficient,then

be

factor

of

formula

m-valuation

consider

this

all

ideal

of

this

is

that

e(!'(V)Qv,Mv).The

to

L ( M v)

artinian

where

Mv

ring. Hence

is

the

=Zd(!,v)L(Mv)V(J)

proper

m-valuations.We

be m - p r i m a r y . H e n c e , i f

providing

that

and

we

can

the

set

of

ideals

an

a generalisation

Q,and

multiplicity

[10].Hence

the

to

the

multiplicity

to

xQ)

is

Teissier

non-zero

refer

that mixed

consider

t=0,i.e.L(v)

m-valuation.

over

to

is

we

with

e(!,J;M) sum

by

the

Ii,...,Id_l.Since

will

,and

implies as

used

paper

a Qv-mOdule,and

formula

this same

m-valuation

We

empty

as

the

the the

possible

a proper

considered

[9]

in

an

dim['Q/L(v))=d.

I'(v)

in

function. Next

v be

occurs

is

Q and

set

proved

{Xl,...,Xd_l,y)

take

dimQ/xQ

y=x. N o w

x is

an

= d-l(since consider

note element

XlX

the

that

is

ring

of

we

do

Q,we

can

a general

Q/xQ

and

not take

element

write

-I- X

of

for

I.+xQ/xQ.Then the i m a g e s of X l , . . . , X d _ 1 in ~ / X ~ g J set of g e n e r a l elements of I . H e n c e we h a v e

independent

--X

e(!,xQtM) which

we

denote

by

d(l

,x,M).We

d(!,x,M) This in

formula,and

greater

M=Q,we

consists we

the

l.Let

also

main

of

just

ideal

one

a special and

such

number ideal

case

XQg

of

in

theorem

the

of

ideal

Then of Q

results

Q has will

of

is

definition

adopted

here,e(IQp,M

where

a general

of

this

paper

1 are

are

all

dimension denote we

good

l.lf

will

prime

dim(Q/p)

2.1n

by

treated

equal,and

this I is

suppose

ideals

-~- I and

p,x

a general

pg

and

being so of

e(J+p/p)e(IQp,Mp). ) has

to

is

equal

to

terms

with

be

are

it.

only

interpreted

as

a

minimal

of

Hence

L(v)=p

that,according

I is over

good

element

the Note

that

there

L(v)=p. Further,the

in

case m-primary

minimal

Q with

contribution

formula

M)

formula

ll,...,Id_

above.Hence set

contains

4.2,the

-e(Ix,(0:x)

the

[6].

we

the the

meeting

l-dimensional,and

proof

of

where

m-valuations

P of

the ideals

which

consider

a prime

of

the

case

obtain

~d(!,v)L(Mv)V(X). of

result

the

p be

prime

[S].If

consider

m-primary

finite

in

now

=

fact,most

we

obtain

not

sum

detail

obtain Now

in

e(!x;M/xM)

=

l,is ,by

to to

the

the

the

e(x(Qp)g,Mpg~

P is Now

x

is

a reduction we

consider

formula. These

of

I

the must

element and

g

other be

so

of

l,but,since

this

possible

proper

definition

Qp

reduces

valuations

valuations

is

.Hence

l-dimensional,X(Qp)g to

the

v occurring the

formula

usual in

one.

the

reduces

to

360

e(l,J;M)

=~e(IQp;M

--

the

first

sum

being

second sem being sum

could

be

over

over

reduced

the

ideas

was

written,and

behind

author

this the

Stockholm

were

a lecture

in

the

Research

first the

August

Symposium

final,presented

minimal

over

l,and

Q(note

that

the

on

simplest

like

to

during

of

during

ideals

seems

hospitality

paper

in

not

1 prime

m-valuations

would

organisers

Symposium

version,still

ht

proper

further,but

its

+ Ld(l,v)L(Mv)V(J)

P

the

the

In c o n c l u s i o n , t h e Institut,Aarhus,for

)e(J+p/p)

~

in

thank

the the

April

and

developed

and

Nordic 1983

Summer

for

which

an

above

May

1979,when

the

first

School

the

form).

Matematisk

and

invitation

formed

the

first

basis

to of

draft Research give the

here.

References I.

S.Abhyankar.

On

the

valuations

centered

Amer.J.Math.78(1956) 2.A

.Grothendieck.

Elements

de

Geom~trie

P u b l . M a t h . IHES 3.D.G.Northcott.

A genera]

theory

P roc.G1asgow 4

"

"

No

in

a local

Algebrique.

11(1961) of

one-dimensional

Math. A s s o c i a t i o n

A generalisation

domain.

pp321-348

of

local

2(1956)

a theorem

on

the

rings.

pp159-169. content

of

polynomials. Proc. Cam. P h i l . S o c . 5 5 ( 1 9 5 9 ) 5

"

"

Lectures

on

Cambridse 6

7

D.Rees

"

Degree

"

University

Functions

ppI-7

pp

Asymptotic

"

,Hilbert

Properties 1983)To

Note

be

of

note

Cycles

of

published

General

P.Samuel

no72(1982)

Multiplicity

Commutative D.von

in

the

given

L.M.S

and

Mixed

planes

et

a Cargese

Conditions 1972

7-8(1973) Theory

Proc. L o n d o n . Math. S o c . ( 3 ) and

1981")

London. Math. Soc.29(1984)397-414.

de W h i t n e y . I n " S i n g u l a r i t e s

12 O . Z a r i s k i

degree

Ideals(Lectures

evanescents,sections

Asterisque D.J.Wright

series

Reductions

Multiplicities.Jour.

ii.

and

Algebra;Durham

series

Generalisations

10.B.Teissier.

Functions

(in"Commutative

Math. S o c . ] e c t u r e

in N a g o y a

"

in

70-78.

Lecture 9.

Press,Cambridge(1968)

Proc. Cam. P h i 1 . S o c . 5 7 ( 1 9 6 1 ) Multiplicities

"

multiplicities.

rings.

London

"

pp282-288 and

Local

functions

8

Rings,modules

Algebra.volume

Nostrand

15(1965) II

(Princeton)f960.

pp269-288

COHOMOLOGIE DE HARRISON ET TYPE D'HOMOTOPIE RATIONNELLE

Daniel TANRE ERA C.N.R.S. O~ 590 U n i v ~ s i t ~ des Science~ e~ Techniques de LILLE U.E.R. de Math~matiques Pur~ et Appliqu~es 59655 - VILLENEUVE D'ASCQ CEDEX (France) I II

-

III-

COHOMOLOGIES DE HOSCHSCHILD ET DE HARRISON. THEORIE DE L'OBSTRUCTION D'HALPERIN-STASHEFF. COHOMOLOGIE DE HARRISON ET FORMALITE INTRINSEQUE.

APPENDICE : MODELE'DE L'ESPACE PROJECTIF TRONQUE

¢P(m)/g~(2).

La th~orie de la deformation permet l'Etude des types d'homotopie rationnelle g alg~bre de cohomologie (ou alg~bre de Lie d'homotopie) rationnelle fix~e ~-S].

Halperin et Stasheff LH-S] ont obtenu les premiers r~sultats dans

ce domaine ; rappelons d'abord la terminologle utilisEe : un espace dont le type d'homotopie rationnelle est entigrement dEterminE par la donn~e de son alg~bre de cohomologie (resp. alg~bre de Lie d'homotopie) rationnelle est appelE formel (resp. coformel). Un espace est intrins~quement formel si son alg~bre de cohomologie est rEalis~e par un seul type d'homotopie rationhelle.

Nous montrons ici que les obstructions d'Halperin-Stasheff ~ la formalitE s'interpr~tent comme classe de cohomologie de Harrison. Cette dernigre semble ~tre le cadre le mieux adaptE ~ cette situation ; g partir du rEsultat ci-dessus, elle a permis ~ D. Merle EM~ Stasheff ~ - ~

et FElix EF~

d'unifier les theories d'obstructions d'Halperinet celle introduite par Lemaire et Sigrist ~L-~

dans le cadre des modules de Quillen. La construction du modgle bigradu~ EH-~

est illustrEe par un exemple

cohomologie non bornEe : ¢P(~)/¢~(2). Nous rendons triviale la premigre d~formation possible de ce modgle. La cohomologie Etant non bornEe, il existe une infinit~ de deformations possibles. Seule l'utillsation de la cohomologie de Harrison permet d'obtenir leur triviaIitE. La demonstration compl~te passe par une determination explicite de tout le module bigraduE et par l'interprgtation de ¢I~(~)/~(2)

cormne espace total d'une fibration ; elle fera l'objet d'une publi-

cation ult~rieure. Plus gEnEralement, le rEsultat obtenu concerne les espaces projectifs tronquEs

g~(~)/C~(n)

;

il s'~nonce ETa I~ :

362

: Rationnellement,

Th~or~me

il existe deux espaces

m~me alg~bre de Lie d'homotopie pas coformel

;

E

rationnelle

que

est l'espace coformel associ~.

n

¢~(~)/¢~(n)

~P(~)/¢~(n).

et E ayant n ~P(~)/~(n) n'est

Ils sont tous deux intrinsgque-

ment formels. Ce texte reprend une partie de ma th~se d'Etat soutenue ~ Lille, 26 janvier

1982.

Notations : gradugs signe tion

Nous emploierons

apparalt.

d'objets

le paragraphe

rationnels. gne par

Si

V

a(o)

x

consid~rfis

gradufi, de base

T(V) = T(Xl,...,x n) dual est notfi par

@ V,

le

o.

gradu~e commutative.

sont sur le corps (Xl,...,Xn), libre,

@

des

on d~si-

~(V) = ~(x I .... ,x n)

l'alg~bre tensorielle, la suspension

q,

~ toute permuta-

et appelg signe de Koszul de

l'alg~bre gradu~e commutative

engendrges par

sV : (sV) n = V n+l,

le

Ixl.

D'une mani~re g~n~rale,

les notations

utilis~es

sont celles de ~T~ .

DE HOCHSCIIILD ET DE HARRISON.

