Buchsbaum rings and applications: an interaction between algebra, geometry and topology

JU:rgen StUckrad Wolfgang Vogel

.Buchsbaum Rings and Applications An Interaction Between Algebra, Geometry and Topology

•

Springer-Verlag

Jiirgen Stiickrad

Wo1fgang Vogel

Buchsbaum Rings and Applications An Interaction Between Algebra, Geometry and Topology

Springer. Verlag Berlin Heidelberg New York London Paris Tokyo

/

Dr. Jurgen Stuckrad KarI-Marx-Universitat Leipzig Department of Mathematics D DR - 7010 Leipzig

Prof. Dr. Wolfgang Vogel Martin-Luther-Universitat Halle- Wittenberg Department of Mathematics DDR-4020 Halle

L( 1 E With 3 Figures

Sole distribution rights for all non-socialist countries granted ,to Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Mathematics Subject Classifikation (1980): 14M05, 13H1O, 13H15, 05A20, 55U99

ISBN 3-540-16844-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16844-3 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data Stiickrad, Jiirgen, 1948Buchsbaum Rings and Applications. Bibliography: p. Includes index. 1. Buchsbaum Rings. 2. Geometry, Algebraic. 3. Algebraic topology. 4. C.ommutative algebra. 1. Vogel, Wolfgang, 1940II. Title. QA 251.3.877 1986 512'.24 86-17880 ISBN 0-387-16844-3 (U.S.) by VEB Deutscher Verlag der Wissenschaften, Berlin 1986 Printed in the German Democratic RepUblic

@

Bindearbeiten: K. Triltsch, Wiirzburg 2141/3140-543210

Preface Da die algebraische Geometrie weaentIich yom Fundamentalsatz der Algebra ansgeht, den man nur deshalb in der gewohnten allgemeinen Form aussprechen kann, weil man dabei die Vielfachheit der LOsnngel'll in Betracht zieht, so muB man anch bei jedem Resultat der algebraischen Geometrie beim Znriickschreiten die gemeinsame QneUe wiederfinden. Das ware aber nicht mehr moglich, wenn man auf dem Wege das Werkzeug verlore, welches den Fundamentalsatz fruchtbar nnd bedeutnngsreich macht. Francesco Severi Abh. Math. Sem. Hansischen Univ.

15 (1943), p. 100

This book describes interactions between algebraic geometry, commutative and homological algebra, algebraic topology and combinatorics. The main object of study are Buchsbaum rings. The basie underlying idea of a Buchsbaum ring is a continuation of the well.known concept of a Cohen-Macaulay ring, its necessity being created by open questions of algebraic geometry and algebraic topology. The theory of Buchsbaum rings started from a negative answcr to a problem of David A. Buchsbaum. The concept of this theory was introduced in our joint paper published in 1973. In presenting our treatment of algebraic geometry, it is a pleasure to acknowledge the help and encouragement which we have had from all sides. Some decisive results eame from the applications of homological algebra to derived categories, an approach we learned in joint discussions with Reinhardt Kiehl. A further development of these ideas, with a view towards their topological applications, came in our long collaboration with Peter Schenzel; to both colleagues go our special thanks. In addition, without using the theory of derived categories, we will describe a different approach to these applications with respect to liaison and combinatorics. This book has profited from the research investigations of a number of doctoral theses. In particular some ideas used here first appeared in the theses of Markus Brodmann, Juan C. Migliore, Peter Schcnzel, Philip W. Schwartau and Ngo Viet Trung. They are presently widely available in standard mathematical journals. Our treatment has also profited from many publications of our Japanese colleagues. ~'or example, we have gained greatly from the work of Shiro Goto, Yoichi Aoyama, Shin Ikeda, Yasuhiro Shimoda and Naoyoshi Suzuki. Among others whose suggestions have served us well, we note David A. Buchsbaum, David Eisenblld, Heisuke Hironaka and Balwant Singh. To all these and others who have helped us, we express our sincerest thanks.

6

Preface

In the late stages of polishing the manuscript wc received valuable suggestions from Henrili; Bresinsky and GUnther Eisenreich. We would like to thank the staff of Deutscher Verlag der Wissenschaften, especially Erika Arndt for her enormous patience and skill in converting a very rough manuscript into book form .

•

Leipzig and Halle, Spring, 1986

JUrgen StUckrad Wolfgang Vogel

Table of Contents

Preface . . . . . . . . . . . .

5

introduction and some examples .

9

Chapter 0 Some foundations or commutative and homological algebra . §t. § 2. § 3. § 4.

§1. § 2. § 3. §4.

45 52

Chapter I Characterizations of Buchsbaum modules

62

Characterization of Buchsbaum modules by systems of parameters Cohomological characterization of Buchsbaum modules Graded Buchsbaum modules. . . . . . . . . . Segre products of graded Cohen.Macaulay modules

Chaptern Hochster·Reisner theory for monomial Ideals. An Interaction between algebraic geometry, algebraic topology and comblnatoric8 §t. § 2. § 3. § 4.

§t. § 2. § 3. §4.

§t. § 2. § 3.

21 21

Local algebra and homological algebra Graded modules and Kiinneth formulas . I..ocal duality . . . . . RelilOlutions and duality. . . . . . . .

33

62 70 95 99

106 107

Foundations. . . . . . . . . . . . . . . . . . . . The homological Cohen-Macaulay criterion of Reisner. The topological Cohen· Macaulay criterion of Schwartau . Further applications to algebraic topology and combinatorics

123 132

Chapter m On liaison among curves in projective three space

155

On liaison among arithmetical Buchsbaum curves in p3 On liaison addition and applications . . . . . On curves linked to lines in p3 and applications On self-linked curves in p3 . . . _ . . . . .

174 181 192

Chapter IV Rees modules and associated graded modules of a Buchsbaum module

199

Some preliminary results . . . . . . . . . . . . . . . . . . . . The Buchsbaum property of Rees modules and associated graded modules . Blowing-up characterization of Buchsbaum modules . . . . . _ _ . . _

201 207 223

115

156

8

§1. § 2. § 3. § 4. § 5.

Table of Contents

Chapter V Further applicatiollll and examples

229

A Buchsbaum criterion for affine semigroup rings . . . . . . Some examples related to problems of Hironaka and Seidenberg On Buchsbaum rings obtained by glueing . . . . . . . . . . Construction of Buchsbaum rings with given local cohomology . Some examples of Segre products. . . . . . . . . . . . . .

229 234 238 242 245

Appendix On generalizatiollll of Buchsbaum modnles

252

Bibliography.

268

Notations

282

Index . .

284

Introduction and some examples

The aim of this introduction is to preEent the idEaS of our theory of Buchsbaum modules with a wealth of background material. The different viewpoints which are treated in the works of Laskcr-Macaulay-Grobncr and Severi-van der Waerden-Weil concerning the multiplicity theory for Bezout's theorem are basic to the understanding of the theory of Buchsbaum modules. The simplest case of Bezout's theorem is the following very simple but fundamental principle in the field of complex numbers. Fundamental principle. The number of roots of a polynomialj(x) in one variable, counted with their mUltiplicities, equals the degree of j(x). The definition of this multiplicity is well-known and clear. The next simple case to consider is that of plane curves. The problem of the intersection of two algebraic plane curves was already tackled by Newton; he and Leibniz had a clear idea of "elimination" processes which describe the fact that two algebraic equations in one variable have a common root. Using such a process, Newton otlerved in his "Geometria analytica", published in 1680, that the abcissas (for instance) of the intersection points of two curves of respective degrees m, n, are given by an equation of degree m· n. This result was gradually improved upon during the 18th century, until Bezout, using a refined elimination process, was able to prove in general that the equation giving the intersection had exactly the degree m· n; however, no general attempt was yet made during that period to attach to each intersection point an integer measuring the "multiplicity" of the intersection, in such a way that the sum of multiplicities would always be m . n. Therefore the classical theorem of Bezout states, that two plane curves of degree m and n, intersect in at most m . n different points, unless they have infinitely many points in common. In this form the theorem was also stated by Maclaurin in his "Geometrica organica'" published in 1720 (see p. 67/68) but the first correct proof was given by Bezout. An interesting fact, usually not mentioned in the literature, is that Bezout proved in 1764 not only the above-mentioned theorem, but already the following n-dimensional version: Let X be an algebraic projective variety of a projective n-space. If X is a complete intersection of dimension zero then the degree of X is equal to the product of the degrees of the polynomials defining X. The proof can be found in the papers of Bezout [1,3] and r2]. In his book "Theorie generale des equations algebriques", published in 1779, a statement of this theorem can be found already in the foreword. We quote from page XII: "Le degre de l'equation finale resultante d'un nombre quelconque d'equations complettes, renfermant un pareil nombre d'inconnues, & de degres quelconques, est egal au produit des exposans des degres de ces equations. Theoreme dont la verite n'etoit connue et demontree que pour deux equations seulement."

10

Introduction and 80me examples

The theorem appears again on page 32 as Theorem 47. The special cases n = 2, 3 are interpreted geometrically on page 33 in Section 3° and it is being mentioned there, that these results are already known from geometry. (For these historical remarks see also Renschuch [1], Dieudonne [1], Vogel [6].) To-day we have the following modern statement of Bezout's Theorem, where the degree of a variety X P" of dimension d, denoted by deg(X), is the number of points in which almost all linear subspaces L c: P" of dimension n - d meet X. Bezout's Theorem. Let X, Y be unmixed (i.e. each component of X and r h.as the dimensiA:m of X or Y, resp.) varieties of the projective n-space P'k over an algebraically cWsed field K 8uch that dim(X n Y) dim(X) dim( Y) - n. Letting 0 run over all proper components of the intersection X n Y (i.e. (~rreduc~"ble) components with dimension equal dim(X n Y» we get that there exist '~1Uersection multiplicities', 8ay i(X, Y; 0), of X and Y along 0 such that the following 'Anzahl-Formel' i8 true:

+

deg(X) • deg( Y)

= E i(X,

Y; 0) • deg(O). c Here the number i(X, Y; 0) itself measures the degree of contact of X and Y along O. It represents fairly sophisticated concepts in full generality and it has taken a century or two and a lot of work to be developed, see also S. Kleiman [1, 2]. To get equality in the above equation one may follow different approaches to arrive at several different multiplicity theories. It is well-known that there is no loss of generality in assuming for projective varieties that one variety is a complete intersection in order to define local intersection multiplicities. This statement does indeed follow from Samu~l's book [1], p. 81. However, it is not clear to us that it is possible to prove a global statement like Bezout's Theorem by reducing in a simple fashion to the case in which one of the two intersecting varieties is a complete intersection. Therefore we would like to present the high points of the proof of the following theorem: Theorem 1. Let X, Y be arbitrary varietie8 of Pic with dim(X n Y) :;::: O. Then there exi8t varietie8 X', Y' of p~+l such that one variety, say X', i8 a complete intersection with

deg(X) . deg( Y) = deg(X') . deg( Y') and there i8 a 1-1 corresporulR:nce between the components 0 of X n Y (in Pic) with dim(O) = dim(X n Y) and the components 0' of x' n Y' (~n p~+1) with dim(O') = dim(X' n Y') and that this correspondence preserves d~mensions and degrees. Proof (see also Vogel [6]): We will apply idealtheoretic methods. Let X, Y c: Pic be our projective varieties with defining ideals a and b in K[xo, ... , xnJ =: Rit and dimensions d and 0, resp. We introduce a second copy K[yo, •. 0' Y.. ] =: Ry and denote by b' the ideal in Ry corresponding to b. We consider the polynomial ring R := K[xo, ... , Xn> yo •. Y.. ] and the ideal c = (xo - Yo, Xn - Yft). Let ho( ... ) be the (rectified) leading coefficient of the Hilbert polynomial of the homogeneous ideal (... ). 0

0,

0

•• ,

Claim. ho(a . R:r) ho(b . R It ) = ho(a R + b' . R), where a . R is the extension ideal of a in R. 0

0

This follows from R/(a

+ b') . R '" R:r/a • Rit ®K RyJb' . RII ,

Introduction and some examples i.e. the Hilbert function H(n, (a+o')· R) of (a+o'). the Hilbert functions of a . R" and 0' . R II , that is

I:

H(n, (a+o') . R)

R can

11

be expressed in terms of

H(i, a· R z )' H(1, 0'· R II ).

l+j=n

The degree and the leading coefficient of the Hilbert polynomial of (a + 0') . Rare given by d + ~ + 1 and k o( (a +0') . R), resp. We choose an integer r such that the Hilbert functions H(i, a· R",) =: Hi and H(i, 0' . RII ) =: Hi are given by their Hilbert polynomials hi and hi, resp. for i > r. Then we can decompose for n 0 (n > .27): n

I:" Hi' H~_i

r

hi' k~_i

i=O

+i=O I: (Hi -

Some calculations involving the coefficient of

ita

Hi .

H~_i

ho(a) . ho(o') [ita

= ho(a) • hoW)' (

II

hi) . h~_i

nd+d+l

+i=n-r I: hi(H~_i

therefore yield:

(!). (n ~ t)] + n

d+~+1

)

n:,.-i)·

(other terms)

+ (lower order terms),

and our claim follows. It is well-known that the degree of a projective variety is equal to the (rectified) leading coefficient of its Hilbert polynomial (see, e.g., D. Mumford [5], p. 112, Theorem 6.25). Therefore we get our first statement of Theorem 1 where X' and Y' are defined by c and (a + 0') in R, resp. The second statement follows from the fact that there is a 1-1 correspondence between the isolated prime ideals a + 0 in R", and the isolated prime ideals of (a +0') . R + c . R in R. This correspondence is given by

Rz ::J ~

H>

(~

+ c)

R

and it therefore preserves dimensions and degrees, q.e.d. S. Kleiman [3] pointed out to us that the reduction of Theorem 1 may be accomplished by replacing the original varieties X, Y of P" by the varieties X', Y' of P2f1+1 which may be described as follows. Take three n-planes in P2"+1 in general position. Embbed X in the first, Y in the second and take Y' to be their join. Take X' to be the third plane. The claim in Theorem 1 is apparently proven by showing that the degree of the join Y' is equal to the product of the degrees of X and Y. We note that the reduction of Theorem 1 is not only to the case in which one of the varieties is a complete intersection but even a linear f!pace. At the beginning of this century one investigated the notion of the length of a primary ideal in order to define another intersection multiplicity. This multiplirity is defined as follows: Let X, Y c: P" be arbitrary varieties. Let 0 be a component of X n Y such that dim(O) = dim(X n Y). Denote by A(X, 0) the local ring of X at O. We set /L(X, Y; 0):= length of A(X, O)jI(Y)· A(X, 0),

where I(Y) is the defining ideal of Y. This length /L(X, Y; 0) is well-defined and is called the idealtkeoretic intersection multiplicity of X and Y at O. For instance, this

12

Introduction and some examples

multiplicity is the intersection multiplicity as set forth in the beginning in the case of projective plane curves. Prior to 1928 most mathematicians hoped that this multi- ' plicity would provide for Bezout's Theorem always the correct intersection multiplicity. In 1928 B. L. van der Waerden studied Macaulay's famous space curve (see Macaulay [1], p. 98) to show, that this idealtheoretic intersection multiplicity does not yield the correct multiplicity for Bezout's Theorem to be valid in projective spaces P" with n;;:: 4. We quote van der Waerden [1], p. 770: "In these cases we must reject the notion length and try to find anothcr definition of multiplicity". Nowadays it is of course well-known that p,(X, Y; 0)

=

i(X, Y; 0)

if and only if the local rings A(X, 0) of X at 0 and A(Y, 0) of Yat 0 are CohenMacaulay rings for all proper components 0 of X n Y where dim(X n Y) dim (X) + dim(Y) - n, see J.-P. Serre [2], p. V·20. Without loss of generality we may now suppose by applying our Theorem 1 that one of the two intersecting varieties X and Y is a complete intersection, say Y. With this assumption we get that p,(X, Y; 0)

i(X, Y; 0)

for each proper component O. Let Y be a complete intersection. Then there arises another problem posed by D. A. Buchsbaum [1] in 1965, as follows: Problem (from the viewpoint of the theory of intersection multiplicities). Is it true that p,(X, Y; 0) i(X, Y; 0) is independent of Y; that is, does there exist an invariant, say I(A), of the local ring A := A(X, 0) of X at 0 such that p,(X, Y; 0)

i(X, Y; 0)

I(A)?

From the viewpoint of local algebra we get the problem in its original foml: Let A be a local ring of dimension d 1 with maximal idealll't. Let q be an m-primary ideal which is generated by a system of parameters. Denote by eo(q; A) the multiplicity of the ideal q (see, e.g. Zariski-Samuel [1], Vol. II, Chap. VIII, § 10) and by l(Alq) the length of Alq over A. Is it then true that l(Alq) eo(q; A) is independent of q? For instance, is l(Alq)

eo(q; A)

= dim(A) -

depth(A)?

We will show that this is not always the case. From the viewpoint of the theory of intersection multiplicities we want to study our first counter-example. Example 1. Let X' c

Pk be the non-singular curve given parametrically by

Let Xc Pk be the projective cone over X'. We consider the surfaces Y and Y' in projective 4-space defined by the two hypersurfaces

Introduction and some examples

Let t:' be the defining ideal of X in K[xo,

Xl' X2, Xa, Xf].

13

Then

and Hilbert's characteristic polynomial of t:' is given by H(n,t:')

= 5· (;) + 6· (~)

- 3

(see, e.g. Renschuch [2], 8.2); that is, the degree of X is 5. Now, we see that X n Y and X n Y' intersect at the vertex 0: (0,0,0,0) of X, and Bezout's Theorem therefore gives us i(X, Y; 0)

= 5

and

i(X, Y'; 0)

10.

Since (t:' + (Xl' X,») = (Xl> X" X2Xa, x~, x~) it is easy to see that ,u(X, Y; 0) It remains to calculate the length p,(X, Y'; 0). We have

(t:'

+ (X2' xi + x~») = (X2' xi + x:, XIX., xix;, XIX;, x~)

ql

and we can construct the following chain of primary ideals belonging to ql c:: (ql' xix,) =: q2 c:: (q2, x;x,)

qa

(qa, XIX;)

c:: (q" xixa) =: qs c:: (qs, Xa X4) =: q6

(q6' XIXa)

c:: (qs, X~) =: q9 c:: (q9' Xa) =: qlO

(qIO' xi)

= 7.

(Xl> X2,

Xa,

X4) :

q, : q7 c:: (q7' xi) =: qs ql1 c:: (qw Xl) =: q12

c:: (Xl' X2, xa, X,) =: q13' It follows that p,(X, Y'; 0).= 13; that is, 2

=

p,(X, Y; 0) - i(X, Y; 0) 9= p,(X, Y'; 0)

i(X, Y'; 0)

3.

The theory of Buchsbaum modules started from such a negative answer to the above problem of D. A. Buchsbaum (see Vogel [4]). The concept of Buchsbaum modules was introduced in Stiickrad-Vogel [2] and [3], and the theory is now developing rapidly; see, for example, the following Symposium: Study of Buchsbaum rings and generalized C-Ohen-Macaulay rings. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto 1982. The basic underlying idea of a Buchsbaum module generalizes the well-known coneept of a Glhen-Macaulay module, its necessity being created by open questions in Commutative Algebra and Algebraic Geometry. For instance, such a necessity to investigate generalized Glhen-Macaulay structure occurs when one classifies algebraic curves in p3 or when one studies singularities of algebraic varieties. Furthermore S. Goto and Y. Shimoda [1] discovered that the Cohen-Macaulay property of Rees algebras of parameter systems can be described by certain Buchsbaum rings. Also it was shown that interesting and extensive classes of Buchsbaum rings do exist. We now introduce the definition of a Buchsbaum module and we describe a first geometrical interpretation of Buchsbaum modules. Let A be a Noetherian local ring with maximal ideal m. Let M be a finitely generated A-module.

14

Introduction and some examples

Definition 1. A sequence ai' ... , af of elements of m is a weak M-8eq:uence if for each i = 1, ... , r

(for i

=

1 we set (al> ..., ai_I)

=

(0) in A).

We denote by eo(q; M) the multiplicity of M relative to a parameter ideal q of M, i.e., an ideal of A generated by a system of parameters for M. Then as our first important result which explains the notion of Buchsbaum modules in connection with the above problem of D. A. Buchsbaum (see Chap. I, § 1) we get Theorem 2 and definition. The 10Uowing conditiOnB aTe equivalent: (i) Jf is a BucMbaum module. (ii) The diflerence 01 length lA(M/q . ~W) and multiplicity €o(Q; M) 01 M is an integer,

say I(M), 01 M not dependt'ng on the choice of the parameter ideal q of M. (iii) Every system 01 parameters lor M is a weak M-sequence. Note that a Noetherian local ring is said to be a BucMbaum n'ng if it is a Buchsbaum module over itself. We want to give some simple examples. Example 2. A finitely generated module M is Cohen-Macaulay if and only if M is Buchsbaum and I(M) 0, that is the class of Buchsbaum modules contains the CohenMacaulay modules. Example 3. Let A be a local ring with maximal ideal m. Let M be a Cohen-Macaulay module over A of dimension d > 1. Let N be a submodule of M such that the factQr module MIN is a finite-dimensional vector space over A/m, say EB A/m. Then N is a I

Buchsbaum module over A with the invariant I(N) = (d

1) . t.

Prool: Let q c A be an ideal generated by a system of parameters with respect to N. Take the exact sequence 0 -+ N -+ M -+ MIN -+ O. Since M is a Cohen-Macaulay module and m . MeN we get the following exact sequenee by applying the functor Aiq ®A: 0-+ Tort(A/q, MIN) -+ N/q . N -+ M/q . M -+ 1W/N -+ O. This is an exact sequence of A-modules with finite length. We obtain therefore that q is also a parameter ideal of M and that

lA(Njq· N)

l(Mjq .•W)

(d - 1)· t

since lA(Tor1(Ajq, MIN)) t .lA(Torl(AIQ, Aim)) = t· lA(qjq . m) = t . d. On the other hand the additivity property of the multiplicity symbol (see Cbap.O, § 1) implies 1) . t does not eo(q; N) = eo(q; "Wi. Thus the difference lA(Niq . N) - eo(q; N) = (d depend on the choice of q since eo(q; M) = l(Mjq . M). Therefore N is a Buchsbaum 1) . t, q.e.d. module with I(N) = (d This example has some useful applications. (3a) Let A be a regular or Cohen-Macaulay local ring of dimension d 2. Then the maximal ideal m of A is a Buchsbaum module which is not Cohen-Macaulay.

Introduction a.nd some examples

15

(3b) Take Macaulay famous curve X in p3 given parametrically by {s', sSt, S/3, 14} (see Macaulay [1], p. 98). This curve was studied by F. S. Macaulay as early as 1916. His purpose was to show that not every prime ideal in a polynomial ring is perfect. We will show that X is arithmetically Buchsbaum; that is, the local ring of the (affine) cone over X at the vertex is a Buchsbaum ring. Let K be a field and R K[ s, t] a formal power series ring in s and I. We put S = K[s', s3t, s&, t4] in R. Take the normalization T of S. Then T is a Cohen-Macaulay K[S4, s3t, s2t2 , st3, t4]. It is easily seen that the ring of dimension 2 and we have T conductor of S in T is the maximal ideal of Sj that is, TIS is a vector space generated by the element s2t2 • Hence we get that S is a Buchsbaum ring with invariant 1(S) 1. (3c) (See also Herrmann-Schmidt [1].) Let K be a field with char K =l= 2 and S = K[x, y] a polynomial ring. We put R = {f E S If(l, 0) = f( -1, O)}. Then R is the finitely generated subring K[1 x2, xy, y, x - x 3 ] of S. So X = Spec(R) is realized as a surface in the affine space Ai- which is non-singular in codimension 1, but with an isolated singularity at the origin. Applying the above statement we see immediately that the local ring of X at the origin is a Buchsbaum ring. Note that this example (3c) is a very simple example of Buchsbaum rings obtained by glueing (see (Joto [4]).

Example 4. 111 is a Buchsbaum module over A if and only if $I is a Buchsbaum module over .A in which case 1(111) I($I). Here denotes /\the m-arlic completion. (See also Lemma 1.1.13,) In order to describe examples from the viewpoint of the theory of intersection multiplicities we need some special results on Buchsbaum modules. We summarize these assertions in the following theorem. The proofs are given in Chapter I under even more general conditions. Let A be a local ring and let a A be an ideal of A. Denote by U(a) the intersection of the primltry ideals q belonging to a with dim(q) = dim(a).

Theorem 3. Let A be a local ring of dimension d;::: 1 with maximal ideal m. The following two conditions are equivalent: (i) A is a Buchsbaum ring. (ii) For each part all "., ak of a system of parameters of A we have

m· U{(al' ,." a~J) r: (at> ... , ak) for every k = 0, ... , d - 1. (notice the case d I!) Furthermore, we have the folkrwing statements. (iii) Let d > 2. Lei a, 0 be ideals of A 8uch that the intersection a n V is the zero ideal of A. Assume that A/a and A/v are Oohen-1I1acaulayrings of dimension dand dim{AI(a +0)) = 0, then m a + 0 if and only if A is a Buchsbaum ring. (iv) Let d > depth(A) 1. A is a Buchsbaum ring if and only if there exists in A a non-zero divisor x E m2 such that the ring A/(x) is a Buchsbaum ring. (Lifting Buchsbaum mod a non-zero divisor.) As an application of the statements (i), (ii) of Theorem 3 we get the following examples:

Example o. Let K be any field and set A:= K[x, y]/(x) n (x 2 , y). Then A is a Buchsbaum non-Co hen-Macaulay ring. If A := K[ x, y]/(x) n (xS, y) then A is not a Buchsbaum ring.

16

Introduction and some examples

From the vantage point of the theory of interseetion multiplicities we can construct the following examples by use of the statements (iii) or (iv) of Theorem 3. Example 6. Let X be the union of two planes in four-dimensional affine space A4 meeting at the point P: Xl X2 Xa X4 0; that is the ideal of X is (Xl' X2) n (Xa, X4)' Let A be the local ring of X at P; i.e.

Our statement (iii) implies that A is a Buchsbaum ring. It is well-known that A is not a Cohen-Macaulay ring. Example 7. Let X be the rational twisted cubic curve in p3 with defining ideal p (XoX2 - xi, XoXa XIX2, XlXa x~) in K[xo, Xl> X2' xaJ for any field K. It is wellknown that p is perfect. We will show that p2 is Buchsbaum; that is, we claim that thc local ring of the affine cone over X, counted with multiplicity 3 at the vertex, is a Buchsbaum ring with invariant 1. Prool: Note that p2 is the defining ideal of the curve X counted with multiplicity 3. Furthermore, we have that p2 is a primary ideal (see, e. g., Achilles-Schenzel-Vogel [1}). Therefore we can apply the statement (iv) of Theorem 3. By using localization it is not hard to calculate the intersection U(p2, x~) of the prhnary ideals belonging to p2 + (x~) of dimension 1. It then follows that (xo, Xl> X2, xa) . U(p2, xi) C p2 (x~). The statements (ii) and (iv) yield our assertion. It is also not too difficult to calculate the in"\(ariant by using a system of parameters.

+

Having this example we can construct new irreducible and reduced arithmetical Buchsbaum curves by using the theory of residual intersection. Therefore we will apply liaison among curves in p3 coupled with our deep result from Theorem III.1.2, on liaison, obtained by applying the theory of dualizing complexes. This r~ult, or the theory of residual intersection for the special case of curves, results in the following. Theorem 4. Assume that the scheme theoretic unicm 01 two curves Xl' X 2 01 P::" over an algebraically closed lield K i8 the CfJmplete t'ntersection 01 two hypersurlace8, and that the curve8 have no CfJmponent8 ~n CfJmmon (Xl and X 2 are then said to be hnked geometrically, see C. Peskine, L. Szpiro [3J and A. P. Rao [11). Xl is arithmetically Buchsbaum (i.e. the local ring 01 the alline cone over Xl at the vertex is a Buch8baum ring) il and only il X 2 is arithmetically BW'kbaum. Take, for example, Macaulay's curve X Xl and the curve X 2 with defining ideal (xo, Xl) n (X2. xa). Then it is not hard to show that

a=

pna=

(xoXa -

X1X2' xo~ -

x~xa).

Applying Theorem 4 and Example 6 it therefore follows again that Macaulay's curve is arithmetically Buchsbaum. Notice, that we have used only a simple calculation on p n a. This liaison was discovered by G. Salmon [1], p. 40, already in 1848 and a little later again by J. Steiner [lJ, p. 138, in 1857. Now, consider our Example 7. Let Y be the curve X counted with multiplicity 3. It follows from K. Rohn [21 that Y is linked for instance, to an irreducible and reduced curve Ci~a of degree 43 and genus 213 by two hypersurfaces of degree 4 and 13. We thus

Introduction and some examples

17

get that the liaison equivalence class corresponding to a vector space of dimension 1 contains also the curve 0i~3. Using the theory of liaison and Buchsbaum rings we also obtain new statements concerning the classification of algebraic curves in projective 3-space P~. For instance, let O~ be an irreducible and non-singular curve in p~ of degree 6 and genus 3 with defining ideal 1(0:) in 8 := K[xo, Xl> x 2 , xa]. It follows from M. Noether [1], p. 87, (aa) and (a~) and Theorem 4 that either is arithmetically Cohen-Macaulay, or is arithmetically Buchsbaum. We note that in 1881 F. Schur [1] discovered a first difference between the curves to distinguish between them. Nowadays the resolution of the curve is well-known if is arithmetically Cohen-Macaulay. It is (see also G. Ellingsrud [1] or L. Gruson and C. Peskine [1]):

0: 0:

0:

0:

0->- 83( -4)

->-

0:

8 4( -3) ->- 8 ->- 8/1(0:) ->- O.

0:

If is arithmetically Buchsbaum then we have obtained in addition the resolution of O!: 0->- 8( -6) ->- 8 4( -5) ->- 8( -2HB 8 a( -4) ->- 8 ->- 8f 1(0:) ->- O.

(See also Chap. III, § 1.) For arithmetically Buchsbaum curves 0 in p3 we get that the invariant l(A) of the local ring A of the affine cone over 0 at the vertex is given by l(A) = dim (~H1(P3,

Jdv)})

where J c is the ideal sheaf of the curve O. Having this arithmetically Buchsbaum property of a variety in P" we want to describe another geometric interpretation of the invariant l(A) of the corresponding (local) Buchsbaum ring A of surfaces. If X is any projective variety, the finite-dimensional vector spaces HI(X, Ox) are important invariants of X. One of the most interesting is the alternating sum of their dimensions, obtaining the so-called arithmetic genus:

P.(X)

dim H"(X, Ox) - dim H"-1(X, O.d

+ .. , + (_1)11-1 dim H1(X, Ox),

where n is the dimension of X. One drops the term dim HO(X, Ox) advantage that for X a curve,

Pa(X)

= dim

H1(X, Ox)

=

1, which has the

usual genus of X.

But, when X is a surface, we get

The point here is that the Italian geometers regarded dim H2(X, Ox) as the dominant term, and called it, for non-singular surfaces, the geometric genus; while dim H1(X, Ox) was considered a "correction" term, and was called the irregularity. Now, consider any arithmetical normal irregular (I.e. the irregularity =1= 0) surface F, such that we have H1(F, OP(p») =1= only for P O. Let A be the local ringofthe vertex of the (affine) cone over F. Applying our cohomological investigations from Chapter I, § 2, we get: A is a normal non-Cohen-Macaulay-Buchsbaum ring where the invariant l(A) of A is given

o

2

Buchsbaum Rings

18

Introduotion and some examples

by the irregularity of F, that is I(A)

=

dim Hl(F, OF)'

It is not 'hard to construct an arithmetically normal irregular surface F free of singularities. For instance, this will be the case if F is the Segre embedding of pairs of points EX G of any two plane curves E, G free of singularities, where at least one of them has a positive genus. For example, let E pI, and let G be the cubic defined by x~ x~ x~ in pl!. Let F be EX G in its Segre embedding in po. For such surfaces F we get that the vertex of the affine cone over F is a" normal Buchsbaum singularity which is not a Cohen-Macaulay singularity. This gives an answer to a question posed by H. Hironaka in a discllssion (at the University of Halle in 1974), who asked whether we can construct normal non-Cohen-Macaulay-Buchsbaum singularities, and which aroused our interest in the subject of singularities (see also Chap. V, § 2). To motivate the study of other Buchsbaum singularities wc quote the following remark made by D. Mumford in [2) on p. 42: "Incidentally, one should regard the depth of 0 (a local ring) itself, for example, as a measure of the topological complexity of the singularity at the closed point of Spec(O): if the depth is maximal, i.e., equals the dimension of 0, then 0 is in a weak sense, nonsingular, while if the depth is much less then the dimension, the singularity is very bad." As mentioned before, D. A. Buchsbaum (see the above problem) and also A. Seidenberg [2], on p. 620 considered the difference between the dimension and depth of any local ring taking it as a measure of the deviation from the Cohen-Macaulay property. With our theory of Buchsbaum rings we will study examples of projective varieties in Chapter I (see Examples 4.14) and Chapter V, § 2, which show that this measurc does not describe the non-Cohen-Macaulay property satisfactorily. Shiro Goto and Yasuhiro Shimoda [1] have introduced another aspect to the study of Buchsbaum singularities. Here certain Buchsbaum rings are characterized by the beha:viour of Rees algebras relative to parameter ideals. We conclude our considerations by briefly discussing a main result from Goto-Shimoda [11.

+ +

Theorem o. Let A be a local ring with maximal ideal m. Let H~(A) denote the ith local cohomology module oj A with respect to m. The jollowing two conditions are equivalent: (i) A is a Buch.ibaum nl/{/ and H~(A) (0) jor i =1= 1, dim(A). (ii) The Rees algebra R(q) = q" is a Oohen-Macaulay rt'ng jor every parameter ideal q oj A. fl~O This striking theorem gives a complete answer to a problem of Rees algebras of powers of parameter ideals. It is stated as follows (see Goto-Shimoda [1]):

Corollary 6. Let A be a local ring and assume that depth(A) 1. Then A is a Cohenil1acaulay ring ij and only ij the Rees algebra R(q") is Cohen-Macaulayjor every parameter ideal q oj A, jor all integer n > O. Also, Shiro Goto [5] obtains the following result. Let A be a local ring of dimension d> O. Then A/H~(A) is a Buchsbaum local ring if and only if Proj R(q) is a CohenMacaulay scheme for every parameter ideal q auf A. In a letter to one of the authors, dated February 8, 1980, he underlines its significance: "I believe that the above blowingup characterization of Buchsbaum rings really clarifies .the importance of the concept

Introduction and some examples

19

of Buchsbaum singularities." Therefore the main object of our study of Chapter IV is the stability of Rees rings and form rings with respect to the Buchsbaum property of parameter ideals. This chapter has its origin in the effort to extend Hironaka's desingularization to a more general situation due to G. Faltings [1] and M. Brodmann (1] (see also the introduction of Chapter IV). We conclude our introduction by briefly discussing an application of the theory of Buchsbaum modules to the so-called Upper Bound Conjecture which establishes some interesting connections among algebraic topology, commutative algebra and combinatorics (see Chap. II). Let A denote an abstract finite simplicial complex with vertices Xl, ••. , X•. That is a family of subsets of {Xl' ••• , X.} such that if a E ,,1 and 1:' C a then 1:' E A, and such that the vertices are in A. We call the elements of A laces. If the largest face of A has d elements, then we say dim A = d - 1. The I-vector of LI is I (f -1> 10, ... , Id-d where exactly Ii faces of A have i + t elements and 1-1 = 1. The Upper Bound Conjecture now states that the number of i-dimensional faces Ii of A is less than or equal to a certain number ci(n, d). In 1975 R. P. Stanley (1] solved the Upper Bound Conjecture for spheres and, more generally, for all simplicial COIllplexes A such that the assoeiated graded k-algebra k(Al for an arbitrary field k is a Cohen-Macaulay ring. In 1976 G. A. Reisner described those simplicial complexes for which k[A] is a Cohen-Macaulay ring. We now define A to be a Buchsbaum complex if k[ A] is a Buchsbaum ring. For instance, if the geometric realization X = iA i of A is a connected manifold, then L1 is a Buchsbaum complex. I.et IAI, for example, be the torus. We know that a major objection to the liRe of simplicial complexes in computing topological invariant of compact polyhedra is that the dissection of the polyhedron may require an uncomfortably large number of simplexes. Thus the r.lOst obvious dissection of the torus, which is pictured as

(see Hiiton.Wylie [1], p. 49)

requires 9 vertices, 27 edges and 18 triangles. But we do get from such triangulations our graded k-algebras k[Al. We obtain for the torus: k[Al =

k[xI' • '"' xg]/a,

where a

= (XIX"

X1X9, X2X6, X 2X" Xa X 4' XaXg, X4 X S, X 5X 9, XsX" X1X2Xa, X4X5XS' x,XSXg, X 1X 6 X S,

X2X4X9, XaX,x" X 1X 4X 7' X 2X 5X 9, XaXsXg) •

We can see that the torus is a (non-Cohen-Macaulay) Buchsbaum complex. For the invariant J(k[A]) of the Buchsbaum ring k[A] we have J(k[A]) = 2 since there are 1-cycles z~, zi in any simplicial decomposition of the torus, and as it turns out that {z~} + 0 and {zn 9= 0 generate the homology group Hl(A) freely. 2*

20

Introduction and some examples

Applying some fundamental principles of Buchsbaum rings our investigations now result in a particular solution of the Upper Bound Conjecture, which was extended to arbitrary manifolds by V. Klee [1] in 1964. For instance, we will prove the following statement in Chapter II, § 4: If the geometric realization X = ILII of LI is a connected manifold then we have:

E .v ( +d)1 .-0 z+ 1 v-1 (

)

dimk jji(LI ; k)

t'

for v = 0, 1, ... , d 1, where jji(LI; k) denotes the reduced simplicial cohomology of LI with coefficients in an arbitrary fixed field k.

Chapter 0 Some foundations of commutative and homological algebra

§ 1.