Soient

~

un corps et

D~fi~o~.~ entiers

le signe correspondant

est un espace vectoriel

L'espace vectoriel

I - COHOMOLOGIES

est permut~ avec un ~l~ment de degrg

adgc signifie alg~bre diff~rentielle

AV = A(xl,...,Xn)

degr~ d'un gl~ment

usuelles de signe pour les objets

I, les espaces vectoriels

l'alg~bre de Lie libre, V.

p

En particulier,

gradu~s est not~

L'expression Hormis

les conventions

: si un ~l~ment de degr~ (-I) pq

o

le

Un

{l,...,p+q}

V

un

~-espace vectoriel gradu~. est une permutation

(p,q)-mixage

de l'ensemble des

telle que :

~(i) < o(j)

si

D@f~on.-

L'espace gradu~

ou

1 @ i < j ~ p

p+l ~ i < j ~ p+q.

T(V)

est une alg~bre gradu~e commutative

pour le pr0duit mix~ d~fini par : o ... o v _: (v I ~ ... @ Vp) *- (Vp+ I ~ ... ~ v n) = X ~ ( o ) v u (n) o - ] (1)

o~

~

parcourt

~l~ments

v. 1 Soit

pour tout

p),

les

(p,n-p)

mixages,

g(O)

est

l e s i g n e de K o s z u l de

o,

les

s o n t homog~nes.

A

une

connexe,

Ik-alg~bre gradu~e commutative, (A° = ~), et soit

M

un

de type fini,

A-module gradu~.

(dim A p

finie

363

La cohomologie

de Hochschild,

provient du complexe suivant

Hoch(A;M),

de

A

~ coefficients

dans

M

: 11

si

a. e A, ]

a I @ ... @ a

I +

n

HomP(~ A,M) que

est mnni du degr6

est l'ensemble des applications

f(a I O ... @ a n ) = O

si

a.i = I

~ (lajl - ]) j=l

~-lin~aires

;

de degr~

p

telles

•)

(@f) (a] @ ... @ an+ l) = alf(a 2 @ ... @ an+ l) + (-l)V(n)f(al

0 ... @ an).an+ I

n

+

~ (-l)~(J)f(al j=!

avec

~(j) =

zn'P(A;M)

@ ... @ aj.aj+ 1 O ... O an+l) ,

J [ (;ail - 1) i=l

est formg des

Hochn'P(A

;

6-cocycles de

; M) = Zn'P(A

n-I ; M)/6 HomP-l( O A,M)

La cohomologie de Harrison, s'obtient

Harr(A;M),

~ partir d'un sous-complexe

HomP(~ A,M)

HomP(~ A,M)

de

A

~ coefficients dans

du complexe de Hochschild,

est formg des gl~ments de

HomP(~ A,M)

M

dgfini comme suit

s'annulant sur les d~com-

S

p o s a b l e s du p r o d u i t

Zn'P(A;M)

mixfi ; i l

est

stable

pour la difffirentielle

~

;

= Zn'P(A;M) A Hom~(~ A,M)

S

Harrn'P(A

; M) = Zn'P(A S

n, A,M) I

• M)/6 HomP-l( @ )

S

1

Elle est reli@e ~ la cohomologie de Hochschild par : Th~o~m£ l)application

(M. Barr ; ~ B ~ ) . -

naturelle

Harr(A;M) est injective.

Si

~

: +

Hoch(A;M)

est un corps de caract@ristique

O,

Pour terminer ristique

p

(~ droite) complexe

du foncteur

cotangent

II - THEORIE

V,

H(O)

0

÷

A

et Stasheff

suppl~mentaire

sur

et v~rifie

: dV p C

AV

eat un isomorphisme

de Sullivan

[H-~ V,

[Su] d'une alg~bre

d~finissent,

V =

@ V p~O P

(AV) p-1

," H+(AV,d)

de

Ltappendice

illustre

la construction

de

'•

pour un bon

;

p

celle-ci

s'gtend

eat bihomoggne

de

(A,d A)

du mod6le

(D-d)(V n) C

= 0

A. des premiers

g~n&rateurs

du modgle

H(¢~(~)/¢~(2);~).

construisent

le ~

le modgle minimal

Halperin

bigradu~

Si EH-S]

d6riv6

eat le

D'HALPERIN-STASHEFF.

eat le module

bigradu~

partant

~

Harr(A;A)

;

: Ho(AV,d)

(AV,d)

connexe.

d'alggbre

eat faux en caractgpas d'un foncteur

EQu].

une graduation

en graduation

ne provient

Avec un saut d'un degr6,

O : (AV,d) * (A,O)

commutative

choix de

degr~

D(A/~;A)

notons que ce r6sultat

de Harrison

HomA(A;-).

DE L'OBSTRUCTION

Soit gradu~e

ces rappe]s,

et que la cohomologie

eat une adgc,

un module

de Sullivan

bigradu6

@ (AV) m m&n-2

cohomologiquement

connexe,

(non minimal)

si

v e Vo,

et Stasheff

~ : (AV,D) ÷ (A,dA) ,

@ : (AV,d) * (H(A,dA),O).

;

Halperin

Ce module

la classe de cohomologie

v6rifie de

en

:

~(v)

eat 6ga-

~H-S] .

Appelons

p(v). (AV,D)

eat le mod6le

filtr6

Chacun de ces deux mod61es TJ-graduation

et

TJ-filtration

de

(A,dA).

v6rifie

un th6or~me

(pour Tate-Jozefiak)

d'unicit6

lea graduation

et filtration

suppl6mentaires. De6i~O~w~ que

(H(A,dA),O).

d'alg~bre r6alis6e

L'alg6bre

de cohomologie

Soit

de

0.

H

(A,d A)

eat formelle

eat intrins~quement

isomorphe

g

H

si elle a m~me modgle minimal

formelle

sont formelles

si toutes

; autrement

lea adgc

dit,

H

eat

par un seul type d'homotopie. R6sumons

bigradu6

: Une adgc

la thgorie

d'obstructions

~ : (AV,D) + (A,d A)

p : (AV,d) Dgfinissons

÷

(H(A,dA),O)

maintenant

~ la formalit6

un mod61e = (H,O)

une application

filtr6 ;

d'Halperin-Stasheff.

construit

notons

~ partir du

q : H ÷ AV °

une section

365

: Hom

o(Vp_l,H)

d6rivation

{)u7 = rl~/

de

÷ Homl(Vp,H) (AV)gp_1,

sur

de degr6

O,

= O @~dv

d6finie

par

p

d6finit

:

le premier

un 616ment

de

indice

tel que

Homl(v

,H),

D-d

~7 = O

est l'unique

sur

soit non nulle

on note

0 (D)

P Homl(Vp,H)/Im

I.

Si

DD = ~d

tel que

O~

oO

V
Vp_l.

Soit (D-d)

I%(~)(v)

par

Op(D)

sur

= O,

(fiV)&p.

sur

sa elasse

P

@w

il existe un automorphisme On remplace

alors

D

V . P dans

par

> = e ~ ]ffID~

de

AV

pour construire

Op+1(D).

de

AV

O (D) P tel que

repr6sente D~ = ~d

est formelle.

construit

sur

[H-~ .-

Th~or~me p, (iV,D)

l'obstruction

sur le modgle

g l'existence

d'un automorphisme

(fiV)~p.

Supposons

H(AV,d)

de type fini.

Si O (D) = O pour tout P bigradu6 (fiV,d), H(AV,d)

p

Si Op(D)

et tout module

de degr6,

de 3 unit6s.

la premi6re

d6formation

Elle est donn6e par 4 D3v I = ~ x 3 x 4

o3

~, 8, 7

de

d

baisse

D3

se prolonge

4 ~(z 1) = ~x 3

:

;

;

en

Z 4 , il faut et il suffit que 3 P(z 2) = ~x3x4, on obtient :

4 = pe~(yly 2 + x4z I + x3z 2) = 2~x3x 4 = 2D3v 1

I(P)(v2)

= 2D3v 2

La d6formation

; D3

modgle bigradu~.

peut donc ~tre rendue triviale

d~velopp6e

La difference

considgrent

toutes

par F61ix entre

EF~

sont identiques,

est ~galement

les deux approches

les applications

en compte que celles se prolongeant

les r6sultats

~ = ~ = T.

l(P)(v 3) = 2DBV 3. par un automorphisme.

L i a ~ o n entre les obstructions d ' H a l p e r i n - S t ~ h e ~ f ~ La th~orie

ne prenant

Pour des

la TJ-graduation

D3v 3 = T x~ ,

~(~)(Vl)

et Stasheff

(iV,D)

formelle.

: 4 D3v 2 = ~ x3x 5

;

filtr6

sont des rationnels.

Pour que En posant

possible

= O pour tout

est intrins6quement

Exemple : Illustrons cette th~orle ~ l'a[de de l'appendice. raisons

D

comme le montre

de degrg g

Vp+ 1 .

de F~lix :

men6e

~ partir

du

tient au fait qu'Halperin I de V dans H, F61ix P Au niveau des obstructions,

le eorollaire

page 26 de

~F~.

366

III - COHOMOLOGIE DE HARRISON ET FO~MALITE INTRINSEQUE. Pour la fin de ce paragraphe,

A

est une alg~bre gradu~e commutative,

connexe, de type fini. La liaison entre les obstructions prgcgdentes et la cohomologie de Harrison passe par l'utilisation d'un module particulier : l'alg~bre des cochalnes sur le module de Quillen ou mod~]e FHS (ECHO, EFe ~ , D~crivons-le pour

~Ta~ page 67).

(A,O) :

p : (AZ,d) ÷ (A,O)

est un morphisme d'adgc induisant un isomorphisme en cohomolo-

gie avec : (AZ,d) = (As -I @

L(W),d I + d2),

W @ ~ = s

-I

@

A,

dI

est lin~aire en

Z

et

d2

quadratique. L'injection canonique de l'alg~bre de Lie libre tensorielle

T(W)

L(W)

dans l'alggbre

fournit par dualit~ et d~suspension :

j : s-IT(@ W) ~ s-I @ T(W) ÷ s-I @ IL(W). Si

(Yi)iei

est une base homog~ne de

@ W,

not@

J(s-l(yi @'''@Yi ))' o p ; l'application s-I ~ ~(W)

Yi ...i fournit un syst~me de g~ngrateurs de o p induite j : A ÷ AZ est une section de p (j@ = id). O

Soit

yiYj = k~!li,jl cij (k)Yk,

e.. (k)lj e Q,

la loi d'alggbre de

A.

En dgtaillant la d~finition de l'alg~bre des cochaTnes, on obtient (ETa] page 71) : m-! d2Yi ...i = ~ Yi ...i Yi ..i o m p=O o p p+ l " m

dlYi

;

m-2 i = ~ ! c.. (k)(-l) x)(j) j=O k lij,lj+ll ljlj+l Yio...i._ik ij+2 .... o "'" m " j

(AZ,d2)

est le module bi~radu@ du bouquet de spheres d'homologie

im "

~ A

(ETa] page 66). La TJ-graduation correspond ~ la longueur des crochets par orthogonalit~ :

Z

= S-I(~P+2~(W)) ~. P

Le lien avec la cohomologie de Harrison appara%t dans la :

Proposition.d~signe par

une application lin~aire de degr~ l, on l m+l la d~rivation d'alg~bre associ~e et par ~ e Homs( @ A,A) le com-

pos~

Si

@ Y y = O Yj.

@

Soit

~ : Zm ÷ AZ o

est la diff~rentielle du complexe de Harrison, alors :

367

(Yi 8 . . . .

i)

67(s

2)

7 peut ~tre fitendue 5 Zm+ 1 ssi

3)

~ peut ~tre rendue triviale par un automorphisme de

o

telle que

~

=

@ Y l m +))l = POy(d2+dl)Yio'''lm+l 6~ = 0

; AZ

ssi il existe

~'

~ y% '.

Dgmons~aZZon

:

I) Par dfifinition, on a : 6~(s-l(YioS"''SYim+Â)) = Yio~(s-l(Yil@'''OYim+l)) +

(-1)

lYi ...iml o ~(s-l(Yio@"'@Yim))Yim+l

( =

lyi 1

-I)

o

+ (-I)

.

P~7

m-I j=O

~(J)% -I y(s (YioS...Syi yi @ .... ) j j+1 ~Ylm+l )

]Yi ...i ] o m Yio ° • "imY im+ 1 +

YioYil " " "im+ 1

m-1

(-1)

|

(-])v(J) ~i

c. (k)Yi o ' ' ' l j -"1 keli j ,ij+ll l j i j + I

j=O

k lj+ ' 2" ..im+l~J

= p@ (d2+dl)Yio...im+ 1" .

2)

y

peut ~tre ~tendue g

(d2+dl)-cocycle,

3)

i.e.

Zm+ 1

pey(d2+dl)y i

.

o'''lm+l

ssi Q (d2+dl)y i . . o ''im+l

est un

= 0.

La derni~re proprigt~ se d~duit directement de la comparaison de %

l(y')

et

~%'

;

la construction de

1

et le th~or~me d'Halperin-Stasheff

se transcrivent tels quels au module FHS. De la proposition ci-dessus et du th~or~me d'Halperin-Stasheff, on d~duit directement : Thgo&~me.- Si

Harrm'l(A;A) = 0

pour tout

m > 2,

alors

A

est

intrins~quement formelle. RemaYcque

l'hypoth~se

: La r~ciproque du th~orgme est fausse en g~n~ral. En effet,

Harr%l(A;A)

= 0

signifie que toute application

l m+l

~ e Homs( 0

A,A)

368

prolongeable Or,

en colonne

d2+dl+7

particulier

Zm+ 1

ne donne pas n6cessairement

Si

Harrm'2(A;A)

sont 6quivalentes (i) (ii)

l'alg~bre

Nous laissons

tre la remarque

A

Harrm'l(A;A)

page 21) s'adapte

une diff6rentielle,

= 0

pour

m > 4,

est intrins~quement = 0

pour

la d6monstration

formelle,

au lecteur

De mSme,

; celle faite par F61ix

l'exemple

de l'annexe

(EFt,

1 de EFe] illus-

ci-dessus. Un simple calcul donne la description

de l'alg&bre de cohomologie

Ixi [ = 2i,

les propri6t6s

m > 2.

rationnelle

de

i(x 3, x 4, Xs)/R o3

sauf dans un cas

:

ici sans probl~me.

Appe~d~ee~ : relations

par un automorphisme.

:

Propos~on.suivantes

peut ~tre rendue triviale

R

est l'id6al engendr6

Les premiers g6n6rateurs Zo

Z1

Z2

Z3

Z4

par

x~-

par ggn6rateurs

£P(~)/K]P(2)

et

:

, 2 x3x5, x4x 5 - x~, x~ - x3x 4.

du module bigradu6

s'gcrivent

:

6

dx 3 = 0

8

dx 4 = 0

10

dx 5 = 0

15

dy 1 = x~ - {x 3 x 5}

17

dy 2 = x 4 x 5 - {x~}

19

dY3 = x ~ -

24

dZl = Yl x5 - Y2 x4 + {Y3 x3}

26

dz2 = Y2 x5 - Y3 x4 - { Y l

31

dv I = Yl Y2 + x4 Zl + {x 3 z 2}

33

dv2 = Zl x5 + Yl Y3 + x 4 z 2

35

dv3 = z2 x5 + Y2 Y3 + {x23 z 1 }

38

dWl = Yl Zl - x4 Vl + {x 3 v 27

40

dw2 = Yl z2 - x4 v2 +

40

dw5 = Zl Y2 + Yl z2

{x~ x4}.

-

x23 )

v I x5 x 4

v 2

+

(x 3 v 3}

369

42

dw 3 = v2 x5 - Zl Y3 - x4 v3

42

dw 6 = Y2 z2 + zl Y3 - v2 x5 + {x~ v I}

44

dw 4 = v3 x5 - z2 Y3 - {x~ v2}.

45

du I

wl x4 - vl Yl + {w2 x3}

47

du 2

w2 x4 + Wl x5 - v2 Yl

47

du 7

x4(w5-w2 ) - Vl Y2 + {x3~w3+w6 )}

47

dUlo

49

du 3

x 4 w 3 - v3 Yl + vl Y3 + w2 x5 + (x~ w I + x 3 w 4}

49

du 6

x 4 w 6 + z I z 2 - w 5 x 5 + {x~ w I + x 3 w 4}

49

du 8

v2 Y2 - Y3 Vl + x5 w2 - x5 w5 - x4 w6 + {x3 w4}

51

du 4

x 4 w 4 + v2 Y3 + w3 x5

51

du 9

51

dUll

53

du 5

w 6 x 5 - v2 Y3 - !/2 z~ - {x~ w 2} 2 Y2 v3 + x 5 w 3 + x 5 w 6 - {x3(w5-w2)} 2 v3 Y3 + w 4 x 5 + {x 3 w3}.

Z5

1/2 z 2

I - v2 Yl + w5 x4 + {x3 w3}

Comme annonc~ dans l'introduction, aux techniques

des modules minimaux

lit~ intrins~que

de cette alg~bre.

la cohomologie

(KS-modUles,...)

de Harrison

permet d'gtablir

allige

la forma-

370

BIBLIOGRAPHIE

[B~

Michael BARR

- Harrison homology, Hochschild Journal of Algebra

LCh~

Kuo Tsai CHEN

-

Extension

of

C~

8, (1968), function

homology and Triples, 314-323.

Algebra by Integrals and

Malcev completion of

1' in Math. 23, (1977),

Advances ~

Yves FELIX

- D~nombrement

des types de

la d6formation,

M@moires

181-210.

K-homotopie. SMF, nouvelle

Th6orie de s6rie n ° 3,

(1980). - ModUles bifiltr6s.

LFe I] Yves FELIX

Can. J. Math. 33, n ° 26, (1981),

1448-1458. ~e

~

Yves FELIX

-

Espaces Luminy

Steve HALPERIN,

formels et (~ para~tre

James STASHEFF - Obstructions Advances

D.K.

HARRISON

-

HOCHSCHILD

-

in Math.

Commutative (1962),

G.

~-formels.

Conf6rence Marseille-

SMF). to homotopy

32, (1979),

equivalences,

233-279.

algebras and cohomology T.A.M.S.

104,

191-204.

On the cohomology groups of an associative

algebra.

Ann. of Math. 46, (1945), 58-67. Jean-Michel

LEMAIRE,

Fran@ois

SIGRIST - D6nombrement

rationnelle.

[M£

Pierre MERLE

- Formalit6

C.R.A.S.

des espaces et des applications

Th~se de 3~me cycle, Nice, Daniel QUILLEN

des types d'homotopie

t. 287 A, (1978), Paris.

- On the (co)-homology

of commutative

Proc. Symp. Pure Math.

continues.

(1983).

17, A.M.S.

rings,

Providence

(1970),

65-87.

Is-s]

Michael

SCHLESSINGER,

James STASHEFF - Deformation homotopy

[Su]

Denis SULLIVAN

Infinitesimal

computations

Publ. I.H.E.S. Daniel TANR~

ETa I]

- Homotopie Sullivan. Verlag.

47,

in Topology,

(1977), 269-331.

rationnelle : ModUles de Chen, Quillen, Lecture notes in Math. 1025, (1983), Springer

!

Daniel TANRE

theory and rational

type (~ paraTtre).

- Th~se, Lille

(1982).

COHOMOLOGIE DE L'ESPACE DES SECTIONS D'UN FIBRE ET COHOMOLOGIE DE GELFAND-FUCHS D'UNE VARIETE par Micheline VIGU~E-POIRRIER(*)

R~sum~

Soit

•

F ~+ E ~

sont connexes par arcs, nilpotents

X

un fibr~ nilpotent,

oN les espaces

et ont le type d'homotopie

de C.W. complexes

+ de type fini. On suppose que le type d'homotopie d'homotopie

d'un complexe

k.+1 r 1 V S i=]

de

H (X,@) # O, qu'il existe

oil

sections continues du fibre. r~elle

C > |

cas suivants

tels que si : ou bien,

type d'homotopie dimension

.< n

et

inf(k.) I

n,

>. n. Soit

tel que et

r

N e ~

F

type

et une constante

~ dim Hi(F,(~) ~ C p dans les deux i=O le fibr~ est trivial (i.e. F = F X), ou bien X ale Sd V y

de Gelfand-Fuchs

(o~

Y

est un complexe

>. 2

d'une vari~t~

telle que

simplicial de

AMS

:

M,

des grou-

C ~, compacte,

con-

H+(M,fR) # O, et dont toutes les

sont nulles est ~ croissance

CLASSIFICATION

exponentielle.