Local algebra and homological algebra

Chapter 0 contains the fundamental tools needed for the following chapters. In § 1 we will assume familiarity with the baeic techniques of local algebra and homological algebra. Since notation and terminology vary from one source to another, we will assemble in this paragraph (more or less without proofs) the basic definitions and results needed. More details can be found in standard sources such as: Atiyah-Macdonald [1], Matsumura [1], Zariski-Samuel [1], Cartan-Eilenberg [1], Grothendieck [3] and Serre [2]. All rings are tacitly assumed to be commutative and Noetherian with unit element. Let A be a "ring. By an A -module we mean a unitary module over A. A Noetherian A-module is then a finitely generated A-module.

1.

Associated primes

Let M be an A-module. We say that a prime ideal p of A is an as.,ociated prime of M, if one of the following equivalent conditions holds (i) there exists an element x E M with Ann xA = p where Ann N {a E A I aN = O} is the annihilator of the A-module N. (ii) M contains a submodule isomorphic to Ajp. The set of associated primes of M is denoted by ASSA III or by Ass M. If M is an A-module, the 8upport of M, written SUPPA III or Supp M, is the set of prime ideals \J of A such that the localization Mp of Mat p is 9= O. The Krull dimension of M, written dimA M or dim lll, is defined to be the supremum of length of chains of prime ideals of Supp lJI if it exists, and 00 if it does not. We have Ass M C Supp M, and any minimal element of Supp M is in Ass M. For example, let a be an ideal of A. Then the minimal associated primes of the Amodule A/a are precisely the minimal prime over-ideals of a. If M is a Noetherian A-module then Ass M is a finite set. An A-module M is said to be co-primary if it has only one associated prime. A submodule N of M is said to be a primary submodule of ill if MjN is co-primary. If Ass MjN = {p}, we say N is p-primary or that N belongs to \J (as a submodule of M). The connection with the classical definition of a primary ideal q of A is given by the following equivalence: (i) the module ill is co-primary,

*

0, and if a E A is a zero divisor for M then a is locally nilrotent on 1ll; that (ii) M is, for each x E M there is an integer n > 0 such that n"x = O.

22

O. Some foundations of commutative and homological algebra

I.et N be a su bmodule of 111. A primary decomposl:tion of N is an equation N Ql n ... (1 Qr with Qi primary in llf. Such a decomposition is said to be irredundant if no Qi can be omitted and if the associated primes of M/Qi (1::;: 2'::;: r) are all distinct. Any primary decomposition can be simplified to an irredundant one. If N = Ql n ... (1 Qr is an irredundant primary decomposition and if Qi belongs to ~i' then we have Ass MIN {\'l> .•. , \'r}. If ~i is an embedded prime of MIN, that is ~i is not minimal in Ass MIN, then the corresponding primary component 'Qi is not necessarily unique. On the other hand if Vi is minimal in Ass IlfjN then the primary component Qi is uniquely determined by N and by ~j' \Ve have the following well-known main result: If ~M is a Noetherian A-module then any submodule N of M has a primary decomposition. Having a primary decomposition N = Ql n ... n Qr we put U(N) = n Qi such that dim lll/Q; dim MIN. Finally, we set Spec A {prime ideals of A}. X Spec A is an affine scheme:

Spec A

=

As a point set, the set of primes of A. As a topological space a basis of open sets is given by the subsets XI {p E Spec: A I / ~ ~} for all / E A.

I

As a locally ringed space, its structure sheaf is defined by F(Xf , Ox) localization of M at the multiplicatively closed set (1, /, /2, ... ).

2.

AI

Systems of parameters and multiplicity

A ring A which has only one maximal ideal m is called a local ring, and Aim is called the residue field of A. If A is a local ring then the Krull dimension of A is finite. Let A be a local ring; m its maximal ideal. Let M be a Noetherian A-module of dimension d:2:: O. A family (xl> "" Xd) of elements Xl' "., Xd of m is said to be a system 0/ parameters 0/ M if dim M/(xl>" " Xd) ill O. If no confusion is possible wc denote a system of parameters (Xl> ... , xdl of M simply by its elements Xl> ... , Xd' We call an ideal q of A a parameter ideal of ~M if there is a system of parameters Xl> ••• , Xd of ill contained in q such that q . M = (Xl> ., " Xd) • ill. If d = 0, any ideal contained in Ann M is a parameter ideal of M; espeeially, the zero ideal is a parameter ideal. We have the existence of systems of parameters in any loeal ring and for every Noetherian A-module. Notice that if Xv .,., Xd is a system of parameters for M, then the dimension of M j(Xl> ... , Xi) ill is d - j for all j 1, ... , d. A system of elements Xl> ••• , Xj, j ~ d, of m is said to be a part of a system of parameters for M if dim M/(x l , ... , Xi) M = d - j. Let xl> ... , Xn be a sequence of elements of A. Reeall that if jlf is an A-module, then Xl' ••• , x. is an M-sequence if 1) Xi+l is a non-zero divisor on Mj(Xh ... , Xi) M for i 0, ... , n 1, and

2) M oF (Xl' ••• , xnl M. This property does not depend on the order of Xl' .• " X.' Every ~M-sequence is a part of a system of parameters.

§ 1. Local algebra and homological algebra

23

Let M be a Noetherian A-module. The depth of M, denoted by depth M, is defined as the supremum of all integers r such that there exists an "H-sequence Xl> ••• , X,. If X E Tn is a non-zero divisor on M then depth M ixM = depth M

1.

Furthermore, depth ilf::;: the infimum of dim Ai+:! as +:! runs through Ass M. In particular we get depth M

dim M

if

M =1= O.

111 is said to be Cohen-Macaulay if depth M = dim M. The ring A is said to be Cohen-Macaulay if it is a Cohen-Macaulay A-module. A is a regular local ring if and only if m is generated by an A-sequence. Renee a JOegular local ring is Cohen-Macaulay. A local ring A is called a (lowl) Gorenstein n'ng if A is Cohen-Macaulay, and whenever Xl' ••• , Xn is a maximal A-sequence, then the ideal (Xl' .•• , xn) A is irreducible. We have the following useful

Lemma 1.1. Let M be a Cohen-Macaulay A.module. Then: (il M is equidz"mensional (i.e. dim Alp = dim 1'1 for all mznz"mal prz"mes p in Supp M) wz"thout embedded primes (i.e. 0 M has no embedded primes). (ii) Let X be an element 0/ m such that dim Mjx.H = dim M - 1. Then X 1:~ a non-z!:TO dz'visor on M and MJxM is Cohen-Macaulay. Because of our investigations in Chapter I we need to characterize the C',ahen.Macaulay property by using local multiplicities. For this we examine the HilbertSamuel function. Let M be a Noetheri.an A-mo'dule of dimension d O. Let q be an ideal of A sueh that the length, IA(Mlq4Y), of MjqM over A is finite. Then we define the so-called Hilbert-Samuel function, denoted by PQ,M(n), as follows: PQ,M(n)

IA (1lfjqfl+ 1M)

for all integers n

O.

It is well-known that there is a polynomial in n, denoted by PQ,ltl(n), such that P q,M(1I pq,M(n) for all large n. The polynomial Pq,M(n) is the so-called (characteristic) Hilbert Samuel polynomzal of the ideal q with respect to M. There exist integers eo eo(q,lY (> 0), e1 := e1(q, M), ... , Cd := ed(q, M), where d = dim M, such that Pq,M(n)

€o(q, M)

(n : d) + el(q, 1'1) (n ; ~ 11) + ... + ed(q, M).

The leading coefficient €o(q, M) of Pq,M is called the mulHpUcity 0/ q 'U'z'th respect to M. When q m and M = A, we simplify the notation as follows: eo(q, A) = eo(A) and eo(A) is called the multipl1'ciiy of A. For our purposes we obtain the following main result:

Theorem 1.2. Let A be a local ring, let M be a Noetherian A.module, The following properties are equimlem: (a) M is a Oohen-Macaulay module. (b) There exists a parameter ideal q (c) For every parameter ideal q

eo(q, M)

=

l(lY jqM).

0/ M

0/ M

in A such that eo(q, M) in A,

IUlfJqM).

24

O. Some foundations of commutative and homological algebra

M. Auslander and D. A. Buchsbaum [1] used the methods of homological algebra to give an explicit expression for a general multiplicity in terms of the Euler-Poincare characteristic of the graded homology module of a certain Koszul complex. In particular, they gave an axiomatic description of multiplicity. This development has opened the subject to a much simpler treatment. An example of this may be found in D. J. Wright's paper [lJ. Here an inductive definition is given for the so-called general Illultiplicity symbol for n elements, relative to an arbitrary module over a commutative ring, which is suggested by Auslander-Buchsbaum [1], Theorem 3.3. We will have to use some well-known properties of this multiplicity symbol. It should be noted that the Recount of the general multiplicity theory in this section is restricted to what is required for our immediate application. However, the theory is more extensive than indicated here, and some of the results are valid under wider hypotheses, see, for instance, D. G. Northcott [lJ. Let A be a local ring and let M be a Noetherian A-module. Let Xl, ••• , Xn be elements of A such that the A-module M/(x I , ... , x,,) M has finite length. When n = 0 this condition is to be understood as meaning that lA(M) is finite. The definition of the mul1iplicity symbol of Xl> ••• , Xn with respect to M, denoted by e(xI,"" x.IM), useR induction on. n. First suppose that n O. In this case, by our convention, lA(M) is finite. We may therefore put e(0plf) Now assume that n>- 1. We set O:M Xl {m E M I mX I = O}. Since lA(M/(xl> ... , x,,) M) < (Xl it follows that lA((O :.u XI )/(X2, •.• , x n )· (O:M Xl») Accordingly, by our assumptions, e(x2 , ••• , x" I M/XIM) and e(x2'"'' x. ! 0 are both defined and so we may put

<

00.

:.11 Xl)

We collect some elementary properties that we need.

Lemma 1.3. (il

If Xl, ••• , Xn Z8 a system of parameters for M, i.e., n q := (XU"'' x,,) A

=

dim M, then we have for

0

eo(q, M)

e(XI' ... , x .. IM) ,

(ii) 0':;: e(X11 ..• , x.IM)

(iii) (The additimcy property) Let Q ~ Mp ~ ... ~ .ltl ~ Mo ~ 0 be an exact sequence of Noetherian A-modules and suppose that the A-modules MJ(x I, ... , x.) M j have finite length for all i 0, 1, p then we have 000'

p

E

;=0

(-l)i e(x11'''' x.IM j ) 0

= O.

(iv) Assume that for some particular value of i we have xr Jf integer, then we get e(x1' ... , xnlM)

O.

=

0, where m is some posihove

§ 1. Local algebra and homological algebra.

25

(v) Let r I , r2, ... , rn be posi#ve £ntegers. Then

(vi) lA(M/(xt> ... , xn)M) - e(xlJ .•. , xnM)

• .1: e(x;_l> ..• , Xn I (Xl' ... , Xi_I) M

:M X;j(Xl> ... , Xi_tl .H).

'=1

The next lemma is useful, apart from its intrinsic interest. It enables us to prove some statements in Chapter I. It is precisely the property (i) of Lemma 1.3 and the following lemma which are of interest in the general multiplicity theory to our investigations in Chapter I. The point is to have a criterion for a multiplicity to be zero.

Lemma 1.4. II M and n

>

XI, ... ,

Xn are as above, then e(xl> ... , xnI1\f) = 0 £1 and only il

dim~H.

Sketch 01 the prool (see also D. J. Wright [1], I..emma 9, or D. G. Northcott [1], Proposition 7 on p. 334): Note that we have always n:2 dim 1lf. Let dim M n. Then Lemma 1.3 (i) implies that e(xh ... , Xnl1lf) > O. Let n > dim M. We use induction on dim M. If dim ~lf = 0, then .XI is nilpotent with respect to M and so x'{' M 0 for a suitable integer tn. Accordingly e(xI' ... , xnlM) = 0 by Lemma 1.3, (iv). From now on assume that dim.H 1. If Xl is nilpotent, then we get the desired result by Lemma 1.3, (iv). We shall therefore suppose that XI is not nilpotent. Since . ~H is a Noetherian A-module we can choose an integer m so that XI is not a zero divisor on M' :='Mj(O:,u x'{'). Now, by Lemma 1.3, (iv), e(xlJ ... , XnIO:M x'{') O. Consequently we have

e(xlJ ... , xnlM)

=

e(xv ••. , x.IM'),

and XI is not a zero divisor of M'. This shows that for the remainder of the proof wc may assume that XI is not a zero divisor of M. But then we get e(xh ... , xnl.H) = e(x2' ... , x.,H/XI1lf).

Since dim MjxlM

< dim M the assertion follows by the inductive hypothesis, q.e.d.

As indicated, some important results in Chapter I will be established with this useful criterion for a multiplicity to be zero.

3.

Local cohomology theory and cohomology 0/ the K08zul complex

A recent addition to local algebra has been provided by Grothendieck's local cohomology theory; basic facts about this theory are available from Grothendieck [3] and Sharp [IJ. If A is a local ring, then any ideal a of A determines the following additive, Alinear, covariant, left exact functor r~ (on the category of all A-modules and homomorphisms) called the local cohomology junctor with respect to a: Let N be a submodule of an A-module M, let N

:M

(a) := {m E M

I there exists an integer n > 0 such that a" • m b

and N :M a := {tn E ~H I a • m b N}.

N) ,

26

O. Some foundations of commutative and homological algebra

For an A-module ill, we define co

O:M

(a)

=

U (0

:.It

ak ).

k=l

If ill is an A-module and i is an integer;;::: 0, then we denote by ll~(ill) the module obtained by applying to iI-I the right derived functor of Fa (ra(Jf) ~ H~Ulf)). Suppose again that A is a local ring having maximal ideal m. It is well-known that if ill is a Noetherian A-module having Krull dimension d"2': 0 then all the modules H:nUI.f) for i 0 are Artinian A-modules and H'fn(ill) O. Moreover, if d > 0 then H'fnVlf) is not Noetherian. Furthermore, H;n(ill) = 0 for all l' > dim ill, and depth ill is the least integer i for which H:n(ill) =t= O. We can give yet another description of local cohomology as follows: The positive integers with the usual ordering form a directed set 1. If i, j E I with 1::::;: j, then ai ai, and the natural A-homomorphism A/a i ->- A/ai induces, for an arbitrary A-module ill, an A-homomorphism ;'l;ij(ill): Ext"(Ajai , ill) ->- Ext"(A/a i , ill). Also, the Ext"(A/a i , ill) and ;'l;jj(ill) form a direct system of A-modules and A-homomorphisms over I, and so the direct limit lim Ext"(A/ak , Jf) can be formed. It is k now easy to check that ~ Ext"(Ajak , ) becomes a covariant, additive, A-linear k

functor on the category of all A-modules; one can show that the functors Il~()

and

lim Ext"(Aja k , k

are naturally equivalent for each n

)

O. Therefore we get isomorphisms for all n

0:

lim Ext"(Ajak , ill).

k

In particular, we obtain canonical maIlS cp~w:: Ext~(A/a, 1'1-1) ->- H~(ill)

for each i

O.

Now, we' want to examine how this relates to the cohomology of the Koszul complex. Let A be a local ring. For an element x of A we define the K08zul complex K(x; A) generated over A by x as follows: Kj(x; A) = 0

Ko(x;

A)~

for all i

K1(x;

0,1,

A)~ A

and a map d l ' Kj(x; A) ->- Ko(x; A)

defined by dj{a)

-=-.

xa for all a E A;

that is, K(x; A) is the complex 0 ->- A A ->- O. I,et ill be an A-module and let Xl> ,. " Xr be elements of A. We define the Koszul complex K(Xb ••• , Xr ; Lll!) generated over A by Xl> •• " Xr with respect to ill! by: K(Xj; A) @ ... @ K(xr; A) @ ill.

§ 1. Local algebra and homological algebra

Its homology we denote by H;(xj, ... ,

Xr ;

27

J11). We now put

and Hi(Xh ... , Xr ; J11):= Hi(K'(Xh ... , Ir; 111)) = Hr_i(XI, .•. , Xr ; M).

Clearly, Ki(Xb"" Xr ; A) is a free A-module of rank

el•... I" 1

II

< '" <

li

(ri)' We take free generators

r, of Ki(Xh ... , Xr ; A) such that

where d' denotes the differentiation in K'(x l , ••• , X r ; A). 'rhen similar formulas hold for K'(xJ> ... , X r ; J11). Let YI, .,., y. be elements of A such that (Yi>"" Ys) A (Xl' ,.,' x r ) A, i,e, there r

are

aii

E A, 1

i

8,

1·::;: j::;:

r,

with Yi =

E nijxj for i

1,

",,8,

Let

j~l

If 1 lJ , .. < l. ::;: 8, I::;: i1 < , .. < i. ::;: r, denote by L1{::::{: the minor of consisting of the elements of the lIth, "" lnth row and the 11th, .," inth column (set LI 0 if n > r or n > 8). Then it is easy to see (using well-known methods of linear algebra) that we have a homomorphism of complexes ~

defined by (denote by

eh•.. I, the free generators of Ki(yj, ... , Ys; A»:

Tensoring with J11 we find a homomorphism of complexes

Clearly, lJIo is an isomorphism. We consider two special eases:

1. Letr 8 and assume that a := (x j , . . . , x r) A = (Yi> ... , Yr) A with r rankAlma/mn. Then ~ is invertible over A since ~ mod m is invertible over A/m, i.e. det ~ El 'm. Therefore rf> and hence lJI is an isomorphism of complexes. We define K'(a; 111) := K"(Xl' ... , Xr ; J11) and Hi(a; J11):= Hi(xv , .. , Xr ; J11) for all A-modules J11, If we choose another set of generators of a, the Koszul complex and hence its cohomology is unchanged (up to an isomorphism). Since HO(a; J11) HomA(A/a; J11), we get from Cartan-Eilenberg [1], Chap,IIJ, Proposition 5.2, canonical homomorphisms 1j!~: Ext~(A/a; J11) -'>" Hi(a; J11) for all i 0,

28

O. Some foundations of commutative and homological algebra

2. Consider the sequence of ideals

from which a direct system of complexes K(Xl' ... , Xr ; ~M)

-:>- K"(x~,

•.. , x~; M)

-'r •••

(M an A-module)

is obtained. The direct limit of this system is denoted by K;"'(Xl' ... , X,; M).

Since direct limits commute with exact sequences, we get Hi(K;"'(XlJ .•• , Xr ; M)) ~ ~ Hi(x~, ... , x;; M),

"

in particular

Hr(K;"'(Xl' ... , xr ; ~M)) ~ ~ M/(x~, ..• , x;) M,

" where the maps of the last right hand direct system are given by multiplication by Xl' ••.• X r •

a

Furthermore, HO(K;"'(xlJ •.. , x,; M)) !i!r: (0:1U (x~, ..., x~) A) = H~(M), x,) A. n Now, for r = 1 it is easy to verify that K;"'(x; M) is the complex

where

:== (Xl1 •. "'

O-+M!:.. M",-'>-O,

where M", is the localization of JI. with respect to the set II, x,

X2, ••• J

and h",(m)

m

(Observe K;"'(Xl' ... , x,; M) ~ K;"'(Xl; A) ... (8) K;"'(xr ; A) M, since tensor products commute with direct limits.) If M is an injective module results of Matlis [1] show that hr is always surjective, Le. IJ1(K;"'(x; ~M)) = O. Using an exact sequence defined by Corollary 1.7 below an easy inductive argument (on r) shows that Hi(K;"'(xlJ ... , x,; M)) = 0 whenever M is injective and 1. Hence by Cartan-Eilenberg [lJ, Chap. III, Proposition 5.2, and Chap. V, Proposition 4.4, we get:

for every A-module M and with a

(Xl' ... , X r )

A.

1.et us denote by Ak- the canonical homomorphism (defined by direct limit) JJi(a; M) -+H!(M), where a is an ideal of A. Then we have by the above remarks

Lemma 1.5. For all i (i)

:2: 0

we

have commutative diagrams

§ 1. Local algebra and homological algebra

(ii) II 6 is another ideal

01 A

29

with a C b then there are lor all i commutative diagrams

H'(6; M) ---+ H'(a; M)

1

1

H~(M)

---+ H:(M)

where the vertical maps are the canonical ones and the other homomorphisms are ~'nduced by 'P.

Next, analyzing the proof of Proposition 1 of Serre [2], IV-2, we get the following Lemma 1.6. Let L be a complex of A -modules and let x be an element of A. Then we have fot all p 1 commutative diagrams with exact rows:'

0 ....... Hl(x;H1H(L)) ....... HfI( K'(x; A) @ L) ....... HIl(X; HP(L)) ....... 0

'1

pI

r1

0 ....... H!A(HfI-1(L)) ....... HfI(K;,,(x;A)@L) ....... H~A(HP(L)) ....... O

where IX and y are ~'nduced by the canonical map K(x; ) ....... K~(x; ) which also defines a map K(x; A) @ L ....... K;"'(x; A) L which specilies p.

From Lemma 1.6 we obtain a very useful corollary. Corollary 1.7. Let A be a local ring and M an A-module. Let XI' •.• , x" r;;::: 2, be element8 of A. Then we have for all p 1 commutative diagrams with exact rows:

o . . . . Hl(X

1;

HP-l(X2 ,

''',

x r ; M)) ....... HP(xl> ••. , x r ; M) ....... HO(x 1 ;HP(x2 ,

t II x l M

t

x r ; M)) ....... 0

t

= 0, we get the following commutative diagram

o . . . . /IP-l(X2' .••, xr ; M) ....... HP(x}> ... , x r ; M)

t 0.......

••• ,

0

....... Hp(X2' .•. , x r ; M) ....... 0

t .......

t

Hr"', ...%rlA(M).......

Hrz ....ZrlA(M) ....... 0.

Prool: The second commutative diagram is a consequence of the first since for an A-module N with XIN = 0 Hl(Xl; N)

f

HO(Xl; N) = N,

H;',A(N)

= 0,

H~,A(N) = N

follows. Thus we need to prove the commutativity of the first diagram. We set L = K'(X2> ••• , x r ; M) and apply Lemma 1.6, Then the top row of the commutative diagram of Lemma 1.6 agrees with the top row of the diagram under consideration. Now we set for n 1

Ln := K(X;, .. ,' x;; M) and

30

O. Some foundations of commutative and homologieal algebra

and denote by (En) the bottom row of the commutative diagram of Lemma 1.6 if iJ := L. and by (Bex,) for L := Loo. Then we have a direct system of exad sequences (E l ) (E 2 ) -i>- ••• Since direct limits commute with exact sequences and with the local cohomology functors, we get (En) = (Eoo) and the map (E l ) -i>- (Eoo) (into the direct limit) describes a commutative diagram (with top row (E I ) and bottom row (Eoo». But (Eco) is just the bottom row of our commutative diagram. Thus, combining the commutative diagram constructed above and this commutative diagram we find a commutative diagram of the desired type, q.e.d.

Lemma 1.8. Let a c A be an zaeal and S A a multiplicaUvely dosed set with 1 E S, 0. Let M be an A-module and lor all i?:::. 0, in abreviated notation, Hi(M) l~ Ext~(a"; M). Then we have:

an S

(i)

"There

o

is an exact sequence H~(M)

and lor all

-i>-

M

i?:::. 1 there

are z80morphisms

Hi(M) "'-' H~+l(M). (ii) The lollowing condiHon8 are equz'valent: (a) Supp M C Via), (iii)

H~(HO(M»)

=

(b) HO(M)

H~(HO(M»)

H~(HO(M»)~ H~(M)

=

=

0,

(c) Hi(M)

=

0

lor all

i?:::. O.

0,

lor all i

2.

(iv) There is a (natural) homom()rphism (01 A-modules) g: HO(M)

-i>-

Ms

such that the composition gl is just the nautral m(.tp M

-i>-

Ms.

(v) g is z'njective if and only if S n l' = 0 lor aUl' E Ass M" V(a). (vi) (Formula of Deligne) There is a natural (A-)isomorphism H{J(M)::::::~Ma, Gal

where Ma denotes the localization 01 M at the multiplicatively dosed set {1, a, a 2 , •• •J. The (A- )modules Mao a E a, lorm an inverse .system: We deline jor a, b E a a honwmorphi8m lJa.b: Mb

-i>-

11fa

il and only il a E VbA, i.e. il at

me"

bc lor tEN, c E A by setting

(m E M, n EN).

Prool: Apply the functors Ext~( ; M) to the exact sequence 0 a" -i>- A -i>- A/a" 0 and take the direct limit of the resulting long exact cohomology sequence for n 1,2, ... This is again an exact sequence and (i) follows by virtue of HomA(A; JJJ) M and Ext~(A; M) = 0 for 1.

§ 1. Local algebra and homological algebra

31

(ii) (a) =? (b): Let Supp M C V(a) and take a E IlO(1I1). Choose s E Hom(a fl ; 111) representing a. Then for all a E an there is a p > 0 with aPs(a) = O. Since an is finitely generated, there is a q 0 with aqs(a) = 0 for all a E an. Therefore sian'. 0, that is, a O. (a) =? (c): Take a minimal injective resolution of 111

o

M

--'>-

1o --'>- 11

--'>- ...

Since Supp 111 V(a), Supp Ii module of the complex

o

IlO(/o)

--'>-

HO(/ 1 )

t;;; V(a)

for all

~.~

O. Now Hi(11f) is the £th cohomology

--'>- •••

But by the first part of the proof HO(/i ) 0 foralli ~ 0 which proves this assertion. (b) (a): If 1l0(lIf) 0 the exact sequence of (i) yields an isomorphism 1l~(M) ~ 111. 'fherefore each element of M is annihilated by some power of a, i.e. Supp iJ-f t;;; V(a). The implication (c) (b) is trivial and (ii) is therefore proven. (iii) From (i) and (ii) we get: Supp .tI

V(a) if and only if 1l~(M) '"'" M and HWlf) = 0

for all i~ 1.

We split the exact sequence of (i) into two exact sequences: (a) 0

Ilg(M)

~M

111'

--'>-

0

and (b) 0--'>-11/' IlO(M) 1l~(M)--,>-O. Since Supp 1l~(M), Supp 1l~(1I1) V(a) we get H~(M)

1l~(1I1')

for all

m(M')

1l~(IlO(1I1))

for all

1l~(M)

1l~(HO(.M))

for

1

and i~

2,

alll:~

2.

hence Also (al yields H~P{') (H'!,.(H!(M)) ~ llWW)):

o --'>- /l~(HO(1I1))

0 and thus (b) gives rise to the following exact sequence H~(M)

--'>-

HWW)

--'>-

m(IlO(M))

--'>-

O.

Now it is not difficult to see that the middle homomorphism is nothing but the isomorphism llk(.llf) H~(M/) obtained froll! (a) and this proves (iii). (iv) Leta E HO(M). 'fake 0 and s E Hom,((a n ; .:tIl representing a. Choose a E a nS. We define

g(a) Straightforeward calculations show that g is a well-defined A-homomorphism. Finally, for all m E M the homomorphism h: a M defined by h(al am for all a E a represents j(m) E HO(M). Therefore, if a E a n S, proves (iv).

gj(m)

h(a) am = -

a

a

m

= -. 1

This

o.

32

Some foundations of commutative and homological algebra

(v) Assume g is injective. For Ass M ~ V(a) let lJ E Ass M" V(a). Then there is a monomorphism A/lJ ...... M giving rise to the following commutative diagram

Alp -4 HO(AllJ) 4 (Alp)s

t

t

HO(M) 4

Ms

where all homomorphisms are injective (HO( ) is left exact and H~(Alp) 0 since lJ ~V(a». Therefore (Alp)s =l= 0, Le.lJ n 8 = O. Assume now :p n S = 0 for all :p E Ass M" V(a). Let a E Ker g and choose 8(ult )

.

E Hom(alt ; M) representmg a. Then for some a E a n S we get - - = g(a) 0, a lt i.e. there is abE S with bs(a") = O. Then bE P n S foralllJ E Ass(A/Ann,8(a n ») Ass M. Therefore Ass(AjAnn s(alt ») ~ V(a), i.e. there is an m:2: 0 with a"'s(a") O. Assume without lOBS of generality that m n. Then alts(c) = ca(a") 0 for all c E am. Hence we have for all c E am:

8

for all:p E Ass(AjAnn 8(cl) ~ Ass M.

a E :p n S

This implies Ass(AIAnn 8(cl) ~ V(a), Le. there is a q 0 with aIl8(c) = O. Since a'" is finitely generated we can find an r 0 with a'8(c) = 0 for all c E am. Therefore 810'"+" = 0, which means (J 0, i.e. !I is injective. We now prove (vi). We see that the maps HO(M) ...... Ma (a E a) given in (iv) are compatible with the homomorphisms ea.b' Therefore we have a homomorphism !p: HO(M) ...... ~Ma· aea

If

(J

E Ker!p, choose an

8

E Hom(a", M) representing

8(alt)

(J.

Then - a lt

=

0 (in Mal

for all a E a, i.e. for fixed a E a there is an mEN with 8(amH ) am . 8(a") = O. Since a is finitely generated, we find apE N with 81al> 0, j.e. (J = 0 and !p is injective. Now let p, E ~ Ma : N and let £Pa: N ...... M a, a E a, denote the canonical maps. aEO m/. If a (at> ..• , at) A, write £Pa,(P,) - ' E Ma with m; E M, n E N. Then for all . 1, .. "' OJ t : ai f 't,

i.e. there is an lEN with (ajaj)1 (m;aj ml~l'l O. m· We put m.:= aIm; and p := n l. Then £Pa (p,) = - ' and miaf = mjaf for all , a~ 1, ... , t. Let L denote the submodule of M generated by mb ... , mi' Assume

+

riaf = 0 with rl> ..., rl E A. Then for all j = 1, ... , t: I

a~

E rimi i=1

=

E r,mia1= E r,afmj = 1

0,

•

I.e. E rim, E 0 :L (11). Choose a q EN v.ith aqL n (0 :L (11» 0 (which exists by the lemm!l of Artin-Rees since L is finitely generated). We define a map 8: (af+q, ... , afHJ) A ...... M

by

8(U~+q) := aTmj,

i

=

1, ... , t.

§ 2. Graded modules and Kiinneth formulas

33

If E 8ia~Tq = 0 with s}) •.. ,81 E A then by the preceding we get E sja?m, E (O:L (a») n aqL 0, i.e. s is a homomorphism. Choose n E N with a" <;;;;; (af+ q, ... , af+q) A. Then the restriction of s to a" represents an element IJ E 11°(M) and it is easy to see that 'P(IJ) Il, q.e.d.

Remarks. 1. The lemma is true for arbitrary Noetherian rings A. 2. The lemma also has a graded version: It remains true if A is a Noetherian graded ring and M a graded R-module. The homomorphisms occuring here are all of degree zero (compare with the notations of § 2). 3. Let X := Spec A, U X" V(a). Then (vi) says that HO(J.11) ~ r(U, ii), where 1ft is the sheaf on X associated with M. This appears in the literature as "Formula of Deligne" (c.f. R. Hartshorne [4], Chap. III, Exercise 3.7).

§ 2.

Graded modules and Kiinneth formulas

In this paragraph we will develop basic facts and notations concerning graded rings and modules which we will need in the sequel. A graded ring R is always understood to be a commutative Noetherian ring with unit 1 which is (considered as an abelian group) a direct sum of subgroups R;,i E Z, such that Ri . R j <;;;;; Ri+1 for all i, j E Z. A graded R-module J.11 is a unitary R-module which is (also considered as an abelian group) a direct sum of subgroups Mil i E Z, such that R i • M, <;;;;; MI+I for all i, j E Z. An element m E J.11 is called hmnogeneous if m E M; for some i E Z. For m =F 0 this i is uniquely determined. We call it the degree of m 'and writei:= deg m. We say that a graded R-module N EB Ni is a submodule of M if NI <;;;;; .llf, for all i E Z. 'EX If ~11 is a given graded R-module and if n E Z let [M].

{m EM I mOor deg m

=

n}

(Le. [M].

=

M. if M

= EB Mi)' 'EZ

Let p E Z. By M(p) we denote the' graded R-module which is given by [M(P)Jn = [M]PH for all n E Z. We say that M(p) is obtained from M by shilting of degrees. By an ideal of R we always understand a graded R-submodule of R, i.e. an ideal (in the usual sense) which may be generated by homogeneous elements of R. By a Noetherian graded R-mooule we mean a graded R-module satisfying the ascending chain condition for gradcd submodules. Since R is Noetherian itself, a graded R-module M is Noetherian if and only if M is finetely generated by homogeneous elements.

1.

Associated primes

Let R be a graded ring and let M be a graded R-module. A homogeneous prime ideal pnme of M if one of the following equivalent conditions is fulfilled: (i) There is an homogeneous element x E 1'J. with Ann xA .):I. (ii) M contains a submodule isomorphic to (R/:p) (n) for some n E Z.

.):I of R is called an associated

3 Buchsbaum Rings

34

O. Some foundations of commutative and homological algebra

The set of associated primes of M is denoted again by AssR M or Ass M if no confusion is possible. Let T denote a multiplicatively closed set of homogeneous elements of R with 1 E T. Then the localization of M at T (denoted by T-IM) is·in a natural way a graded T-IRmodule where 'l'-lR is a graded ring (let deg ~:= deg m deg t, where mE M is a t homogeneous element and t E T). II/ E R is a homogeneous element let T := (1, /, /2, ... J and denote by RJ (MJ ) the localization T-IR (T-1.J.lI). By R(f) (M(/)) we denote the subring of RJ (submodule of M J ) consisting of homogeneous elements of degree zero. Clearly, M(/) is a (non-graded) R(J)-module. If V is a homogeneous prime ideal of R then there are three different localizations of Rand M at V : 1. Rp := T-IR, Mp T-IM, where T:= R" V. This is the usual localization where we forget about the grading of Rand M. 2. By Rll,homog or Rll,A and M p,homog or M Il,A we denote the localization at the multiplicatively closed set of homogeneous elements of R not contained in V. The ring Rll,A is graded and Mll,A is a graded Rll,A-module (see the above remarks). 3. R(ll) (M(ll)) denotes the subring of Rll,A (submodule of M Il,A) consisting of elements of degree zero. We have the following inclusions R(p) ~ R p.1I. ~ R", and a natural homomorphism of graded rings R ....... Rll,A' It is clear that R(ll) and Rll are local rings. Rp,A is a graded ring with exactly one maximal homogeneous ideal (the ideal VR",.h)' The following lemma relates these localizations: Lemma 2.1. Let .p

R be a homogeru;oWJ prime ideal. We have

(i) Mil ~ (M""A)PRp,h and (ii) .As8Ume that [Rh g;;

.p. Then lor some /

Mll,A = M(p)[f1

~M(Il)[X]x,

E [Rh

where X

".p;

u; an ~ndeterm~nate.

The proof is easy and we omit it. Remarks. 1. Using this lemma we will show in Chapter I (Lemma 2.27): R ll :::: R(oP)[X]mlxl for all homogeneous primes vcR with .p 11 [R]n where X is an indeterminate and m denotes the maximal ideal of R(ll)' (This implies MtJ""" M(~:))[X]mlxl') Also a consequence of Corollary 1.2.28 will be that for all homogeneous primes V and R with .p [Rh we have;

M., is a Cohen-Macaulay module (over R Il ) if and only if M(ll) is a Cohen-Macaulay module (over R(tJ)}. 2. Statement (ii) of Lemma 2.1 has a more general version: Assume T c R is a multiplicatively closed set consisting of homogeneous elements 0. Let S denote the subring of T-IR of elements of degree with 1 E T and T n {Rh zero (i.e. S = [T-IR]o)'

'*

§ 2. Graded modules and Kiinneth formulas

35

If t E 'P n [Rh and if X is an indeterminate then we have T-IR

S[tJt~S[Xlx =

s[x, ~l

If M is a graded R-module and if we put N;= [T-IMJo, then T- 11Y

N[tJ/ ~N[XJx

N

lx, ~

J

X1 .

Set SUPPn M;= (I' c:: R I I' prime, M'p =1= 0). The Krull dimen8ion of M, written dimB M or for short dim M, is defined to be the supremum of length of chains of prime ideals in Supp M if it exists and 00 otherwise. The same assertions made in § 1, 1. of this chapter on the existence of primary decompositions of submodules of Noetherian modules hold word for word in this situation. Therefore we omit an explicit discussion. Note that for any graded submodule N of a Noetherian graded R-module M there are graded primary submodules Ql' .. " Q, of M belonging to homogeneous primes Ph ' .. , 1', of R where {Ph"" Pr) = Ass M /N such that N Ql n ". n Qr.

2,

GTaded k-algebra.s, 8'gstem8 of parameters and Hilbert functions

Let k be any field. By a graded k-algebra we understand a graded Noetherian ring R with [RJi = 0 for all i < 0 and [RJo = k. The ring R has a unique homogeneous m~ximal ideal rnB E:B [RJi' ;~l

Let M be a Noetherian graded R-module. The same results as in § 1, 2. of this chapter on systems of parameters and M -sequence are true also in the graded case. Therefore we give only a short summary. We set d:= dim M. Homogeneous elements Xl> ... , Xd of rnB are called a system oj parameters of M if dim M/(Xh ... , Xd) M = O. An ideal q for which there is a (homogeneous) system of parameters Xl>"" Xd of M such that q. M = (Xl> ... , xd) • M is called a parameter ideal of M. Homogeneous elements Yh ... , Yr of rnB are called an M-sequence if for i 0, 1, ... , r - 1 (Yl' •.. , Yi) M :M Yi+l

(YH"" Yi) M.

As before depth M is defined to be the supremum of all integcr~ r such that there is an M-sequence consisting of r elements. Clearly any M-sequenee is part of a system of parameters. Therefore we get depth M

dim M.