55 P 62, 55 R 05, 57 R 32

MOTS CLES : ModUle minimal de Sullivan, fibr~ nilpotent, cohomo]ogie de Gelfand-Fuchs. (~) ERA au CNRS 07 590

a

p ~ N, on a

de dimension

classes de Pontryagin

ale

X

l'espace des

d ~ I). On en d~duit que la suite des dimensions

pes de la cohomologie nexe, nilpotente

r > 2

de dimension

On d~montre qu'il existe

d'un bouquet et

simplicial

n >. I

372

O. Introduction. Dans

[16], Thom ~tudie le type d'homotopie

cations continues d'un espace

X

dans un espace

de l'espace des appli-

F, homotopes ~ une applica-

tion donn~e. Dans commutative,

[]4], Sullivan d~crit une alg~bre diff6rentielle

module de l'espace des sections d'un fibre alg@brique donn~. Dans

l'espace tent

F

[5], Haefliger d~termine

connexes par arcs, tel que (A,dA)

tel que

alg~bre diff~rentielle de l'espaee

le type d'homotopie

rationnelle de

des sections homotopes ~ une section donn~e pour un fibr~ nilpo-

E : E + X. Si on a un tel fibr~

module

gradu~e

H (X,~)

dim A n < ~

~ : E + X

tels que les espaces soient

soit de dimension pour tout

gradu~e commutative

finie, et

X

a un

n, il d~montre qu'une certaine

(ASZ,D)

est un module de Sullivan

F. Tousles

espaces consid~r~s

dans ce papier sont connexes par arcs,

nilpotents et ont le type d'homotopie de C.W. complexes de type fini ; ce qui nous permettra d'utiliser,

de mani~re biunivoque,

Sullivan entre la topologie et l'alg~bre, Nous nous int~resserons la base

X

et la fibre

ale F

le dictionnaire

~tabli par

voir §.I.

~ des fibres nilpotents

F~+ E

E ~ X

type d'homotopie d'un complexe simplicial de dimension

o~ n ~ ]

est n-connexe.

Utilisant les r~sultats de [5] et [18], nous montrerons

T h ~ o r ~ e 3.3.

Soit

X

un espace n i l p o t e n t a y a n t

le type d'homo-

topie rationnelle d'un complexe simplicial de dimension n ~ 1 et tel que + H (X,~) # O. Soit F un espace ayant le type d'homoto~i e rationnelle d'un k.+l

Vr S 1 o~ r ~ 2 e t inf(k i) ~ n. Alors, si F X est i=! l'espace des applications continues de X dans F muni de la topologie

bouquet de spheres

eompacte ouverte, il existe N e ~ P p ~ N, on a ~ dim H:(FX,~) ~ C p 0

et une constante r~elle

C > |

tels que

373

Th~or~me 3.4. Soit un fibr@ nilpotent propri~t~s

suivantes,

I) 1 .
et

X Y

2) r V i=l

il existe

ale

n ~ 1

~ ~ X

F

type d'homotopie

ale

ayant les

tel que : rationnelle

d'un bouquet

est un complexe simplicial de dimension type d'homotopie

rationnelle

Sd V Y

o~

$ n.

d'un bouquet de spheres

k.+l S

1

o~

r ~ 2,

inf~k.) ~ n.

- -

i

Alors, existe

N e ~q

si

F

est l'espace des sections continues du fibr~ il

et une constante r~elle

I dim Hi(F,~) i=O

C > ]

tels que si

p >. N, on a

>. C p.

En corollaire,

on obtient un r@sultat prolongeant

Th~or~me 3.7. qoit tente de dimension

> 2

M

une vari~t~

e_t telle que

C=

M

des formes multilin~aires

c.ontinues s u r

sur

et une constante r~elle

M, il existe

N ~ IN

~ dim Hi(c~(LH)) i=O

compacte,

ceux de Ill]. connexe, nilpo-

H+(M,~R) # O. On suppose clue toutes les

classes de Pontryagin de

on a

F*-+ E

sont nulles. Alors,

si

C~R(L M )

est 1 ! alg~bre

..........

l'alg~bre de Lie des champs de vecteurs A > |

tels clue si

p >. N,

>. A p.

I. Th~orie du module minimc~ de Sullivan. Nous rappelons bri~vement qui seront n~eessaires

les r~sultats de la th~orie de Sullivan

dan~ la suite. Les d~tails se trouvent dans

[]4],

[9],

[6], E7], [17]. Les alg~bres consid~r~es oO

k = ~

ou

IR, commutatlves

dans le sens suivant

b.a = (-l)Pqa.b. On notera

lal = p

deux alg~bres eommutatives

gradu~es,

par : (a O b)(a'Ob')

le degr~ de

= (-])Ibl'la'laa'

A ° = k. Une alg~bre diff~rentielle A.D.G.C.)

sont des k-alg~bres : si

gradu~es

~ An n~O b e B q, alors

a e A p,

a e A p. Si

la multiplication

A =

dans

A

et

A O B

B

sont

est d~finie

O bb'. Une alg~bre est dire connexe si

gradu~e commutative

(A,d)

(en abr~g~

est une alg~bre 8radu~e commutative munie d'une diff~rentielle

d

374

de degr~

+I

v~rifiant

A.D.G.C.

(M,d)

d(a.b) = (da).b+(-l)]ala.(db).

est un module de

d'A.D.G.C.

: (M,d) ÷ (A,dA)

Une A.D.G.C.

(A,d A)

(A,d A)

On dit qu'une

s'il existe un homomorphisme

induisant un isomorphisme

en cohomologie.

est dite libre s'il existe un espace vectoriel gradu~

V =

@ V n tel que A = AV est le produit tensoriel de l'alg~bre ext~rieure n>,! construite sur @ V 2n+| et de l'alg~bre sym~trique construite sur @ V 2n. n n On d~montre, ~ ] , que toute (A,dA) telle que H°(A,d A) = k possgde un module minimal unique ~ isomorphisme une A.D.G.C.

libre Dans

A(),

(AV,d)

pr~s. Dans le cas oh

caract~ris~e

HI(A)

par le fair que

d(V) C

A~2V.

[14], Sullivan d~finit un foncteur contrevariant,

not~

de la cat~gorie des ensembles simpliciaux dans celle des A.D.G.C.

~. Si

X

est un espace topologique,

cial des simplexes singuliers

de

X

on consid~re

l'int~gration

morphisme d'alg~bres

gradu~es de

H~(A(X))

singuli~re

H~(X,~).

Une A.D.G.C.

(A,d A)

si

est un module de

(A,d A)

Sing X, l'ensemble

et on note encore

A(Sing X). De plus,

A(X)

des formes diff~rentielles

type d'homotopie espace

sur

simpli-

l'alg~bre d~finit un iso-

sur la cohomologie

rationnelle

est appel~e module de l'espace

X

A(X).

Si on se restreint ~ des espaces topologiques

nilpotents

ayant le

d'un C.W. complexe de type fini, on peut associer ~ un tel

X, un ~-espace

gie rationnelle que

X~

ayant m~me homotopie

X. Le foncteur de Sullivan

de categories entre la cat~gorie homotopique sont les Q-espaces), que

= O, c'est

rationnelle et m~me cohomoloA

rationnelle

et la cat~gorie des @-A.D.G.C.

dim Z n < ~ pour tout

induit une ~quivalence (dont les objets

libr~

(AZ,d)

telles

n, et il existe un ensemble bien ordonn~

I

tel

n que

Z =

$

Z

; pour tout

est une fonction croissante d'un espace nilpotent En particulier,

on a :

X

e, il existe de

~ ," d(Za ) C

n

e N A( @

tel que

n

(X),~).

Z e ;

n

ZB). Le module minimal

correspond ~ la d~composition Z n = Hom(H

Z C

de Postnikov de

X.

375 Soit maintenant

F jr-j-+ E _~N÷ X

un fibr~ dont t o u s l e s espaces

sont connexes par arcs. On suppose que

H (X,~)

dimension finie en chaque degr#, et que

HI(X)

sur

H (F). Soit

(B,d B)

espace vectoriel gradu6 me d'A.D.G.C.

~

A(p)

(B,dB) ~

L'inclusion @

i

i

dule minimal de

d

sur

B

du fibr~

,

A(E)

A(j)

÷ (B @ AZ,d)

~. On montre, dans

E

> A(F)

q

q

,~

~],

que

(AZ,~)

sont des morphismes d'A.D.G.C. ~ : (AZ,d) + A(F)

(AZ,d)

(B,dB)~-+ (B @ AZ,d)

n

~

÷ A(F)

l'appli-

est le mo-

est le module minimal de

~ ~ X.

De plus, il existe un ensemble bien ordonn6 Z =

B @ AZ, et un morphis-

tels que le carr# suivant commute :

et la projection

F. On dit que

sont de

X, alors il existe un

induit un isomorphisme en cohomologie. Soit

cation induite par

base

un mod&le de

~ : (B @ AZ,d) + A(E)

H (F,~)

agit de mani~re nilpotente

Z, une diff~rentielle

A(X)

et

~ A(X)

ou bien

I

tel que

@ Za, d ( Z ~ ) C B ® A(8<~ ~ ZB)' les ~l~ments de Z~ sont de m~me degr~ ~el et n est une fonction croissante de a (Propri~t~ "N"). Un fibr~ nilpotent poss~de un tel module minimal

~],

(remarque 6.7).

2. ModUle de Sullivan de l'espace des s e c ~ o ~ d'un ~ibr~ d'apr~s Haefliger

([3], ~], ~], D~). Soit

F ~+ E

~ ÷ X

un fibr~ nilpotent ayant une section

s

dont

t o u s l e s espaces sont connexes par arcs, nilpotents, et ont le type d'homotopie d'un C.V. complexe de type fini.