M is called a Cohen-Macaulay module if depth M = dim M. Now we define the Hilbert function of a Noetherian module. Let M be a Noetherian graded R-module where R is a graded k-algebra. We set HM(n) := rankk([MJ.) (note that for all n E Z[MJ. is a k-vector space of finite rank). HM is called the Habert junction of M. There is a numerical polynomial hM (E Q[TJ, T an indeterminate) such that HM(n) = hM(n) for all sufficiently large n. The polynomial hM is called the 3*

36

O. Some foundations of commutative and homological algebra

Hilbert polynomial of ill. We write hM = ho(.M)

(~') + h (M) (a 1

'1'

1) + ... +

hdUI1)

with d := deg hM (considered as a polynomial in '1'), and where the integers ho(~~f) > 0, hl(JI), ... , hd(M) are called the Hilbert coellicienl n}, if M is a Koetherian Rmodule. riM) is called the index 01 regularity of M. If M 0 is Noetherian (Artinian) then a(M) (e(M») is finite.

+

3.

Proj and cohomology

Let R denote a graded k-algebra (k a field) with maximal homogeneous ideal mR. By Proj R we denote the projecti1'e spectrum of R. It is the following scheme: As a point set, the set of homogeneous primes V of R with mR g;; V·

I

As a topological space, defined by a basis of open sets X/ := {V E Proj R

Proj R =

I

1 g;; Vi for all homogeneous 1 E mR'

I As a locally l := R U)'

ringed space with structure sheaf defined by r(X"OProiR)

As is well known Proj R is a scheme. Note that in our notation dim R

1

+ dim Proj R,

where dim Proj R denotes the dimension of the topological space Proj R. Put X := Proj R. There are two functors which relate graded modules over R with sheaves of modules over X. One is the "sheafification" functor, which associates to each graded R-module M a (quasi-coherent) sheaf of modules over X:

MH-M. This functor is exakt. For caeh point p of X, the stalk PW)., of M at .p is simply M(!;l)' the degree 0 localization of M by the homogeneous prime ideal V c: R. If M R, M is precisely the structure sheaf of X and is denoted by the symbol Ox. More generally, the sheaf R(v) is denoted as Ox(v). In the opposite direction we have the "twisted global sections" functor, which assoeiates to each sheaf of modules J over X (quasi-coherent or not) a graded Rmodule:

§ 2. Graded modules and Kiinneth formulas

:17

This functor is not exact. It is left exakt, however. Therefore it is natural to consider its right derived functors, which are the higher cohomology modules H!(X, ) and H;(X, ), ... In contrast to the analogous affine case, the functors,....., and I1~ do not quite establish an equivalence of categories between graded R-modules and quasi-coherent sheave>; of modules over X. In one direction the two functors do compose to give the identity: if J is a quasicoherent sheaf of modules over X, then the sheaf H~(X, c'F) is canonically isomorphic to :T. However, in the other direction, if M is a graded R-module, the module I1~(X, .~1) is not isomorphic to M in general. In fact H~(X, M) need not even be finitely generated if M is finitely generated. But there are important cases in which I1~(X, M)~ M namely for R = k[Xo, ... , X,,], K a field, X o, ... , Xn indeterminates, n 1 and a) M any free graded R-module, b) M any homogeneous ideal I c::: R which is saturated (that is, I :R mR = I), c) M = I1~(X, :T) where c'r is any quasi-coherent sheaf of modules over X. In this situation (i.e. R = kfX o, ... , Xn]) it is a basic fact that,....., and H~ define a 1-1 correspondence: {closed subschemes of Xl ........ {saturated homogeneous ideals of R} ,

where d v is the ideal sheaf of V and

Proi (RjI, RjI)

+-!

I.

If a homogeneous ideal I = R is not saturated and not equal to an mwprimary ideal, it is contained in a unique homogeneous ideal J of R which is saturated and which defines the same closed subscheme V as does l. This latter ideal is called the total ideal defining the closed subscheme V and will be denoted by Iv. We denote that H~(X, dv), b) I v = dV' We observe also that b) is a consequence of a) since the ideal sheaf dV of a closed subscheme V is always quasi-coherent. We shall use the above 1-1 correspondence to translate all our statements about projective subschemes of X into ring-theoretic statements about R and its homogeneous ideals. Let now R be again an arbitrarity graded k-algebra. It is a basic fact of commutative algebra that in many important respects R may be treated as a local ring with mR as its maximal ideal. For example, Nakayama's Lemma admits a graded version which holds for (R, mR)' , Now we want to establish the connection between these concepts and our discussion of the local case. Let V denote a k-vector space. By f we denote the graded R-module given by [fl" = 0 for all n =F 0 and [flo = V. Let M, N be graded R-modules. HomR(M, N) is the k-vector space of all R-homomorphisms I: M -7 N of degree zero, i.e. 1([M]n) ~ (N]n for all n E Z. We denote by HomRUlf, N) the graded R-module given by a) Iv

(Homn(M, N)]n

=

Homn(M, N(n)) '='" HomR(M( -n), N) for all n E Z.

38

O. Some foundations of commutative and homological algebra

Note that HomR( , ) and HomR( , ) are left exact functors from the abelian category of graded R-modules with R-homomorphisms of degree zero as morphisms to the category of k-ve<.:tor spaces and graded R-modules, resp. We denote by Ext~( , ) and Extk( , ) the right derived functors of HomR( , ) and HomR( , ), resp. Extk(M, N) again are graded R-modules with

[Extk(M, N)]n:= Ext~(Jf, N(n»):= Extk(M( -n), N) for all graded R-modules M, N and all n E Z. Next, we define for a graded R-module M Hi _mR (M):= lim _ _ Ext' _ R (Rim"R, M)

(local cohomology),

n

--"

fii(R, M) := lim Extk(mR' M)

(Serre cohomology).

If no ambiguity exists we will write fi~(M) for H'(R, M). fi'm,,(M) and fii(R, M) are related by the exact sequence

o -* fi~R(M) -* M -* fiO(R, M) -* fi;,,(M) -* 0 and by isomorphisms fii(R, M) c::::.fi:ri;(M)

for all

i> 1

(c.f. Lemma 1.8(i».

Notice that fi'm,,( ) and fii(R, ) are the right derived functors of the left exact functors = 0 :M (mR) for any graded R-module M. If M is Noetherian then fi'mR(M) is Artinian for all i. Now we have

fi~n,,( ) resp. fiO(R, ). It is clear that fi~,,(M)

Proposition 2.3. Let X:= Proj Rand aS8Ume that M is a graded R-nwdule. Then tor all i here are natural i8omorphisms H~(X,

Ai) ~

fi'(R, M).

Proof: We prove Hi(X, M) c::::. Hi(R, M) (then taking the direct sum over all shifts of degrees the result will follow). For i = 0 we get HQ(X, M)

~ r(XI> M) = XJ<;;;X

lim M(/l IEm"

~

HomR(mR' M) = HO(R, M)

n

(ft is obtained from Lemma 1.8(vi), c.f. the remarks of Lemma 1.8). This isomorphism is a natural one. If M is an injective object in the category of graded R-modules, M is flasque and hence Hi(X, Ai) 0 and Ht(R, M) 0 for all i> O. Therefore we get the desired isomorphisms of all i> 0 using Cartan-Eilenberg [1], Chap. III, Proposition 5.2, and Chap. V, Proposition 4.4, q.e.d.

Like in the local case we have for a Noetherian graded R-module M dim M

sup {n E N Ifi'mll(M) =t= O}

and depth M = inf {n E N Ifi'm)M) =t= OJ. The same relations between the Koszul cohomology and the local cohomology examined in § 1, 3. hold in the graded case. Therefore we mention only some differences in the

§ 2. Graded modules and Kiinneth formulas

definition of graded Koszul complexes. Let x E R and set d we denote the complex

39

deg x. Then by K.(x, R)

O..-;..R..!.... R(d) ..-;..0.

For t homogeneous elements Xl, .,., X, E R and a graded R-module 11I we set

K,(XbM) ®M.

K.(x l , R)

K.(Xl' •• " XI> M)

We now define again the graded Koszul cohomology !.!i(mR' M) of M to be the (t i)th homology of the Koszul complex K,(x v ... XI> M) where {Xl>"" x,) is a minimal basis of mR consisting of homogeneous elements, c.f. Lemma 4.2. Then we obta.in the following (see also Lemma 1.5) j

Lemma 2.4. For all i

4.

0 we have a commutative diagram:

Segre products

Let k be a field and let Rb R2 be graded k-algebras with maximal ideals ml and m2, resp. Definition 2.5. (i) A graded k-algebra R is called a Seyre product 0/ RI and R2 over k. denoted by R = 11t(Rl> R 2 ) or R = (f(RI' R2 ), if for every pEN , [R]p = [R1]p

®k

[R2 ]p'

(ii) Let Mv ~M:2 be graded R 1-, R 2 -modules, resp. Let R be the Segre product of RI and R2 (if it exists). A graded R-module M is called a Seyre product 0/ MI and M2 over k, denoted by 11I = (f,,(M I • M 2 ) or M = (f(MI' M 2 ), if for all p E Z [M]p = [M1)p

®k

[M 2 1p,

As is known Segre products exist and are uniquely determined (up to isomorphisms) by definition. Also since a sequence of graded modules over a graded k-algebra is exact if and only if the corresponding sequences for each grading are exact (as sequences of k-vector spaces), the Segre product is an exact covariant functor in each variable. Using this fact it is not difficult to show that the Segre product of Noetherian graded modules is again a Noetherian module. Note that the converse is generally not true. For example, if one takes an Artinian non-Noetherian graded RcmoduleA 1 (for instance Al = !.!:r:,(Rtl, d l := dim R 1 ) and considers (f(AI' ~(d)) where d is chosen such that [Ad_ d =f: O. Then (f(AlJ k(d)} ~ [AILd (d) and this is a finite-dimensional k-vector space and hence a Noetherian R-module.

40

o.

Some foundations of commutative and homological algebra

If RI and R2 for-example are generated (over k) by XI, ... , x. E [Rlh, YI, ... , Ym E [R 2]1' resp., then R := a(R I, R 2) is generated by XI ® Yh ... , x. ® Ym. If MI and M2 are finitely generated by m n , ... , m lr and mw ... , m 28 (over RI resp. R 2) then M := a(M I, M 2) is generated (over R) by the following elements: mli ® n2m2j

for all 1 s:;: is:;: r, 1 s:;: is:;: s with deg mli ~ deg m 2j and n2 running over a basis of the k-vector space [R2 ]degmu- d egm.;'

nlml/ ® m 2•

for all 1 s:;: l s:;: r, 1 s:;: n s:;: s with deg m 2• ~ deg mIl and nl running over a basis of the k-vector space [R I]degm •• -deg ml l .

and

Hence M is again finitely generated, i.e. M is a Noetherian graded R-module. This explicit description is useful for studying some specific examples.

5.

K unneth relations for Segre products

In this section we want to give some Kiinneth relations for Segre products., They allow us to "compare" the "cohomology of the Segre product" and the "Segre product of the cohomology" . The main point of interest in the following is that our Kiinneth relations are given by algebra homomorphisms and not only by k-linear maps (k denotes again the ground field) as would be the case using results of Grothendieck [2]. First we prove several lemmas. Let R I, R2 be graded k-algebras.

Lemma 2.6. Let M; a,nd Ni be gmded Ri-modules lor i = 1, 2. Then there is a, natuml R.homomorphism

where R = ak(R I, R 2). Prool: We only give a short outline since all constructions are canonical. Let p E Z and Ii E HOlliR,(N i, Mi(p)) = [HomR,(N j , M;)]p for i = 1,2. Then we define for all ni E [Ni]q, q E Z and i = 1,2: ,°(/1 ® 12) (nl ® n 2) = II(n l ) ® 12(n2)· It is clear that ,°(/1 ® 12) is a well-defined R-homomorphism

i.e. an element of HomR(a(NI' N 2), a(MI' M 2)) of degree p. Hence we have a k-linear map .0: a(HomR1(N I, M I), HomR,(N 2, M 2)) -* HomR(a(NI' N 2), a(M I, M 2))

preserving degrees. It is not difficult to verify that ,0 is even an R·homomorphism. To this end we choose p, q E Z, Ii E [HomR,(N j , M;)]p, rj E [Rd q for i = 1,2 and prove

,o(h ® r2) (/1 ® 12)) =

(rl ® r2) ,°(/1 ® 12)·

§ 2. Graded modules and Kiinneth formulas

Let

nj

E [N i ].,

(1 0((rl

8

E Z, i = 1,2. Then we get:

r2) (/1

12») (nl

(TO(rtfl ® r2/2») (nl

n2)

n 2) = (rtfll (nl) ® h/2) (n2)

rl(tl(n l ») ® r2(12(11 2 »)

h ® r2) (t°(/1

=

41

M~»)

(r1 ® r2) (11(111)

12) (n1

n2»)

r2) to(h ® 12») (nl ® n 2 ),

((r 1

which finishes the proof. Finally, the naturalness of TO is evident, q.e.d.

Lemma 2.i. Let TO be as in Lemma 2.6. (i) II lor i = 1,2 Ni = R1'(p) utah Sl, 82 E N, P E Z, then TO is an i.sorrwrphism. (ii) II lor i = 1,2 Ni are linitely generated Iree R;-modules and il H~,(Mi) 0 lor i = 1,2, then to is injective. Prool; By the naturalness of TO we may assume that Ni = Ri(Pi) with Pi E Z (p = PI = P2 in case (i)~. Now we h~ve natural isomorphisms for i 1,2:

for~' =

1,2

¢i: HomR.(Ri(Pi), Mi):::: M i ( -Pi)

sending each lPi to lPi(l) where 1 E [Ri(p;)]_p,' Let # TO(a(¢l> ¢2»)-1. Since a(¢l> ¢2) is an isomorphism it is sufficient to prove (i) and (ii) for # (instead of to). We have for m l ® m2 E [a(M l(-Pl), M 2(-P2»)]t ann r 1 ® r2 E [a(R I(Pl), R 2(P2»)]. with 8, t E Z:

(#(m i If now PI = P2

m2») (ri ® r2) P and if

1: mli

rIm}

r2m2'

m2 i is a homogeneous element of Ker #, then we

i

obtain for 1 ® 1 E [a(RI(p), R 2(p»)]_p:

o

(#(1: mu ® m2i») (1

1) =

1: mH

m 2 i,

i.e.

Ker # = O.

-----

~--~' If I E HomR(a(R1(PI), R 2(P2», a(Ml> M 2 »), then I = #(1(1 ® 1») where 1(1 ® 1) denotes that we consider 1(1 1) as an element of degree deg I in a(Llfl( -p), M 2( -p»). Hence I is surjective and (i) is proven. m2i To prove (il) we initially assume that Mb M2 are finitely generated. Let 1: mli be a homogeneous element of Ker # where the elements mIl are linearly independent. By our assumption we can find an sEN (sufficiently large) such that there are r l E [Rll., r 2 E [R2 ]HP,-Pl with 0 :M, ri 0 for i 1,2. Then the elements 'imli are again linearly independent and we have:

1: 'lmH

'2 m 2i'

i

Therefore injective. If now generated limits the

we get r 2mji

= 0 for all z', i.e.

m2 i E 0

:M.

r2

= O. Hence Ker #

0, i.e. 11, is

M l , M2 are arbitrary modules then they are direct limits of their finitely graded submodules. Since all functors which occur commute with thes{' statement is true in general, q.e.d.

42

O. Some foundations of commutative and homological algebra

TO be as in Lemma 2.6. Additionally 8Uptpose that Nh N2 are finitely generated. If there i8 apE Z such that [Ndp generates N. as an R;-module for i = 1,2 then TO is ~'njective. If also !!..~,(Mj) = 0 for i = 1,2 then TO is an isamorphiBm.

Lemma 2.8. Let

Proof: Take the exact sequence Gj -+ F j -+ N j -+ 0 with F j , Gj free and F. = R,/'(p), ni E N, for i = 1,2 which ill possible by our first assumption. Set Hi := HomR.(N;, M i), N a(N}, N 2 ), ~M aUlfl> M 2 ). Then we have exact sequences 1,2, (i) 0 -+ Hi -+ HomR,(Fj, M i ) -+ HomR,(G;, M i) for i and

(ii) a{G I , F 2 ) Ef) a(Fl> G2 ) -+ a{Fl> F 2 ) -+ N -+ 0 (see Cartan-Eilenberg [1], Chap. II, Prop. 4.3. (c)). From this we obtain a commutative diagram with exact rows (the exactness of the top row follows from (i) and CartanEilenberg [1), II, Prop. 4.3 (c) and the exactness of the bottom row by (ii)):

0-+

a(Hl> H 2 ) -+ a(HomR,(Fh M I), Homn,(F 2 , M2))-4 !~

!r

0-+ HomR(N, M)-+

-4 a(Homn,{Fl> M I), Homn,(G2 , M 2 )) Ef) a(Homn,(Gl> M I), HomR,(F 2 , M 2 ))

!1°EB-r

O

where T°, TO, ~o denote the appropriate natural homomorphisms. By Lemma 2.7. TO is an isomorphism and therefore TO is injective. Furthermore, if !!~,(Mi) = 0 for i 1,2 then again by Lemma 2.7 i O Ef) TO is injective and therefore TO is an isomorphism, q.e.d.

Lemma 2.9. Let It, 12 be injective graded R I -, resp., R 2-modules. Then !!;:'(a(1t, 12 )) and !!"(a(1I, 12 )) are zero for all n 1. Proof: Since for n 1 !!"(a(/I' 12 ))::::::' !!;:'Tl(a(/I' 12 )) it is sufficient to prove !!;:'(a(/I' 12 )) for n 1. By results of Matlis [1) we obtain that'li is a direct sum of modules of the form I(Ri/'fJi) (Pi) where fJ/ E Proj R j u {mi}, Pi E Z, j = 1,2 and where I(J1) denotes the injective hull of Mi' Since a and !!;:, commute with direct sums we may assume without loss of generality that Ii I(Ri/fJi) (Pi) for j 1, 2. If fJI Uti (or fJ2 = m2) then Supp II = {mIl (Supp 12 = {m2}) and in each case Supp a(Il> 12 ) = {m}. But this implies !!;:'(a(Il' 12 )) 0 for all n 1. If fJI 4= ml and fJ2 =t= lll O. Let x := Xl ® X2 E [R]p. Each element Xi gives rise to an isomorphism

o

Therefore X defines an isomorphism I all n 2:: 0 isomorphisms

I(p) where I

a(I h 12 ), Hence we have for

!!;:'(I) ~ !!;:'(/(p)) "" !!;:'(I) (p). Since Supp !!~(/) ~ {m}, this implies H;:'(I) = 0 for all n

0, q.e.d.

§ 2. Graded modules and Kiinneth formulas

43

It is not true in general that the Segre product of two injective modules is injective. A counter-example is in Chapter V, § 5, Example 5.6.

Now we can prove the following statement: Proposition 2.10. Let Nl> n;:::: 0 we get: -

MI

and N 2, M2 be graded R I -, reap. R 2-module8. Then for all

(i) There are natural R-homomorphi8m8

E8

T":

p+q=n

a(Ext~,(Nl> MIl, Ext'k.(N2, M 2))

-+

Ext~(a(Nl' N 2), a(MI' M 2))

(generalized Ktinneth relations). (ii) There are natural R-isomorphi8m8

,,": E8 p+q=n

a(!!p(M I),Hq(M2))-+!!"(a(Ml>M2))

(Ktinneth relations). Proof: We first prove (i): For i = 1,2 let I; be injective resolutions of Mi and K; := Homn,(N i , IJ Since a is an exact functor we have by Cartan-Eilenberg [1], Chap. IV, Theorem 7.2 (see also Chap. IV, Prop. 6.1):

E8

a(Ext~,(Nl>MI),Ext'k.(N2,M2)) =

p+q=n

E8

a(HP(K~),Hq(K~)l~ H"(a(K~,K~))

p+q=n

where HI denotes the jth cohomology of the underlying complex. By Lemma 2.6 we have a natural homomorphism of complexes

Now a(I~, I;) is a (not necessary injective) resolution of a(Ml> M 2) (see, for example Cartan-Eilenberg [1], Chap. IV, Theorem 7.2) and therefore we have natural homomorphisms H"(a(K~, K;)) -+H"(Homn(a(N I,N2 ), a(I~, I;)))-+Ext~(a(Nl> N 2), a(Ml> M 2)).

Putting together everything we obtain (i). Now we prove (ii). There are an integer p and for i = 1,2 mi-primary ideals qi which are generated by their homogeneous elements of degree p. For example, take the ideals generated by suitable powers of the basis elements of m I and m 2, resp. Then q: will be generated by homogeneous elements of degree pt for i = 1,2. Now we set in (i) Ni = q: for i = 1,2, t;:::: 1 and take the (direct) limit over all t. Since a(q~, q~) = (a(qI' q2))1 we get natural homomorphisms

,,": E8

a(!!P(M I), !!q(M2 ))

p+q=n

-+

!!"(a(M I, M 2)).

To prove (ii) it is sufficient to show that ,,0 is an isomorphism for any modules Ml> M2 and that ,," is an isomorphism for all n > 1 whenever M I, M2 are injective (see CartanEilenberg [1], Chap. V, Prop. 4.4). But the last statement is true by Lemma 2.9 (both sides are zero). Therefore we have only to verify that ,,0 is an isomorphism.

44

O. Some foundations of commutative and homological algebra Assume first that H~Pli)

=0

for i

1,2. Then by Lemma 2.8 we have for each

t'? 1 isomorphisms (q:= a(qb q2)}: a(Homn,(qL Mil, Homn.(q~, M 2 )} -+ Homn(ql, a(Mb

Mi»)'

Therefore in this case U O is an isomorphism. M;/!l~,(M;) for z· Let now lVI' M2 be arbitrary modules. We put Mi Then !l~(Mi) = 0 f~r i = 1,2 and since !l°(!l~,(Mi)} = 0 (note that Supp !l~,(Mi)

1,2.

{m,})

we obtain isomorphisms

!l°(Mi) ~ !l°(Mi)

for i

1,2.

Let;' denote the natural projection a(Mb M 2 ) commutative diagram:

-'>-

a(M~, M~).

Then we get the following

a(!l°(M1 ), !l°(M2 )} ~ !1°(a(M}, M 2»)

V

tirO)

a(HO(M~), HO(M~») ~ HO(a(M~, ~V~») We also have an exact sequence (see Cartan-Eilenberg [1], Chap. IV, Prop. 4.3(c»:

a(!1~(.iWI)' M 2 }

e a("Vv H~.(M2») -+ a(M}, M

2)

~ a(M~, M~)

Therefore Supp Ker 1 ~ {m}, i.e. H'(Ker 1) = 0 for all morphism, i.e. UO is an isomorphism, q.e.d.

-+ O.

O. Hence HO(l) is an iso-

Using the natural maps

ifk,: Ext'R,(m., M i )

-+ !18 (M i)

for i

1,2

and we get Corollary 2.11. Let M 1, M2 be graded R I " resp. R 2,moaules. Then there are for each 0 ccnnmutative diagrams n

e O'(Ext~,(m!> M

p+q~

I ),

Ext~.(m2' M 2 )} -+ ExtMm, O'(MI' M 2 )}

..

tIl

...1

p-tq=n

a(ipP

e a(!1"(M}), Hq(M ») 2

p+q~

~q

Ml' IW •

_" 1

)

~a(M"MtI

...

~ HfI(a(Mb M i )}

..

Finally we prove Corollary 2.12. Let Mb M2 be Noetherian graded R1"resp. R 2,moaules. Then

depth a(M!> M 2 ) '? inf (depth M I , depth Mil.

§ 3. Local duality

Proof: Using the canonical maps Mi --+ HO(M i ) (i tative diagram with exact bottom row:

45

1,2) and ,,0 we obtain a commu-

a(M!> M 2 ).!...t. a(HO(M 1 ), HO(M 2 ))

t"·

V

0--+ H~(a(1!fu 1112 )) --+ a(M 1> M 2 ) --+ HO(a(Ml' M 2 )) --+ H'm(a(Ml> M 2 )) --+ O.

If depth Ml 0 or depth M2 0 there is nothing to prove. Hence we assume depth Mi ~ 1 for i 1,2. Then p, is injective and therefore H~(a(M1' M 2 )) 0, i.e. depth a(M}, M 2 ) 1. Therefore if depth "'!f1 = 1ordepthM2 1 the conclusion follows. Assume now depth Mi > 2 for i 1,2. Then p, is even an isomorphism since M, =+ HO(M i) for i 1,2. Therefore l!~(a(Ml> "'!f2 )) = l!.k(a(M 1, M 2 ») = 0 and we get:

1 + inf{n E N I It''(a(M}, M 2 ))

depth a(M!> M 2 )

1 + inf{n E N

=

I

O}

a(HP(M 1 ), l!.Q(M2 )) =1= O}

p+q"~ll

~ 1 +inf{inf{n EN ll!.tI(M}) =l=0}, inf{rnE N ll!.m(M2 ) =l=0}}

inf{depth Mb depth M 2 }, q.e.d.

§ 3.

Local duality

In this paragraph we shall review further basic facts on homological algebra, In particular we shall introduce the notion of the dualizing complex which is a very useful concept of homological algebra. For our purposes here we shall need the dualizing complex to give a homological description of Buchsbaum modules. First of all we need to define some terminology. By a complex we shall understand a complex of modules over a fixed (commutative and ~oetherian) ring A. Let X': ... --+ X" --+

XfHl

--+ ...

denote a complex. We write X" for its nth cochain module and 8": X" --+ nth differential. If X' and Y' are complexes we define complexes Hom'(X', Y')

resp.

X'

Y'

given by Hom"(X', Y')

= [J Hom(Xi,

YH,,)

iEZ

with differential 8"(/)

= 8f -

(-1)" f8,

resp,

with differential o"(x' n E

yi) = ol(X') @ yi + (_1)i xi @ 8i (yl) ,

Z.

Recall that Xi @ yi --+ (-1)# yi X'@Y'=+Y'

X'.

Xi induces an isomorphism

X"~l

for its

46

O. Some foundations of commutative and homological algebra

Also, for n E Z

Y"[n] denotes the complex whose ith cochain module is yH" and whose ith differential is given by (-1)" 81+", that is Y"[ n) denotes the complex Y" shifted n places to the left. A morphism of complexes f' : X' -?> Y" is called a qua.si-isomorphism if the induced homomorphism on the cohomology

is an isomorphism for all i E Z. A complex X' is called bounded below (resp. bounded abo've, resp. bounded) if X" = 0 for all n ~ 0 (resp. n }> 0, resp. n 0 and n }> 0). All results which we list in the following are well-known. For proofs we refer to R. Hartshorne's lecture notes [2), R. y, Sharp's more elementary exposition in [1, 3,4), or B. Iversen's preprint [I), Let E' be a complex of injective modules which is bounded below. Then for any quasi-isomorphism j' : X' -?> Y" Hom'(f', 1): Hom'(Y", E')

-?>

Hom'(X', E')

is a quasi-isomorphism, This may be reformulated as follows: Let j': X' -?- Y" be a quasi-isomorphism and g': X' -'>- E' a morphism into a bounded below complex of injective modules, Then there exists h': Y" -'>- E' such that the following diagram is homotopy commutative

,.

Moreover, h' is unique up to homotopy, From this it follows that any quasi-isomorphism between bounded below complexes of injective modules is a homotopy equivalence. For any bounded below complex Z' there exists a bounded below complex of injective modules E' and a quasi-isomorphism f': Z' -?- E', Furthermore, E' may be chosen as a minimal injective complex, i.e. Ker(8f1 )

-?-

E"

is an essential extension. The above-mentioned results have the following "dual" form. Let P' be a bounded above complex of projective modules. A quasi-isomorphism X' -?- Y" induces a quasiisomorphism

A quasi-isomorphism between bounded above complexes of projective modules is a homotopy equivalence, For bounded above complexes X' there exists a quasi-isomorphism P' -'>- X' where P' is a bounded above complex of projective modules. In the case X' has finitely generated cohomology, P' can be chosen as a bounded above complex

§ 3, Local duality

47

of finitely generated projective modules, I..et F' be a bounded above complex of flat X- --+ yo of bounded above complexes, modules, Then for any quasi-isomorphism

r:

r ® 1: X' @ F' --+ yo ® F' is a quasi-isomorphism, Here we are mainly interested in complexes X' such that their cohomology modules H'(X'), ~'E Z, are finitely generated A-modules. Regarding this we remark that if X' is a bounded above complex and E' is a bounded below complex Qf injective modules, and assuming that both complexes have finitely generated cohomology modules, then Hom'(X', E") has finitely generated cohomology modules, too, Given a complex E" of injective A-modules and an A-module M, we set for i E Z Exti(M, E')

H'(Hom(M, E')).

I..et X', E' be complexes of modules over a fixed ring A. For n E Z consider the map X" --+ Hom"(Hom'(X', E'), E")

which assigns to x" E X" the element (It)'e% E Homtt(Hom'(X', E'), E") where It:

n Hom(XI, EH/)

= Hom'(X', E') --+ E"H

j€%

is defined by li( (gj)jE%)

(-1)i"g,,(x,,)

for all (gj)j€%

En Hom(Xi, EHi), jE%

In fact, this defines a map of complexes e: X' --+ Hom'(Hom'(X', E'), E') which is called the evaluation map. In particular, if X' is the complex Xi and XO = A we get a map

=

0 for i::f: 0

e: A --+ Hom'(E', E').

Now we shall state the definition of a dualizing complex.

Definition and Theorem 3.1. Let D' be arbounded complex 0'1 injective modules with linitely generated cohmrwlogy modules, Then the lollowing conditions are equivalent: (i) The evaluation map X- --+ Hom'(Hom'(X', D'), IT) is a qua8i-isomorphism lor any bonnded complex X' with I~"nitely generated cohomology. (ii) The canonir-al homomorphism A --+ Hom'(D', D') is a qua8i-i80m0rphism, If D' satisfies one of these equivalent conditions we call it a dualizing complex for A.

In particular it follows that if A has a finite injective resolution E" then E" is a dualizing complex, This means that if A is a Gorenstein ring then it possesses a dualizing complex since A has finite injective dimension,

48

O. Some foundations of commutative and homological algebra

For the proof of this result see Hartshorne [2J, Chap. V, § 2, or Sharp [3J. Let A be a ring and D' a dualizing complex for A. For any ideal a in A we have that Hom~(Ala, D') is a dualizing complex for Ala. Let A be a local ring which is a quotient of a local Gorenstcin ring. Then A possesses a dualizing complex. By the theorem of Cohen we know that any complete local ring is a quotient of a regular ring. Hence it follows that any complete local ring admits a dualizing complex. On the other hand there are local rings which don't have a dualizing complex, compare Hartshorne [2], Chap. V, Proposition 10.l. Let A be a local ring with maximal ideal m. If D' and D are dualizing complexes then there exists an n E Z such that D' and D'"[n] are homotopy equivalent, see Hartshorne [2J, Chap. V, Theorem 3.1, or Sharp [3]. Since k = Aim itself is a dualizing complex there exists an integer d E Z such that t

•

Hom'(k, D')

= k[ -d)

for the local ring A admitting a dualizing complex D'. We call D' rwrmalized if the integer d = 0, Since the translate of a dualizing complex is again a dualizing complex we can normalize by translation. In the sequel a dualizing complex is assumed to be normalized,

Proposition 3.2. Let A be a ring admitting a dualizing complex D', (a) For any prime ideal.).) the locah"zation A~ has a dualizing complex, (b) Let D~~ denote the normalized dualiZing complex

0/ AI>' Then there

'18 a homotopy

equivalence D' ®A AI'

-l>-

D~I>[dimAI.).)].

Proof: Let X' be a bounded above complex and E' a bounded below complex of in-

jective modules and suppose that both complexes have finitely generated cohomology modules. Then the canonical map

is a quasi-isomorphism, Furthermore, D' AI> is a complex of injective A~-modules whose cohomology modules are finitely generat~d over AI>' The quasi-isomorphism (*) and the definition of the dualizing complex imply that D' C8JA AI> is a dualizing complex for AI>' Hence we have proved (a). Since D' C8JA AI> is a dualizing complex there exists an integer s E Z such that Hom~~(k(.).»), D'

®A A I1 ) = k(.).») [s]

where k(.).») = AIJ/.).)A~. The assertion (b) follows if we can show that s = dim AI.).), Our last condition gives . ExtA~(k(.).»), D' C8JA AI') 4=

o.

By slightly modif.ying results of Bass [1] to the case of a bounded below complex of injective modules having finitely generated cohomology we get ExtA8+tllmAflJ (k', D')

4= O.

Since D' is a dualizing complex of A we have Hom'(k, D') -s + dim AI.).) 0, This concludes the proof, q.e.d,

k and we therefore obtain

S 3. Local duality

49

Corollary 3.3. Swppose that A has a dualizing complex, Then A admit8 a dualizing complex D' with Di =

E9

E(A/'p).

IlESpecA dimAill=-i

Proof: We take a minimal injective complex which is dualizing and normalize it. Since D' A Il [ -dim A/'p] is homotopy equivalent to the normalized dualizing complex of Ap it suffices to show for a local ring A with maximal ideal m the following: J)O = E(A/m) and E(A/m) does not occur in Di for i =l= O. Since D' is chosen to be minimal Hi(Hom'(k, DO))

Hom(k, Di)

follows. Hence Hom(k, Di) = k if i = 0 and 0 otherwise. Since Di is isomorphic to a direct sum of E(A/'p), 'p E Spec A, we obtain the required result, q.e.d.

In the sequel we will assume a dualizing complex D' of this form. For a bounded below complex X' we denote by Rrm(X') the complex obtained as ' follows: Choose a quasi-isomorphism X' -i>' E' where E" is a bounded below complex of injective modules, and let

This complex is unique up to homotopy and we define

In particular, for a module M, considered as a complex, we recover the local cohomology modules considered in § 1. Now we state and prove the main result of this paragraph. Local Duality Theorem 3.4. Let A be a local ring with maximal ideal m, Swppose that A has a dualizing complex D'. Let X' be a bounded below complex wz~h ftnuely genemted cohomology modules, Then there exists a qua8i-ismnorphism Rrm(x')

where E

=

-i>'

Hom(Hom'(X', D'),

E),

E(k) denotes the injective h1tll of the residue field k.

Proof: First we remark that for any bounded above complex L' of finitely generated free modules and any bounded below eomplex of injective modules E" we have an isomorphism of complexes

H we now start with a bounded above complex Y' with finitely generated cohomology modules we can choose a quasi-isomorphism L' -i>' Y' where L' is as above. This induces. a quasi-isomorphism Hom' ( Y', E')

-i>'

Hom'(L', E'),

We observe that the complex on the right consists of injective modules whence

4 Buchsbaum Riugs

50

0, Some foun~ations of commutative and homological algebra

Note also that the quasi-isomorphism L' Hom'( Y', rm(E'»)

-)0-

-)0-

Y' induces a quasi-isomorphism

Hom'(L', rm(E',)

by using that rm(E') consists of injective modules. Putting this together we obtain a quasi-isomorphism Rrm(Hom'(Y', E'»)

-)0-

Hom'( Y', rm(E'») ,

For the normalized dualizing complex D' we obtain rm(D') and therefore we have Rrm(Hom'( Y', D'»)

-)0-

=

E (see R-B. Foxby [1])

Hom'(Y', E).

If now X' is a bounded below complex with finitely generated cohomology modules we get for Y' Hom'(X', D') Rrm(Hom'(Hom'(X', D'), D'»)

-)0-

Hom(Hom'(X', D'), E).

By using the quasi-isomorphism e: X'

-)0-

Hom'(Hom'(X', D'), D')

we have proven the statement, q.e.d. The particular case of a local Gorenstein ring is of some interest since in this situation the local duality is of quite a simple form.

Corollary 3.5. Let A be a, fa,ctor of the local Gorenstein ring B with dim B ha,ve for a,ll i E Z na,tura,l isomorphisms H~PI1.) '" HomA(Ext~-i(M, B),

: n, Then we

E),

where E denotes the injective hull of A/rnA (consider M a,s a, B-module a,nd Ext~-i(M, B) as an A-module), Proof: Since B is a Gorenstein ring the minimal injective resolution E' of B is a dualizing complex. Hence ' B[ -n]

-)0-

E'[ -n]

is a normalized dualizing complex. Then D' Taking the cohomology in Rrm..(M)

-)0-

HomB(A, E') is a dualizing complex of A.

HomA(Hom~(M, D'), E)

and taking into account that Ext~-i(M, B) '" H-i(Hom~(M, E'[ -nD)""'" H-i(Hom~(M, D'[ -11,]))

we get the corollary, q,e,d. Let M be a finitely generated A-module. Then we obtain by the Local DU!1lity Theorem H!(Hom'(1lf, D'») 9= 0 if -dimA1lf i, and

§ 3, Local duality

This follows since i of M, We call

51

" is the highest non-vanishing local cohomology module

= dim" M

the allwnical module of the A-module M, Next we examine the complex Hom'(llf, D'). Since Di

=

e

E(A/p)

for 0

i

dim A

Il€Spec" dim"/Il=-i

we have

e

Homi(M, D')

Hom,,(M, E(AM),

I1€SPCC"
Furthermore, we have Ass HomA(M, E(A/.»))

Supp M n {.»},

From this we obtain that Hom'(M, D') is a bounded complex with finitely generated cohomology modules such that Homi(M, D')

0

for i> 0 and i < -dimAM,

Therefore there is an injection of complexes

where we regard KM as a complex concentrated in degree zero, Next we recall some basic facts about KM and the cohomology modules Ext~(M, D') for -dimAM < 0, For this we introduce the following notation: Let Z be a subset of Spec A, Then we denote by Zi> i E Z, the set of primes in Z of dimension i, We collect some useful data which we need later, Proposition 3.6. Let M be a finitely generated A-module, Then we get the following properties: (a) AssAKM = {AssAM)d' where d dimAM,

0 z' dimA.U, (b) (ASSAM); = (Ass"Ext"A'(M,D'))i' (c) dimAExt7(M,D')S:i, OS:i inf{2, dim"KM ). Proof: The assertions (a) and (c) follow immediately from the definition of the dualizing complex D', Next we shall prove (b), l!"or this let p E (Ass"MJi, It follows that

.»AiJ E ASSAil Mil

and

Hg"p(M v )

The Local Duality Theorem for AI> hnpJies HO(Hom"IlUlfll' D~Il)) =\= 0, 4*

0,

52

o.

Some foundations of commutative and homological algebra

Since HomA;.(M;.,

D~;.) ~

(HOlllA(M, D') [-dim A/V]) @A A;. we get

H-dlmAII1(HomA(M, D')) ®A A;.

*' 0,

that is V E Supp ExtAdlmA/;,pl, D'). From (c) we get the inclusion (AssAM)j

(ASSAExt'Ai(M, D'));.

The other inclusion can be proved similarly, It follows by (a) that dilllAM = dimAK3[ and that depthAKJ[?: 1. Without loss of generalit,y we can assume the existence of an ,lll-regular element x which is therefore also K,l["regular. The short exact sequence O--l>M":""'M --l>M(xM -,..0

induces an exact sequence

o --l>K

M ":"'" KM

KM/%M --l> •••

by applying the functor Hom( ,D'). Therefore KM/xKM is isomorphic to a submodule 1 we obtain the property (d), q.e.d. of K M /zM • Since K MlzM has depth Corollary 3.7. Let A be a local complete n'ng. Let ill be a Noetherian A-module. Then we have

(ASSA1'1l)j

for i

(AssAHomA(H:"(M), E))i

0, ... , dim M - 1.

Proof: Using local duality we get the corollary from Proposition 3.6(b), q.e.d.

§ 4.

Resolutions a.nd duality

Let R denote a Noetherian graded k-algebra (k a field) with maximal ideal m, i.e. we have [R]j = 0 for all i < 0, [R]o = k and m = EB [R]i (althought many ofthe following ;21

facts remain true in more general situations). For the notation compare § 2. Our aim is to recall basic facts and some applications regarding the following topics: - free resolutions of (Koetherian) graded R-modules; - duality similar to the local duality studied in the previous section. For the proof of the graded version of the Local Duality Theorem 4.14 we use the methods of R. Y. Sharp [2]. This may be considered as an alternative for proving these results without using dualizing complexes.

1.

Graded modules of finite length

We shall often be concerned with graded R-modules of finite length. Recall that a module ill has finite length if it has a finite composition series; then the length lR(M) is the common length of all composition series of M. (If M is a graded module then for any composition series there is a composition series of the same length consisting of

§ 4. Resolutions and duality

53

graded submodules of M.} For graded R-modules we have the following characterizations of finite length:

Lemma 4.1. Let M be a graded ii-module. Then the following conditions are equivalent: (i) M has finite length. (ii) M isfinite-dirnensional as a k-vector space. (iii) J.lf is Noetherian and m"M = 0 for some n. (iv) M ~8 Noetherian and Ass M <::;;;; {mi. (v) Jf is Noetherian and Supp M {mi. (vi) .ill is Noethenan and Supp M rftJniains no homogeneous primes other than m. 0 for all .p E Proj R. (vii) M is Noetherian and M(Pl We note that modules of finite length represent a somewhat pathological case, for if M has finite length then .IW = 0 on Proj R. This is beeause the stalks of j1f are the .W(PP .p =1= m, which are zero by (vii) of I~emma 4.1. 2.

.Minimal sets of generators

Let M be a Noetherian graded R-module. Then M := Mlm· J.lf is a finite-dimensional vector space over the residue field k::::: Rim. A set {m l , ••• , mil of elements of M is called a minimal set of generators for M if the residues mH ••• , mt in M form a vector space basis of M. The graded version of Nakayama's Lemma guarantees that a minimal set of generators is in faet a set of generators for M as an R-module. We have the following basic lemma:

Lemma 4.2. Let.ill be a Noetherian graded R-module. Then every set of generators for M as an R-module contains a mimmal set 01 generators lor M; the number 01 elements in (J minimal set of generators is uniquely determined; it is possible to choose a set 01 homogeneous generators lor M which is a minimal set 01 generators I~r M, and the degrees 01 such minimal set 01 generators are ·uniquely determined by .ill (up to order). If R is a graded or a local ring and M a Noetherian (graded) R-module then we denote by fhR(M} the number of elements of a minimal set of generators of M. (We note that fhR(m) is called the embedding dimension of R, denoted by emb R, if R is local with maximal ideal m.) Using Lemma 2.1 we obtain:

Lemma 4.3. Let M be a Noetherian graded R-module and .p =1= m a komogenecnls 'prime ideal of R. Then fhR(PIU1f(Pl)

fhRj),A(Mp,h)

=

fhRp(Mp)

fhR(J.ll).

This will allow us to show that a projective subscheme is a local complete intersection, or a generic complete intersection, by checking ordinary loealizations instead of degree 0 localizations.

3.

.Minimal free re8olutions

Let M be a graded R-module. Then there is a Bet B of homogeneous elements of M generating M (as anR-module). Therefore we can define an epimorphism 11': R( -deg b) -r M

54

O. Some foundations of commutative and homological algebra

sending 1 E R( - deg b) to b. The module

EEl R( -deg b)

is even a free graded R- module

VEB

and Ker rp is again a graded R-module. So we can repeat this procedure using Ker rp instead of M and obtain an exact sequence (with degree zero homomorphisms) -)0- ••• -)0-

F0

-)0-

ill

0,

-)0-

where all F i , 0, are free graded R-modules. This exact sequence is called a free re8olution of M. "Ve note that we work in the category of graded R-modules with degree zero homomorphisms as Illorphisms. This is a complete (and cocomplete) Grothendieck category, (d. Schubert [1], Def. 14.6.1.). Direct and inverse limits are formed "degreewise", i.e. in the category of k-vector spaces. This category possesses enough projectives (as we have seen before) and therefore enough injectives. This fact we will use in Section 4 of this paragraph. For any graded R-module M we have projective and injective resolutions (even free resolutions). Therefore we define the project~'ve and the znjective dimension of M: pdR M := inf{n E N I there is a projective resolution

o inj dimR M

-)0-

pn

-)0- . . . -)0-

Po

-)0-

Jf -)0- 0 of M}

inf{n E N I there is an injective resolution

o

-)0-

M

-)0-

10

-)0- ••• -)0-

In -)0- 0 of M}.

Assume now that M is a Noetherian graded R-module and let (ml' ... , m/} be a minimal set of homogeneous generators for 11f. Let d i := deg mj for i t, ... , t. Using I

the above we obtain an epimorphism rpo:

EEl R( -d;}

I

-)0-

M, where F 0

i=l

EEl R( -d i ) i=l

is a finitely generated free graded R-module. Hence Ker rpo is again finitely generated, i.e. Noetherian . .\:Ioreover, by the Lemma of Nakayama, Ker rpo m· Fo. Therefore there is a graded free resolution

of M such that all F;, i'?:. 0, are finitely generated free graded R-modules and 1m rpi+l = Ker rpi <;;;; m . Pi for all O. Such a resolution we call a minimal graded free resolution of M. Using standard techniques (first of all Nakayama's Lemma) we see that two minimal free resolutions F, F' of M (M finitely generated) are isomorphic, i.e. there are for all isomorphisms 1J1i: F i -)0- P~ such that rpi1J1; 1J1i-llPj forI all i'?:. 1 and 97~'IJ!0 = 970' Especially, any two minimal free resolutions of Jl have the same number of terms, thus pdR M may be calculated by computing some minimal free resolution. We recall that by the famous Syzygy Theorem of Hilbert pdR M is always finite if R is a polynomial ring over k.

°

Using the above remarks we get the following well-known result: Lemma 4.4. (Criterion of minimaIity) Let M be a finitely generated graded R-module and let

G: ...

-)0-

Gi ~ Gi _ 1

•••

~

Go ~ 111

-)0-

0

§ 4. Resolutions and duality

55

be a graded free resolution of M (not necessarily minimal) where all Gil 0, are finitely 0, and matrices representing Pi, Z' 1. generated. Assume we are given bases of Gj, i Then G is minimal it and only if all the maJrix entries lie in m. Corollary 4.5. Let M be a Noetherian graded R-module with p Extk(M, R) 0 for all i > P and Ext~(M, R) =F O. Proof: Let

F: 0

-'>-

Fp ~ F p_1 -'>-

~ Fo ~ M

...

-'>-

0

be a minimal free resolution of M. Let Gt:= HomR(F;, R) and 'ljJi:= HomR(tpi+l' il): Gt -'>- GHl for all O. Then Extk(M, R) is the ith cohomology of the complex

G: 0

-'>-

GO~

G11

QI-,>- •••

-'>-

0,

i.e. Extk(M, R) 0 for all i > p. Since 1m P11 ~ m· F p _ 1 , 1m 1pp-l ~ m· Gp. But Gp =F 0 and we get an epimorphism Ext~(M,

R) cy Gp/lm 'ljJpl

~

Gp/m· Gp =F 0,

by the Lemma of Nakayama, q.e.d. We now want to describe an application of Proposition 2.3 for curves in P~. Definition 4.6. By a curve C c PZ we mean a closed subscheme of P~, given by the homogeneous ideal l(C) of S := k[XQ' " ' J X a], satisfying any of the following equivalent conditions: (i) C is a one-dimensional scheme, (ii) Sll(C) is a two-dimensional graded ring, (iii) l(C) has height 2. Furthermore, we will assume that C is locally Cohen-Macaulay and equidimensional.

Pro

Let C be a curve in Let A be the local ring of the vertex of the affine cone over C with maximal ideal m; that is A (Sjl(C)}n' where n (Xo, ... , Xa) S denotell the maximal homogeneous ideal of S. Then we get the following corollary: Corollary 4.7. Let C be a curve in P~. The following c(mditions are equivalent: (i) The local cohomology module lI1n(A) is annihilated by m.

(ii) The gruded S-module

ED Hl(p~, 1(0) (m)) is annihilated by n. m€Z

Proof: We have by Proposition 2.3 (and the isomorphisms given there):

ED Hl(pZ, l(C) (m)) '" !!~(l(C)). mEZ

It follows immediately from the exact sequence 0

~

l(C) ~ S ~ S/l(C) -+ 0 that

~(l(C)) '" lIA(S/l(C)).

Let M be a graded S-module with Supp ~lf ~ In}. Then for any mE ]I{ there is an n with n"m O. Hence we obtain an isomorphism between (non graded) S-modules:

56

O. Some foundations of commutative and homological algebra

J.lf =:::: Mn. Therefore we have: . H:n(A) '" (!t.MS/I(O)))n '" !t.MS/I(O)) =:::: E8

Hl(p:, 1(0) (m))

meZ

and this proves our Corollary 4.7. q.e.d. Using the notation from Chapter III we therefore have given a new proof of the following corollary (see our Theorem III.1.2 and the rlOte in the preface):

C.orollary 4.8. The arithmetical Buchsbaum property for curves in

P2 is preserved under

liaison. Proof: The property of Corollary 4.7(ii) is preserved by shifting degrees and dualization. Therefore we get our assertion from our observations on liaison among curves in pi of Chapter III, Proposition I.2.12 and Corollary 4.7, q.e.d. 4.

Dualization

If M is a graded R-module, the k-vector space J.lfv

E8 Homk([llf]_n, k) is in a natural neZ

way a graded R-module with [MO]. := HOID/c([M]_n, k). We say that M" is the dual (or sometimes the k-d·ual) of M. If f: ilf ~N is an R-homoIDorphisID (of degree zero) of graded R-modules, then f defines an R-homomorphism f": N" ~ M". The functor llf H> M", f H> f" is an exact contravariant functor from the category of graded RIDodules to itself. We have:

Lemma 4.9. Let M, N be graded R-modules. Then (i) In the subcategory of all graded R-modules with rankk["lf]"

< 00 for all n E Z (especially, in the 8ubcategory of Noetherian or Art~'nian graded R-modules) the duality is perfect: There is a canonical isomorphism M"" M. (ii) There i8 a natural isomorphism of graded R-modules )'M.N:

HomR(M, N') ~ (M ®n N)·.

Proof: (i) and the construction of ;'M.N in (ii) follows from standard techniques of linear algebra. (For p E Z the k-vector space [M ®n N]p has a basis of elements of the form m (8) n with m E [M);, n E [N)j) ~. + i p and therefore we can define for all p E Z / k-linear maps

Ap: HomR{M, NV(p)) ~ [(ilf

N)v)p = Homk([J.lf ®R N)_p, k)

in the usual manner. Some straightforeward calculations sbow that

)'A/,N

E8 }'p PEZ

is a natural R-bomomorphism.) It is easy to see that A"t.N is an isomorphism whenever )'11 is a graded free R-module, i.e, if M ~ E8 R(ni) where I is a set (of indices) and

n, E Z. Then consider an exact sequence F

iff

~

G ~ J.lf ~ 0, where F, G are free graded R-modules, This gives rise to a (;ommutative diagram with exact rows

o ~ HomR(llf, NV) ~ Homn(G, N°) ~ Homn(F, N°) Sin(;e

).G,lV, ).r,lV

are isomorphisms,

;'M,lV

is an isomorphism too, q.e.d.

§ 4. Resolutions and duality

57

Corollary 4.10. R" is an injedive objed in the category of g·raded R-modules. More precisely, R" is the £njective hull of ~ (or better, of ~V ~ ~). Proof: By Lemma 4.9(ii) we have for all graded R-modules M: HomR(M, R") ~ M", i.e. the functor HomR( ,R") is exact and therefore R" is injective. The epimorphism R ~ ~ gives rise to a monomorphism !;V ~ R" and it is easy to

eheck that this is an essential extension, q.e.d. R-module.~ M, N and all i morphisms (of connected sequences of derived functors):

Corollary 4.11. For all graded

0 there are natural iso-

ExtkOlf, N°) ~ Torf(M, N)". Proof: Note that {Torf(M, N)"liEN are the right derived functors of the left exact contravariant functor ( ®R N)", N fixed. For i = 0 we take the isomorphism of Lemma 4.9(ii). Since Torf(M, N)' = 0, Extk(M, N°) 0 for all i> 0 whenever M is projective, the result follows from Cartan-Eilenberg [1], Chap. III, Prop. 5.2,

and Chap. V, Prop. 4.4, q.e.d. The local version of the following very useful lemma can be found in R. Y. Sharp [2], Lemma 3.2:

Lemma 4.12. Let T be a right exact, covariant additive R-linear functor from the category of graded R-modules to itself which respects shifts of degrees. Then there is for every graded R-module M a natural homomorphism ;,v: M which

~8

®n T(R)

~

T(M)

an isomorphism if M is f%ilitely generated. If T commutes with dired i£mits,

;M is always an isomorphism. Proof: Let M be a graded R-module. For any homogeneous element m E M let h m : R ~ M(deg m) denote the homomorphism defined by hm(r) := rm for all r E R. Then T(h m }: T(R) ~ T(M(deg m») = T(M) (deg m) is an R-homomorphism. Define ;M: M ®R T(R) ~ T(M) by ;M(m e) := T(h m) (e) for all m E [MJi> f! E [T(R)]J' It is

easy to verify that ;M is a natural R-homomorphism.

Now;M is an isomorphism if M R(p) with p E Z. Therefore ;.u is an isomorphism if M is a finitely generated free graded R-module, since T commutes with finite direct sums. If T commutes with direct limits, ;M iF; an isomorphism if M is an arbitrary free graded R-module, since tensor products also commute with direct limits. Now we can take an exact sequence F h G.f!....t, M ~ 0, where F, G are (finitel~' generated) free graded R-modules. Then we obtain a commutative diagram with exact rows T(R} g0id • M

F

l'(F}

Since

~F' ~(;

~'G T(G)

T(R}

~O

~ '-" T(M}

~O.

are isomorphisms, ;.u is an isomorphism by the 5-Lemma, q.e.d.

58

O. Some foundations of commutative and homological algebra.

Using the definitions of Section 3 of this paragraph we define:

Definition 4.13. A graded k-algebra R is called a graded Gorenstein k-algebra, if inj dimR R < 00. The graded versions of the result of H. Bass [1] yield that inj dim R = dim Rand that R is a graded Cohen-Macaulay algebra, i.e. dcpth R dim R. Now we are able to state and prove the following. Theorem 4.14. (Duality Theorem). (i) 8uppose that R i8 a graded Gorenstein k-algebra of dimension n. ]1'or any graded R-module ~W and all i E Z there are natural R-isomorphisms !!..!n(M)"", Ext~-i(M, R) (r

where r

1),

r(R) is the index of regularity of R (d. Def.2.2).

(ii) If R is arbitrary and if 8 i8 a graded Gorenstein k-aigebra of dimension n such that R i8 a factor of 8, then we have for all graded R-module8 M and all i E Z natural

isomorphism8 !!..!n(M)" "-' Ext~""'(M, S) (8 -

1)

where s r(8), Mis con8idered as an 8-module and Ext~-i(M, 8) a8 an R-module. (iii) If under the assumptions of (ii) ~W is Noetherian, we get !!..!n(M)~ Ext;-i(M, S)· (1

8).

Proof: By Corollary 4.11 we get with N Rv (Rv. ~ R by Lemma 4.9(i): ExtiRUtf, R) Torf(M, RO). for all j. To prove (i) we establish the following claim:

~

Claim. Let R be a graded Gorenstein k-algebra with n:= dim R. Then we have R" ~ !!..~(R) (r - 1), where r := r(R). We use induction on n. If n 0, !!..~(R) = R and therefore this is an injective graded e(R) (d. Del. 2.2). R-module. Thus we have to show that R" '" R(r - 1). Now r = 1 Take a E [R]r-l'" (O}. Then m . a 0, i.e. we can find a monomorphism !f --+ R(r - 1). Since R(r - 1) is injective, this induces a monomorphism Rv --+ R(r - 1) (R" is the injective hull of !f). Therefore we obtain for all i E Z:

+

rankk[R"li s:;; rankk[R]r-1+i>

rankk[Rl-i

and replacing i by - r + 1 i we find rankk[Rlr_i-i-l rankk[R(r I)J; for all i E Z and this gives

rankk[Rl_i, i.e. rankk[R"J,

R·~R(r-l).

If n > 0, we take a homogeneous non-zero divisor x E m with d : = deg x. Since R := RjxR is again a graded Gorenstein k-algebra of dimension n 1 with r(R) = r(R) d, we have

+

!!..~-I(R) (r

+d -

The exact sequence 0

--+

0--+ !!..':n-1(R) (r y

1) '"

H':n-:I{R) (r

R( -d) ~ R --+

+d

+d -

1)~

R".

R --+ 0 induces a diagram with exact rows

1) --+ H':n(R) (r -

1)"':-+ H':n(R) (r

+d -

1) --+ 0

!

--+

O.

§ 4. Resolutions and duality

59

Since R" is injective there is a homomorphism f: H':n(R) (1' - 1) --+Rv such that the diagram is commutative. Now f(d): !1':n(R) (I' + d 1) --+ RV(d) makes the whole diagram commutative. Therefore Ker f (Ker f) (d) and· Coker f ~ (Coker f) (dl. Both Ker f and Coker fare Artinian graded R-modules and hence Ker f = ('A)ker j = 0 by these isomorphisms, i.e. f is an isomorphism and our claim has been proven. ~ow

we have with Lemma 4.12

(!1':n( ) is right exact):

(11f®R")"~(J1{®!1':n(R)(r

1))"

(M

!1':n(R))v(1-r):::::!1':n(J1{)V(I-r),

hence !1::'(J1{). ~ HomRC,1{, R) (I'

1).

The right derived functors of the left exact contravariant functor H':n( ). are just the functors !1::'-'( )", i> O. If F is a free graded R-module then !1':n-i (F) = 0 for all i> 0 since F is a direct sum of the Cohen-Macaulay modules R(m), m E Z. If P is a projective graded R-module then there ill a free graded R-module F such that P is a direct summand of F, hence !1~i(Pl 0 for all ~. > O. Since Extk(P, ill = 0 for all i> 0, we find by (,.,artan-Eilenberg [1], Chap. III, Prop. 5.2, and Chap. V, Prop. 4.4, for all i"20 natural isomorphisms !1::,~i(M)· ~ Extk(M, R) (r - 1) which proves (i). To prove (ii), let n denote the maximal homogeneous ideal of 8. Then !1~(iIf)" ~ Exts-i(M, 8) (s - 1) where M is considered as an S-module. But H~(M)V ~ !1:n(M)", considered as S-modules. (iii) follows from (ii) by dualization and Lemma 4.9(i) since H:n(M) is an Artinian graded R-module for all i, q.e.d.

Corollary 4.15. Let M be a Noetherian graded R·module. Then H:nUIf) is of finite length for all ~. =1= dim M if and only if M is locally Cohen-J~facaulay and equidimen.sional, i.e. M(1J) ~8 a Cohen-Macaulay R(1J)-module jar all l' E Proj R and dim 11{ = dim Rjl' for all m~'mmal primes l' E Supp M. Proof. Let 8 be a graded Gorenstein k-algebra suth that R is an epimorphic image of 8. If Mis lotally Cohen-Macaulay then we have for all l' E Proj 8 with M(1J) =1= 0:

o=

Ext~{l.lpfUl» 8(1J))

for all

~.

=1= dim 8(.\1)

(Extk(M,8))(1J) dim M(Il)

dim S - dim M.

Therefore Ext~(M, 8) is of finite length for all i =1= dim 8 - dim M by Lemma 4.1. Thus !1:n(M)" and hence !1:n(M) is of finite length for all ~. =1= dim M by Theorem 4.14. The converse is also true, q.e.d. For another important consequence of Theorem 4.14 we need the following notation: Let M be a Noetherian graded non-free R-module. Then take a finitely generated free graded R-module F and an epimorphism :n:: F --+ 1lf suth that Ker:n: <;;;; m . F (F is up to an isomorphism uniquely determined by M, see our investigations in a.). Then let R)) = Coker Hom R(:n:, R) .

MX is uniquely determined by Jf up to an isomorphism and MX =1= O.

60

O. Some foundations of commutative and homological algebra

Corollary 4.16. Assume that R is a graded Gorenstein k-algebra with '1': r(R) and n:= dim R. Let M 0 be a Noetherian graded R-TlUJdule with dim M = dim R = n such that H:n(M) is of finite length for all i n. Then we have for all i 0, n isomor-

*

*

pMsm8

*

!!.:n(MX) ~ H:!t-i(M)" (1 - '1').

Proof: By Theorem 4.14 we have to showthatH:n(MX) ~ Extk(M, R) for all i 1, ... , n - 1. By our assumption the Ext~(M, R) are modules of finite length for all z' 21 (use Theorem 4.14 and Corollary 4,15), Take a minimal free resolution,., -;.. Fi -;.. F;_l -;.. ..• -;.. Fo -;.. M -;.. 0 of M. Applying the functor HODlR( ,R) we obtain a complex

F': 0 -;.. MX ~ FI!!.:+ F2 -;.. ... , where Fi

I!omR(Fj, R) for all i Hi(F')

{ 0 ExtkPf, R)

1. Then for for

i i

0, 1.

By using for all i > 1 the exact sequences

o -;.. 1m dH

-;..

Ker di

-;.. Ext~(M,

°

R) -;.. 0,

o -;.. Ker d i -;.. Fi -;.. 1m di -;.. !!'~(Fi) = 0 for all i n, !!.~(ExtkPf, R)) =

and foralli=l, ... ,n

1:

Ext~(ilf, R)

*

0 for all

!!.~(Ext~(l\tr, R)) ~ !!.Mlm di -

l

i =!= 0,

1, we obtain

)

"" !!.;(Ker di-I)~ ... ~ Mk(Im dO) ~ Hk(M X), q.e.d. If R is a graded Gorenstein k-algebra and a R is a homogeneous ideal with dim Ria dim R, such that R('f),/a('fJ) is a Cohen-Macaulay ring for all 1:1 E Proj Ria, then we 1, . '" n - 1 have for all i !!.k(RjO :R a) ~ !!.:!t-i(Rla)< (1 - '1'),

since (RjoY RjO :R a. If R = k[Xo, ... , XmJ!(Fl> ... , F s), where X o,.", Xm are variables (of degree 1) and F I, ... , F. are forms such that n:= dim R = m - s + 1 then R is a graded complete intersection, hence a Gorenstein k-algebra with r(R) = -m + deg FI + ... + deg F •. Using again the notation of Chapter III we get a further application of Theorem 4.14:

< m, locally Cohen-Macaulay subscheme which is linked by a complete intersection C to a 8ubscheme W c: Pt'. If C Hln ... nH., s=codimpZC, Hl>_ .. ,H. hypersurfaces of P~ of degree dl> ... ,d, (thus n = m s) we have for all i = 1, ... , n isomorphisms

Corollary 4.17. Let V c: PZ' be a pure n-dimensional, n

EEl H'(P';, P€Z

dy(p)) ~

(EEl Hn-.+I(pZ', dW(P)))" (m + 1 P€Z

where dy, dw denote the corresponding sheafs of ideals.

dl

§ 4. Resolutions and duality

61

Proof: Let S:= k[Xo, ... , X".], n:= (Xo, ... , x".) S, R:= S/(Fl> ... , F q ), m:= (Xo, ... , X".) R, Fi = 0 the equation of Hi for £ 1, ... ,8. Then by Proposition 2.3 and the isomorphisms given there: '

EEl H'(P,!:, 3 v(P)) ~ H~+1(I(V))

!.f.1t(S/I(V))~ H:n(RII(V) R)

l'E%

~ H~-;+l(RII(W)

~

+ 1-

d1

-

(EEl H ..-i+l(P'!:, 3 w(p)))" (m + 1 PE%

q.e.d.

R)" (m

... -

d1

d8 ) ... -

d.),

Chapter I Characterizations of Buchsbaum modules

§ 1.

Characterization of Buchsbaum modules by systems of parameters

We will always denote by A a local ring with (unique) maximal ideal m. First we will recall the problem of D. A. Buchsbaum expresscd in the language of local algebra (see Introduction). To do this let M be a Noetherian A-module and q a parameter ideal of M. We will use the notion of the Hilbert-Samuel polynomial Pq,M and its leading coefficient co(q, M), the multiplicity of q with respect to M (see Chap. 0, § 1, 2.). D. A. Buchsbaum's original questions was as follows: Does there exist a natural number I(M), such that the difference of "length" and "multiplicity" of every parameter ideal q of M is equal to I(M), Le., it is true that

lA(M/qM) - co(q, ilf)

I(lll)

for all parameter ideals q of M?

Already in the Introduction we gave examples which showed that this will not be true in general (see Examples 1 and 5). But we have also seen (see Examples 2-7 of the Introduction) that there are a lot of in this sense "good" examples, see also Chapter V. The main purpose of this paragraph is to give a first characterization of those Amodules M, for which the above qnestion has a positive answer. We start with some definitions and easy lemmas. Definition 1.1. Let III be a Noetherian A-module. A system of elements Xl' ... , X, E m is called a weak M-sequence, if for each i = 1, ... , r (Xl> ••• , Xi-I) •

III : Xi =

(Xl> ••• , Xi-I) •

III : m.

Remark 1.2. For r = 1 we have 0:MX 1 = 0 :Mm

or, equivalently, m· (O:M

Xl)

= 0

(since

Xl

Em).

Consequently, if Xl is a non-zero divisor of M, Xl also forms a weak M-sequence (consisting of one element). We therefore see that the weak M-sequences are a direct generalization of the well-known lW"-sequences (see Chap. 0, § 1, 2.). We know that every ilf-sequence forms a part of a system of parameters of ilf. For weak ilf-sequences we have: Lemma 1.3. Let III be a Noetherian A-module of positive dimension d. Then every wea,k M-sequence Xl> " ' j X, with r S d is a part 0/ a system of parameters 0/ M .•

§ 1. Characterization by systems of parameters

63

Proof: Clearly, X2' "., x, forms a weak M/xlM-sequence. Therefore, by an easy induction argument, it is sufficient to prove the lemma for r = 1. Let x E m be an element 0 :11{ m. Then x q p for all p E Ass M" {m}, particularly, x fi p for with 0:11{ x all p E Supp M with dim Alp = dim M (> 0). But this means dim Mix' M d - 1, Le. x forms a part of a system of parameters of M, q.e.d. d for every Remark 1.4. It was stated in Stiickrad-Vogel (2], Corollary 4, that r weak M-sequence Xl> ••• , XT • This is not the case, however, as was pointed out by Balwant Singh with the following example: Let k be a field and set A k kx with X2 = O. Then x is a weak A-sequence but dim A = O. Also, we note that two maximal weak M-sequences (Le. the number of elements in such a sequence is eqtial to or less then dim M and it is not possible to lengthen it) have not always the same number of elements. To see this, we refer to Proposition 2.1 of this chapter and Example 1.2.5 or Example V.5.4. These examples render modules M fulfilling statement (iii) of Proposition 2.1 which are not Buchsbaum modules. By Proposition 2.1 every system of parameters contained in m 2 is a weak M-sequence. On the other hand it is easy to see that every element x with dim MIx, M = dim M - 1 forms a weak M-sequence, i.e. every weak M-sequence of maximal length consists of at least one element. But not every system of parameters is a weak M-sequence since 111 need not be a Buchsbaum module, see our next definition.

e

Now we are able to define Buchsbaum rings, resp. modules. Definition 1.6. A Noetherian A-module M is called a Buchsbaum module if every system of parameters of M is a weak M-sequence. A is called a Buchsbaum ring if it is a Buchsbaum module as a module over itself. From the Introduction we already have quite a number of examples of Buchsbaum modules. Another easy consequence which enhances our knowledge of Buchsbaum modules is the following statement. Lemma 1.6. Assume that A is an epimorphic i'fTU1{Je of the local ring B. An A-module M is a Buchsbaum rnodule over A if and only if it is a Buchsbaum module considered as a B-module by "restricting scalars". . Proof: Clearly, dimB M dim,!. M and every system of parameters of M in A may be obtained by restricting a system of parameters of M as a B-module. Conversely, the restriction of any system of parameters of M in B to A is again a system of parameters of M as an A-module. Therefore the validity of the statement of our lemma becomes clear by the definition of weak M-sequences, q.e.d. Before continuing we state a seemingly technical yet in the sequel very useful result. To this end we define: Definition 1.7. Let a A be an ideal and M a Noetherian A-module with dim Mfa·.L11 O. Letd:= dim M. A system of elements Xl> ••• , XI of A is called an .LV-basiS of a if the following conditions are fulfilled: (i) Xl,. '" Xt form a minimal basis of n. (ii) For every system il>"" id of integers with 1:::; t'1 < ... < i d ::::- t the elements Xi" ••• , Xi. form a system of parameters of M.

640

I. Characterizatiolll! of Buchsbaum modules

Remark 1.8. 1. Since 0 dim Mia· jJl = dim M/(xl> ... , XI) . M dim M d. 2. If dim M = 0 any minimal basis of a is an M-basis.

it follows that

t

We now prove:

Proposition 1.9. Let a c A be an ideal and M l , ... , Mfl NoetMrian A·modules with dimA M;/aMj = 0 lor all i 1, ... , n.· Then there are a1> ... , al E a lorming an M;-basiB 01 a lor all i 1, ... , n. Prool: Clearly, we. can omit zero-dimensional modules from our collection, i.e., we may assume d j := dim M; > 0 for all i = 1, ... , n. We now prove by induction on m, 0-:;; m -:;; t, t := rank A/m alma, a (bl> ... , bl ) A: There are elements a1> ... , am of a with

(i) a (al> ... , am, bm+l> ... , bl ) A and (ii) for all I = 1, ... , n,all j 0, "', min(m, dl ) and all 1-:;; i l < ... < i j m,ai" ... , ai, is a part of a system of parameters of MI' This is trivialfor m = O•.Let therefore be 0 < m t and assume that aI' ... , am-l (E a) fulfil (i) and (ii). Define

I\)

L :=

E Spec A

I there are 1, j, i1> ••. , i j E N, 1 1-:;; n, 0-:;; j < min(m, dl)' 1 i l < ... < i j <

m

with \) E Supp Mt/(ai" ..• , aj) Mt> dim A/\) {~ ELI bm ~

Ll

If L2

=

q for all q E L with ~ ~ q}

and

=

dl

j},

L 2 := L " L l .

0, set am := bm • If L2 =1= 0, we have (aI' .•. , am-I> bm+1> ... , bl ) Ann ~ Q;; q for

all

q E L2•

j;EL,

Therefore we can find an

x E [(aI' ... , am-I' bm+h

••• ,

bl) Ann ~] " [ U j;EL,

and we define am :=

X

QU.

q]

+ bm • Then aI, ... , am-I> am satisfy (i) and (ii), q.e.d.

Now we state a proposition which enables us to examine the Buchsbaum property in several different ways.

Proposition 1.10. Let M be a Noetherian A-module with d:= dim M properties are equivalent:

>

O. The lollowing

(i) M iB a BucMbaum module, i.e., every system 01 parameters iB a weak M-sequence. (i)' For every system 01 parameters Xl, ••• , Xd 01 M we have (Xl .... ,

Xd-l)' M,:Xd

=

(Xl' ••• ,

(ii) For every system 01 parameters (Xl' ••• ,

such that

Xi)' M:Xi+1

Xl, _. _, Xi,

=

Xl. ___ ,

Xd 01 M and all i M:x

=

0, _.. , d - 1 we have

lor all X Em

X lorm a part 01 a 8Y8tem 01 parameter8 01 M.

Xd-l) . M :Xd =

such that

Xl' ••• ,

(Xl> ___ , Xi) -

(ii)' For every system 01 parameters (Xl> •• _,

Xd-l)' M:m.

Xl> _._,

(Xl" _,

Xd 01 M we have

Xd-l) . M:x

lor all X E m

Xd-l> X lorm again a 8ystem 01 parameter8 01 M.

§ 1. Characterization by systems of parameters

(iii) For every system

0/

111: Xi+l

(Xl' 0", X;) •

(iii)' For every system (Xl' ',., Xd-l) •

(iv) For every part

parameters

0/

=

Xl, "" Xd 0/

(Xl' ... ,

M we have lor all ~. ~ 0, "" d - 1:

xJ 0 jlf: Xf+l '

0/ parameters Xl, 0", Xd 0/ M ilf :Xd =

(Xl' "" Xd-l) •

a sY8tem

0/

U((xv ... , Xi)' M)

=

65

parameters

we have

M: x~, Xl, .,., Xi 0/

M with i

<

d we have

(Xl> "', Xi)' M:m.

Proof: The implications (i) =? (i)', (i) =? (ii), (il' (ii)', (ii) =? (iii), (Hl' =? (iii)', (iii) =? (iii)' are trivial. It remains to prove (iii)' =? (iv) and (iv) =? (il. (iv), To prove (iii)' =? (iv) we show (iii)' =? (iil' and (iiy Let Xl> •• " Xd be a system of parameters, (iii)' implies that (Xl> ... , Xd-l) ' Jf:Xd = (Xl>

"0,

Xd-l)oM:x~

(Xl' .. "

Xd_l)'~H:x~ = .'"

i,e, (Xl' ... , Xd-l)·M:Xd = U((XI' ''', Xd-I),M) since dimM/(xI, ... , X£1-l)' M = 1. Consequently, (Xl' .. ,' X£1-I)' M :X£1 does not depend on Xd and this implies (ii)'. Assume now that (ii)' is satisfied. Let Xl> ... , Xi be any part of a system of parameters of M, i < d. Then we choose Xi+I' •• " Xa such that Xl> Xd form a system of parameters. For all integers n ~ 1 the sequence Xl' ... , Xi> X~+l' ... , Xd-l' Xd is also a system of parameters. By Krull's Intersection Theorem we obtain for all X E m such that Xl> ••• , Xd-I> X is again a system of parameters of M the following: "'j

(Xl' ... , Xi)'

1H:Xd

= (n

(Xl' .. " Xi>

X~+1' .. " Xd_l) , M) :Xd

ft~l

= n ((Xl>

... ,

Xi, X7+1' ... , X;;_l) , M:Xd)

ftZl

=

n ((Xl> "., Xi, X!'+l' ... , X;;_l) , M:x)

(Xl' ... , Xi)' ~H:x.

ft~l

Amongst the elements X we find an (lH/(xt> .. " A (see Proposition 1.9) and this gives (Xl' ... ,

Xi)

,Jf :Xd

=

(Xl> ... , Xi)'

Xd-l) ,

M)-basis ofthe maximal ideal m of

llf:m,

On the other hand it is easy to see that Xd can be chosen in such a way that we have (Xl> ... , Xi) , M :Xd U((XI' , •. , xd' M) and therefore we obtain (iv), Finally, we verify (iv) =? (i). To do this let Xl, ,." Xd be an arbitrary system of parameters for M, Then we have for all i = 0, "" d 1; (Xl' ... ,

xd' M:m ~

= i.e. Xl' .. "

Xd

is a weak

(Xl' .'" Xi)'

M:Xi+l ~ U((XlJ

"'J

x;l' .H)

(xl".·,xj).M:m,

~H-sequence,

q.e.d.

Corollary 1.11. Let lrl be a Buch8ba,um module, Assume XI, "') Xr is a part 0/ a SY8tem 0/ M with r < dim ilf. Then (il M/(XI,'''' x r ) , M and (ii) MfU((xl> "') Ir) 'M) are Buch8baum module8 (0/ dimens70n dim M - r), 1Jarameters

5 Buchsba.um Rings

of

66

1. Characterizations of Buchsbaum modules

Furtherrrwre, the localizaticm8 M),) are Cohen-Macaulay module8 for all pnme ideals

:p =1= m of Supp M. Proof: (i) is a direct consequence of the definition of weak M-sequences.

Since MjU((XI, .•. , xr), M) = (_Mj(XI' ••• , xr)·M)jU(O) (U(O) in Mj(xl> ••. , x,). M) it is sufficient to prove (ii) for r = O. Let Yl' ... , Yi> i < dim MjU(O) = dim M be a part of a system of parameters of MjU(O). Then it is a part of a, systelu of parameters of Mas well and we have by Proposition 1.1O(iv) (U(O)

+ (Yl> ... , y;). M):m

U(U(O)

+ (YI' ..., Yi)' M)

U((Yl> ... , Yi)' M) <;;; (U(O)

(YI' ... , Yi)' M:m

+ (Yl> ... , Yi)' M) :m.