376

On va expliciter pace des sections de

X

tel que

les techniques

du fibr@ homotopes

les r~sultats

(C,dc) L+ (C @ AZ,d) + (AZ,do) C

et

O

sur

mension

ainsi

; de plus,

alg@briques

D'apr~s

minimal

(C,dc) U+ (C @ AZ,d) + (AZ,d o)

fin du §.I. Dams

~

de l'esun module

[4] et

de

d'A.D.G.C,

[63, chapitre

[5], Haefliger

I=(C @ Z), construite

un module minimal

qui est l'identit~

s. aussi dams

[13]) peuvent

on se donne une A.D.G.C.

et un morphisme

les r~sultats

le morphisme,

de [5] (d~crits

A.D.G.C.

o2

¥ : (C,dc) ÷ A(X)

de la section

: plus g@n~ralement,

totale finie,

libre not@e

s. Soit

un module

du §.I, le fibr~ poss~de

$, est un module

Les calculs se r~sumer

[5] donnant

dim C < ~.

D'apr~s

sur

g

de

ayant

de

(C,dc)

~

sur l'espace

a un module

"N"

(C @ AZ,d) vectoriel

de di-

dams une autre

6, ce morphisme

la propri~t~

associe

(C,d C)

d@crite

g la

une A.D.G.C.

gradu~

W =

$ Wk keg k dim W <=,

=

@ (Hom(Ci,~) @ zJ). Si dim Z p < ~ pour tout p, on a j-i=k pour tout k. En g@n~ral, W contient des ~l~ments de degr~s ~ O, sauf s'il existe

n > O

tel que

Z p = O,

L'application d~finie en posamt c " e C',

c'(c)

p ~ n

et

C

et

(el)

p > n.

E' : C @ AZ ÷ C @ Z~(C ~ Z) = Hom(C = e

si

c e C,

~'(z)(c *) = c ~ @ z

z e Z, s'~tend en un C-morphisme

une base de

Cp = O

la base Nu~]e,

E'(z)

d'alg~bres

,Z (C ~ Z) pour tout

gradu~es.

Si

(el)

est

on a :

= ~,. c i 8 (c.~ @ z). 1

Ii est clair que diff~rentielle

gradu~e

C ~ = Hom(C,~)

a une structure

si on la munit de la diff~rentielle

de coalg~bre d~

de degr~

d~finie par :

< d C ,c> = (-1) Ic~I

si

c e C,

c

e C

,

-!

377

II existe une unique diff~rentielle de

E' : (C @ AZ,d) + (C,d C) @ (E~(C @ Z ) , D ' )

On a , si

c

D'

sur

E*(C @ Z)

faisant

un morphisme d'A.D.G.C.

~ C , z e Z

D'(c ~ @ Z) = E'(dz)(c ~) + (-|) ]zl d~c ~ @ z

Consid~rons dans des

I~

de degr~s

(I~(C @ Z),D')

$ O, et la partie lin~aire

rentielles. L'A.D.G.C. quotient notre Th@0r~e ayant une section finie, soit

Q(D')( $ W k) k$O

(E~(C @ Z,D)

2. I. IIaefliger [5] : soit

s. Soit (C,dc) + A(X)

(C,d C) + (C @ AZ,d)

Alors I'A.D.G.C.

l'id~al engendr~ par les ~l~ments de leurs diff6-

est connexe.

F ~+ E ~ X

un module de

X

un fibr~ nilpotent de dimension totale

le module minimal du fibre, de base

(i~(C ~ Z),D)

(C,dc).

d~erite ci-dessus est un module de l'espace

des sections du fibr~ homotopes ~

s.

2.2. Changement du module de l a base. Soit

(B,d B) ÷ (B @ AZ,d)

d'A.D.G.C, de source une A.D.G.C. que

Zp

B

de dimension totale finie. On suppose

est de dimension finie pour tout On se donne une A.D.G.C.

un morphisme

o~

le module minimal d'un morphisme

p, et que

(A,d A)

Z 1 = O.

de dimension totale finie et

~ : (B,dB) + (A,dA). On en d~duit un diagramme commutatif :

(B,d B)

+ (B @ AZ,d)

(AZ,d o)

(A,d A)

~ (A @ AZ,~)

+ (AZ,d o)

6z = (a@Id)(dz), pour tout

z.

378

Par dualit@,

on d@finit une application

lin~aire

:

( ~)i @ Id : (Ai) ~ @ Z j ÷ (Bi) ~ @ Z j

pour tout

i, tout

j, qui se prolonge en un morphisme

~(~)'

:

(Z~(A 8 Z),~') + (Z~(B @ Z),D')

o~ les diff@rentielles

A'

et

de Haefliger d@crite ci-dessus.

D'

sont celles d~finies dans la construction

On v@rifie que

~"(~)'

commute aux diff~ren-

tielles et que Z~(~)'(Q(&')(

~ ((Ai) ~ @ zi)) C i

Donc, par passage au quotient, morphisme

Q(D')( 8 (Bi) ~ 8 zi). i le morphisme

8

> (B,dB)

e > (A,dA)

alors

D~monstration

en cohomolo~ie,

:

Elle g@n@ralise

de

dB @ I

Dans le cas g@n@ral, tel que

alors

celle de la proposition

q > I

et

tel que D

alors le lemme est vrai, car les diff@rentielles

I

:

en cohomolo~ie.

§.5.5 de E4]. On remarque que s'il existe

tivement aux transpos@es

d'A.D.G.C.

Z~(~ o 8) = Z~(8) o Z (~).

Le,me @.2. S i e induit un isomorphisme

Z*(~) induit un isomorphisme

donn~

d~finit un

= (E~(A @ Z),A) ÷ (Z:(B 8 Z),D).

~(~)

Ii est clair que si on a des morphismes (C,dc)

Z~(e)'

et

Z = Zq A

3 du

et

dZ e B,

sont ~gales respec-

d A @ 1.

rappelons qu'il existe un ensemble bien or-

Z =

8 Z ; pour tout ~, il existe ~el Z ~ et la fonction ~ + n ~ est croissante ~ " d(Z ~ ) C

n

tel que

n

Z C

~o e I

Soit (B,dB) ~

(B @ A(

~

fix@, l a c o n s t r u c t i o n

Z ),d)

est une A.D.G.C.

~+

(~<(~ o

appliqu@e ~

de la forme

~o

o

B @ A( ~<~ $ Z~)."

379

o~

Z+(B @ ( @ ~<~

Haefliger

2+

Z ))

est stable par

D

et appara£t comme la construction de

o

appliqu~e N

(B,dB)'-+ (B ~ A( ~ Z a ) , d ) , o

Si on q u o t i e n t e

( ~ ( B ~ ( ~ Z )),D)

par l ' i d g a l

engendrfi par les

~o ~l~ments de degr~s

tient

(2+(B @ Z

> 0

),D o )

de

~ +( B

~ ( @ ~<~

Z )), on obtient une A.D.G.C.

quo-

o

qui e s t la c o n s t r u c t i o n de H a e f t i g e r r e l a t i v e m e n t ~

o l'inclusion

:

(B,d B) ~+ (B @ AZ

d B 8 0). o

Les A.D.G.C.

consid~r~es

les th~or~mes d'isomorphismes Z~(~)

de

sont toutes connexes

c~
d'apr~s

[6], si le lemme est vrai pour

, il est vrai pour

IZ~(A e ( $

;donc,

~(~)

Z c~)

. On conclut

IZ~(A 8 ( 0 oc~
On se restreint m a i n t e n a n t

Z c~)

~ des fibres n i l p o t e n t s

F~+ E

H>

X,

tels que : I) n > I

et

X

ale

type d'homotopie

d'un complexe

simplicial

de dimension

H+(X,~) # O. 2)

F

est n-connexe.

Sous ces hypotheses, vide, connexe et nilpotent.

l'espace

F

des sections

du fibr~ est non

380

Lgmme f.3. Soit d'homotopie

d'un complexe

x

simplicial

Soit (AU,6) le module minimal (A,d A) et un morphisme en cohomologie

un espace

~ : (AU,6)

H+(X,~)

le type

¢ O.

> (A,dA)

induisant

un isomorphisme

tels ~ue:

D~monstration: pour (A,dA) , I'A.D.G.C. p > n, I n e s t

ja I = inf { p > 0 1 A p # 0 } , dAa = 0

Ce lemme est bien connu, quotient

des lemmes

[5]! on peut supposer

minimal

> n.

:

il suffit de prendre

de (AU,6) par l'id$al

un suppl@mentaire

Compte-tenu

un module

n,

ayant

de X, alors il existe une A.D.G.C.

ii) il existe a C A +,

de

nilpotent

de dimension

i) dim A < ~, A p = 0 s i p

(AU) p

connexe

de Ker 6

2.2. et 2.3.,

dans

I d@fini par I p = (AU) n e t

Ip = 0 s i p

et de la remarque

que le fibre nilpotent

v@rifiant

(A,dA) ÷ (A O AZ,d) ~ (AZ,d o)

finale

I) et

2) poss~de

o~ :

+

i)

dim A < =, A p = 0

si

p > n, il existe

lal = inf{i > 0 I Ai # O} ii)

Zp = 0

p ~ n.

< n.

a e A , da = O,

381

Explicitons Soient gradu@

ao = |'

A, num~rot~e

une base homog~ne

~ij E ~

par

de l'espace vectoriel

telle que

dAa ] = O. On

dAa i = .~. Sij a..j (On a j>1

Blj = O, pour

j ). Notons

Soit

al,...,am

par ordre croissant des degr~s,

d~finit des constantes tout

les calculs du th~or~me 2.1.

(a~)

ai = fail

le degr~ de

la base duale de

a.. Comme 1

constantes de structure ainsi

A

l

l

(SZ) k =

Yij

3-uplet

S.Z = $ (S.Z) k, on a l k l Comme implique que

O"

aI

$ ~om(Ai,~)

et

~'~"

, et

SZ = @ (SZ) k.