Therefore MjU(O) is a Buchsbaum module by Proposition 1.1O(iv), The last statement follows from Proposition 1.1O(iv), q.e.d. We note that the converse of (i) or (ii) of Corollary 1.11 is not true in general. We will return to this topic later when we discuss the so-called "lifting property", see Proposition 2.19 and Proposition 2.23. Now we come to the main result of this paragraph. It shows the connection between the Buchsbaum modules defined above and the original problem of D. A. Buchsbaum (see Theorem 2 of the Introduction). Theorem 1.12. Let M be a Noether~'an A-module with d:= dim M > O. Then M is a Buch8baum rruxlule if and only if there is an integer J(M) 0 such that

l(Mjq . M) - eo(q, M)

J(M)

for all parameter ideals q of M.

Proof: For every parameter ideal q of M we set c(q, M) := l(Mjq . M) - eo(q, M)

0

(see Lemma O.1.3(ii».

Assume first that there is an integer J(M) such that c(q, M) = J(1lf) for all parameter ideals q of M. Let, Xl> •• " Xd be an arbitrary system of parameters for M, q := (Xl> .• ,' Xti) . A. ]'or i = 1, ... , d we let 1'1f. := (Xl' ... , Xi-l) • .M :Xij(Xl> ... , Xi-l) . M

and ei := e(Xi+1' • ,., XdIMi)'

Then by Lemma O.1.3(vi) c(q, M)

=

Now we replace Xti by

ei'

x~

and we let (1

q':= (Xl' ... , Xd-l, X~) • A,

=

2e;

1)

ei:= e(Xi+l' ... , Xti-l> x~IMi)'

Then by Lemma O.1.3(v) we have for i ei

d-

1, ... , d - 1:

§ 1. Characterization by systems of parameters

67

and this results in a-I

+ 1: e,

l((Xl> •.. , Xd-l)' M :X~/(Xl> . '" Xd-l)' M :Xd)

.=1

i=l

= c(q', M) - c(q, M) = I(M) - I(M)

d-l

d-l

+ 1: e;

l( (Xl' ... , Xd-l) . M: X~/(XI' "., Xd-!) . M: Xa)

l(Md) -

1: ei i=l

O.

Since all terms of the left sum are non-negative integers, l( (Xl' ... , Xd-l) . M :x~/(Xl> ... , Xd-l) . M :Xd) = 0,

i.e. (Xl' ... ,

Xd-l)' M :Xd

=

(Xl' ... ,

Xd-l)' M :x~.

Hence, by Proposition 1.10 (iii)', M is a Buchsbaum module. Now assume that M is a Buchsbaum module. Let Xl' ••• , Xd be a system of parameters for M. If we use the same notations as defined above, we have for i = 1, ... , d: m . Mi

0,

especially,

dim M;

O.

Therefore by Lemma 0.1.4 e(xi+l' ... , xdIM.) = 0

for i

1, ... , d

1

and we have c(q, M)

l(Md)'

We notice that the left part of this equation does not depend on the order of the elements Xl> ... , Xd and, consequently, the right-hand term does not depend on this order. Let now q':= (Yl, •.. , Yd) . A be another parameter ideal of M. We show by induction on d that c(q, M) = c(q', M). If d = 1, c(q, M) l(O:MXI) l(O:Mm) = l(O:MYI) = c(q', M) and we are done. Suppose now that d 2. 'Ve choose an element z E m such that Xl, •• ,' Xd-l, Z and Yt> •.. , Yo-I, z are again systems of parameters with respect to M. Then by Pro~sition 1.10 (ii)' and the induction hypothesis we have with q := (Xl> ••. , Xd-l)' A, q' := (Yl> ... , Yd-l)' A, Xl := M/z. M: c(q,M)

l(q.M:Xd/q·M) =

l(q·M:z/q·M)

l( (z, Xl' ... , Xd-2)' M :Xd-l/(Z, Xl, ... , Xd-2) . M) l((XI' .•. , Xd-2)' M :Xd-l/(XI ,

= c(q, M) = =

... ,

Xd-2)"

Xl)

c(q', M) = ... = l(q'·M :z/q' . M)

l(q'·M:Yd/q'·M)

=

c(q', M),

since M is a Buchsbaum module of dimension d - 1 by Corollary 1.11 and are parameter ideals for M, q.e.d.

q resp. if

Now we are able to.state the following

Lemma 1.13. Let M be a Noetherian A-module 0/ positive dimension. M is a Budl.sbaum module if and only if tke m-adw completion this case I(M) = I(if). 5*

if of M

is a Buchsbaum module over

A.

In

68

I. Characterizations of Buchsbaum modules

Proo/: Let It! be a Buchsbaum module and let tJ denote a parameter ideal of :it. Then there is a parameter ideal q of M with tJ· if = q .:it and we have ' lA(:itjtJ·:it) - eo(tJ, if) = lACitjq . if)

eo(q, if)

= (t(MIq . M) -

eo(q, M)

--

lA(M/qM) - eo(q, if) l(M).

This is independent of the ehoice of tJ, i.e.,:it is'a Buchsbaum module with l(Jl) = liM) by Theorem 1.12. Conversely, if q is a parameter ideal with respect to M then q is a parameter ideal of :it and the same reasoning as before provides our statement, q.e.d. Next we prove two lemmas needed in the sequel. Lemma 1.14. Let M be a Noetherian A-module and a

A be an ideal8UCh that MJa .M is a Buchsbaum module 0/ positive dimension. Then /or every part 0/ a system 0/ parameters xl> ... , x, o/Mja . M we have with b := (Xl' ... , xr)·A:

Proo/: Assume the statement of the lemma is false. Then there is a maximal member among all ideals a for which the assumptions are fulfilled but the statement is not true. Let it be ao. Then there is an r 2 0, a part of a system of parameters Xl> ... , Xr of M lao·M and an integer k 1 with

We also may assume that k is minimal with respect to this property. We set

ao·M:m = U(ao' M)

U Clearly, r

(see Proposition 1.lO(iv)).

0 is impossible (since then b

= 0). If r

U n x~ ·M = x~ . (U:.I{ x~) = x~ . U

1,

Xt-l • Xl'

U ~ X~-l • ao . M,

a contradiction. Therefore r 2 2. Choose a E (U n b k .M) "- ao . b k - 1 • M. Then we can write a u. E (Xl' •.• , Xi)k-l. M. Notice that m . a ~ ao· M. Set b

Now, for all

Xi • Uj

X

E b'k . ill,

E m we have

bE ((a o

X •

b

,

= I: Xi

• Ui

with

;=1

where b'

+ X • x, . U

r

X •

a E ao . 1,{, i.e.

+ X,' A) .M:m) n b'k ·M.

Since l"'f/(a o + Xr ' A)· M ~ (M/ao ·M)ixr . (M/ao ·M) is a Buchsbaum module of positive dimension (r 2) by Corollary 1.11(i) and since b' is generated by a part of a system of parameters with respect to M/(a o X r ' A)·M, we have by the maximality of ao that b E (a o

+

Xr •

A) . b'k-l . M.

§ 1. Characterization by systems of parameters

Hence a

b

69

+ Xr ' Ur E (a o ' b'k-l + Xr ' b'k-l + Xr ' bk-l) ·ld n U ao • b'k-l . M + (x r • bk- ~~f n U) ao ' b'k-l . ~~f + x, . (bk - 1 • M n U :MX ao • b'k-l • M + (bk - l . M n U). 1 •

r)

Xr •

If k

2, by the minimality of k, we obtain

a E ao . b'k-l . M

+

Xr •

ao . bk - 2 • iII

=

ao . bk - 1 • M

which is not possible. Hence k 1 and this implies a E ao . M + Xr • (M n U) which is also impossible. This contradiction proves the lemma, q.e.d.

=

ao . M

An easy consequence of this is Lemma 1.15. Let.M be a Buchsbaum module over A 01 positive dimeMion. Then lor every system of parameters Xl' ••• , Xa of M we have for q := (XV"'' Xa) . A: (qk+I.M:x.)nq.M=qk·M If depth M

1.

forallk

1 then

qk+l . 11I : Xa Proal: Set q'

qll • M

~

1.

(Xl> " ., Xa-I) . A. Then we have by Lemma 1.14 (a

qhl '111 :Xa

(qk • Xa • M qk • 111 qk. M

If depthM If depth M

for all k

+ q'k+1 • M) :Xa =

qk • M

+ (q'k+1

+ ((Xd' M: m) n q'kTl . 111) :Xd + O:MXt/.

qk . M

Xa • A) : • M :Xd)

+ Xd' q'k • M :Xd

1,O:.I(Xd O:Mm = 0 and we have obtained the required equality, 0, we have by Lemma 1.14:

(qk+l.~lf:Xd)

n q. M

(qk. M qk, M

since O:Mm n q.111

0 (a

+ O:Mm) n q. M + (O:Mm n q. M) =

qk.

~Ilf,

0 in Limllna 1.14), q.e.d.

Remark. Our characterization of Buchsbaum local rings A resulted in the notion of weak A-sequences. Several authors have studied further generalizations of a regular sequence (for example, M. Fiorentini [1J, C. Huneke r1J, N. V. Trung [to], or P. Schenzel [3]). We will show that these generalizations coincide in a Buchsbaum local ring. First we recall some definitions: Definition 1.16. Let A be a local ring of dimension n system of parameters of A. Then: 1.

Xl> " ' ,

>

0, Suppose that

XI' •. " X"

is a

Xn is a weftk A-sequence if (Xl> "', Xi)' A :Xi+1

(Xl' ... , Xi)' A:m

for every i

=

0, "" n

I,

2, XI, "" Xn is said to be a d-sequence if for each subset {iI"'" ii} (possibly 0) of {I, ''', n} and all k, m Il, ..., n} " {iI' ... , i j } we have ((Xi", '"

Xij ) '

A :xk ' Xm)

(Xi" .. " Xi,)' A :Xk •

1. Characterizations of Buchsbaum modules

70

3,

Xl> " "

x. is a relative regular sequence if for every integer i

((xt " 4. An element

•• , Xi-t, Xi+t, .,., X

Xn) ,A :Xi) n (Xl> ••• , xn) . A

=

=

1, ... , n we have

(Xl> •.• , Xi-I'

Xtf-t> ••• ,

Xn) ,A.

in an m-primary ideal q is an absolutely superficial element for q in A if

(qk+t:X) n q

=

qk

for all integers k> 1,

XI, .,., Xn is said to be an absolutely superficial system of parameters if the element Xi is an absolutely superficial element for the image of (Xl> ., " x n )· A in A/(xt> .. " Xi-I)' A for all integers i 1, ... , n. 5. Xl> •• " Xn has property (F), if

((Xl' "', xi-d· A :Xi) n (Xl> ••• , x n)· A = (Xl> ••• ,

xi-d· A

for every integer i with 1 :s;; n. (Note for i 1 we obtain (O:xd n (Xl> ... , x.)· A = 0.)

Proposition 1.17, A is a Buchsbaum ring if ami only if one of the five conditions of Definition 1.16 hold8 for all BysterruJ of parameter8 of A. In this case all five conditions are equivalent. Proof: By definition of course A is a Buchsbaum ring if and only if L holds for every , system of parameters of A. The equivalence of 1. and 2. follows from Proposition LlO(ii) and (iii). Now we prove the implications 1. :::;. 4. :::;. 5. :::;. 3. :::;. 1. (for all systems of parameters of A). 1. :::;. 4. follows from Lemma 1.15. 4. :::;. 5.: Let B := A/(x!> ... , Xi-I) • A (1 :s;; n). Then (O:BX;) n (x;, "', x.)· B 5,;; ((x;, "', xn)k+l. B:x;) n (Xi> ... , x.)· B

(x;, •.. , xn)k. B by

4. Since n (Xi'

for all k

~

1

••. , xn)k. B = 0, we get for A :

k~1

((Xl' ... ,

XI-i)'

A :x.) n (Xl' .•. , Xn) . A

(Xl' ... ,

Xi-d·

~.

5. :::;. 3. is trivial. 3. :::;. 1.: For t' n we obtain from 3. : ((XI' ... , x._ I), A :xn) n (Xl> ... , x.)· A (XI' ... ,

x.-tl· A

= (Xl' ... , X.-l) . A

hence (Xl>"" Xn-l)' A ;x~ 1.lO(iii)', q.e.d.

§ 2.

(Xl> ''',

+ Xn . A n ((Xl> .. " X._l) . A :xn) + Xn ' ((Xl> ... , x n- 1) . A : x;),

x.-tl· A :X. and 1. now follows by Proposition

Cohomological characterization of Buchsbaum modules

The aim of this paragraph is to establish "parameter-free" cohomological criteria for the Buchsbaum property of modules over a local ring. The main result describes a cohomological characterization of Buchsbaum modules without the usc of systems of parameters.

§ 2. Cohomologicai characterization of Buchsbaum modules

71

An essential role in our investigations will be played by the local cohomology modules with support in the maximal ideal m of A, i.e. the modules H~(M) (see Chap. 0, § 1, 3.). Let A denote again a local ring with maximal ideal m, and residue field k A/m. We know that a Noetherian A-module M is a Cohen-Macaulay module if and only if H~(M) = 0 for all ~. with S i < dim M (see Chap. 0, § 1,3.). Related to this is our next result which gives a first indication that local cohomology is an appropriate tool for studying Buchsbaum modules.

°

Proposition 2.1. Let M be a Noetherian A-module oj positive dimension. The jollowing conditions are equivalent: (i) There it! a system oj parameters jor M contained in m2 which 18 a weak M-sequence. (ii) Every system oj parameters jor M in m2 it! a weak M-sequence. (iii) m· H~(M) = 0 jor all i with Os i < dim M. Furthermore, ij one oj these conditions it! juljilled, we have (iv) l(M/q· M) - eo(q, M) it! independent oj q jor all parameter t'deals q m2 • Prooj: The implication (ii) =? (i) is triviaL First we prove {i) =? (iii). Let d := dim M 1. Let Xl' ... , Xd be a weak M-sequence in m 2 • Since Xl E m 2 we have O:,I{m2 ~ O:MXl = O:Mm ~ O:Mm 2 , i.e. O:Mm = O:Mm2. This implies O:Mm O:Mm" for all n;:::> 1 and consequently H:ilM") = U O:Mm" O:,l{m, i.e. m·H:it(M) O. n:
1 the proof has been given. Suppose d

Now we use induction on d. For d have two exact sequences

2, We

and

o -+ Xl . M ~ M -+ M' -+ 0 with .lit' SUPpO:MXl = SuppH:it(M) ~ {m), l:fk(O:.lfXl)

Since Hk(p) is an isomorphism for all 1

0 for all j

1. Therefore

1 and we have O:Him IM)Xl'

The long exact cohomology sequence obtained from (E2 ) yields epimorphisms H{;l(JJ') -+ Ker Hk(i)""'" O:Hi (M)X l , m

By the induction hypothesis (dim M' d 1 and X 2 , ... , Xd form a weak M'-sequence 1. Thus m· (O:Hi (M)xd = 0 for all in m2) m· Hi;;;l(M') = 0 for all j with 1 S 1 d m j = 1, •.. , d - 1. Since Xl E m 2 we again obtain

s

Hk(M) = H:it(Hk(ll{»)

= U O:Hi (M)mn n~l

m

O:Hi

(M)m,

m

Le. m' H~(M) = 0 for all 1 = 1, .. " d 1. Since the case '1 settled, (iii) is proven. To prove the implication (iii) =? (ii) we need the following

=

° has already been

Lemma 2.2. Let M be a Nretherian A-module 01 positive dimension such that H~(M) are modules oj linite length jor all i < dim M. Then jor every part 01 a system oj parameters

72

1. Characterizations of Buchsbaum modules

Xl' ••• , Xr with re8pect to M the 8ubmodule (Xl> ••• , x,). M oj M i8 unmixed up to m-primary component8, i.e., ij p E AssM/(Xl> ... , x r) . ill, then either p = m or dimAlp dimM - r.

Prooj: Let d:= dim M. We use induction on r. If 'I' = 0 we have to show that (Ass M)j := {p E Ass M I dim Alp i} 0 for all i, 0 < i < d. Since lA(H~(.it))

--

lA(H!n(M») = lA(H!n(M») < 00 for all i < d we can assume (using Proposition 3 and Lemma 8(i) of the Appendix) that A is complete. Then Corollary 0.3.7 yields (Ass M); = (Ass Hom(H!nOlf),

E))i

for all i

< d.

Since H!n(M) are modules of finite length for each i < d, the same holds for the module Hom(H~(M), E), i.e., Ass Hom(H!n(M), E) <;;;;; {m} for all i < d and the lemma is therefore proven for this case. Suppose now '1';;::: 1. Since 0 is unmixed up to m-primary components, Supp 0 :.11 Xl {m}, i.e., Hk(O:MXI) = 0 for all j 1. Therefore the two exact sequences (E I ) and (E2) of the first part of the proof of Proposition 2.1 yield for each jOan exact sequence

Hk(M) ---+Hk(Mlxl' M)

H{;I(M).

This shows that Hk(Mlx l . M) is of finite length for all j < d - 1 our induction hypothesis the lemma is therefore proven, q.e.d.

dim Mixi . M. By

Continuation oj the prooj oj Proposition 2.1. Since (iii) is satisfied, H!n(M) is a module of finite length for all i = 0, ... , d 1. Thus we can apply Lemma 2.2. Let Xl> ... , Xd denote an arbitrary system of parameters of M in m 2• Then m· (0 :MXI) m· H~(.M) = 0, i.e. O:MXI O:Mm and Xl is a weak M-sequenee. 1 we are already done. Now we use induction on d. If d Therefore let d;;::: 2 and 1 ::;; i < d. We need to show (Xl' ••• , Xi) •

llf : Xi+l

(Xl' ••• , Xi) • M: m.

We proceed in two steps: (a) Xl x" with X E m, n;;::: 2. We have 0:Mm<;;;;;0:Mx<;;;;;0:MX2 and this results in H~(llf)

= O:,\fm = O:MXt

...

<;;;;;~(M)=O:Mm (since m·~(M)=O)

for all t

1.

Now let M' := jlf/H~(jlf), jn: M' ---+ M defined by jn('m) := x" • m for all all n 1. If n: M ---+ M' denotes the canonical ]Xojection,

Un' n) (m)

=

x" . m

for all n

m E M'

and

1.

Since H!n(H~(M») = 0 for all t';;::: 1, H:n(n) is an isomorphism for all 1. Furthermore, x" . H!n(M) = 0 for all n 1 and i < d. Hence H'm(fn) 0 for all n 1 and all i < d. Thus the commutative diagram

0---+ M' ~-+ M ~ llflx" . .M ---+ 0

1 I xn

-,

1 An

0---+ M' ~ M.!!!....+ Mix, M

0

§ 2. Cohomological characterization of Buchsbaum modules

73

(rli denotes for each 1 the canonical projection and hi is defined by the inclusion M ~ X • M) gives us for all z", 1 :::;: i:::;: d - 1, commutative diagrams

Xi •

1H~-l(h~)

11 .

o

-l-

Hi-1tu,). m ~ Il';;;I(Mjx

Il';;;I(M)

1

xn-l

.

. M)

-l-

Il!n(M')

-l-

O.

Since xl . Il~(M/) xl . Il~(M) = 0 for all d 1 and all j n > 2 and all d 2 uniquely determined homomorphisms rli,.: Il:n(Mlxfl

•

M)

-l-

with

Il:n(M)

Il:n(YI) . fl.,.

=

1, we have for all

Il:n(h.).

It is not difficult to verify that with flH,. the top row of our last commutative diagram

becomes a "split sequence", i.e. Il:n(Mlx fl • M) '" Il:n(M)

for all

EB Il:;I(M')

Il:n(M)

EB Il:';I(M)

2 and all n > 2. This means

d

m . H:n(.J.1:fjx" . M)

=0

for all

2

d

(n::2:2).

Since dim Mix" • M d - 1, the induction hypothesis proves our statement for this case. (b) Xl is an arbitrary element (in m2). By an easy induction argument (induction on z") we may assume that Xl' ••• , Xi is already a weak M-sequence, in particular, (Xl> ... , Xi-I)' M:Xi

=

=

(Xl' ... , Xi-I)' M:m

(Xl> ... , Xi-I)' M:(m),

Xi E m 2 •

where the last equation follows since Now, let y E m, z E m 2 be elements such that system of parameters of M. Set

N Since N:m (1)

Xl> ••• , Xi-I,

y, z is again a part of a

(Xl> ... , Xi-I) . M.

N:y

N:m

~

N:y

N:z c N:(m) (Lemma 2.2) we get

N:(m) and N:m

= N:z = N:(m).

From this we obtain (y.M+N):(m)

(y.M+N:y):(m)

=

+ N) :(m»):y ((y2. M + N):y):m

((y2. M

+ N) :y) :<m) ((y2. M + N):m):y (by step (a)) (y. M + N:y):m = (y. M + N:m):m.

((y2 . M

Furthermore, we have (z E m2) (y. M

+ N:m):m ~ (y. M + N):m2 ~ (y.

M +N):z

~

From all this we obtain (2)

(y.M+N):z=(y.M+N):(m)

•

(y.M+N:m):m .

(y' M

+ N):(m).

74

1. Characterizations of Buchsbaum modules

Now, let mE (Xl> "" Xi)' ~y :Xi+l = ((Xl> .. ,' Xi-I)' M:m + Xi' M):m (using equation (2) with Y = Xi> Z Xi+l)' Choose an UW/(xt> ,.,' Xi) • M)-basis JIt, ... , Yt, of the maximal ideal m of A. Let y E {YI' ... , Yl}' Then

Y' m

u

=

+ Xim'

with

u E (Xl>

••• ,

Xi-I)' M:m,

m' E M

and

y2 . m y , u

+ Xj .y . m' ,

y . u E (Xl'

Hence, by step (a) and equations (1) and (2) with m' E (y2, Xl> .. " Xi-I) ,M :Xj) : y

Z

Xi-I) . M. Xi we get

«y2, Xl' .'" Xi-I) , ..elf :m):y

= (y2, Xl' ... , Xi-I)' M:y):m = = (y,

••• ,

(y. M

+ (Xl' .. " Xi-I) .M:m):m

Xl> "., Xi-I)' M:Xi'

Therefore u = y. m -

Xi'

m' E (y, Xl> ... , Xi-I)·M n (Xl' ... , Xi-I)' M:m) (Xl>""

Xi--I)' M

+ y. M n (Xl> ••• , Xi-I)' M :m)

(Xl> , •• ,

Xi-I) . AI

+- y. (Xl' ... , Xi-I) . M:y. m) +- y. ((Xl> ' •. , Xi-I)' M :m)

=(Xl> ••• , Xi-I)' M =

Thus y. m

u

(Xl' ... , Xi-I) . M,

+- Xj . m' E (Xl' ... , xd . M, and consequently

mE (Xl' ... ,

Xi) •

M:y

for all y E {YI' ... , y,}.

But this implies m E (Xl' ... , Xi) • M: m and Xl' ... , Xi+l is therefore a weak M-sequence. This proves (ii). Finally, the implication (ii) ::;> (iv) is obtained using similar arguments (with some obvious modifications) as was in the corresponding part of the proof of Theorem 1.12, q.e,d. Remark 2.3. Condition (iv) of Proposition 2.1 does not imply (i) or (ii) or (iii), general. For example, let k be any field and X, Y indeterminates. Set R := k[ X, Y], A := k[X, Y]/X, R n (X3, Y)·R,

III

m = (X, Y) ·A.

Then H~(A) X . RjX . R n (X3, Y) . R and therefore m· ~(A) =l= O. But for all parameters Z of A in m 2 we have (where Z is the picture of Z ERin A) l(AJz . A)

eo(z . A, A)

= l(O:AZ) = l((X . R n (X3, y). R) :ZjX . R n (X3, = l(X . RjX . R n (X3, y). R)

y). R)

and this last number does not depend on z.

•

Also, we note that N. Suzuki [5] and S. Goto [10] have studied the class of local rings for which condition (i) of Proposition 2.1 is satisfied. These rings are called q'uasi,Buchsbaum rings. S. Goto [10] has established the ubiquity of quasi-Buchsbaum rings which are not Buchsbaum rings.

•

§ 2. Cohomological characterization of Buchsbaum modules

75

Corollary 2.4. II M is a Buchsbaum module then rn· H~(M) = 0 lor all i =!= dim M. In particular, the local cohomology modules are modules oll~'m'te length ~n this case. Unfortunately, the following example will show that the converse of this statement is false. Nevertheless Corollary 2.4 gives it first necessary condition for Buchsbaum modules independent of systems of parameters. Example 2.0. Let k be a field and XI, X 2 , X a, X 4 indeterminates. Take

A := k[ Xl' ... , X4]j(Xlo X 2 ) n (Xa, X 4) n (Xi, X 2 , X 3 , X~). Then U(Oj = Xl' X 4 . A and therefore rnA' H?nA(A) = rnA . U(O) = O. The exact sequence

o -+ H~)A) -+ A

-+

B

-+

0

with B: = AjU(O) = k[ Xl' ... , X4]j(Xlo X 2 ) n (Xa, X 4) results for all

i>

1 in isomorphisms (H~A(H~)A)) = 0 for

i> 1):

H~A(A)~ H~A(B).

Therefore (see Example 6 of the Introduction or Proposition 2.25 which show that B is a Buchsbaum ring and hence a Buchsbaum module over A by Lemma 1.6) rnA . H~JA) = O. Notice that dim A = 2. We next show that A is not a Buchsbaum ring. Take z:= Xl X 4 mod A. Clearly, dim Ajz . A = 1 and we have

+

Therefore Xl . U(z· A) ~ z· A, i.e. rnA . U(z. A) we find that A is not a Buchsbaum ring.

~

z· A. Thus by Proposition 1.1O(iv)

It is possible to construct similar examples which will have depth zero. In Chapter V (see § 5, 3.) we will give an example with depth greater then zero (Example V.5.4). The usefulness of local cohomology for our purposes will next be demonstrated by the following result which enables us to give a "parameter-free" expression for the invariant I(M) of a Buchsbaum module M. Proposition 2.6. For any Buchsbaum module M with d : = dim M we have

I(M)

=.r (d -. 1) .l(H~(M)). d-I

,=0

~

Proof: We already know from the proof of Theorem 1.12 that

I(M)

= l((xl> ... , Xd-l)' M:Xdj(Xlo ... , Xd-l)' M) = l((XI' ... , Xd-l)' M:rnj(xlo ... , Xd-l)' M),

where Xlo "" xdis some system of parameters of M. Let us writex~:= xi, M' := Mjx~ ·M. We use induction on d. If d = 1, I(M) = l(O:Mrn) = l(H~(M)), since rn· H~(M) = 0 by Corollary 2.4 which finishes the proof. If d ~ 2 we have already seen in the proof of

•

76

I. Characterizations of Buchsbaum modules

Proposition 2.1 (impl1cation (iii) =? (ii), step (a)) that H~(M') ~ H:nUJI)

E8 H;;I(M)

and we oqtain with the induction hypothesis (dim M' I(lll) = l((x~, x2,

••• , Xd-l)'

lJI:m/(xi, x 2 ,

d - 1):

... , Xd-l)'

= l((X2' ... , Xd-l)' M' :mJ(xz, ... , Xd-l) . M')

M) I(M')

.X/ (d . 2) .l(H:n(M')) .};2 (d ~ 2). (l(H:nUJI) + l(H~:l(M))) =

.=0

'=0

t

=i~l ((~ =~) + (d =d-l

i

t

2)) .l(H:n(M))

(d i 1) .l(H:n(M)) ,

q.e.d. Now we want to state a result which shows first that for a Buchsbaum module the Hilbert-Samuel function Pq,M(n) and the Hilbert-Samuel polynomial Pq.M(n) coincide for each parameter ideal q of M and all n O. Secondly, the Hilbert-Samuel coefficients ei(q, M) are shown to be independent of q for all 1. We obtain an expression for I(M) using these ei(q, M). Finally we find for each t>- 0 non-negative integers It(M) such that

J}[

i>-

l(MJqt+l . M) for all t

0 and all parameter ideals q of 111, where d := dim M

>

Proposition 2.7. Let J}[ be a Buchsbaum module with d:= dim M parameter ideal,q of M (i)

l(M jqt+1 . M)

;=0

(il)

i)

= Ed (t+d . ' e.(q, M) d

for all

O.

>- O. Then lor every

t> O.

t

for all i = 1, ... , d,

ei(q,1}[)

for

for

P 4= -1, P = -1.

Ii.

(iii)

I(M)

=

ei(q, M).

Proof: (iii) is a consequence of (i) if we set t = O. We prove (i) and (ii) by induction on d. For d = 0 there is nothing to prove (q = 0). Assume d>- 1. Let q = (Xl' ... , xa) . A be any parameter ideal of M and let q'

•

§ 2. Cohomological characterization of Buchsbaum modules

:= (Xl' ... ,

t>

77

Xd-l) . A, M':= MjXd . M. Then we have an exact sequence for., each

1:

0--+ qt+l . M :Xdjqt . M

--+

f

Mjqt . M -+ Mjqt+1 . M

--+

M'jq't+l . 1',f'

--+

0

where t is obtained from multiplication by Xd' As in the proof of Lemma 1.15, we obtain

qt+l. M:Xd = qt. M

+ O:MXd =

qt. M

+ O:,Hm

and therefore

since O:Mm n qt. M = 0 by Lemma 1.14. Also l(qt. MjqH1. M) = l(Mj qt+l. M) - l(Mjqt. M)

= -l(H::'(M))

+ l(M'jq't+1. M').

By our induction hypothesis we have

=};l (t + d -. i - 1) .ei(q', M')

l(M'jq't+l. M')

i~O

d - z-

1

and

ei(q', M') =

d1-

l

j~O

(d ~1 -i -1 2) . l(H~(M'))

for all z' = 1, ... ,

d-

1.

From the exact sequence g

0--+ MjO:Mm -+ M

--+

M'

--+

0

where g is induced by multiplying the cosets by Xd, we find for all j S d - 2 short exact sequences (see also the proof of Proposition 2.1(iii) :::::} (ii), step (a)):

o --+ H~(M) --+ Hk(M') --+ H{;l(M) --+ O. Therefore for i = 1, ... , d - 1

=dlj~O

= =

l

(d ~1 -i -1 2) .l(Hk(M)) +j~l X/ (~~1 -i -2 2) .l(Hk(M))

X/ ((d ~1 -

i -1 2)

j~O

1j~i (d 7~ ~ l(H::'(M))

+ (d ~1 -z' -2 2)) .l(Hk(M))

1) .l(H{,(M))

+ l(H}.,(M))

for

z'S d- 2,

for i

=

d - 1.

78

1. Characterizations of Buchsbaum modules

Now for all t 2 0 I

I(Mjql+l . 11{)

I(Mjq. M)

+ .E l(qi. Mjqi+l . Ml 1=1

t

l(M/q· M)

+ .E l(M'/q'id. M')

t .l(H~(M))

;=1

= I(M/q' M) +;~ j~1 = I(Mlq' M)

(d ~ iii

+ £1 ((t + d --: i) d-1

j=O

= l{M/q· M) - iE1e/(ql, M')

+( ~

1).

(ea-I(q', M')

11).

ej(q', M') - t

.1(H~(M))

_ 1) .ei(q', M') _ t .l(H~(M))

+ I(H~(M)) +

7;: e~ ~ ~ i) .

ej(q', M')

-1(H~(M))).

For sufficiently large t this polynomial in t coincides with Pq,M(t) and comparing coefficients we obtain

ej(q, M) = ei(q', M')

for

i

0, ... , d

2,

ea-I(q, M) = ed-I(q', M') - I(H~(M)), a-I

ed(q, M)

=

l(M/q ..Jf)

.E ej(q', M') + l(H~(M)).

j=O

But this proves (i) since the above equation is true for all t for ej(q', M') and a-I l(M/q . M) = I(MI/q' . M') = .E ej(q', M')

O. From our expressions

j=O

(see (i) applied to M' and q', t

0) we also obtain (ii), q.e.d.

Coronary 2.8. Let M be a Buchsbaum module 01 dimension d > O. Then lor every t there i8 a natural number [,(M) 8uch that jor every parameter ideal q oj M '

l(MjqH. M)

(t ~ d) . eo(q, M) =

0

[I(M).

This follows immediately from Proposition 2.7(i) and (ii). Next we will prove a first sufficient "parameter-free!' criterion for the Buchsbaum property which allows us to find many examples for Buchsbaum modules and which is also a necessary condition if A is a regular local ring. First we need the following

Lemma 2.9. Let 0 ->- M' ->-llf -4 Mil ->- 0 be an exact sequence oj A·modules with m· Mfl O. Then the sequences

o ->- Ext~(k, M')

->-

Ext~(k, M).f.!...". Ext~(k, Mil) ->- 0

(k

A/m)

§ 2. Cohomological characterization of Buchsbaum modules

with Ii := Ext~(k, I) are exact lor each i.e. the sequence

i>

°

il and anly il we have exactness lor

79 t'

= 0,

0--+ HomA(k, M') --+ HomA(k, M) --+ M" --+ 0 is exact.

Prool: The only if part is trivial. Assume now that 0 --+ HomA{k, M') --+ HomA(k, M) ~ M" --+ 0 is exact (note:

HomA(k, ll-f") '"'":: O:M"m = M" since m· M" --+ 0 and get a commutative diagram

=

0). We apply HomA(k, ) to M.1.. M"

HomA(k, M) ~ Mil --+ 0

nl

II

M

- - t o M"--+O.

Next, by using Ext~(k, ), we obtain commutative diagrams Ext~(k, HomA(k, M)) -+- Ext~(k, M") --+ 0

I

.J.-

Ext~{k, Mil) --+ O.

Ext~(k, M)

Since M" is a direct summand of HomA{k, M) (by virtue of 10), the top row is exact. Therefore the bottom row is also exact, q.e.d. Theorem 2.10. Let M be a Noetherian A-module with d:= dim M maps (see Chap. 0, § 1, 3.)

1. II the canonical

IPk: Ext~(k, M) --+ H:n(M) are surjective lor all

t'

=f: d then M is a Buchsbaum module.

Prool: By the surjectivity of IPk the H:n(M) are modules of finite length and we conclude by Lemma 2.2 that for every part of a system of parameters Xl' ••• , Xr of M with r < d the submodule (Xl> ••• , x r )· M of M is unmixed up to m-primary components. Now, IP~ is the inclusion HomA(k, M)~ O:,itm c O:M(m) = H:h(M) and therefore we have for every parameter X (i.e. dim Mix. M = d - 1)

m· (O:MX) ~ m· (O:M(m»

m· H~n(M) '" m· HomA(k, M)

0,

i.e., x is a weak M-sequence. 1, we are already done. We use induction on d. If d Assume d>- 2 and let Xl' •.. , Xd be an arbitrary system of parameters for M. Suppose first that depth M > 0. In this case, since O:.IIIX I = 0, we have an exact sequence 0--+ M ~ M --+ Mlx l

This gives for i Mixl' M):

= 0, ... , d

•

M --+ O.

- 2 commutative diagrams with exact rows (set M'

o --+ Ext~(k, M) --+ Ext~(k, M') --+ Ext~+l(k, M) --+ 0

1. Characterizations of Buchsbaum modules

80

By hypothesis q;k,
Xl' M ~ M is the factorization of Xl through its image. Since Hk(O:Mm) where M for all i 1, H~(n) is an isomorphism for each 1. We now prove that Ext~ (k, n) is surjective for all i d 1. '1'0 establish this we prove the following

o

Claim. The homomorphisms Ext~(k, <X) are injective for all i ~ d. Let

be minimal injective resolutions of 0 :Mm and M, respectively. {m}, we have Ii Em" where E denotes the injective hull Since Supp O:Mm of k and mi l(Ext~(k, O:Mm)). It is well known that Ii J, EI" where J; is a direct sum of copies of injective hulls of Alp, p E Supp M '" {m}, and li l(Ext~(k, M)). The homomorphisms Ii ---* Ii are induced by <X. Since HomA(k, H~()) HomA(k,), we have Ext~(k, M)

H'(HomA(k, I~)t

H'(HomA(k, H~W)))

=

Hi(HomA(k, El.))

(Hi denotes the ith cohomology of the underlying complex).

We write the complexes Em. and El. in the following manner; O~O :.\1 m~/~'m'---?>Z~~h''''I-7'

0-,>0

±m~L~l£:~l-,>

... ~"JII""~Z!~Em, ~ ••.

. . ~l~:(.~i-J_. .

where Em, ---* El, H~(li Ii) which induces the other homomorphisms. Since the resolutions were minimal, we have , HomA(k, Z~)

Ext~(k, O:Mm),

HOIUA(k, <X')

Ext~(k, <X).

Next we show that Ext~(k,

<xi)

HOIUA(k, Zl)

is injective for all j

rv

Ext~(k, M),

0 and all i

0, ... , d - 1.

§ 2. Cohomological characterization of Buchsbaum modules

81

The following commutative diagrams with exact rows (£:2 1)

o -+ Z~-l -+ Em.I a'-

1

-+ Z~

+

o -+ Zl-l -+ EIH

-+ B;-l -+

show that Ext~(k, fJI) Ext~,"l(k, the induced homomorphisms Ext~(k, M)

ro...-

-+0

ttl·

t

1

0 for all j

(Xi-l)

0 and

HomA(k, Z;) -+ Hom,.(k, H:n(M))

£> 1.