(Siz)k = Q a~ Q Z

"

; il

Yji = (-l)l J Yij 1

k+=.

i > O,

[],~3

l

; on a

@ Z j]

< ~l < ... ~ a = n " m on d~finit des

(i,j,l) £

J-i=k On pose, pour

O = ~

est une alg~bre,

: pour tout

existe Yij e ~ tels que a i a. = ~ 3 £ Yij = 0 si £ ~ j ou £ $ i. On pose

a., on a 1

i

I

k et

S Z = Z. o

Zp = 0

si

p .< n

et que

~i "< n, alors

(Siz)k # 0

k >. I. On a des isomorphismes

lin~aires

de degr~

-~. : 1

k+c~.

k 1 +( SiZ )

S. : Z I d~finis par

Si(z) = a! @ z,

pour tout

i e [O,...,m] (S ° = Id). m On peut ~crire : (SZ) k = @ (SiZ) k et SZ = Z @ S+Z, m i=O s+z = ® s.z, sz = AZ ~ A(SIZ) ... e A(s z). i=l i m On d~finit un h o m o m o r p h i s m e

de A-alg~bres

o~

:

¢ : A @ AZ ~ A O ASZ par : e(a @ ]) = a 0 I e(|

@ ~) =

m ~

i=0

pour tout

a.

i

@

(a~

i

~ ~)

= ~ +

a e A m ~

i=]

a.

I

~ S.(~),

i

si ~ e AZ.

382

Remarque : que, pour tout cation

~

soit un homomorphisme

£ e [I, .... m], l'application

AZ ÷ AZ 0 A(SIZ ) 0 ... O A(S Z) m

~ AZ, pour tout (I)

Le fait que

Sl(~')

+ (-I)

telle que, pour tout

~ Sl(~') +

~i,j$£-I

~ij Si(~) Sj(~')

z e Z :

dz = doZ +

m [ a i 0 Oi(z) i=l

On d~finit ainsi des d~rivations D'apr~s

Par d~finition de

oil

8 i : AZ + AZ

de degr~ D

soit un morphisme

sur

-(el-l). ASZ

d'A.D.G.C.

e, on a : m

m

m

~ a. O @.(z) + ~ (a. O I) ~ a. O S (0i(z)) i=l I l i j i=I j=l 3

m e(dz) = d oz + i=l[ a i 0 [Si(doZ)+Oi(z)]

D'autre part,De(z)

Dc(z) = DE +

l al @ Sj + i,j,f [ Yij (0i(z))"

m = Dz + D( ~ a. 0 S.(z)) i= 1 I I

m m e. ~ dAa i O Si(z) + ~ (-I) z a. 0 DS.(E) i=l i=l i l m

De(z) = Dz +

d'A.D.G.C,

O.(z) e AZ. i

[5], il existe une unique diff~rentielle

e : (A 0 AZ,d) ÷ (A,dA) @ (ASZ,D)

E(dz) = e(doZ ) +

le module minimal du fibr~

(A,d A) + (A 0 AZ,d) + (AZ,d o)

on peut ~crire, pour tout

telle que

se prolonge en une appli-

~' e AZ, on a :

= Sl(~)~'

Soit

S1

~quivaut ~ dire

~.

m

[ ~ Bija 0 S,(z) + ~ (-I) : a i 0 DSi(z). i=l j>i j 1 i=!

L'unique diff~rentielle

D

est donn~e par la formule

:

telle que

e

soit un morphisme

383

" Dz = d z o

si

z e Z

(2) (-I) a£ DS£(z) = S£(doZ ) + el(z) +

-

< ~ < £ Sj(@i(z)) l.i,j.£-I Yij

[ BiiSi(z) 2~i$l-I

ProposfJt£o~ 2.4. (Haefliger, ~ ] ) un module de l'espace

F

:

des sections du fibr~

si

z e Z

et 1 ~ [I ..... m].

(ASZ,D) = Z:(A @ Z),D) E ÷ X

est

v~rifiant les hypo-

theses 1) et 2).

R~m~que

2.5. On a un fibr~ alg~brique

(AZ,d o) ÷ (ASZ,D) = (AZ e ^S+Z,D) o~

q

est la projection de

ASZ

sur

q ~ (AS÷Z,D)

ASZ/A+Z, et

D

est d~finie par

Dq = qD. Soit le fibr~ une application

F~

E

~ + X

~ : F ÷ F = p-l(xo)

donn~ ; soit

par

@(s) = S(Xo)

x

o

e X

fix~, on d~finit

pour toute section

s,

alors le fibr~ alg~brique d~crit ci-dessus est un module au sens de Halperin, ~ ] , d e l'application

~.

Rem~que f.6. tels que X a l e + et H (X,~) # O, continues de x° e X de

X

et dans

X

X

et

F

des espaces connexes nilpotents

type d'homotopie d'un complexe simplicial de dimension F

est n-connexe, et soit

dans

Yo e F F

Soient

F

FX

et notons

telles que

(F,Yo~X,Xo)t

l'espace des applications

s(x ) = Yo" Alors on a un fibr~ : o

(X'Xo)~+ FX ~(S) = S(Xo)

pour

l'espace des applications

muni de la topologie compacte ouverte. Fixons

(F,Yo)

O~

n ~ |

s e F X.

> F

s

384

Ce fibr~ satisfait aux hypotheses du th~or~me 20.3 de ~ ] du th[or~me 6.4 de ~],

eta

pour mod[le, le fibr~ alg@brique

(AZ,do)~-+ (ASZ,D) = (AZ e AS+Z,D)

q

ou

:

~ (AS+Z,D)

(X,xo) En particulier,

(AS+Z,D)

est un module de

les formules (2) se simplifient puisque La diff~rentielle

D

I

(F,y o)

e. = O, pour tout I

.

Ici,

1,...,4.

i £

est donn~e par :

Dz = d z o

(2') (-1)~£ DSl(z) = Sl(doZ) -

~

6i£ Si(z)"

2~i~£-1

Exemple de de Rham

~(X)

2.7.

x = s n,

n ~ ]

et

poss~de le module r~el F Sn

lul = n. L'espace

F

est n-connexe, alors l'alg~bre

(A,d A) = (A(u)/u2,dA = O)

des applications continues de

Sn

dans

F

o~ a pour

module :

(AZ ~ A(SnZ),D)

(-l)nD Sn(Z) = Sn(doZ) -n

o~

o~

S n : AZ ~ AZ O A(S Z)

prolongeant l'application

~nF ÷ F Sn + F

a

(Sn Z) k = Zk+n

i d e n t i q u e de

Z

si

k ~ 1

est la dgrivation de degr~

dans (S Z). Le f i b r g n

pour module :

(AZ,d o) + (AZ ~ A(SnZ),D) ~+ (A(SnZ),O).

3. D [ m o ~ o n

des t h [ o r ~ m ~ 3.3. e£ 3.4. Soit

F ~+ E

÷ X

un fibr~ nilpotent dont t o u s l e s

espaces

sont connexes par arcs, nilpotents, ont le type d'homotopie d'un C.W. complexe de type fini. On suppose : ]) n

et

X

H+(X,~) # O.

ale

type d'homotopie d'un complexe simplicial de dimension

385

2') F r k.+l V S 1 i=l

o~

ale

r ~ 2,

type d'homotopie

d'un bouquet de spheres

inf(k.) ~ n. I

Soit

H (F,~) = H = ~ $ H , la cohomologie

mologie rationnelle de D'apr~s

F

et soit

[lO], [15] chapitre II, la module de Quillen de L(s -! H+)

(s-I H+)n = Hn~l )" Si

C~

munie de la diff6rentielle

Cm(L(s -I H+),O) = (AZ,do)

F

est

nulle

gradu~es dans A.D.G.C., alors

est minimale et munie d'une graduation inf~rieure

~ la suite centrale descendante

plus, par adjonction,

H l'ho-

d~signe le foncteur "cochaines" de la cat~-

gorie des alg~bres de Lie diff6rentielles

correspondant

de

F.

l'alg~bre de Lie libre (oN

rationnelle

on d~finit un morphisme

de l'alg~bre de Lie libre. De surjectif

P : (AZ,d o) ÷ H

et

on d~montre, dans [15], qu'on obtient ainsi le module minimal bigradu~ de au sens de Halperin-Stasheff,

Z =

et

d

est donn~e par

$ p>n k~O

H

[8]. On a

Z~

,

do
2 p+l

(A Z)k_ l

:

0



= ± <doZ , sy A sy'>

^

si

zeZ

;

y,y' e L(s -l H+) ; (si

l'~l~ment ~gal ~

y

mais

l'espace des gan6rateurs Lie libre

L(s -I H+), et

est de degr~

p, alors

est de degr~_ p+|). De plus,

sy s

y

--1

^

H+

~(~

sur lequel est construite

zk)

sy

Z

O

est

correspond

l'alg~bre de

=o.

k>O II s'en suit que l'application tive, oO

Pk

est la projection

de

(A2Z)k/

Z

(A2Z) k

@

,.
Pk o d O : Zk+ I * Zo-Z k sur

Zi

Zk-i

est injec-

386

Soit lemme 2.3., et

(A,d A)

un module de

X

satisfaisant aux conclusions du

(A,dA)~+ (A @ AZ,d) ÷ (hZ,do)

le module minimal du fibr~

de base

(A,dA). Quitte g faire un isomorphisme, on peut supposer que

(AZ,d o)

est le module minimal bigradu~ de

(ASZ,D)

le module de l'espace des sections du fibr~ donn~ par la proposi-

H

au

sens de [8]. Soit

tion 2.4. D'apr~s les r~sultats de Shibata, D l ] page 404, on peut d~crire (ASZ,D)

ainsi : il existe une structure de

gradu6e sur

A O L(s -I H+)

telle que le complexe de cocha£nes

C~(A ~ L(s -I H+))

est un module de

C~(A O L(s -I H+))

est exactement

Le.mme 3.1.

[II] : Soit

E

et le complexe de cocha~nes

(ASZ,D). (A,d A) + (A @ AZ,d) ÷ (AZ,d o)

minimal d~crit ci-dessus. Alors, pour tout existe alors

a

~ (A+ O AZ) p

A-alg~bre de Lie diff~rentielle

tel que si on pose

d'z = d o z + 7- b J O z., J b.J c A + , 3

z e Z p, pour tout ~(z)

=

Z-az, et

le module

p > O, i l d'

=

~-I

odo

z.3 ~ Z.

Compte-tenu du lermme 3. I, on supposera donc, dans la suite que, m

pour tout

z ~ Z,

dz = doZ +

~ a:~ O 0i(z) i=l

o~

e.l(z) ¢ Z.