For each i~ d - 1

H:n(M)

in the commutative diagram O-+Z~ =Z~--+O-+O

t PI

tal

o -+ BH ..:4 ZiI I

t -+

are just the canonical maps the sequences

Him (M)

-+ 0


o -+ Ext~(k, B;-l) -+ Ext~(k, Z{) -+ Ext~(k, H:n(M)) -+ 0 are exact for each j 0 and i ~ d j 0 and £~ d - 1. Also, we have

1. In particular, Ext~ (k, til is injective for all

Ext~ (k, <x') = Ext~ (k, til . Ex~ (k, (J') = Ext~ (k, til . Extil(k, <xi-!). Since <x0 is the identy map (of O:J,fm) an easy inductive argument (induction on £) now shows that Ext~ (k, <xl) is injective for all j 0 and d - 1. In particular, 0 we have that Hom,.(k, <xi) is injective. But since for j HomA(k, <x d ) = HomA(k, t d ) • Hom,.(k,

Pd ) =

HomA(k, t d )

•

Ext~(k, <xd- 1)

is also injective, our claim is therefore proven. Now we continue with the proof of Theorem 2.10. Since 0 Ext~(k, Xl) Ext~(k, t) • Ext~(k, n) and Ext~(k, n) is surjective for each i ~ d - 1, we obtain Ext~(k, t) = 0 for all i d - 1. Analogously H:n(t) = 0 (since H:n(n) is an isomorphism for £> 1) for all i~ d - 1. Therefore we have for i 0, ... , d 2 commutative diagrams with exact rows

o -+ Ext~(k, M) -+ Ext~(k, M/) -+ Ext~+l(k, Xl' M) -+ 0

l~~

HI

l~~'

1

~Xl'J,f

H;:I(XI' M)

Furthermore, we have commutative diagrams Ext~(k,") Ext~(k, M) -~--+l Ext~(k, Xl' M)

l~~

l~~"M

H:n(M)

H:n(XI • M)

6 Buchsbaum Rings

-+0.

82

1. CharacterizatiollS of Buchsbaum modules

Thu~ the surjectivity .of IPk implies the surjectivity .of IP;,.M f.or all i < d - 1 and this means the surjectivity .of IPk' (use the 4-Lemma), i.e. by the inducti.on hyp.othesis M' is a Buchsbaum 'm.odule. Since m· (O:MXI) 0, Xl' "', Xd is a weak M-sequence, q.e.d. Remark 2.n. It is n.ot true in general that this sufficient criteri.on f.or Buchsbaum m.odules is als.o a necessary c.onditi.on. This is sh.own by the Example V.5.7. But as menti.oned already, this criteri.on is necessary if A is a regular l.ocal ring, see Corollary 2.16. We n.ow give the f.oll.owing imp.ortant c.onsequence .of this criteri.on.

Proposition 2.12. Let M be a Noetherian A·nwdule with, r:= depth M and Il!n(M) 0 for all i =1= r, d. The following properties are equivalent: (i) M is a Buchsbaum nwdule. (ii) m· Ilin{M) (iii) Let Xl' ••• ,

< dim M

: d.

O. Xr

(Xl' ... ,

m2• Then

be an M-sequence in

Xr ) '

M:(m)

=

(Xl' ... ,

X r )'

M:m.

Proof: (i}:::} {ii} f.oll.ows fr.om Cor.ollary 2.4 and {ii} :::} (iii) is a c.onsequence .of Pr.opositi.on 2.1. T.o pr.ove (iii) {i} it is sufficient to verifythesurjectivity.of IP~: ExtA{k, M} -'>- Ilin(M) since the .other IPk's, i =1= dare .obvi.ously surjective and theref.ore Theorem 2.10 implies (i).

T.o this end we pr.ove the m.ore general statement:

f.,'laim. Let M be a N.oetherian A-m.odule with r: M-sequence Xl> ... , Xr in m 2 such that (Xl' ... ,

Xr )'

M:(m)

=

(Xl' ... ,

X r )'

depth M

< dim M.

If there is an

M:m

then IP~ is an is.om.orphism. F.or the proof we use inducti.on.on r. If r 0, IP~ is the embedding O:Mm ~ O:M(m). Als.o O:M(m) O:,ym by (iii) and IP~ is theref.ore an is.om.orphism. If r > 0 we have an exact sequence

o -'>- M

~ M

-'>-

M'

-'>-

0

with M':= MlxI . M.

This gives rise t.o a c.ommutative diagram

19'~-;1 0-'>-

Il';;"I(M'}

19'M --+

Ilin(M}

24 Ilin(M).

By the inducti.on hypothesis IPM-I is an isom.orphism and thus IPM splits into an is.om.orphism 9?~: ExtA(k, M} -'>- O:Hf (M)X I Il';;"I(M') and the embedding t: O:HT (M)X I ~ Il':n(M).

m

m

§ 2. Cohomological characterization of Buchsbaum modules

83

and consequently

O:H'm(M)X l

=

O:H'm(M)m

H';,,(M)

O:H'm(M)(m)

=

which shows that £ is the identy map. Hence

9'~

is an isomorphism, q.e.d.

We are now able to give our first class of examples for Buchsbaum modules. To do this we define: Definition 2.13. Let r, d be integers with 1::;; r ••• , Yd indeterminates. We set

<

d. Further let k denote a field and

Xl' ... , X d , Y 1,

R d := k[Xl> "" X d, Y ll ... , Ydk..

where md is the ideal generated by the indeterminates or

We define by induction on r what is meant for an ideal a Rd to be of type (r, d): 1. a is said to be of type (1, d). if a (Xl> ... , X d ) Rd n (Y l , ... , Y d ) R d ; 2. a is said to be of type (r + 1, d) with r + 1 < d, if a al n a2 and a) RaJal a Cohen-Macaulay ring with dim Rd/a l d, b) the automorphism of Rd given by exchanging the indeterminates (Xi __ Y;) carries a l into a2 , c) a l + ~ (Xd' Ya) , Rd + Rd, where Rd- l is an ideal of type (r, d - 1). By Definition 2.13 it is not difficult to give an explicit description of ideals of type (r, d) for arbitrary integers r, d with 1::;; r < d (d. Proposition V.2.7).

°.

°

For these ideals the following statement is true: Lemma 2.14. Let a c Rd (see Definition 2.13) be an ideal 01 type (r, d) with 1

r

<

d.

Then

H:n,.(Rd/a) = 0

for all i

k.

r, d and H';".(Rd/a)

Therefore Rd/a is a Buch8baum module (over R d ) of depth r and dimension d. Proof: We have an exact sequence (recall: a

=

al n a2, see Definition 2.13)

where f(a mod a) = (a mod all a mod ( 2 ), g((al mod all a2 mod ( 2)}

a E Rd ,

(al - a2) mod (a l

+(

2 ),

all a2 E Rd'

But Rd/ah i = 1,2 are Cohen-Macaulay rings (and therefore Cohen-Macaulay modules over R d) and Rd/(a l + (2 ) :::-:: Rd-l/O. Hence H'm.(Rd/O l 6*

EB R d /( 2 ) "-' H'm.(Rd/al) EB H'm.(Rd/a 2 )

=

0

64

I. Characterizations of Buchsbaum modules

for all i

d

1 and

H~.(Rd/(al for aU

~'.

+a

2 ))""'"

H~'_1(Rd-d6)

Thus the long exact cohomology sequence renders for

d

1 isomorphisms

H~.(Rdla) ~ H;;':,(Rd-l/b).

Now, induction on d proves thc lemma, q.e.d. In order to obtain a necessary and sufficient characterization of Buchsbaum modules one has to look for "better" criteria. We will show that this is possible if we replace the "Ext"-functors by the "cohomology modules" of the Koszul complex in the sense which was made precise in Chapter 0, § 1, 3. The following theorem is the main result,of this chapter. First it gives a complete co homological characterization of Buchsbaum modules. Secondly it shows how to verify the Buchsbaum property by considering only a finite set of weak M-sequences. We note that Theorem 20 of the Appendix gives another approach for proving this Theorem '2.15. Theorem 2.16. Let M be a Noetherian A-nwdule of positive ditMnsion d. The followi'T/.{/ propertie8 are equivalent: (i) M U! a Buchsbaum module. (ii) The canonical maps

lk: Hj(m, M)

-* H~Ulf)

(cf. Lemma 0.1.5)

are surjec#ve for all i < d. (iii) Let Xl, ... , X, be an M-basUs of the maximal ideal m of A (d. Definition 1.7 and Proposition 1.9). For every system iI, ... , it!. of integers with 1 i l < ... < it!. t the sequence xi:, ... , xi: U! a weak M-sequence for all rb . '" ra E {I, 2).

s::

Proof: (i) =? (iii) follows from the definition of Buchsbaum modules (Definition 1.5) and M-bases (Definition 1.7). , We now prove (iii) =? (ii). 1~ is (up to a. natural equivalence) the embedding O:.wm C O:M(m). But we have O:Mm 2 ~ O:Mxi O:J![m ~ 0 :Mm 2 and, consequently, O:Mm O:Mm2 .. , = O:M(m), i.e. 1~ is an isomorphism. Now we use induction on d. If d 1 we are done. Assume d > 1. Since xi, ... , x= is a weak M-sequence in m2 , m . H~(M) 0 for all i < d by Proposition 2.1. Assume first depth M > O. By Lemma 2.2, (0) is unmixed up to m-primary components and hence unmixed. Therefore we have an exact sequence 0 -* j1f ..:'4 M -* M' -* 0 with M':= Mlxl . M. This gives rise to commutative diagrams with exact rows 0-* HI-I(m, M) -* Hi-1(m, M') -.,.. HI(m, M) -* 0 (D j )

l,~l 0-.,.. H;;;I(M)

for all i < d. (Notice for all i < d.)

iA~l

-*

Xl'

H;;;I(M')

Hi(m, M)

iAk -* H~l(M)

-*

0 for all ~. and

0 Xl'

H~(M) ~ m . H:n(M)

0

§ 2. Cohomological characterization of Buchsbaum modules

85

Let A' A/Xl' A, m' := m· A. Then x2"'" XI is an M'-basis of m' (considered as A'-modules; we write 53 for the image of X E A in A') satisfying the hypothesis of (iii) (with respect to A' and M/). Therefore the natural homomorphisms ),~~,: Hi(m', M') -~ H:",(M')

are surjective for all i < d - 1 by the induction hypothesis. Let n := (X2' ••• , XI) • A. Then obviously as A-modules H'(m', M') ~ H'(n, M')

and (for instance by R. Y. Sharp [IJ, Theorem 4.3) H:",(M') ~ H~(M').

Hence the corresponding natural homomorphisms l~w : H'(n, M/)

-)<-

H~(M/)

are surjective for all i < d 1. By use of the commutative diagrams of Corollary 0.1.7 we get:

;.:v,: H'(m, M/)

-)<-

H:,,(M')

are surjective for all i < d 1. Now the commutative diagrams (D;) imply (ii) in this case. Assume next depth M = O. The exact sequence 0 -)<- O:M m -)<- M -)<- N -)<- 0 with N:= MjO:Mm MjO:M(m) gives for each i> 0 rise to a commutative diagram (H:,,(O:Mm) = 0 for all~' > 0): H'(m, O:M m) 4

H'(m, M)

lA~

(Di)

4

H'(m, N)

ll~

H:"(M) ...::...- H:"(N).

We have depth N > O. Further N also fulfils the assumptions of (iii) too, since Xl> ••• , XI is of course also an N-basis of m. For each 1 ~ i] < ... < id ~ t, rl> ... , rd E {I, 2}, o~ j < d we have

+ (x[;, : .. , x~:). M):xi::: ~ ((O:Mm) + (x;;, ... , xi;) . M) :(m: (xi;, ... , xi;) . M :(m) (xi;, "', xi;) . M:m ((O:M m) + (xi;, ... , xi;) . M):m

((O:Mm)

(notice

(xi:,. 0., xi:) . M : m2 ~ (xi:, .. 0, xi;) 0M: xL

(x~;, ... , xi,) .

M:m implies

(xi;, ... , xi;) . M :(m) = (xi;, .0., x~:) . M :m).

Therefore ;.~v is surjective for aU i < d. We now prove that also g' (in the diagram (Di» is surjective for all ~. < d. Then the proof of (iii) ::::} (iI) will be complete by the diagrams (D/). ~ To this end we prove the injectivity of /i for all i d. We use the notation introduced in Chapter 0, § I, 3.

86

I. Characterizations of Buclulbaum modules

Let Ki Ki(x}> ... , X,; M), ili;= Ki(XI' ... , X,; O:Mm) and let Hi, Hi denote the cohomology modulcs of these complexes. Since (Xl"'" X,) (O:M m) 0, the differentiation of j(i is zero, i.e. fl· j(i for all i. Let d t denote the differentiation H of K. We have to show that itt" 1m d = 0 for all t''S d (considered as submodules of Ki). Let X E j(i :l 1m di - 1, i.e. x di-l(y) with y E K,-1: We write

Then we obtain X = di - 1(y)

E

=

(

lS;n1.< ...
1:

(_I)P-1 xnpmn1 ... ;;P .....) en, ... n"

IJ=l

hence u·'····' E (0:.11 m) r: (x." ... , x.,) 111 for all 1 < n 1 < ... < nj t. Let U '11."""'" (n1, ... , nj fixed) and choose 8 minimal with u E (x. , ... , Xn ) JII, 0<8 i. Sincei d, XI! , ... , x" ,x2" forms a weak M-sequence. We write • 1

~

If 8

1I-1

S

1, multiply by x n, and obtain (xn,u

x;,ms E (x." ... , XII,) M

0):

and thus

ms E (x." ... , xn._.l J1: m

by our assumption on Xl>"" x" Therefore wE (XII" •• " x n ,_,) • M which contradicts the minimality of 8. Hence 8 = 0, i.e. u = O. Since this is true for all integers i), ... , ij with 1 i1 < ... < ii t, we get x = 0 and (ii) is proven. Finally, we show (ii) =? (i). If A denotes the rn-adic completion of A, M := ~'l:f @A A then

A ~ H~(.Iff), i.e. the canonkal maps

A~: Hi(m, M)

--'?-

H~t(M)

are surjective for all i < d = dim) M. By Cohen's structure theorem for complete local rings there is a regular local ring R with maximal ideal n such that A is an epimorphic image of R. Without loss of generality we may assume that dim R = rankk(m/rh2) Then Hi(m, J'1) natural maps

rankk(mjm2)

(k

= Aim = Aim).

Hi(n, M) and H~(i1) "" H~(M) as R-modules. The corresponding

Ak: Hi(n,M )

--'?-

H:1(M)

therefore are surjective for all i < d dimR M). Using the commutative diagrams of Lemma 0.1.5 we see that CP~({: Ext~(k, M) --'?- H~(M) are surjective for all i < d since n is generated by an R-sequence ,and the canonical maps 1p~ are therefore isomorphisms. Thus M is a Buchsbaum module over R by

§ 2. Cohomoiogical characterization of Buchsbaum modules

87

Theorem 2.10 and by Lemma 1.6 a Buchsbaum module over A. Now 1Jf is a Buchsbaum module over A by Lemma 1.13, q.e.d. Corollary 2.16. Let A be a regular local ring and M a Noetherian A-module 01 positive dime'fUJion. M is a BucMbaum module if and only if the canonical maps IPk: Ext~ (k, M) -+ H~(M) are surjective for all i

<

dim M.

Remark 2.17. Our initial investigations of Buchsbaum modules made it necessary to study all systems of parameters (see, for example, Proposition 1.10 and Theorem 1.12). Using the cohomological characterization of Buchsbaum modules we are able to reduce these considerations to a well-defined finite set of systems of parameters, see Theorem 2.15. In special situations Proposition 2.12 enables us to work with only one system of parameters. Therefore we would like to pose the following Problem. Is there a criterion for Buchsbaum modules which uses only one fixed system of parameters1 One might be tempted to try to answer the following Question. Let M be a Noetherian A-module of dimension d:2: 1. Suppose tQa.t there are elements Xl' •.• , Xd of m such that

is a weak M-sequence for every permutation 8 on {I, ... , d} and for all integers n l , > O. Is M then a Buchsbaum module~

... ,

nd

The following (unpublished) example due to S. Goto shows that this question has a negative answer. Example. Let R: k[X I , ••• , X a, Yv "', Ya], d:2: 3, be the formal power series ring in the indeterminates Xl' ... , X a, Yl> ... , Y a over an arbitrary field k. Put a:= (Xl' ... , Xa) R (j (Y I , ' ' ' , Y a)· R, q:= (Xi, X 2 ,

Fi := Xi

••• ,

+ Y.

A := R/((a

(j

q)

Xd,

Yi, Y

for i

+ Ff· R)

2 , ••• ,

Y d )· R,

1, .,', d, with n

3.

Then dim A = d 1 and A is not a Buchsbaum ring since m· U(O) =f: 0 in A. It is now easy to see that the images of F 2 , ••• , Fa in A form a system of parameters for A and have the required property. We note that the images of F 2 , .,', Fa are even a-part of an A-basis of the maximal ideal m of A since they form a part of a minimal basis of m. A first and important application of Theorem 2,15 is the solution of the so-called lifting problem for Buchsbaum modules, i.e. the possibility of lifting the Buchsbaum property by a non-zero divisor. M. Hochster asked the following related question:

88

I. Characterizations of Buchsbaum moduleS

Let A = Ria be a local ring where R is regular and a is an ideal of R. Suppose that: (i) All is a Cohen-Macaulay ring for all .\) E Spec A " {m}. (ii) there exists a non-zero divisor x of A such that A/x· A is a Buchsbaum ring. Is it true that then A is a Buchsbaum ring? The following example shows that the answer to this question is negative. Example 2.18. Take A : = k[ Xl, X 2 , X a, X 4 ]/(Xi, Xi) II (Xs, X 4 ) where k is an arbitrary field and Xl • ... , X, are indeterminates. Then we get the following: (i) All is a Cohen-Macaulay ring for all .\) E Spec A " {m}. (ii) A/(X l + Xa) . A is a Buchsbaum ring (of dimension one). (iii) A is not a Buchsbaum ring. (iv) m· H~(A) =l= O. The statements (i), (iii), (iv) are clear (see e.g. Proposition 2.25). To prove statement (ii) it is sufficient to show that (X 1,

••• ,

X 4 ) • U(a

+ F . R) ~ a + F

.R

(by Proposition 1.10) where

a

=

(~, Xi) . R II (Xa, X,)· R

F:= Xl

+ Xa,

(Xi' Xa, X~. X" X 2 • Xa,X i

•

X,) . R,

R:= k[XlJ X 2 , Xa, X,].

But this follows from U(a

+ F . R)

(Xl' X 3 , X 4 ) • R

II

(Xi, X 2 , Xi, F) . R

(Xi, X;, X l X 2 ' X 2 X" F). R. What we can prove is the following: Proposition 2.19. Let M be a Noetherian A·module with depth M > O. The /ollowi1UJ . candz'tions are equivalent: (i) M is a Buchsbaum module. (ii) There is a non-zero divisor x E m 2 0/ M such that Mix, M is a Buchsbaum module. (ii') M/x· M is a BucMbaum module lor every non-zero divisor x E m2 0/ M. (iii) Thue is a non-zero divisor x E m 01 M such that: a) MIx, M is a Buchsbaum module. b)x· H:n(M) 0 lor all i < dim M. (iii') For all non-zero divisors x E mol M a). and b) 01 (iii) are true. (iv) There is a non-zero divisor x E m 01 M such that: c) M /x . M is a Buchsbaum module. d) x· H:n(M/x 2 • M) = 0 lor all i < dim M 1. (iv') For all non-zero divisors x E m 01 M c) and d) 01 (iv) are true. Prool: The necessity of the conditions (ii), (ii'), (iii), ... is obvious by Corollary 1.11(i) and Corollary 2.4. Therefore it remains to prove the implications (ii) :=;, (i), (iii):=;, (i), (iv) :=;, (i).

§ 2. Cohomological characterization of Buchsbaum modules

89

We start with the exact sequence O-+M

M -+M' -+0,

where M':= Mjx. M.

From it one obtains a commutative diagram with exact rows 0-+ Hi(m, M) -+ H'(m, M') -+ HHl(m, M) -70

Now in each case ).,k. is surjective for all i < dim M' = dim M - 1 by Theorem 2.15. We want to show that we have always x . H{.lI.Y) = 0 for all j < dim M. Then by the commutative diagrams (Dj-l) ).k: is surjective for all j < dim 111 and (i) follows by Theorem 2.15. In case (iii) this is clear. In case (ii) we obtain by the 'exactness of the bottom row of (D j - 1 ) an epimorphism Hi;;;l(M') -70 :Hi (M)x. m

Since m . Hi;;;l(M') = 0 for all j < dim M (Corollary 2.4), we get m.(O:H~(M)X)

0

for all

i


But x E m yields O:H~(M)X = O:H~(!.f)m 2

If (iv) holds we obtain for all

=

O:H~(M)<m)

H~(M) and we are done.

i by the exact sequence

z·

0-+ M -+ M -+ Mjx 2 • M -70

epimorphisms Hi;;;1(Mjx2. M) -+O:Hi (Mlx2. m

Hence, by d) X· (0;H~(MIX2) O:Hi (MIx m

=0

and this implies

O;Hi (M)x 2 = ... = H~(M), m

since every element of H~Ull) is annihilated by some power of x. But this also implies x . H~(M) 0 for alIi < dim M, q.e.d.

I

Remark 2.20. We can apply this lifting property to the classical investigations on hyperplane sections. This is connected with Bertini's Theorems on linear systems. In 1977 H. Flenner [1] proved local analogues of these theorems which answer a conjecture of A. Grothendieck. These striking results also improve the observations on the hyperplane sections of normal varieties (see, e.g., Seidenberg [1, 2] and Kuan [2J). Let us consider a hyperplane section through a rational normal point of an algebraic variety. To do this let V be an irreducible algebraic variety of dimension ~ 3 defined over an infinite field k in affine n-space over k, and let H be the generic hyperplane defined by Uo Ul • Xl Un • Xn = 0, where U o, ••• , Un are indeterminates over k. Let P E V be a rational point, that is the coordinates of P are elements of k. Then one is interested in the intersection of V with H. If V is normal at P over k then it is not true in general that V n H is normal at P over k(uo, ... , Un); however, V n H is normal at P if the local

+

+ ... +

90

1. Characterizations of Buchsbaum modules

ring of V at P is a Cohen-Macaulay ring (see Kuan [2]). Therefore Trung [1, 3J has studied the hyperplane section through a Buchsbaum (and Cohen-Macaulay) point of an algebraic variety and the theory of specializations of Buchsbaum points. From the view-point of local algebra these results follow from our Buchsbaum lifting property, Corollary 1.11 and Lemma 2.26 below. Using geometric language wc will discuss one result of a generic hyperplane section through a Buchsbaum point. Let Q be a universal domain and k b;:; Q, that is Q has infinite degree of transcendence over k. By affine n-Il'pace An we mean the n-fold cartesian product of Q: An=QX ... XQ.

A point in n-space, or briefly a point, is an n-tuple (all"" an) with components ai E Q. If k is our fixed ground field we denote the ring of polynomials in n indeterminates by

k[XJ k[XlJ ... , Xn]. Let V An be an algebraic variety over k denoted by V/k; that is, the defining ideal of V is an ideal in k[XJ. Let u l , ••• , U T be elements of Q algebraically independent over k. Let fl' ... , fT be polynomials of k[X]. We consider the hypersurface F tI over k(u) = k(u l , ... , u T) in An defined by the equation u 1 • fl U T ' fT = O. We want to examine the general hypersurface section V n Fa over k(u). I~et P be a point of Vjk with defining ideal lJ c k[X]. We denote by A the local ring of Vjk at P. We state without proof:

+ ... +

Proposition 2.21. Assume P E V n Fa and height (iI, ... , fT) . A > 1. If P is a Buchsbaum point of Vjk then P is a Buchsbaum point of V n FIIJk(u). The wnverse is true if grade (/1' ... , fT) . A > 1 and (/1' ... , fT) • A b;:; lJ2 . A.

The next result was proven by N. V. Trung, see [2], Theorem 4. It gives informations about the lifting property in case depth M = O. Proposition 2.22. Let M be a Noetherian A -module of positive dimension d and depthM = O. M ~'s a Buchsbaum module if and only ~'j the following conditions are satisfied: (i) m· H~(M) = O. (ii) MIH~(M) is a Buchsbaum module. (iii) There is an ~V-ba8is Xl, ••• , XI of m such that H~(M) n (Xi, ... , Xid)'"V

0

for all I S il

< ... <

idS t.

Proof: Let ~V be a Buchsbaum module. Then (i) follows from Corollary 2.4, (ii) from Cmollary 1.11 and (iii) from the proof of Theorem 2.I5(iii) =:;> (ii).

Conversely, the proof of Theorem 2. 15(iii) =:;> (ii) shows that the canonical maps 41: Ht(m, llf) -i>- H~t(M) are surjective for all i < dim ~V, i.e. -LV is a Buchsbaum module by Theorem 2.15, q.e.d. The next proposition which has its origin with M. Brodmann [4], gives a further criterion for "lifting" the Buehsbaum property. It is closely related to the statements of Proposition 2.19. Proposition 2.23. Let M be a Noetherian A-module with d: dimM? 2 and depthM O. M is a Buchsbaum rnodule if and only ~1 mn:n(llf) = 0 and MjxM :(m) is a Buchsbaum module for all X E m with dim llfjx1lf d - 1.

§ 2. Cohomological characterization of Buchsbaum modules

91

Proof: If N: is a Buchsbaum module then mH:n(M) 0 by Corollary 2.4 and JfjxM :(m) = IlfjU(Xllf) is a Buchsbaum module by Corollary 1.11. Assume mH:n(M) 0 and that 1lfjxM:(m) is a Bm·hsbaum module for all x E m with dim MjxM dim MjxM:(m) = d - 1. If d 2, there is nothing else to· prove. Let d > 3. Choose x E m 2 with x It .):l for all .):l E Ass M. The exact se- M ~ ilf -)- MjxM -'>- 0 induces for all {epimorphisms

Since mHf;;l(MjxM) mH:;;l(ilfjxM:(m») = 0 for all i with 2 i < d (this follows from 111 jxM: (m) being a Buchsbaum module, compare Corollary 2.4) we have m(O:Hi (M)X) 0 for 2 i < d. Since x E m 2, this implies O:H i (M)m 2 ~ O:H' (M)X m

m

m

~ O:H' (M)m ~ O:H i (M)m 2 • Therefore

m

m

O:H:n(M)m

0:H~(M)m2

= ... =

O:H~(M)(m)

Htn(M)

and this shows that H:n(M) is annihilated by m for { 2, ... , d 1. Since mH:'t(M) 0 by our assumption (and H~(M) = 0), we have by Lemma 2.2 that for every system of parameters Xl' ••• , Xd of M the submodule (Xl> ..., xi)M is unmixed (in M) up to m-primary components for all i = 0, ... , d - 1. H x E m with dim MjX1lf = d - 1, X is a non-zero divisor of M (since 0 is unmixed in M). Therefore the exact sequence 0 -'>- M ~ M -'>- MjxM -'>- 0 induces an isomorphism H~(MjxM) ~ H:n(M), i.e. we have mH~(1ffjxM) = O. But this means m(xM:(m») xM or xM:(m) = xM:m. In particular for every y E m with dim M/(x, y) M d 2 we have xM:y xM:m. Let now Xl> ••• , Xd be an arbitrary system of parameters of M. 'Ve choose an x E m such that x, Xl> ••• , Xd-l is again a system of parameters of M. Since IJf/X"~Y :(m) is a Buchsbaum module for all n 1, it follows for all j = 1, ... , d I and all n 1 that (x",

Xl, ... ,

+ (Xl' ... , Xi-I) M):Xi = (x"M:(m) + (Xl' ••. , XH) M):m.

xH)M:Xj ~ (x"M:(m)

If we take the intersection over all n -;::: 1 we get by Krull's Intersection Theorem

in particular

+

Claim. x1M:m n (x2M:m (X3' ... , Xi) M) ~ xlM for all j For i 2 we have to prove: xIM:m n T2M:m TIM. Now x l x 2M:m XIX2.lf:X2 = XliII, hence x 1x 2M:m

(X 1x 2M :m) n xliff X1((X l X21lf:xl):m)

=

X l ((X I X 2 .1J.{ :m) :X I )

=

Xl(X2 M

: m ).

2, ... , d - 1.

92

1. Characterizations of Buchsbaum modules

Since dim M/X 1X2M = d

1, we conclude xIM:m nx2M:m 2

x1M:m n x2M:m

= (XI(x 2M:mx 1»):m X1X2M:m Let 3:;; j

(xiM n (x2M:m)):m

= (X 1(x 2 Jf:m)):m = (x1x 2M:m);m

XI(x2)1I:m)

xiM.

d - 1 and mE (xIM:m) n (x2M:m + (xa, ... , Xj) ill). Then m = U2 U 2 E x2M: m, ma, ... , mj E M. Therefore

+ X3ma + ... + xjmj with mj E (xIM:m

+ X2M:m + (X3' ..• , xj-l) M):x;

(XI1 X2, ••• , Xi-I) M: (m) = (Xl' ..• , x;-d M : m,

+ ... + mj_1Xi-1' But this implies m~xI = U2 + ~X2 + (ma + m~)xa + ... + (mj_l + mj_I)Xj_l E x 111I:m n (x2M:m + (xa, ... , xj-l) M) x1Jf,

, i.e. we have xjmj =m~xI

m

i.e. m E x1M which proves the contention. Now we have (M/x1M;m is a Buchsbaum module): m( (Xl' ... , Xd-l) M :Xd) ~ m( (xIM; m ~

and exchanging Xl and

xlM; m

+ (x

2 , ... ,

Xd-l) M) :Xd)

+ (X2' •.. , Xd-l) M

X2

m((xI' .•. , Xd-l) M: Xd) ~ x2M: m

+ (Xl' xa, ... , Xd-l) ill.

Hence

m((Xl' ... , Xd-l) M:Xd)

+ (X2' ... , xd-d M) n (x 2Jll:m + (Xl' X Xd-l) M) (X2' X3, ... , Xd-l) M + ((xIM :m) n (x2111:m + (Xl' xa, ... , Xd_1M») (XI1 X2, ... , Xd-I) M + ((xIM:m) n (x 2M:m + (xa, •.. , Xd-l) "If))

~ (xIM:m = =

3 , ••• ,

(Xl' ... , Xd-l) M.

Therefore by Proposition 1.10 M is a Buchsbaum module, q.e.d. Corollary 2.24. Let M be a Noetherian A-module with d;= dim M 3 and depth M > O. Assume lurtherrrwre that either A Z8 an epirrwrphic im,age 0/ a Gorenstdn n'ng or Hin(M) is a Noetherian module. Then M i8 a Buchsbaum module il and only il MJxM:(m) is a Buchsbaum module lor aU X E m with dim M/xM = d 1.

Prool: The "only-if-part" needs no further elaboration. Assume that M /xM: (m) is a Buchsbaum module for all X E m with dim 11l/xM d - 1. Suppose A is an epimorphic image of a local Gorenstein ring. If there is a ~ E Ass M with 1:;; dim Aj~ < d - 1 then (O:M.\.l)V O:Mv.\.lAv O. We choose an X E q with X ~ q for all q E AssM with dimAjq = dand xMv n (O:Mv.\.lAv) = O. Then

'*'

Hom(Av/)'Av, xMv) '" O;.Mv.\.lAv = xMv n (O:Mv.\.lAv) = 0

§ 2. Cohomological characterization of Buchsbaum modules

93

and the exact sequence 0 -+ xMlJ -+ MlJ -+ (M/xM)lJ -+ 0 induces a monomorphism Hom(AlJ/vAlJ' MlJ) -+ Hom(AlJ/pAlJ, (M/xM)lJ)' Since Hom(AlJ/pA lJ , MlJ)::::::: 0 :MlJVAlJ i= 0, it follows that Hom(AlJ/VAlJ, (MlxM)lJ) i= 0, i.e. VEAssM/xM. Since AssMlxM:(m) = AssM/xM'-.{m}, we obtain fJ E AssM/xM:(m). Also since dim M/xM d - 1, M/xM:(m), is a Buchsbaum module, hence dim Alp = d - 1 2. Let A be an epimorphic image of the local Gorenstein ring B. Then by the local duality theorem (cf. Corollary 0.3.5), H:n(M) Hom,4(Ext~-1(M, B), E), where E denotes the injective envelope of A/m (as an •. A-module) and n dim B. Let H:n(M) i= 0 and V E ASSAExt~-l(M, B). Then dim A/V :S: 1 by Sharp [2], Proposition (3.8) and Theorem (2.3), and we have with q denoting the inverse image of fJ in B): r-.J

Ext;-l(M, B)lJ ~ Ext~;l(2l!lJ' Bq) where Bq is a Gorenstein ring of dimension n 1 and AlJ is an epimorphic image of B q • If I denotes the injective envelope of the AlJ-module AlJ/VAlJ' then by local duality HOmAlJ(Ext~;l(MlJ' Bq), I)~ H~AlJ(MlJ)

0

if dim A/V

=

1

as was shown previously. Hence dim Alp 0, i.e. p m. Thus Ext~-l(M, B) is a module HomA(Ext~-l(M, B), E) is a module of finite of finite length and therefore H:n(M) length, i.e. a Noetherian A-module. Next assume that H:n(M) is Noetherian. Take an element x E m with dim M/xM = d 1 and xH:n(M) = O. Then from the exact sequence 0 -+ M":"'" M -+ M/xM -+ 0 we get a monomorphism H:n(M) -+ H:n(Mjdf) H:n(lrf/xM :(m»). The last module is annihilated by m since lrf/xM :(m) is Buchsbaum and 1 < d 1 dim M/df:(m). Hence mH:n(M) 0 and the Corollary follows from Proposition 2.23, q.e.d. r-.J

r-.J

Another application of Theorem 2.15 gives informations on the Buchsbaum property of a local ring whose zero ideal is the intersection of two "perfect" ideals. We note that the ideals "of type (r, d)" defined above (Definition 2.13) belong to this setting, see also Lemma 2.14. Additionally the following statement has some useful applications with respect to liaison among arithmetically Buchsbaum curves in pl!.

Proposition 2.25. Let A be a local ring with d dim A 2 2 and a, b ideals of A with a n b = 0, dim A/a + b < d. Assume A/a and A/b to be Oohen-Macaulay rings of dimension d. Then A i8 a Buchsbaum ring if and ooly if either a + b = m or B A/(a + b) ~'s a Buchsbaum ring of dimeru:;wn d - 1. Proof: As in the first part of the proof of Lemma 2.14 we have an exact sequence:

O-+A -+A/a(f)Alb-+B-+O. Since depth A/a

depth Alb

HH(m, B)

lA~-1

~

= d, we have for all i < d commutative diagrams:

H'(m, A)

lA~

H:;;-l(B) ~ H:n(A).

By Theorem 2.15 A is a Buchsbaum ring if and only if ).~ is surjective for all i

d - 2.

94

I. Characterizations of Buchsbaum modules

If dim B 0, i.e. RJ.n(B) = 0 for all j 1 this is equivalent to the surjectivity of ).~. But RO(m, B)~ O:Bm~ (a + 0) :m/(a + 0) and R?n(B) B = A/(a + 0) and thus A is a Buchsbaum ring if and only if a + 0 = m. If dim B > 0, ).ltm B cannot be surjective since R'!t:mB(B) is not a module of finite length. Since dim B < d - 1, A is a Buchsbaum ring if and only if dim B d - 1 and ).1 are surjective for all j < d - 1, i.e. if and only if B is a Buchsbaum module over A by Theorem 2.15 and hence a Buchsbaum ring itself by Lemma 1.6, q.e.d. •

Another application of Theorem 2.15 was proven by U. Daepp and A. Evans in [1]. In order to formulate this result we need to introduce some further notions. Let M be an A-module and X an indeterminate. Then we set M*:= 1'1:f[X]m[xj,

where m[X] denotes the kernel of the map A[X] -+ (A/m) [X] given by the canonical projection A -+ A/m, i.e. m[XJ is a prime ideal of A[XJ. Now, it is clear that M* M @AA* and that the natural map A -+ A* is a local .flat homomorphism. For every A-module N of finite length lA.(N*) = lA(N). This implies for example eo(q*, M*) = eo(q, M) for every Noetherian A-module M and every ideal q c A with l(Mlq· M) < 00. • For each ideal a c A we have a* = a A* a· A* and thus for the Koszul complex K.(a· A*, M*)

~

K.(a, M)@AA*

(as complexes).

Therefore Ri(a*, M*) ~ Ri(a, M)@AA* and R~.(M*) R~(M) @AA* since tensor products commute with direct limits. Consequently, (m* is the maximal ideal of A*) dimAM = dimA.M*. Now, we are in position to prove.

Lemma 2.26. Let M be a Noetherian A-module 0/ positive dimension. M is a Buchsbaum module over A if and only if M* is a Buchsbaum module over A. Moreover, [(M*) = [(M). Proof: We have ).it.:::::: ).it®AidA• and since ®AA* is an exact functor, ).it. is surjective if and only if ).it is surjective which proves (using Theorem 2.15) the first statement. Next, let M (and therefore M*) be a Buchsbaum module. Let q be a parameter ideal of M. Then q* is a parameter ideal of M"" and [(M"") = l(M*/q*' At*} - eo(q*, M*)

=

l(Mjq. M)

eo(q, M)

=

l(Mlq· M)*) - eo(q*, M*}

[(M) ,

q.e.d. The previous lemma has an interesting consequence which was first proven for a special case by Daepp and Evans in [1]. To establish this we need a lemma (for notations see Chap. 0, § 2, 1.):

Lemma 2.27. Let R be a graded ring and aS8ume that ~ c R is a homogeneoos prime ideal with [R]l g;; ~. Then Rp""",Rtp)

§ 3. Graded Buchsbaum modules

95

and hence we have for every graded R-module M: MlJ~MrlJ)'

Proof: This follows immediately from Lemma 0.2.1. (It is sufficient to prove RlJ ~ RrlJ)' From Lemma 0.2.1 we obtain RlJ ~ (RlJ.h)lJ.RlJ.h ~ (R(lJ)[X]x)m[x]x ~ R(lJ)[X]m[x] = Rrw where m is the maximal ideal of R(lJ)')

Corollary 2.28. Let R be a Noetherian graded ring and let M denote a Noetherian graded R-rrwdule. Then we have for every homogeneous prime ideal tJ c R with [RlI g;; tJ: M lJ is a Buchsbaum rrwdule if and only ~'j M(lJ) ~'s a Buchsbaum module. In this case, I(MlJ) =I(M(lJ))'

§ 3.

Graded Buchsbaum modules

The goal of this paragraph is to expand the results of both previous paragraphs to graded modules. The geometric background and the motivation for this is to get information about the Buchsbaum property of the local ring at the vertex of the affine cone over a projective variety. Throughout we use the concepts introduced in Chapter 0, § 2. In addition we always suppose that our graded k-algebras (k a field) are generated by their homogeneous elements-of degree one, i.e. they are of the form k[Xo, ... , Xn]ja where Xu, ... , Xn are indeterminates (of degree 1) and a is a homogeneous ideal of k[Xo, ... , X n]. Let R always denote such a graded k-algebra with the maximal (homogeneous) ideal m = ffi [R]n. n:2:1

Definition 3.1. Let M be a Noetherian graded R-module of positive dimension. M is called a Buchsbaum module if Mm is a Buchsbaum module (over Rm)' M is called an h-Buchsbaum module if every homogeneous system of parameters with respect to M is a weak M-sequence. Thereby weak M-sequences are defined analogously to the local case. It is clear that M is an h-Buchsbaum module if it is a Buchsbaum module. Our goal is to study the converse of this statement. We are able to prove it if k is an infinite field. This means, geometrically speaking, that the Buchsbaum property of the local ring of the affine cone at the vertex over a projective variety may be verified by regarding only homogeneous systems of parameters. In a similar way as in § 1,3f-bases consisting of homogeneous elements (M a Noetherian graded R-module) of homogeneous ideals a with dim Mja . M are defined. Therefore we omit an additional definition. But in contrast to the local case such bases ~ay not exist. We give two examples:

1. In a homogeneous basis of a appear elements of different degrees : We choose R = k[X, Y], a = (X, Y2) . R, M = RjX . R (X, Y indeterminates). 2. The field k is finite: We choose a = m and M := Rjp . R where p denotes the product of all elements of degree one in R. If we exclude these two cases we are able to prove the existence of M-bases. The proof is essentially the same as in the local case (Proposition 1.9). In addition we need here the following easy

96

1. Characterizations of Buchsbaum modules

Lemma 3.2. Assume that the ground lield k is inlinite. II Yb ... , Yt are elements 01 [RJ/, 1 > 0, and il there are homogeneous ideals hI> ... , h. z'n R with (Yt> ... , Yt) R g; h; lor all i 1, ... , s, then there are elements lXI' ••• , IX, 01 k with IXI •

Yl

+ ... +

<XI'

Yt

q hi

U ...

u h •.

Prool: Let V t;:;;;; [RJ/ be the vector space spanned by the Yi'S. We put Vi V n hi' i = 1, ... , s. Since (YI' ... , ytl . R hi for all i, Vi is a proper subspace of V for each i. Since k is infinite, this implies V n (hi U ... u h.) = VI U •.• u V. ~ fT, hence we get V hI U ... u h., q.e.d.

Now we can" prove: Proposition 3.3. Assume that the ground lield k is z'nlinite. Let a R be an ideal possessing a basis whz'ch consists 01 homogeneous elements 01 the same degree r and M I , " ' j Mn Noetherian graded R-modules with dimR MdaM; 0 lor all i = 1, "', n. Then there are all ... , at E a lorming a homogeneous M;-basis 01 a lor all i = 1, n. "'j

Prool: We can use the proof of Proposition 1.9 word for word with only one modification: when constructing am we simply choose an element of degree r" not contained in (at> ••• , am-I) R u U V (apply Lemma 3.2), q.e.d. PEL

Also we have: Proposition 3.4. Let M be a Noetherian graded R-module with d: dim M > O. The lollowz'ng conditz'ons are equivalent: (i) There Z8 a homogeneous system 01 parameters 01 M contained in m2 whwh is a weak M -sequence. (ii) Every homogeneous system 01 parameters 01 M contained in m 2 is a weak M-sequence. (iii) m· !l:n(M) 0 lor all i =F d.

Proal: (ii) =} (i) is trivial and (i) =} (iii) may be verified as in the proof of Proposition 2.1. We only have to pay attention to the necessary shifting of degrees. Finally, we obtain the implication (iii) (ii) by localizing at m and applying Proposition 2.1, q.e.d.

Likewise by localizing at m and applying Theorem 2.10 we obtain Theorem 3.5. Let M be a Noetherian graded R-module with dim M maps (! Rim)

> O.

II the natural

glk: Extk(!, M) .....,. !l~(M)

are surjectz've lor all i

< dim M then M

is a Buchsbaum module.

Corollary 3.6. Let M be as z'n Theorem 3.5. II in addition r:= depth M < dim M and !l~(M) = 0 lor all r, d then the lollowz'ng conditions are equivalent: (i) M is a Buchsbaum module. (ii) M is an h-Buchsbaum module. (iii) m· !l~(M) = O. Prool: (i) =} (ii) is clear, (ii) =} (iii) follows from Proposition 3.4 and (iii) if we localize at m and apply Proposition 2.12, q.e.d.

Now we prove the main result of this paragraph:

=}

.d

(i) we obtain

§ 3. Graded Buchsbaum modules

97

Theorem 3.7. Assume that k is an infinite field. If M is a Noetherian graded R-module w#h d := dim 1W > 0, the following corul#ions are equ~'valent: (i)

M is a Buchsbaum module.

(ii) M £s an h-Buchsbaum module.

(iii) Take a homogeneous llf-basz8 Xl, ••• , XI of m. Then for each system'il' ... , Zd of £ntegers until, 1 £1 < ... < ia - t the sequence x'\ . .. , X~d is a weak M-sequence for all ~ ~ rl, ... , rd E {I, 2}.

<

(iv) The natural maps

lie: lli(m, M) -l>-ll~(M) are surject£ve for all i

<

d.

If R is a free k-algebra then (i)-(iv) are equ£valent to (v) The natural maps

-ll~( M) are surjecl£ve for all £ < d.

The proof is not difficult. The implications are either self evident or follow by localizing at m and applying Theorem 2.15 and Corollary 2.16. The following example shows that our assumption on k is necessary. It implies that (for finite k) the conditions (i) and (ii) of Theorem 3.7 are not equivalent. Example 3.8. Let k be a finite field. Choose a polynomial ring R k and a Noetherian graded R-module M such that: (i) M is not a Buchsbaum module, dim M

3 and depth M

>

k[Xh ... , Xn] over

O.

(ii) m· ll~(M)

0 for all £ < dim M. We note that such Rand M exist (see Example V.5.4). We have to distinguish the following cases: 1. Each linear form of R is not a part of a system of parameters of M, i.e. all homo geneous systems of parameters of M are contained in m 2 • Then M itself is an h Buchsbaum module by applying Proposition 3.4. 2. There is (at least) one linear form l of R with dim Mil· M < dim M. Let p be the square of the product of all these linear forms and set N := Mjp· M. We note that p is a non-zero divisor of M contained in m 2• Let Xl' "0' Xa be a homogeneous system of parameters of N. Then Xl> •• " Xd E m 2 and by Proposition 3.4 p, Xl, ••• , Xd is a weak M-sequence. Therefore Xl> ••• , Xa is a weak N-sequence and N is an h-Buchsbaum module. If N would be a Buchsbaum module, i.e. N m is a Buchsbaum module over Rm , the lifting property (Proposition 2.19) implies that Mm is a Buchsbaum module. This is a contradiction and we have found an h-Buchsbaum module which is not a Buchsbaum module. As an application of our last theorem we will state a new sufficient criterion for graded Buchsbaum modules using only local cohomology modules. To this end we need: 7 Buchsbaum Rings

,

98

1. Characterizations of Buchsbaum modules

Lemma 3.9. Let R:= k[Xh .••, X,,] (Xl' ... , Xnindeterm~'nate8) and let H be a graded R-module with m . H = O. Then

Extk(~, H):::: HomR(R(r)( -i), H)

for all i> O.

H(7)(i)

Proof: The graded Koszul complex K.(Xh ••. , X .. ; R):

o -+R(:)(-n) -+ ••• -+R(;)(-2) -+ R(7)(-l) -+R-+O provides a free resolution of (m.HomR( ,H)

~

Rim. Applying HomR( ,H) we get for all i> 0

= 0):

Extk(~, H)

HomR(R(r)( -i),

"-J

H):::: HomR(R(r), H(z,») '"'-' H(f)(£) ,

q.e.d. For abbreviated notation we define for each graded R-module M the following set of integers: g(M) := {i E Z I [M]j 9= O}. Proposition 3.10. Let M be a Noetherian graded R-module with d:= dim M > 0 and m . !!:n(M) = 0 for all i < d. If for each pazr of integers i, j with 0 ~ i < j < d and all p E g(lt:nCLlf»), q E g(l:!~(M»), (z'

+ p)

(j

+ q) 9= 1

then M is a Buchsbaum module. Proof:IFirst assume R = k[X}> ... , X .. ], Xl' ... , X" indeterminates. If 0< i then for q E g(ltfn(M» (by Lemma 3.9):

[Extk-i+l(!, It:n(M) )]q

[It:n(Mh-r+I)]q+j-i+l i + 1 Et g(lt:n(M»).

< j
0,

, since by our assumption q + j Now let 0 -+ 1° -+ [1 -+ '" be a minimal graded injective resolution of M and set Ji:= lt~([i). We examine the corresponding complex O-+JO~Jl~ ...

Set Bi:= 1m d i , Zi:= Ker d i • Then there are exact sequences

o -+ Bt-l -+ ZI -+ ltfn(M) -+ 0

(for all j

and isomorphisms

I ,

HomRC!, Zl) The map Zi

"-J

-+ lt~(M)

Ext~(!, M)

0)

ExtiRC!, M)

(since the resolution was minimal).

induces a homomorphism ni:

"-J

HomRC!, Zi) ~ HomR(!,

ltfn(M») '" ltfn(M).

The composition of these maps is just the natural map p.L-. By induction on j we show the surjectivity of ni. Since m ·lt~(M) = 0, nO is even an isomorphism. Let 0 < j < d. Then we have for each q E g(ll!n(M») an exact sequence

0-+ [HomR(!, BH)]q -+

[ExtM!, Bi-1)]q.

-+

[HomR(!, ZI)]q ~ [HomR(!,

ltfn(llf»)]q

§ 4. Segre products of graded Cohen-Macaulay modules

For i < j we have by the induction hypothesis and by Lemma 2.9 for each l sequences o ....,. Ext~(!, BH) ....,. Ext~(!;, Zi) ....,. Ext~(!, H~(M)) ....,. O. Since Ext~(!;, BH) c:::::: Ext~l(!, Zi-l) for alIl > 0, i (Ext~(!;,

[ExtM!;, BH)]q

99

0 exact

> 1 we get

ZH)]q c:::::: [Exti(!, BH)]q

'" [Ext~(!, ZH)]q '" ... c:::::: [ExtkI(!, H~(M))Jq

[Ext~(!, Jj6)]q

O.

Moreover, for r ~ g(H~(M)) we have [HomR(!;, H~(M))lr £H~(M)]r = 0 and therefore (ni]; is surjective for all z· E Z. Hence ni is surjective and M is a Buchsbaum module by Theorem 3.5. ' Now letR be an arbitrary graded k-algebra which is an epimorphic image of S k[X 1 , ... , X,,]. Letn:= (Xl' ... , X,,)· S. ThenH~(M) and H~(M) are isomorphic as S-modules. Hence M is a Buchsbaum module over S and consequently also over R by Lemma 1.6, q.e.d. We note that this criterion is not necessary. To show this we will provide an example using Segre products of graded modules, see Example V.5.5.

§ 4.

Segre products of graded Cohen-Macaulay modules

In this paragraph we will investigate Segre products of graded Cohen-Macaulay modules in order to get additional examples of Buchsbaum modules. In this context we obtain as an easy consequence the results of W. L.'Chow on the Cohen-Macaulay property of such Segre products (for our special graded k-algebras), see Chow (1]. Throughout this paragraph we make the following assumptions: 1. Our ground field k is infinite. 2. Cohen-Macaulay module always means Noetherian graded Cohen-Macaulay module. 3. Our graded k-algebras are finitely generated by their homogeneous elements of degree one. Always let R I , R2 denote two graded k-algebras with maximal homogeneous ideals m!> reap. m2' We set R a(RI' R 2 ) and m := a(ml> m2)' m is then the maximal homogeneous ideal of the graded· k-algebra R, see also Chapter 0, § 2, 4. First we state and prove some preliminary results: Lemma 4.1. Let R be a graded k-algebm. Suppose that M i8 a Noetherian graded R·module with d := dim M > O. We hwve: (i) If depth M > 1, then [M]" =t= 0 for all n a(M).

for all n < e(H::"UW)) (a(M), e(H::"(M)) are defined in Chapter 0, Definition 2.2).

(ii) m· £Hiit(M)]" =t= 0

Proof: (i) If [M]p

0, then we have for all q < p:

[R]p_q' [M]q <;:;; [M]p since depth M 7*

1. Hence p

0, i.e.

< a(M).

[M]q

O:JdR]p-q

=

O:Mm p- q <;:;;

T

1. Characterizations of Buchsbaum modules

100

(ii) Since d 1, !l'fn(M) !l'fn(MI!l~(M)), i.e. we may assume depth M 1. Let x E [Rh be a non-zero divisoroLif (kisinfinite!). For each t 1 let M t Mix!, M. Then dim Me = d 1 and the exact sequence 0 -)- M ~ M(t) -+ Mt(t) -+ 0 yields an epimorphism

!lifn(M) ~ !lifn(M(t)) <= !lifn(M) (t). From this we obtain for each n E Z epimorphisms

[!l'fn(M)],. ~ [!l'fn(l1f)],,+t· If ill' [!l~(M)]p

0, then we have for all q 0,

[!l'fn(M)]q

p:

e(!l'fn(1lf)) ,

p

i.e.

>

q.e.d. Corollary 4.2. Let M I , Ma be Noetherian graded R r , resp. Rt-modules with d j := dim 1lf; > 0 lor i = 1, 2. Then

dim a(M1 , M i ) = d 1

+d

1.

2

Prool: Let M:= a(M 1, M a), d := d1 Proposition 0.2.1(ii)) !lfl(M)

EB

+ d2 -

1. Then by the Kiinneth formulas (see .

a(!lP(M1 ), !lq(M2 ))

for all n.

p+q~n

Since !lP(M1) 0 for all p > d1 1, !lq(1lf2 ) 0 for all q > d2 1, we get !lfl(M) for all n > d l + d a 2, i.e. dim M d. Now by Lemma 4.1 there are e1, e2 E Z such that [!ld,-I(M;)]", =F 0 for all nj i = 1, 2. Hence

=0

!ld-l(M) ~ a{!ld,-l(M1), !l'io-I(M2)) =F O. If d

>

1 this implies !lifn(M) =F 0, i.e. dim M = d. Let d 1. If dim M = 0, then M and !lfn(M) = O. Therefore !l°(M) 0 which is impossible, q.e.d.

!l~(M) ~

Lemma 4.3. Let R be a graded k-algebra and M (( Noetherian graded R-module with d ;= dim M > O. 111W is a Cohen-l1facaulay module then

HM(n) - hM(n)

(-I)d rankk[!l'fn(M)]n. !

Prool: By Serre [1], Nr. 79, we have for arbitrary Noetherian graded R-modules M:

E

hii(n)

1)/ rankk(Hi(X, M(n))

for all n E Z

;:?i:O

where X := Proj Rand 1W denotes the sheaf associated to M. For all 0 and n E Z one has (see Proposition 0.2.3)

Hi(X, iJf(n))::::. H((M(n)) = [!l'OW)]" and therefore

hM(n)

hii(n)

=

rankk[!l°(M)]"

+E

1)' rankk[!l::l(M)],.

j~l

=

rankk[M]" -

E j~o

1)/ rankk[!l:n(M)],.

§ 4. Segre products of graded Cohen-Macaulay modules

is obtained. Since rankk[M]n = HM(n) and since by our assumption H:n(M) all i =l= d, the statement now follows, q.e.d.

101

=0

for

Together with Lemma 4.1(ii) this implies Corollary 4.4. Let B, M be a.s in Lemnw 4.3. Then (see Definition 0.2.2): (i) r(M) = 1 + e(ll~{M»). (ii) r(M) = inf{n E Z I hM{n) = HM(n)}.

Our next result enables us t{) calculate the index of regularity rIB) of a graded CohenMacaulay algebra B without knowledge of the Hilbert function Hn(n) of R.

Lemma 4.0. Let B denote a graded Oohen-Macaulay algebra over k with d:= dim B > O. Then we have for every system of parameters Xl, •.. , Xa with respect to R which i8 contained in [Blt: d

+ r{B)

inf{t E H I mt

(Xl' ••• ,

xa) . B} .

inf{t E M I mt <;; (Xl' ... , xa) . B}, r r(R) and S := R/(x l , Proof: Let to Since B is a Cohen-Macaulay algebra we have for all t 0:

• '"

Xd) . R.

{io e

Hn(l) =

and therefore kR(t)

i +d d-l

ifo (t

1) Hs(i) ..

The validity of the first equation follows by an easy induction argument (induction on d). Now, HR(r) = hR(r) and i +dd- 1

Therefore }; (r i;::>r+a

(r _ i

+d-

1)

d- 1

1) Hs(i)

0 for all i

r

+ 1, ... , r + d

O. All binomial coefficients occuring in this sum

have the same sign. Hence Hs(i) 0 for all d + r, Le. mdh Thus to d + r. On the other hand m'· <;; (Xl> ... , Xtl) • B, i.e. Hs(i) Since (to

i-I)

d-1

hR(to - d)

=

0 for all i = to - d I.~a

"'"

(to - d d

i=o

Now Corollary 4.4{ii) implies r

i

1.

C (Xl' ... ,

Xa) . R.

0 for all ito'

1, ... , to - 1, we have

+d 1

to - d, i.e. to

d

+ r, q.e.d.

We are now able to prove the main result of this paragraph:

Theorem 4.6. Let Mv M2 be Oohen-Macaulay modules over Rb resp. B2 with d,:= dim M 2 for i = 1,2. Then (i) I1(Mb M 2 ) i8 a Oohen-Macaulay module if and only if

r(M 1 )

a(M2)

and

r(M2)

a(M 1 ).

(ii) I1(MI' .!.lf2 ) is a Buchsbaum module if and only if r(Md

< 1 + a(~~f2)

and

r(Ma)

1

+ a(M}).

102

I. Characterizations of Buchsbaum modules

Proof: Let M:= O'(Mh M 2 ). By Corollary 0.2.12 we have depth M 2, i.e. H~(M) = Hfn(M) 0, HO(M):::::: M. Also by our Ktinneth formulas Proposition 0.2.10 we have

H"(M) = 0

for all n :9= d l

1, d2

-

-

1, d l

+d

2

2 -

and

H d .-1(M) "-' 0'(Hd .-1(M1 ), HO(M2 )} Hd.-I(M):::::: 0'(HO(M

d I 1 ), H .- (M2 )) ' "

O'(H~l(MI)' M 2 } , } 'f d 'd 1

O'(MI' Hi!:.(M2 J}

I

'1=

2'

H'J-1(M):::::: 0'(HH(M1 ), HO(M 2 )) E9 0'(HO(M 1 ), H d - 1(M 2'r) :::::: O'(H~,(MI)' llf2) E9 O'(MI' H~,(M2))

if d l = d2

:

d, resp.

By the commutative diagrams of Corollary 0.2.11 we obtain commutative diagrams: O'(Ext~~(~,

M I), M2)

-+ Ext~l(~,

lq(~:;l·idM') 0'(H;;:.(M1 ), M 2)

M)

-+

Ext~C~, M)

a(Ml> Hi!:,(M2 )}

-+

H~(M)

1q(ldMl'~~.)

l~:r -+

O'(MI' Ext~:(~, M 2))

ll~(M)

1~~

if dl =1= d 2 and a similar diagram if dl = d2 • Now, M is a Cohen-Macaulay module if and only if H:n(M) 0 for all i < d l + d 2- 1, . i.e. if and only if O'(H;;:,(llfl }, M2) = O'(MI' Hi!:,(M2l) = O. But this is equivalent to (Lemma 4.1(i), (ii) and Corollary 4.4(i))

r(M1} = 1 + e(H~,(Ml})

a(M2)

and

r(M2 )

= 1 + e(Hi!:,(M2))

a(M I}.

Next assume that M is a Buchsbaum module. Then m . H:n(M) = 0 for all i 1 by Proposition 3.4 and, consequently, 2 -

+d

< dl

and Om· O'(Ml> Hi!:,(M2)} But depth

mj'

O'(m l • M I, m2 • Hi!:,(M 2)).

> 1 for i = 1,2 and by Lemma 4.1 1 + e(H;;:.(M I)) 1 + e(m Hf;,(M I)} ~ a(m2 . M 2) = 1 + a(M2)

.ilf i

r(M I) =

l •

and, by exchanging MI and M2

Conversely suppose that these both relations hold. We want to show that the canonical maps tpk are surjective for all i < d l + d2 - 1. By Theorem 3.5 this will prove our statement. According to the above commutative diagrams it will be sufficient to prove the surjectivity of O'(tp'j;., idM .) and O'(idM ., '1'1;,). If a(M2) > e(Hf;.(M I)), O'(H;;:.(M I), M2) 0 and there is nothing to prove. Assume therefore a(M2) = e(H;;:,(M1 )} =: e.

103

§ 4. Segre products of graded Cohen.Macaulay modules

Now, [0'(Hg;,(M 1), M 2)]" = mg;,(M1)]PQ9k[M2 ]p = 0

for all p =1= e,

I.e. [0'(9'~" idM,)]" is surjective. It is also easy to see that Ext'1i,(!,Ml)~Homll,(k,Hg;,(Ml))::::O:Hd'(M,)ml and -m, that 9'~1 is the corresponding embedding. On the other hand mI' [Hg;,(M I )]. 0, i.e. [Hg;,(MI)]e c [O:!!~I(M,)md•. Hence [9'~.l. is even an isomorphism and therefore [a(9'~" idM .)]. is also an isomorphism. Thus 0'(9'~" idM .) is surjective. Exchanging Ml and M2 we obtain the same for a(idM" 9'~.) and the proof is finished, q.e.d.

For dim Ml = 1 or dim M2 We do this in the following

=

1 we need another formulation of our statement.

Lemma 4.7. Let Mv M2 be as in Theorem 4.6, but dim Ml • hat'e for M O'(Ml> M 2): (i) If dim M2

=

1, dim M2

> 1.

Then we

1, M is a Cohen-Macaulay module.

2, M is a Cohen-Macaulay module if and only if r(M I ) M is a Bucksbaum module if and only q r(M I ) 1 a(M2)'

(ii) If dim M2

+

Proof: (i) By Corollary 4.2 we get dim M

=

a(M2) and

1 and by Corollary 0.2.12: depth M

1.

(ii) is obtained by the same methods used in the proof of Theorem 4.6, q.e.d. Next we state two corollaries of Theorem 4.6. The first is a main result of Chow (1] for our graded k-algebras and the second gives a very easy method for calculating the Cohen-Macaulay resp. Buchsbaum property of graded complete intersections.

Corollary 4.8. Let Rh R2 be graded Cohen·Macaulay algebras over k with d; : = dim Ri 2 for i = 1,2. O'(RlI R 2) is a Cohen-Macaulay algebra if and only if RI and R2 are proper k-algebras, ~·.e. there are systems of parameters Xl>"" Xd, of Rl and YI' ... , Yd, of R2 consisting of homogeneous elements of degree one such that m~' <;:: (Xl> ... , Xd.) . Rl and mg· <;:: (Yl> ... , Yd,) • R 2• Proof: By Lemma 4.5 we have for i proof is finished, q.e.d.

=

1,2: d i

+ r(Ri) ::;; d i •

Corollary 4.9. Let R1:=k[XI, ... ,Xn ], R 2 :=k[Y1 , ~1uJeterminates)

and let al = (/1> ... , fr) . Rl of the lJNrwipal cla88 r, re8p. 8 with n - r, m

(i) O'(RIJalJ R 2/a2) is a Gohen-Macaulay algebra if and only if r

8

E deg /; ::;; n

and

;=1

;=1

m.

j=l

r

E degJ.

E deg gi::;;

n

+1

s

and

E deggj j=l

0, the

Y m ] (X 1, ... ,Xn , Y 1 , ... , Ym (gll . '" g.) . R2 R2 be ideals 2. Then we have: .... ,

Rl> a2 8

Since a(R,)

m+ 1.

104

I. Charact.erizations of Buchsbaum modules

Proof: We apply again Theorem 4.6 and notice that for instance r(Hl/a l ) = -n

r

+ 1: deg /;

(see Grabner [1], 142.) ,=1 The modifications of these corollaries to the case dim Rl = 1 or dim R2 1 (Corollary 4.8) or n - r 1 or m s = 1 (Corollary 4.9) are easy to obtain (use Lemma 4.7) and are left to the reader. By reason of the following geometrical discussions we still prove the following

Proposition 4.10. Let llfi' llf2 be Noetherian graded Rr resp. R 2-modules with d,:= dim llf. 2 for i = 1,2 and let llf aUlll> llf2 ). llf is locally Cohen-llfacaulay and equid~men~ional if and only if llfl and llf2 are locally Cohen-llfacaulay and equidimensional. Proof: By Corollary 0.4.15 and the existing relations between the local cohomology modules l!:n(llf) and the cohomology modules !P(llf) := !Jl(R, llf) l~ Ext~(m", llf) n

(d. Chapter 0, § 2, 3.) we have that llf is a locally Cohen-Macaulay module if and only 1) since if !Ji(llf) are Noetherian R-modules for all i < d l + d 2 - 2 (= dim llf Rm is an epimorphic image of a local Gorenstein ring. Further, we recall that the Segre product of two Noetherian graded modules is again a Noetherian graded module and that the Segre product of a Noetherian and an Artinian graded module is a Noetherian (and Artinian) graded module. If 1lfl and M2 are locally Cohen-Macaulay modules these remarks and our Kiinncth 2, formulas (Proposition 0.2.1O(ii)) imply that !In(llf) is Noethcrian for all n < d l + d2 i.e. ill is a locally Cohen-Macaulay module. Conversely, assume that llf is a locally Cohen-Macaulay module. Let 0 i < d l - 1. Then again by our Kiinneth formulas a(!J'(llfl ), !Jd.-l(llf2)) is a direct summand of !Ji+d.-l(llf) and hence Noetherian. But there is an integer e such that [!Jdd(llf2 )]p =4= 0 for all p e (see Lemma 4.1(ii)). Therefore there must be an integer a with [!Ji(llfl)]q = 0 for all q < a, i.e. !Ji(llf 1 ) is Noetherian. Hence llfl is a locally Cohen-Macaulay module. Exchanging llfl and llf2 we obtain that llf2 is also a locally Cohen-Macaulay module, q.e.d. We next state some corollaries and make some comments in a geometric context.

Corollary 4.11. Let V p.. and W <;;;;; pm be arithmetically Cohen-Macaulay varieties with p~"tive dirnensions in projectz've n-space resp. m-space. Then the Segre embedding S( Vx W) of V and W in p,.+m+"·m is an arithmetically Cohen-llfacaulay variety if and only if the arithmeti£ genus of V and W is O. We note that we mean by the arithmetic genus Pa(X) of a projective variety X the following number: Pa(X) = (-1 )dim X (hx(O)

1)

where hx(O) is the constant term of the Hilbert polynomial of if) x (see Chap. 0, § 2, 3. and 2.). Now the corollary follows immediately from Theorem 4.6 and Corollary 4.4(ii). Since for non-singular curves the arithmetic genus agrees with the usual (geometric) genus, we obtain from this corollary better information than does A. Seidenberg in [2] for the construction of arithmetically normal irregular surfaces free of singularities. First, we prove the following result of A. Seidenberg [2].

§ 4. Segre products of graded Cohen-Macaulay modules

105

Corollary 4.12. Let G, H be plane curves without singularities and such that at least one oj them has positive genus. The Segre embedding oj G X H l:S than arithmetically normal non-Coken-Macaulay surjace F jree ojsz'ngularities. Prooj: Corollary 4.11 shows that F is an arithmetically non-Cohen-Macaulay surface. Let A be the local ring at the vertex of the affine cone over F. Corollary 0.2.12 implies depthA 2. Clearly F is free of singularities. Therefore Serre's characterization of normal rings (see, e.g., Matsumura [1], p. 125) shows that F is an arithmetically normal surface, q.e.d.

From the foregoing we obtain immediately an example: Example 4.13. Let G be the cubic curve defined by the equation Xi . X 2 - X~ + Xo . Xi = 0 in and let H = Ill. Let F be the Segre embedding of G X H in lis. Let A be the local ring at the vertex of the affine cone over F. By Corollary 4.12, A is a normal non-Cohen-Macaulay ring. By Corollary 4.9(ii) we obtain that A is a Buchsbaum ring.

'2

These observations render moreover for every given dimension d;:::: 3 and depth t 2, d> t, normal (local) Buchsbaum rings A such that dim A = d and depth A t. This is shown by the following example. Another class of such projective varieties is given in Chapter V, § 2. Example 4.14. Let XC PZ, n:::: 2, char k.f n + 1, be the variety defined by the equation X~+l + ... + X:+l = O. Let Y be the Segre embedding of X X p'm in pn+m+"'m, m> 1. Let A be the local ring at the vertex of the affine cone over Y. By Corollary 4.9(ii) A is a Buchsbaum ring. Since Y is free of singularities we have that A is a normal ring. By Corollary 4.2 dim A = n +m. From our Ktinneth formulas (Proposition t, char k .f t + 1, m = d - t we obtain 0.2.10) we get depth A = n. Choosing n what was claimed before. Furthermore, we note that the Segre embedding S( V X W) of two projective varleties is a locally Cohen-Macaulay variety if and only if V and Ware locally Cohen-Macaulay varieties. Clearly this statement is true because the local behavior of V X W does not depend on the embedding, thus the theorem that the tensor product of two kalgebras is Cohen-Macaulay if and only if the factors are can be applied. Using Proposition 4.10 we have a direct proof: Corollary 4.15. Let V <;;;: p .. and W <;;;: pm be varieties oj posit~ve dimension. The Segre embeddt'ng S(V X W) is a locally Cohen-Macaulay variety ij and only ij V and Ware locally Cohen-Macaulay varieties.

In analogy to Corollary 4.15 one might be tempted to ask whether S( V X W) is a locally Buchsbaum variety, when V and Ware locally Buchsbaum. However, this is not the case as we will show by Proposition V.5.1. We note that the investigations in Chapter II, § 2, also permit us t{) work with arithmetically non-Co hen-Macaulay varieties. For instance, the Segre embedding S(X X X) is an arithmetically Buchsbaum (non-Cohen-Macaulay) variety where X is our well-known curve in 113 given parametrically by (S4, S3 • t, 8 . ea,t4). The proof of this statement follows from Proposition II.2.1O, see also Chapter V, § 5, 2. for generalizations.

Chapter II Hochster- Reisner theory for monomial ideals. An interaction between algebraic geometry, algebraic topology and combinatorics

We shall in this chapter develop a new topic, that of face rings of simplicial complexes. The final paragraph of this chapter will be a brief introduction to Buchsbaum complexes. Following P. Schenzel [1] we will study an expression which can be interpreted as measuring the error when we no longer have the Cohen-Macaulay case of simplicial complexes. Hence we will get an inequality for the so-called h-vector of a Buchsbaum complex which generalizes the inequality for a Cohen-Macaulay complex which implies the Upper Bound Conjecture. The motivation for §§ 1-4 is provided by examples of Chapter III. We will discover through liaison addition that the ideals

never define projectively Cohen-Macaulay but projectively Buchsbaum curves in Pi:. Further, the ideals

1. In this chapter we would define projectively Cohen-Macaulay curves for all n like to understand the reason for this independent of liaison addition. Each of these ideals it generated by monomials. Thus we may try to apply again the Hochster-Reisner theory of monomial ideals, which associates to each such ideal I a simplicial complex 1:[. In particular we have the Reisner homology criterion, which relates the Cohen-Macaulay property of the quotient K[xo,"" xn]/I to the homology of 1:1 and its link subcomplexes. There are two problems with this approach. One is that the Hochster-Reisner theory was developed only for ideals generated by square-free monomials, whereas the above-mentioned ideals are not square-free in general. The other problem is that the computation of simplicial homology is a laborious process, making the Reisner homology criterion difficult to apply in practice. The first problem is solved by Schwartau's polarization, which extends the HochsterReisner theory to the non-square-free case. This will be explained in § 1. Following Schwartau [1] we deal with the second problem by developing new Cohen-Macaulay criteria for monomial ideals of height 2. We discover that in the orientable case it is possible to replace the Reisner criterion altogether by conditions on 1:1 which have nothing to do with its homology but rather its singularities. Also we show that these new criteria may be given in purely algebraic terms, as a condition on the primary decomposition of the polarization of 1. This in turn can be easily measured by means of a certain graph we associate to any monomial ideal of height 2.

§ 1. Foundations

107

The main results of this chapter rest at crucial points on some completely concrete construction in which algebra, geometry, topology and combinatorics are significantly intertwined.

§ 1.

Foundations

We will assume some basic facts about monomial ideals. For example, we will use the so-called splitting lemma: II A, 'r, ml> ••• , m" are monomials and A, 'r are relatively prime, then (A'

'r,

mI ,

••• ,

m,,)

(A, m I, .", m n ) n ("/:, mI' .. ,' m,,).

First we want to discuss Schwartau's polarization, which extends the HochsterReisner theory to the non-square-free case of ideals generated by monomials. Throughout this chapter K will denote an arbitrary algebraically closed field. We shall always write S for K[xo, ... , x n], the ring of polynomials on P~. Definition 1.1. Let m be a monomial xg' ... x:' of S. Then we define the polar m to be the square-free monomial obtained by replacing all repeated variables in m by new variables:

Let 1= (mI' ... , m r ) be a monomial ideal of S, where (ml> ... , mr ) is a minimal set of generators of I consisting of monomials. Then define the "polarization of I" to lk the ideal Polar I

=

(Polar m h

••• ,

Polar mr)'

Note that if I is generated by square-free monomials Polar I = I; whereas in general the square-free ideal Polar I lies in a higher dimensional polynomial ring than the original ring S. We collect some properties on Polar I. These assertions are immediate consequences from basic facts on monomial ideals. Theorem 1.2. (i) Let 1(1) = (m'll), ... , m'III) ), PI Polar (I(l) n ... n 1(6»

••• ,

=

I(e)

(mIle), ••• ,

m!,6) ) be monomial ideals ·00

01 S.

Then

Polar 1(1) n •.. n Polar ](e).

(ii) Let I c:: S be a monomial ideal. Then il I is equidimensional and unmixed, Polar I and both ideals have the same height. (iii) For a primary decomposition we have Polar (xa, Xb)m)

=

n

80

is

(x~), x~).

i+i~m+l l~i.i";m

We will review the correspondence between monomial ideals of S and finite simplicial complexes. Our treatment is based on M. Hochster [4] the reader is referred to this paper and to G. Reisner [1] for more details. Also, the reader may consult to HiltonWylie [1], Spanier [1] and McMullen-Shephard [1] for basic facts of algebraic topology.

108

II. Hochster-Reisner theory for monomial ideals

,

First we want to recall some basic definitions and results from algebraic topology; for the most part we follow Stanley [1] and Baclawski [3]. An (abstract) simplicial complex Ll on a vertex set V is a collection of subsets F of V satisfying: (a) if x E V then {x} Ell, (b) if FEll and G c F, then G E Ll. Elements of Ll are called laces or simplices. If FEll, then define dim F and the dimeruJion of Ll, dim Ll max (dim F).

=

#F - 1

FEJ

If #F q + 1, then F is a q-Iace or q-simplex. We frequently identify the vertex x with the face {x}. Suppose V is finite, say V = {Xl> ••• , x,,}. Let ei be the ith unit coordinate vector in R". Given a subset F ~ V, define

iFi

= cx{e,

I Xi E F} ,

where ex denotes convex hull (see, e.g., McMullen-Shephard [1]). Thus if F is an (abstract) q-simplex, then iFl is a geometric q-simplex in R". Define the geometrz'c realization Ill! of the simplicial complex Ll by

!t11=uIFI· FEJ

Thus !Lll inherits from the usual topology on R" the structure of a topological space. If X is a topological space homeomorphic to ILlI, then we (somewhat inaccurately) call Ll a triangulahon of X. An oriented q-simplex of Ll is a q-simplex F together with an equivalence class of total orderings of F, two orderings being equivalent if they differ by an even permutation of the vertiees. Denote by [vo, ... , vq ] the oriented q-simplex eonsisting of the q-simplex F = {vo, ... , vq }, together with the equivalence class of orderings containing 1:0 < VI < ... < vq. Fix a ring A (commutative with 1). Let Cq(Ll) be the free A-module with basis eonsisting of the oriented q-simplices in Ll, modulo the relations 0"1 0"2 = 0 whenever 0"1 and 0"2 are different oriented q-simplices corresponding to the same qsimplex of Ll. Thus Cq(Ll) 0 for q < 0, and for q 0 the module Cq(Ll) is a free A-module with rank equal to the number of q-simplices of Ll. If Ll is empty, then Cq(Ll) 0 for all q. We define homomorphisms Oq: Cq (L1) --:.. Cq_1(Ll) for q > 1 by defining them on the basis elements by

+

q

Oq[V o, VI,

••• , V q ] =

1:

;=0

where 'Vi denotes that Vi is missing. It is easily verified that Oq indeed extends to a O. The chain complex C(Ll) homomorphism Cq(d) --:.. Cq_ 1(d), and that OqOq+l = {Cq(d), Oq} is the oriented chain complex of d. Define an augmentation e: Co(d) --:.. A by e(x) = 1 for every vertex x E V. The augmented ehain complex (C(Ll), e), is the augmented oriented chain complex of d (over A). Then the qth reduced homology group of L1 with coefficients A, denoted Hq(d; A), is defined to be the qth homology group of the augmented oriented chain complex of Ll over A.