On dgfinit sur et

D'S.(z)t

;

(p e I d )

H @ AS Z une diff~rentielle D' par + DSi(z) si z e Z et i ~: {1 , . . . , m } . A l o r s

0 @ Id : (ASZ,D) + (H O AS+Z,D')

D' = 0 Ii l

induit un isomorphisme en cohomologie. On

a donc H * (r,~) = H*(H @ AS+ Z , D ' )

Le but de cette ~tude est de minorer les dimensions des groupes de cohomologie de

(H O AS+Z,D').

Lemme 3.2. II existe z ¢ Z p, on a :

N ~

tel que pour tout

D'SI(Z) c H + O SIZ, e_~t D,(H + 0 s 1 z P )

= O.

p ~ N, tout

I~

387 D~monstration

:

D'apr~s

(2) page

12, on a :

(-I) ! • DSI(Z) = Sl(do z) + el(Z)

,

d'o~ : (-I)

. D'SI(z)

=

(t),

11, S 1 e s t

D'apr~s AZ O A(SIZ). v. i

et

Si

z ¢ Z~,

page

(P 0 Id)[$1(doZ)

une d~rivation

doZ = E z i u i + ~ v i wi, o~

de d e g r ~ - ~ z i ~ Zo,

1

de AZ dans

u i ~ Zk_ 1 ,

w. e Z ; on montre faeilement que 1 + (P 0 I d ) ( S l ( d o Z ) )

puisque

S1

e p(Z o) 0 SIZ = H+ O S1Z

est une d~rivation et que

p(Z+) = O. Soit

N = a I + sup{d > 0 I H d #

alors si

0},

z ~ Z p, p ~ N, on a

lel(Z) [

donc

+ Of(z) ]

p(el(Z))

= t+p-a 1 ~ 1 + sup{d

t Hd # O} ,

= O.

On va d~montrer la croissance exponentielle Betti de

F

des hombres de

dans des cas particuliers.

Th~or~me 3.3. Soit

X

un espace nilpotent a y a n t l e

topie rationnelle d'un complexe simplicial de dimension H+(X,@) # O. Soit

type d'homo-

n ~ !

e t tel que

un espace ayant le type d'homotopie rati0nnelle d'un k.+| r 1 b o u q u e t de s p h e r e s V S oh r ~ 2 e t inf(k.) ~ n. i=l i 1 Alors si FX e s t l ' e s p a c e d e s a p p l i c a t i o n s c o n t i n u e s de x dans

F

tante

r~elle

F

muni de l a t o p o l o $ i e C > l

tels

compacte ouverte,

que,

pour tout

dim Hi(FX,~) i=O

il

existe

p ~ N, on a :

>. C p

N

e t une c o n s -

388

D~monstration (H @ AS + Z,D')

: D'apr~s la remarque 2.6., un module de

FX

sera

o~

(-i)

D' S£(z) = (O ~ Id) S£(doZ) -

D'apr~s le lemme 3.2., on a : suffisamment grand. Ii existe donc (H+~ S|z)P/(Im

N

[ 6il Si(z). 2~i~£-I

D'(H + @ S|Z) p = 0

tel que, pour tout

pour

p

p ~ N

D' N H + @ SIZ)P~+ HP(F,~), ce qui implique que

dim HP(F) ~ dim(H + ~ SIz)P - dim(Im D' ~ H + @ SIz)P.

II s'agit de majorer la dimension de On a:

H ~ AS + Z

D"

(Im D' N H + @ S|Z)~

> Im D"

T

T

SIZ

~ Im D~NH + ~ SIZ s

On va montrer que D] est surjective, ~ partir d'un certain degr@. Soit

¢ e H @~S~Z, on peut d~composer !

mani~re suivante : et

!

~ = ¢I +

¢" ~ (H + @ ASZ) ~

~ ¢i + ~'' o~ 2 $i.<m

!

¢i £ SiZ

de la i e {l ..... m}

si

A~2SZ.

On volt que D' ¢"

+

@ A >'2

@ A >'2 SZ

2.<16m- I Ecrivons ÷

(H @ ASZ) + = H + @ SIZ • (H + +

~ (H + @ SiZ)+ H + @ A >'2 SZ + A SZ) i>.2

(H @ ASZ) + = (H+ e S!Z) e C

il est clair que

D'¢" +

~ D'¢'i e 2.
C.

389

Soit maintenant = D'~ = D'~I' + (D' ~" +

que et

~ £ Im D' N (H + @ SIZ), il existe donc

D'$" +

~ D'#~), ce qui entraine ~ 25i~m D ' ~ = O. On a donc pour tout p > O : i )

~ i~2

que

~

tel

~ = D '~I'

dim(Im D' O H + @ SIZ)P ~ dim(Siz)P-l.

On en d~duit qu'il existe

dim HP(F,Q)

N e ~

tel que si

p ~ N, on a :

>, dim(H + 8 SIZ)P - dim(SlZ)P-l.

r k.+1 H + = H+( V S i ), on a : i=l

Comme

r p-k. - ] r p+~ -k. - ] ~ dim(SiZ) l = ~ dim Z 1 i i=l i=l

dim(H + 8 SIZ)P =

r d'o~,

p ~ N, dim HP(F,~)

si

i=l La d~monstration est identique

l'espace hypotheses

du th~or~me

~ celle du th~or~me

Remarque.

Le th~or~me

des sections du th~or~me

dim Hp+al_l_k i(F)

d'un fibr~

[18].

3.3. se g~n~ralise, E H

(A,dA)~-> (A @ AZ,d) -~ (AZ,d o)

propri~t~s

suivantes (I)

o O~

1 ~ d @ n

X et

ale

o~

Soit un fibr~ nilpotent

: il existe

Y

~ X

3.3., et le fibr~ poss~de

Th~or~me 3.4.

n ~ ]

type d'homotopie

est un complexe

@ Q.

3.3, ~ l'aide de cette minoration,

4.1. de

F~

8 ~ - dim Hp+al_l(F)

oO

de mani~re X

et

F

~vidente, v~rifient

les

un module minimal

d = dA @ 1 + I @ d o

F ~+ E

H

> X

ayant les

tel que rationnelle

simplicial

d'un bouquet

nilpotent

Sd V Y

de dimension

@ n.

(2) F a l e type d'homotopie rationnelle d'un bouquet de spheres r k.+I V S I o~ r ~ 2, inf(k.) ~ n. Alors, si F est l'espace des sections i=l -C > | tels continues du fibre, il existe N ¢ ~ et une constante r~elle que si

p ~ N, on a :

~ dim Hi(F,~) i=O

~ C p.

390 Dgmonstration mod~le

(B,d B)

l'alg~bre de

de dimension

commutative

le lemme 2.3• on peut supposer

finie tel que

gradu~e

S d. II est classique

dule de

: D'apr~s

Bp = O

(A(u)/u2,ds

que

(A(u)/u 2)

= O),

si

off

que Y a un

p > n. D'autre part,

lu] = d , est un module

~ (B,dB), not@e

(A,dA),

est un mo-

S d V Y, on a : (~u/u 2)

A = ~ @

+ @ B ,

Si on utilise proposition

u.B

+

= O,

les techniques

2.4., on voit qu'un module

dAU = O,

de Haefliger

de

"F

dA(b)

r~sum~es

= dB(b).

par la

est

m

(AZ 8 A( ~ SiZ) @ ASuZ,D) i=l

o2

(SiZ)1
+

est d ~ f i n i

~ partir

d ' u n e b a s e homog~ne

l'isomorphisme

lin@aire

forme lin~aire

sur

On a et

A

de degr6 d~finie

Su(~')

-d

par

(b 1 . .•. .

de

Z

u'(1)

= Su(~).~'

b m)

sur

= O,

de

et

Su

si

u'

est la

@u' @ Z u'(bi)

+ (-l)d]~]

B ,

= O,

~'Su(~')

u'(u)

si

est

= I.

~,~'e

AZ ;

(-I) d DSu(Z ) = Su(do z) + 0u(Z ). Une d@monstration

existe

N e ~

analogue

tel que pour tout

~ celle du lemme 3.2

p >. ~q, on a

montre

qu'il

D'(H + @ S Z) p = O. On v~rifie, U

comme pour le th@orgme (S Z) p dans U

3.3.,

que

pour p assez grand,

[Im D" n (H + ~ S Z)] p+I

est sur~eetive,

Ii

l~'application

D" de

et donc que:

r

si

p >. N,

et on conclut

dim HP(F,~)

comme dans

>. [ dim Np+d_l_k.(F) i=l i [18],

th6or~me

O ~. - dim Hp+d_l(F ) O @

4.1. H

Th~or~me 5.5.

soit un fibr6 nilpotent

r k.+1 I F = V S , r >. 2, q .< inf(ki). Soit i=I tent de dimension n o~ q < n < inf(ki). nue

f : X ÷ Sq

telle que la q

i eme

F ~+ E

o ~ sq o

X

un complexe

simplicial

o2 -nilpo-

On se donne une application

application

induite en homotopie

eonti-

ration-

391

nelle

(fa~:@ ~ ) q

le fibr~

Ho

: ~q(X) @ @ ÷ ~q(S q) @ @

et le fibr~ image r~ciproque

soit

non nulle . Alors, pour

N = f (Ho) : F ÷ E -~ X, la coho-

mologie de l'espace des sections est ~ croissance exponentielle. D~monstration : Soit

A*(-)

le foncteur de Sullivan d~fini de la

cat~gorie des complexes simpliciaux dans celle des A.D.G.C.. On a un morphisme A*(f) : A*(S q) ÷ A*(X). Soit Sq

dimension finie de du fibr~

g

(a

(A(a)/a2,d = O) __mm_+ A~(S q) e s t un g ~ n f i r a t e u r de degr~

un module de

q ) . Un module m i n i m a l

est de la forme :

o

(A(a)/a 2) ÷ ((A(a)/a 2) O AZ,d) ÷ (AZ,do)

Ona:

dz = d Oz + a O ea(Z) L'application

o~

Oa(Z) e AZ.

(A(a)/a 2) A~(f)°m ~ A~(X)

(A(a)/a2)C i ) (h(a)/a 2 @ AU,6) ~ o~

@

est

(f@~ @ IQ)q

un quasi-isomorphisme et est surjective,

~u = ~ u + a @ ~(u) o me d'A.D.G.C, pour tout de

entraine

6 u e AU o

et

u ~ U. On a

Ker 6

A~(X)

@ o i = Am(f) o m. L'hypoth~se que que pour tout

u ¢ U,

par

r(a) = a,

r(u) = O

r o i = Id. Comme dans le lemme 2.3 , soit

engendr~ par les ~l~ments de degr~ en degr~

on a :

~(u) ~ A+U. On d~finit alors un morphis-

r : (h(a)/a 2 @ AU,6) + A(a)/a 2

A(a)/a 2 @ AU

mentaire de

o~

a un module

> n

n . Le passage au quotient

P : (A(a)/a2 @ AU,~) ÷ [(A(a)/a 2 @ AU)/~,~J = (A,d A)

est un isomorphisme en cohomologie. On a :

1

l'id~al

et par un suppl~-

392

A*(S q)

1

( A ( a ) / a 2)

J

(A,dA) =

---+ ( ( A ( a ) / a 2 ) S A Z , d ) ÷ (%Z,d o)

r

((A(a)/a2)OAU,6)

[(A(a)/a2OAU)I,~-

A*(X)

On appelle

j l'inclusion d@duite de

on d~finit un morphisme d'A.D.G.C,

r' : (A,dA) ÷ A(a)/a 2

Un module minimal de base = f.(~o )

est

i ; com ~

(A,dA)

rtl) = O,

tel que r' o

j = Id.

du fibr@ image r@ciproque

:

(A,dA)~+ (A 8 AZ,D) ÷ (AZ,d o)

o~ Dz = doZ + j(a) O ea(Z) = doZ + a O @a(Z)-

Dans ces conditions~ morphismes d'A.D.G.C.

j ® Id

et

les morphismes r' O I d

j

et

r'

s'~tendent en des

rendant commutatifs les diagrammes

suivants :

(A(a)/a2,d = O) :

, (A(a)/a 2 @ AZ,d)

(AZ,d o)

j @ Id (A,dA)

Ir' ( A ( a ) / a 2 , d = O)

..........

.......(A @ A Z , ~

J

r'

@ Id

( A ( a ) / a 2 e AZ,d) --

(AZ,d o)

II (AZ,d o)

393

II est clair que la construction

E

de Haefliger d~crite dans

le th~or~me 2.1

est fonctorielle,

on a donc des morphismes

R = Z~(r ' @ Id)

et

tels que

l=Z~(j ~ Id)

I o R = Id :

(Z'(A 8 Z ) , D ) I ~==g=;

d 'A.D.G.C.

Z~(A(a)/a 2 O Z,d).

R

En particulier, surjective

l'application

; on a donc, pour tout

dim Hn(F,~)

est l'espace des sections du fibr~

on prend

Y

I

en cohomologie est

n e ~ :

~ dim Hn(Fo,~)

Le th~or~me 3.5

induite par

H

oN

r

(resp.

F o)

(resp. Ho).

se d~duit donc du th~or~me 3.4

dans lequel

~gal ~ un point. On est amen~ ~ ~noncer la conjecture

Conjec~e

:

suivante

Soit un fibr~ nilpotent

F ~+ E

]I > X

o__~ X

a

le type d'homotopie d'un complexe simplicial de dimension n >. | e t + r k.+l l H (X,~) # O, F a l e type d'homotopie d'un bouquet de spheres V S i=I o__~ r >. 2 e t inf(ki) > n, alors la suite des hombres de Betti de l'espace des sections du fibr~ est g croissance exponentielle.

Ce r~sultat aurait des applications de la cohomologie de Gelfand-Fuchs Soit soit

LM

M

d'une vari~t~

C~

une vari~t~

int~ressantes

paracompacte

l'alg~bre de Lie des champs de vecteurs

:

de d i m e n s i o n continus sur

resse ~ la c o h o m o l o g i e de I ' A . D . G . C .

C (L M)

continues sur

M, appel~e cohomologie

de Gelfand-Fuchs

le U -fibr~ n

U

:

n

+ EU (2n) n

au-dessus du 2n-squelette BU

n

~ BU (2n) n

du fibr~ tangent de

M

de

n ~ 1, M, on s'int~-

des formes multilin~aires de

M. On consid~re

restriction du fibr~ universel,

de la base

le fibr~ associ~ au-dessus

dans l'~tude

BUn. Soit BU

n

.Vn : EU(2n)n'- ÷ EU~2nn" Xu EUn ÷ n et de fibre EU~2n)t~. Le complexifi~

est classifi~ par une application

f : M + BU . n

394

L'image r~ciproque par On a l e

f

du fibr~

~n

r~sultat suivant d~montr~ par Haefliger

Th@or~me 3.6. Conjecture de Bott [3] : l'espace des sections continues du fibr~ :

On montre que

EU (2n)

pie rationnelle de

S 3 ; si

n ~ 2,

quet d'un nombre fini de spheres en nombre est I'A.D.G.C., non libre :

lhil = 2i-I,

I c i l = 2i,

&l~ments de degr~ Une base de

ale

EU (2)

Yn

ale

type d'homoto-

type d'homotopie d'un bou-

~ 2. Un module de l'espace

l'id~al de

S[c I ..... Cn]

dh i = ci,

o~

engendr~ par les

(voir par exemple,

[12]).

a ~t~ d~crite par Vey [3].

[3] ou [II], un module du fibr~

partir du module de

le type d'homotopie

(E(h I ..... hn) @ Sic I ..... Cn]/l,d)

> 2n, on a d c i = O,

H~(EU(2n),Q) Dans

Iest

est un module de

EU (2n) + E + M.

n = I,

EU (2n)

:

C (LM)

est 2n-connexe, e t a

rationnelle d'un bouquet de spheres. Si

EU (2n)

EU(2n) -> E -~ M.

est un fibr~ :

et du module de

f

EU (2n) + E ÷ M

est donn~

not~

f~ : H • (BUn,~) = R[~ 1 ..... Cn] + fl~(M) o~

Icil = 2i, et

f (c2i_]) = O,

f (c2i) = P i e

ferm~e repr~sentant la classe de Pontryagin

Pie

~4i(M)

est une forme

H4i(M,~).

II est clair, que si toutes les classes de Pontryagin sont nulles, le fibr@

EU (2n) ~ E ÷ M

poss~de un module minimal du type

(a'(M),d M) + (~(~)

o~

d = dM O 1 + 1 0

0 AZ,d) + (AZ,d o)

do.

On d~duit, de la remarque suivant le th~or~me 3.3 suivant :

le r~sultat

395

Th~or~me 3.7. Soit tente, de dimension

~ 2

une vari@t@

et telle que

les classes de Pontryagin de et une constante

M

A > I

M

connexe,

compacte, nilpo-

H+(M,IR) # O. On suppose qua toutes

sont nulles,

tels que, si

I

C~

alors il existe un entier

N

p ~ N, on a

dim H i (C ~ (LM)) > A p .

i=o Le th@or~me 3.7. s'applique en particulier sion ~ 2),

aux spheres

aux groupes de Lie compacts connexes nilpotents,

(de dimen-

et aux produits

finis de telles vari@t@s.

B I B L I OGRAPH .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I E .

[I] GELFAND I.M. and D. FUCHS : The cohomology of the Lie ~ e b r a on a smooth m a ~ f o l d . Funct. Anal. 3 (1969) 194-210. [2] GRIVEL P.P. : Formes d i f f ~ r e n t i e l l e s e t s u i t e s s p e c t ~ a l e s . Annales I n s t . Fourier.

24 (]979)

17-37.

[3] HAEFLIGER A. : Sur l a cohomologie de l ' a ~ b r e te~.

[4]

HAEFLIGER A.

n°484,

de Lie des champs de vec-

Ann. Scient. ENS, 4~me s@rie, 9, (1976) 503-532. :

S ~ l a cohomologie de Gelfand-Fuchs, Lectures Notes in Mat,

121-152.

[5] HAEFLIGER A. : Rational homotopy of t h e space of s e c t i o n s bundle. Trans. Am. Math. Soc. 273 (1982) 609-620. [6] HALPERIN S.

: Lecture

on minimal models.

of a n i l p o t e n t

M@moires de la Soc. Math. France 9/10,

1983.

[7] HALPERIN S. : Rational f i b r a t i o ~ ,

minim~ models, and fibrings of homogeneous spaces. Trans. Am. Math. Soc. 244, (1978), 199-223.

[8_] HALPERIN S, STASHEFF J. : Obst~uctions to homotopy equivalence. Advances in Math. 32 (1979) 233-279. [9] LEHMANND. : Th~orie homotopique des formes d i f f ~ r e n t i e l l e s .

Ast@rique

45

(]977).

[lO] QUILLEN D. : Rational homotopy theory. Ann. of Math. 90 (|969) 205-295. I l l ] SHIBATA K. : On HaeflXger's model for t h e Gelfand-Fuchs cohomology. Japan J. Math. 7 (1981) 379-415. ~12] SHIBATA K. : S ~ l i v a n - Q u i l l e n mixed type mod~l for f i b r a t i o ~ and t h e Haefliger model for the Gelfand-Fuchs cohomology. A s t ~ r i s a u e , 113-114, 1984, 292-297.

396

FI3] SILVEIRA da F. : Homotopie r a t i o n n e l l e d'espaces f i b r e s . Th~se. Universit~ de Gen~ve (1979).

[14] SULLIVAN D. : I n f i n i t e s i m a l computatlo)~ i n topology, Publ° I . H . E . S . 47 (1977) 269-331. [153 TANRE D. : Mod~l~5 de Chin, Qaillen, Sullivan. Lecture Notes in Mathematics, 1025, 1983, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo.

[16] THOM R. : L'homologie des espaces f o n c t i o n n ~ .

Colloque Topo. Alg.

Louvain (1956) 29-39.

[17] VIGUE-POIRRIER M. : R~alisation de morphism~ donn~ en cohomologie e t s u r e s p e c t r a l e d'Eilenberg-Moore. Trans. Am. Math. Soc. 265 (1981) 441-484. ~18~ VIGU~-POIRRIER M. : Homotopie ra~ionnelle e t croissance du nombre de od~sique~ ferm~es. Ann. Scient. Ecole Normale Sup. 4 e s~rie, 17, 1984,

~13-43].

!

Micheline VIGUE-POIRRIER 37, Parc d'Ardenay F. 91120 Palaiseau