§ 1. Foundations

109

Furthermore, the reduced Euler characteristic i(L1) of L1 is defined by

E

x(L1)

(-1)q rank Hq(L1; A).

q2~1

It is independent of A and is also given by i(L1}

=

-1

+ 10 -/1 + ... ,

where Iq is the number of q-simplices in L1. If X(L1) is the ordinary Euler characteristic then X(L1) = X(L1) 1. ' If L1 4= 0, then Hq(L1; A) = 0 for q < O. If L1 = 0, then H q(0; A)~ A for q = 1, and 0 for q 4= -1. In particular1 X(0) = -1. We now wish to define the homology groups of a space X, rather than a simplicial complex L1. Let X be a topological space. Let L1q denote the standard q-dimensional ordered geometric simplex (Po, ... , pq) whose vertices Pi are the unit coordinate vectors in Rq+l, A singular q-simplex in X is a continuous map (1:

L1q -i>- X •

Let Oq(X) be the free A-module generated by all singular q-simplices. The elements of Oq are formal finite linear combinations E ca(J, where (J is a singular q-simplex a

e!:

and Co E A. Given a vertex Pi of L1 q, there is an obvious linear map L1q-l -i>- L1q which sends L1q-l to the face of L1q opposite Pi' The ith face of (J, denoted by (J(i), is defined to be the singular (q - 1)-simplex which is the composite

We now define a linear map (= A-module homomorphism) Oq: Oq

-i>-

Oq_1 by

where (J is a singular q-simplex. It is easily checked that Oq-l Oq = 0, so O(X) = {Oq(X}, Oq} is a chain complex, the singular chain complex of X (over A). Define an augmentation c:: Oo(X) -i>- A by C:«(J) = 1 for all singular O-simplices (J, The augmented chain complex C(X) is the augmented singular chain complex of X (over A). Then the qth reduced singular homology group of X with coefficients A, denoted Hq(X; A), is the qth homology group 01 the augmented 8%1Igular chain camplex of X (over A). Considering this case the reduced Euler characteri8tic i(X) of X is defined by i(X}

E

(-l)q rank Hq(X; A).

q2~1

It is independent of A. If L1 is a simplicial complex and L11 and L12 are subcomplexes of L1, then there is an exact sequence (whose definition we omit)

(with all coefficients A), called the reduced Jfayer- VUtOri8 8equence of L11 and L1 2. Similarly, if X is a topological space and Xl, X 2 are "nice" subspaces (e.g., if Xl U X 2

110

II. Hoohster-Reisner theory for monomial ideals

= (intx,ux, Xl) u (intx.ux. X 2 ), where inty Z denotes the relative interior of Z in the space Y), then we have a reduced Mayer-Vietoris sequence of Xl and X 2 exactly analogous to that of ,11 and ,12' We now come to the relationship between simplicial and singular homology: Let ,1 be a finite simplicial complex and X = [,1[. Then there is a (canonical) isomorphism for all q:

i1q(.d; A) '" iiq(X; A). F6r example, let Sd-l denote a (d l)-dimensional sphere. Then Hq(.d; A) A for q d - 1 and 0 for q =to d - 1. A simplicial complex ,1 or topological space X is acydic (over A) if its reduced homology with coefficients A vanishes in all degrees q. (Thus the null set is not acyclic, since ii_l(0; A)~ A.) Let Y be a subspace of X. Then the singular chain module Oq( Y) is a submodule of Oq(X), so we have a quotient complex O(X, Y) = O(X)/O(Y) = {Oq(X)/Oq(Y), 8q}. Define the relative homology of X modulo Y (with coefficients A) by

We next want to define reduced cohomology of simplicial complexes and spaces. The simplest way (though not the most geometric) is to dualize the corresponding chain complexes. Let 0'(,1) = 0(,1, e) be the augmented oriented chain complex of the simplicial complex ,1, over the ring A. The qth reduced ln1u.Jular cohorrwlogy group of ,1 with coefficients A is defined to be iiq(.d; A)

iiq(Hom,,(O'(.d), A)),

where Hom,,(O'(.d), A) is the cochain complex obtained by applying the functor Hom" ( ,A) to 0'(,1). Exactly analogously define iiq(X; A) and Hq(X, Y; A). Sometimes one identifies the free modules Oq(.d) and Oq(.d) Hom,,(Oq(.d), A) by identifying the basis of oriented q-chains (J of Oq(.d) with its dual basis in Oq(.d). Similarly one can identify Oq(X) with oq(X). There is a close connection between homology and cohomology of ,1 or X arising from the "universal-coefficient theorem for cohomology". We merely mention the (easy) special case that when A is a field k, there are "canonical" isomorphisms

i1q(.d; k) '" Homk(iiq(.d; k), k), i1 q(X; k)~ Homk(iiq(X;

k), k).

Thus in particular when iiq(.d; k) is finite-dimensional (e.g., when ,1 is finite), we have and similarly for X, but these isomorphisms are not canonical. We recall that a topological n-manifold (without boundary) is a Hausdorff space in which each point has an open neighborhood homeomorphic to R". An n-manifold with boundary is a Hausdorff space X in which each point has an open neighborhood {(Xl' ... , X,,) E R" I Xi O}. The boundary which is homeomorphic with R" or R,: oX of X consists of those points with no open neighborhood homeomorphic to R". It follows easily that oX is either void or an (n - I)-manifold.

i1q(.d; k)~ i1q(.d; k)

§ 1. Founda.tions

111

Suppose X is a compact connected n-manifold with boundary. Then one can show H,.(X, A) is either void or isomorphic to A. A compact connected n-manifold X with boundary is orientable (over A, if we have H,.(X, A) = A. (The usual definition of orientable is more technical but equivalent to the one given here; see also Definition 3.13 below.) For example, every compact connect n-manifold with boundary is orientable over a field of characteristic two. H a compact connected n-manifold X is orientable over A, then we have Hq(X; A) '::::: H,,-q(X; A). This is the so-called Poincare D'lMtluy Theorem. An n-dimensional p8eudomani/old withm.a boundary (resp., with boundary) is a simplicial complex A such that:

ax;

ax;

(a) Every simplex of A is the face of an n-simplex of A. (b) Every (n - I)-simplex of A is the face of exactly !wo (resp., at most two) n-simplices of A. (c) H F and F' are n-simplices of A, there is a finite sequence F F l , ... , F m = F' of n-simplices of A such that Fi and Fi+l have an (n I)-face in common for 1< i< m. The boundary 0,1 of a pseudomanifold A consists of those faces F contained in some (n - I)-simplex of A which is the face of exactly one n-simplex of A. Let A be a finite n-dimensional pseudomanifold with boundary. Then H,.(A, 0,1; A) '::::: A or O. In the former case we say that A is orientable over A; otherwise nonorientable. Let I be the unit interval [0, 1]. The 8WJpeWlWn EX of a topological space X is defined to be the quotient space of X X I in which X X 0 is identified to one point and X X 1 is identified to another point. The n-/oid 8USpeWlWn E" X is defined recursively by E"X = E(E"-lX). For any X and q we have

flq(X; A) '::::: Hq+l(EX; A). The purpose of this chapter also is to introduce a new kind of partially ordered set: Buchsbaum poset. The notion of a Cohen-Macaulay poset originated in Baclawski's thesis, see Baclawski [3]. It is now known that this concept provides some interesting connections among algebraic topology, combinatorics, commutative algebra and homological algebra. LetP be a /~nue poset; that is, a partially ordered set. We need some auxiliary concepts. A chmn of P is a totally ordered subset of P. We will usually write Xl < ... < Xn for a typical chain of P. The rank of a chain is the number of elements in it; thus r(xl < ... < xn) = n. More generally, the rank of P, written r(P), is the rank of the longest chain of P. The length of P, written l(P), is given by l(P) = r(P) - 1. The length is a more topological notion whereas the rank seems to be more combinatorial. Apparently topologists start counting at zero while combinatorialists prefer to begin at 1. We will do both. A poset is said to be ranked if every maximal chain has rank r(P). Given a poset P, we will write P for the poset obtained by adjoining a new pair of elements to P, written Ii, t such that Ii < if < t for all x E P. H we only require that Ii or i be adjoined, we will write Po or pI respectively. We use the convention that (j or i is never an element of P. The context should indicate to which poset (j or i is to be adjoined.

112

II. Hochster-Reisner theory for monomial ideals

,

A subset J ~ P will be called an order-ideal if for every x E J, Y x implies y E J. The dual definition gives the concept of an order-filter. The order-ideal generated by a subset 8 P will be noted J(8) or J p (8); while V(8) V p (8) denotes the order-filter generated by 8. The special case J(x) for x E P can also be denoted (6, x]. If P is ranked, then so is every subset J(x), and we write r(x) for r(J(x)). The function r takes values in the set Lr(P)] which by definition denotes {I, 2, ... , r(P)}. The length of an open intervall will be denoted l(x, y) instead of l( (x, y)). We will often use the Mobius function. For a poset P we write ",(P) for ",(0, i) as computed in P. For x E P we will write ",(x) or ",p(x) for ",(J(x)). Finally, for x yin P we will think of ",(x, y) as an abbreviation for "'( (x, y)). . For a finite set 8, let B(8) denote the poset of nonempty subsets of 8. A finite 8implicial complex is an order-ideal of B(8). The minimal elements are called vertices and elements in general are called simplice8. Much of what we do in the sequel may be extended routinely to simplicial complexes. As we have defined it, a simplicial complex is a special kind of poset. However, given a finite poset P, we can define the order complex of P, denoted LI(P), to be the subset of B(P) consisting of the nonempty chains of P. By this device one may view posets as a special kind of simplicial complex. We now review the correspondence between monomial ideals of the polynomial ring 8 = K[xo, ... , x n] and finite simplicial complexes. Let LIn denote the standard n-simplex; that is, the complete simplicial complex on (n+ I)-vertices which we label as xo, ... , x n • Recall that this means that LIn is the set of all subsets of {xo, Xl> "" x n}, Let K be a field and I an ideal of 8. Let V(I) be the subset of Kn+l where the elements of I vanish. If I = (Xi., "', Xi), we refer to V(I) as a coordinate hyperplane. Then we get 1-1 correspondences between: 8 1 = {subcomplexes of LIn}, 82 {ideals of 8 generated by square-free monomials}, 8 a {unions of coordinate hyperplanes in Kn+l}. We describe some of these 1-1 correspondences in detail, The correspondence 8 1 -+ 8 2 is defined by 1:~IJ.:

where 1: denotes a subcomplex of LIm and IE denotes the ideal of 8 generated by the monomials Xi• ••• Xi., io < ... < if) such that the simplex (Xi., , •• , Xi) is not in 1:. The correspondence 8 1 *- 8 2 is defined by 1:[ ~ I where I denotes a square-free monomial ideal of 8, and 1:I denotes the subcomplex of LIn consisting of all simplices (Xi., ... , Xi,) such that the monomial Xi, .' • Xi, is not in 1. For the correspondence

8 2 -+ 8 a ,

I

~

V(I)

we simply associate to I its vanishing locus in Kn+l. The correspondence 8 3 -+ 8 1 is given by H~1:H'

Here H denotes a union of coordinate hyperplanes in K"+l, and 1:ll denotes the subcomplex of LIn consisting of all simplices (Xi" ... , Xi) such that the element of K,,+l whose Xi., •• ,' Xi, coordinates are 1 and whose other coordinates are 0 is in H.

§ 1. Foundations

113

If the field K is the real field R, we may view this last correspondence in terms of ,geometric realizations. First we recall:

Definition 1.3. Let the vertices xo, ... , x" of An be identified with the canonical basis of R"+1. Then if 1: is any subcomplex of A", its geometric realization i1:1 is defined as U cx(O') C R"+1; 11:1 R,,+1 is a topological space (see McMullen-Shephard [1]). Then aEE

R the above correspondence may be viewed as 8 a --"" 8 1 given by

if K

H ___ 1,1,,1 n H = I1:H I.

We now relate these correspondences to primary decomposition. Lemma 1.4. (i)

If P c::: 8 is a 8quare-free monomial ideal of the form (Xi" •.. , Xi,), then the a880ciated Bimplwial complex 1:p c::: A" is a simplex of codimension h := n - dim 1:p ~'n An.

(ii) If I c::: 8 ~'n any 8quare-free monomial ideal, let PI n ... n Pm be an irredundant primary decompoBition of 1. Then each P j i8 of the type described in (i), and we have III

1:1

U 1:p a8 the decompOBition of 1:1 into its maximal8implWie8 . • =1

'

(iii) If I c::: 8 is any 8quare-free monomial ideal, ht I Proof: (i) is obvious from the correspondence 8 2

--""

codim(1:J, An). 8 1 above.

(ii) That each Pi is of type (i) follows from the splitting lemma. The fact that the III

intersection of PI n ... n Pm corresponds to the union ,U 1:p, follows from the cor.~l

respondences 8 2 --"" 8 a --"" 8 1, The maximality of the simplices 1:p , in 1:1 follows from the irredundancy of the primary decomposition. (iii) is a direct consequence of (i) and (ii).

Definition 1.6. If 1: is a finite simplicial complex, we define the codimension of 1: (written codim 1:) to be codim(1:, ,1"...1) where v is the number of vertices in 1:. Lemma 1.6. Let I 8 be any 8quare-free monomial ideal, and let 1:1 8impZwial complex. Then

A" be the a880ciated

ht I.

codim 1:1

Proof: This follows immediately from Lemma 1.4(iii).

The following definitions are fundamental in the sequeL Definition 1.7. If (J' is a maximal simplex in a simplicial complex we call it a facet of 1:. Definition 1.8. If a E 1: is any simplex, we define the 8tar of a to be the subcomplex Starl' a

{s

E 1: I 0' usE 1:}.

Definition 1.9. If a E 1: is any simplex, we define the h'nk of rJ to be the subcomplex Ikl'rJ = 8 Buchsbaum Rings

{8

E 1: I 0' n s

0 and rJ

U

s is in 1:) .

114,

II. Hochster-Reisner theory for monomial ideals

Lemma 1.10. Let E be a simplicial complex. Then: <X) Any Unk IkE 0' is an iterated link 01 vertices (the vertices in 0'). p) II v is a vertex 01 E, StarEv is the simplicial cone with base Ik.z;v and vertex v. 1') Any StarE 0' is an iteratUm 01 cones over IkEa (by the vertices in 0'). Lemma 1.11. Let E be a simplicial complex which is equi-dimensional 01 dimension d. rl'hen: <X) 110' is any simplex 01 E, StarEa is an equi-iUmensional complex 01 dimension d. fJ) 110' is a k-simplex 01 E, IkEa is an equi-dimensional complex 01 dimension d - k - 1. Definition 1.12. H E is a simplicial complex lct Conep(E) denote the simplicial complex given by the simplices {a

I 0' is a simplex of E}

u {(a, P) I 0' is a simplex of E} u {P}.

Note that Conep{E) contains E as a subcomplex. Conep{E) is called "the simplicial cone over E" or "the simplicial cone with base E and vertex P". The following lemmas are of basic importance. Lemma 1.13. E is equi-dimensional more than E).

~

Conep{E) is equi-dimensional (01 dimensUm one

Prool: Observe that ( ,P) induces a 1-1 correspondence {facets of E) ~+ {facets of Conep{E)} • 1-1 Since ( ,P) raises the dimension of any simplex by exactly one we conclude that one set of facets is equi-dimensional if and only if this is true for the other set. Lemma 1.14. Let T be a simplex 01 0 Conep(E). Then there are three possibilities lor IkcT: <X) II T = P, then IkcT = E. fJ) II T Ol P (i.e. T (a, P) lor some simplex 0' E E), then IkcT = IkL'a. (') II T ~ P (i.e. TEE), then IkcT Conep{lkET). Prool: <X) This is obvious from the definition of Conep(E). fJ) Ikc(a, P) = Iklkco P (recall Lemma 1.10)

= IkconepOkEo) P (by 1')) = IkEa (by <x)). (') Step 1: 0' E E and 0' n T 0 ~ (a, P) E 0 and (0', P) n T Step 2: 0' uTE E ~ (0', P) uTE O. We now present two examples of simplicial complexes. Example 1.10. X

E= 1

=

0.

§ 2. The homological Cohen.Macaulay criterion of Reisner

For each Xi =l= X o, IkrXi

= _____ and StarrX. =

115 '

A

XIX.

I I and StarrXo x.x. is the simplicial cone with base I I and vertex XO'

However, for Xi

=

X o, Ik,!;'Xo

=

I.

X,X,

I

x,x,

Example 1.16.

IkX j For Xi

= Xo or Xl>

-----

IkXi=-L

For edges e =l= XOXb For e

XoXl>

I is the simplicial cone over

lk,!;'e = • • •

..l-., q.e.d.

Stars and links are important in the study of simplicial manifolds, as we shall see in the following sections.

§ 2.

The homological Cohen-Macaulay criterion of Reisner

Let I c: A" be any finite simplicial complex. We denote by K[IJ the quotient ring S/lz. We will see that the graded K-algebra K[I] is closely related to the combinatorical and topological properties of I. Therefore we have interactions between commutative algebra and combinatorics. By combining these techniques we open the way to a deeper study of K[I] and I. First, we begin by presenting Reisner's Cohen-Macaulay criterion. ii ;(I; K) will always denote the ith reduced homology group of I with coefficients in K (see, for example, E. H. Spanier [1J, p. 168). In 1976, G. A. Reisner [1] proved the following statement: Theorem 2.1 (the Reisner Cohen-Macaulay criterion). Let I c: A" be any linire simplidal complex. Then the 10l101.MTtg conditions are equivalent: (i) K[I] is Oohen-Macaulay. (ii) K[lkEx;J is Cohen-Macaulay lor every verrex Xi E I; and iii(I; K) = 0, i < dim I. (iii) Hi(IkEO'; K) 0, i < dim IkE 0', lor all simplices 0' E I; and Hi(I, K) 0, i < dim I. Prool: Reisner [1], Theorem 1 (see also Weibel [1)). 8*

116

II. Hochster-Reisner theory for monomial ideals

If the geometric realization 11:1 of 1: (see again E. H. Spanier [1]) is a manifold (with or without boundary) the condition on the links holds automatically. This follows by using excision (see E. H. Spanier [1]). Hence we have; . Corollary 2.2. Ij 1: All is a simplicial complex suck tkat 11:1 is a topological manijoldwitk-a, tken tke jollow£ng conditions are equivalent: (i) K[1:] Us Coken-Macaulay. (ii) 11,(1:; K) = 0, i < dim 1:.

Furthermore, by proving Theorem 2.1 Reisner has also investigated the property for 1: to have Cohen-Macaulay links of its vertices. From the point of view of commutative algebra the key is the following simple lemma. Lemma 2.3. Let R be a Nnitely generated graded K-algebra witk Ro = K. Suppose R = K[xo, ... , x.] wkere xo, ... , x" are jorms oj degree 1. Tken tke jollow~ng conditions are equivalent: (i) Rp Us Coken-Macaulay jor all prime ideals P except, perkaps, tke irrelevant ma~mal ideal (xo, ... , x,,). (ii) R/(xj - 1) Us Coken-Macaulay jor i = 0, ... , n. Prooj: Assume (ii), and let P =F (xo, ... , x.) be a prime ideal. Choose Xj ~ P. It suffices to show that R", (i.e. R localized at the powers of Xi) is Cohen-Macaulay. But this follows from the well-known isomorphisms (see Grothendieck [2], Chapter II, 2.2.5) R",

(R",)o [(l/Xi),

xd,

(R,,)o~ R/(Xi -

1)

and the fact that a ring S is Cohen-Macaulay if and only if S[t, lit] is Cohen-Macaulay with t an indeterminate. . Assume (i). Then the above isomorphisms and since Xi - 1 is not a zero divisor in R If, imply (ii). "-

We will be interested in Lemma 2.3 when R K[1:] for some 1: c An. The preceding will be used in proving the following Buchsbaum criterion. Theorem 2.4 (the Reisner locally 'Cohen-Macaulay criterion). Let 1: c A" be any jinite ~mplicial complex. Then tke jollowing conditions are equivalent: (i) (K[1:J)p Us a Coken-Macaulay r£ng jor all pnme ideals P dijjerent jrom tke irrelevant ideal (xo, ... , XII) • K[1:]. (ii) For all 8£mplices (J E 1: we kave

fi .(Ikz;

(J;

K) = 0

ij

i

=F dim lkz; (J •

. (iii) K[1:] Us a (local) Buchsbaum nng. Prooj: The equivalence of (i) and (ii) is proved by G. A. Reisner [1]. The implication (iii) =} (i) is trivially true because a Buchsbaum ring is locally a Cohen-M!,-caulay ring. The converse (i) =} (iii) results immediately from Proposition 1.3.10 and the following Lemma 2.5(ii).

§ 2. The homological Cohen-Macaulay criterion of Reisner

Lemma 2.5. Let 1: c L1n be any jznite simplicial complex. Set R jar the local cohomology module !1~(R) the jollowing pr'Operties: (i) [!1~(R))n = 0 jor all n > 0 and i E Z. (ii) AS8Uming that !1~(R) is oj finite length then

117

K[1:). Then u'e have

!1~(R) = [!1~(R))o.

Proof: In order to prove these facts on local cohomology we make use of the techniques of Hochster and Roberts about the purity of the Frobenius homomorphism (in characteristic p =l= 0) and the fact that R has a presentation of relative graded F-pure type (in characteristic zero). The key is then the following assertion. (We use terminology from Hochster-Roberts [21.) .

Claim. Let K denote a perfect field of prime characteri:5tic p > O. Then K[1:] is F-pure. For an arbitrary field K, K[1:] haB a presentation oj relative graded F-pure type. Proof oj claim: First let K be a perfect field of prime characteristic p > O. We show that the Frobenius map F: R ~ R is pure. To this end we prove that F(R) is a direct summand of R as F(R)-module. This is equivalent to showing that there is an R-retraction, sayr: R ~ F(R). Here the point is that we can define r for monomials by r(x~'

x~' ... x!"

... x!")

o

{

for k; 0 mod p, O:S:; i otherwise

< n,

and extend it K -linearly to R. Since K is a perfect field we get r(R) F(R). To complete the proof we note that r is the identity on F(R) and r is an F(R)-module homomorphism. For an arbitrary field we get from Z[xo, ... , xn]/h a presentation of relative graded F-pure type, see M. Hochster and J. L. Roberts [2]. Having proven this claim we can now apply conclusions on local cohomology from Hochster-Roberts [2]. Therefore our Lemma 2.5 results immediately as a consequence from this important paper, q.e.d. Analyzing the proof of J.emma 2.5 it becomes desirable to look for more precise information. Before stating results in this direction, we make some general observations and collect some known results from Hochster-Roberts [2J which will be needed. In the following we call a homomorphism of rings R ~ S pure, if

M

~

M

S

via

m

t4

m

1

is injective for every R-module M. A ring R of prime characteristic lJ F-pure, if the Frobenius map F:R~R,

> 0 is called

rt4r P,rER,

is pure. Fe denotes F iterated e-times. We note that R is F-pure if and only if Fe: R ~ R is pure for all e ~ 1. If R is F-pure, it follows that it is reduced. A pure homomorphism of rings R ~ S induces a homomorphism of complexes j':X'

~X'

@RS

for a complex of R-modules X'. In fact, it follows by M. Hochster and J. L. Roberts [1], Proposition 6.5, that the induced homomorphisms Hi(/'): Hi(X')

~

Hi(X' @R. S)

118

II. Hochster-Reisner theory for monomial ideals

are injective for all i E Z. Assume that R -+ S is a pure homomorphism of graded rings such that R -+ S multiplies degrees by d. Let K"(~t; R) be the Koszul complex of R with respect t{) ~t (xL ..• , x~), t> 1, where (Xl>"" x,,) is a system of forms with Rad(Xl' .'O, x,,)R = m, the irrelevant maximal ideal of R. Then the maps

are injective for all i E Z and for all k E Z. Here, ~/ = (x~, ••. , 'x~) denotes the image of ~ = (Xl' ••.• xn) by the pure homomorphism,R -+ S. By taking the direct limit of the cohomology of the Koszul complexes it follows that the maps [H:n(R)Jk -+ [H:n'S(S)]kd

are injective for all k E Z and for all i E Z, where m'S denotes the image of m in S. In particular let R be a F-pure graded ring. Then the maps

are injective for all k E Z, i E Z, and e > 1, since Fe(m) has radical m. This implies the following lemma due to M. Hochster and J. L. Roberts [2]. Lemma 2.6. If R is a graded F-pure ring, then we have [H:n(R)]" = 0 for all n > 0 and i E Z. Suppose that the local cohanwlogy module H:n(R) is of finite length. Then H:n(R) = Lll:n(R)]o also follows. >

= 0 for n ~ 0, the first part of the lemma is obtained by previous considerations. If the local co~ homology module is of finite length, then we have in particular [H:n(R)]" 0 for n ~ O. Thus, the second statement follows the same way, q.e.d.

Proof: Using the fact that H:n(R) are artinian R-modules, i.e. [H:n(R)]"

In characteristic zero M. Hochster and J. L. Roberts [2], Lemma 4.7, proved a corresponding result for rings R which have presentations of certain F-pure types. For the definition and related technical results in particular the definition of "R has a presentation of relative graded F-pure type" - we refer to the fundamental paper of M. Hochster and J. L. Roberts [2]. In fact, Lemma 2.6 and the corresponding result in characteristic zero lead to a number of Buchsbaum rings.

Theorem 2.7. Let R be an equi-dimensiorwl graded k-algebra suck that Rtl is a OohenMacaulay rz'ng for all prime ideals d~fferent fr()'fl1, tke %rrele'l.!ant ideal m. Assume tkat R is F-pure re8p. has a pre8entation of relative graded F-pure type. Then it follow8 tkat R is a Buchsbaum rz'ng.

< dim R, are modules of finite length. the proof will follow. From this property we get by Lemma 2.6:

Proof: If we show that all the local cohomology modules H:n(R), 0 S i [H:n(R)] ..

=

0

for all n

0 and 0

i

< dim R.

Then Proposition 1.3.10 does imply that R is a Buchsbaum ring. Now, it is clear that the finite length of local cohomology modules H:n(R), 0 S i < dim R, is equivalent to

§ 2. The homological Cohen-Macaulay criterion of Reisner

(see also Corollary 0.4.15) , (i) R is equi-dimensional and (ii) R:p is a Cohen-Macaulay ring for all prime ideals Hence Theorem 2.7 is proved, q.e.d.

~

119

'*' m.

Next we investigate a geometrical interpretation of Theorem 2.7. To this end we state an interpretation of Lemma 2.6. Let X c:: Pk P be a projective scheme such that for a coherent sheaf (F and an integer t: (a) The canonical map M" -+ HO(X, (F(n))

'*'

is bijective for all n t, where M denotes the graded module associated to (F, and (b) Hi(X, (F(n)) = 0 for all n t and 0 < i < dim (F. Then J" is arithmetically Buchsbaum, i.e., M is a graded Buchsbaum module. Using this notion, our Theorem 2.7 says: Let R be an F-pure graded k-alfJebra resp. R has a presentation 0/ relative graded F-pure type. 1/ (X, Ox) = Proj(R) i8 a pure dt'mensional (locally) Oohen-Macaulay 8(Jheme, then X is arithmetically Buchsbaum.

'*'

Now we will examine some Buchsbaum rings which arise from the purity of the Frobenius. Example 2.8. Let k be a field of characteristic p > 0 reap. of characteristic zero, and let R be a graded k-algebra which is F-pure reap. has a presentation of relative graded F-pure type. Suppose R is a domain such that R:p is regular for every prime ideal ~ different from the irrelevant ideal m. Let S c:: R be a graded k-subalgebra which is pure in R. Then S is a Buchsbaum ring. M. Hochster and J. L. Roberts [2], § 5, showed that S is F-pure resp. has a presentation of relative graded F-pure type. Also, they showed that S:p is a Cohen-Macaulay ring for all prime ideals ~ different from the irrelevant ideal which follows from the main result of Hochster-Roberts [1], i.e., a pure subring of a regular ring with characteristic p > 0 is a Cohen-Macaulay ring. From this it follows that the ring of invariants of linearly reductive affine linear algebraic groups acting on regular rings are CohenMacaulay rings, compare Hochster-Roberts [1]. Therefore, our Theorem 2.7 shows that rings of invariants of those groups acting on certain singular rings are Buchsbaum rings. Example 2.9. Let k, R be as in the previous example. Let G be a linearly reductive affine linear algebraic group over k acting on R by preserving degrees. Then the ring of invariants S = RG is a Buchsbaum ring. We note that, S = RG is in general not a Cohen-Macaulay ring, if R is singular. For this we consider an example of M. Hochster-J. L. Roberts P]. Let

R

k[Xl' ... , x,,]/(xi

+ ... + x:).

Then R is F-pure for a perfect field k with characteristic p 1 mod n. Therefore we have a presentation of relative graded F-pure type, if k is a field of characteristic zero. Let S k[Yl' Y2]' Let G = Gl(l, k) k" to} acting on R resp. S by multiplication of

120

II. Hochster-Reisner theory for monomial ideals

a form of degree m by gttl resp. g-ttl for 9 E G. The tensor product

R @k S = k[Xl' •.• , XII' y), Y2]/(x~

+ ... + x:)

is F-pure resp.has a presentation of relative graded F-pure type. Furthermore, all the assumptions of Example 2.9 above are fulfilled. Thus (R @k S)G is a Buchsbaum ring. In fact, (R @k S)G is the Segre product of Rand S. Because R is improper in the sense of W. L. Chow [1], i.e., (see Chapter IV, Corollary 4.4 and 4.5) [M~(R)]o =F 0, d = dim R, we obtain (R @k S)G is not a Cohen-Macaulay ring, compare Chapter I, § 4. More generally, we can show that certain Segre products are Buchsbaum rings. For this we extend the investigate of Chapter I, § 4. I~t R;, j = 1, 2, be two graded k-algebras over a field k with dim RI 1. Let S denote their Segre product. Let (Xj,Ox1 ) Proj(Ri ), j = 1,2, and (W, Ow) = Proj(S). Then we have W

Xl Xk X 2

and

Ow(n) = p!Ox,(n) @kP:OX,(n),

where Pi: W -+ Xi> i = 1, 2, denote the canonical projections. Proposition 2.10. Let rObe an integer. Assume that the following conditions are f1dlfiled = 1,2:

for i

a) The canonical map

[RjJn -+HO(X j , Oxj(n)) ,

n =F r,

is bijectz·ve.

b) Hi(Xj,OX/n)) 0 for all n =F rand 0 < i < dim Xi' c) Hdl(X;, Ox/n)) = 0 for all n 0, n =F r and d j = dim Xi' Then W is arithmetically Buchsbaum. Additionally W 8atisfies conditions a), b), c). Proof: By virtue of the above remark, it is enough to show that conditions a), b), c) are

satisfied on the cohomology of W. Using the Kunneth formula H'(W,Ow(n)):::::

EB

Ha(x), Ox,(n)) @kHb(X2, Ox,(n)) ,

O-r b =8

compare Proposition 0.2.10, this follows immediately, q.e.d. Thus by analyzing the proof of Lemma 2.5 we have obtained our results 2.6-2.10 on Frobenius purity and the arithmetical Buchsbaum property. These assertions are also contained in the paper by P. Schenzel [1], 4.4. Now we return to Reisner's locally Cohen-Macaulay criterion of Theorem 2.4. By virtue of this statement it is possible to construct many Buchsbaum rings arising from simplicial complexes. Example 2.11 (Reisner's example from [1]). The fact that the Cohen-Macaulay property of K[E] depends upon K follows now immediately from Theorem 2.4. Take, for example, a triangulated manifold M, whose only nonzero homology is pure p-torsion (for some prime pl. Then K[x), ... , x~]/{u is Cohen-Macaulay if K has characteristic other than P and is not Cohen-Macaulay if char K = p. Examples of such manifolds are Lens spaces (see, e.g. Hilton-Wylie [1], p.223). For a simpler example, one can take M to be the projective plane. In particular, if we consider the minimal triangulation of the projective

§ 2., The homological Cohen.Macaulay criterion of Reisner

121

plane (Fig. 1) then In this case K[.E] is not Cohen-Macaulay if char K = 2 and it is Cohen-Macaulay if char K =1= 2. It will also follow from Corollary 2.12 below that K[.E] is not CohenMacaulay but Buchsbaum for char K 2. We also could have taken LI. to be a finite triangulation of the ~al projective n-space P~. We then get for the reduced simplicial

Fig. 1

homology of LI" with coefficients in an abelian group G (see, e.g., Hilton-Wylie [1], 3.9.4) : H.(LI . G ~ { G/(2) G if i is odd, , ", )T 2(G) . 1'f' ~lseven.

where T2(G) we get for 0

Ig E G such that 2g


OJ denotes the 2-torsion part of G. For a field K

n:

if char K 2, otherwise. Therefore K[LI,,] is a Cohen-Macaulay ring of dimension n+ 1 if char K =1= 2. H char K = 2 then K[LI,,] is a Buchsbaum ring by applying again Corollary 2.12. Furthermore it follows in this case that dim K[LI,,] n 1, depth K[Lln] = 2, and the invariant of the Buchsbaum ring K[LI,,], I(K[LI,,]) 2'* (n + 1).

+

This example therefore shows that the invariant of the Buchsbaum ring K[LI,,] depends on the ground field K. This fact was examined by P. Schenzel [1], 6.2.3, by considering Reisner's example in case n 2. Moreover, S, Solcan [1] showed that the Buchsbaum property of K[.E] depends on the characteristic of the ground field K by extending Reisner's Example 2.11. Other examples are constructed by L. T, Hoa [2] by using Segre products of affine semigroup rings (see also Grabe [1]). Solcan's example is as follows: Take Is (xo, X 2 , xa) II (xo, X 2 , x.) II (xo, X3. xs) II (xo, X4, x6 ) II (xo, xs, xs) II 1M

K[xo, .. "

xsl

where 1M is Reisner's Example 2,11. Introduce a second copy K[x 7 • ... , xIa] of K[xo, ... , xs] and denote by the ideal in K[X7' ••• , x Ia ] corresponding to Is. Let

Is

A

K[xo, .. " xl3l!I

122

II. Hochster·Reisner theory for monomial ideals

where

Then it follows that A is Buchsbaum if char K =t= 2. For char K = 2 we get that A is not a Buchsbaum ring. This fact is obtained from the following localization and by applying Example 2.11: If ohar K =t= 2 then A(z, .....~., .... z..) is a Cohen-Macaulay ring for all i 0, ... , 13. If char K = 2 then A(z......~•.....z,,) is not a Cohen-Macaulay ring for i = 0, 1,7,8. For proving the Buchsbaum property of KPn] the key is the following corollary of Theorem 2.4. It is precisely the phenomenon exhibited here, whioh makes the theory of local Buchsbaum rings interesting in algebraic topology. Corollary 2.12. Let E be any finite simplicial complex. (i) If the geometric realization lEI iB a connected (topological) manifold then K[E] iB Buchsbaum. (ii) The follou-ing conditions are equivalent: (a) lEI iB a homology manifold. (b) K[E] iB Buchsbaum. Proof: (i) If lEI is a topological oonnected manifold then the condition (ii) of Theorem 2.4 regarding links holds automatically; that is, K[E] is Buchsbaum by applying Theorem 2.4. (See also Corollary 2.2.) (ii) This follows immediately from the definition of a homology manifold and Theorem 2.4 since the linked complex for each simplex of lEI has the homology groups of a sphere (see, e.g., Hilton-Wylie [1], p. 156), q.e.d. Finally, let us collect some well-known examples and facts.

Example 2.13. (i) Take for X which defines in K[xo, "0' x 5 ]:

lEI a cylinder. Consider

th~

triangulation in Fig. 2

(see Eisenreich [1]). By Corollary 2.12 it follows that K[E] is a non-Cohen-Macaulay Buchsbaum ring since Jt(E; K) =t= O. (ii) Take for X = lEI the torus and a triangulation which defines in K[xo, ... , X8]

Fig. 2

§ 3. The topological Cohen·Macaulay criterion of Schwartau

123

(see our Introduction). K[E] is a non-Cohen-Macaulay Buchsbaum ring. This assertion results immediately from Corollary 2.12 and the computation of the homology of a torus (see, e.g., Hilton-Wylie [1), p. 64). Furthermore, it follows from the homology groups that the invariant 1(K[E]) of this Buchsbaum ring is 2. (iii) Take for X = lEI the Mobius band. Consider the triangulation in Fig. 3. Here we get in K[Xl' ••• , X6) 1z

(XIX3, XIXS, X~, X2X3X., XaX.Xs,

x.xsx6 ).

K[E] is a non-Cohen-Macaulay Buchsbaum ring with 1(K[E]) 1

2

3

"

4

5

6

1

LZLZLZl

1.

Fig. 3

Example 2.14. (i) Spheres and discs are always Cohen-Macaulay. (ii) If dim E = 0 then K[E] is Cohen-Macaulay. (iii) If dim E = 1, K[E) is Cohen-Macaulay if and only if E is connected. (iv) Let E be the complex of Example 1.16. Then E is not a manifold, but K[E] is Cohen-Macaulay. We also mention the following lemma proved by Hochster, see, e.g., Schwartau [1]:

Lemma 2.16. Let E c L1n be any linite simplicial camplex, and let Xi be any vertex As U8Ual identify Xi with a variable 01 S = K[xo, ... , xn]. Then we have:

01 E.

K[lk,l;xi] '" (K[E])(z,) '" (S/lzk.,). Since Spec(S/hk",) form an open affine cover of Proj(S/hl, Lemma 2.15 shows directly that the subscheme V c Plc defined by h will be locally Cohen-Macaulay if the links lk,l;xi are Cohen-Macaulay for each vertex Xi E E. Remark 2.16. We recall the polarization 1 --+ Polar 1 of Definition 1.1. LOfwal and Weyman have pointed out that this association allows one to extend Reisner's CohenMacaulay criterion to the non-square-free case. The reason for this is that 1 and Polar 1 differ only by a regular sequence (to obtain 1 from Polar 1 simply set the new generations of variables equal to the original generation again). Therefore 1 determines a Cohen-Macaulay quotient if and only if Polar 1 does; but Polar 1 is square-free. Thus we may extend Reisner's Cohen-Macaulay criterion by the association 1--+ EPolarI'

We call

§ 3.

EpolarI

the Rei8ner complex associated to 1.

The topological Cohen-l'lacaulay criterion of Schwartau

First we describe the combinatorical singularities of simplicial complexes. All simplicial complexes are assumed to be finite. We interpret (1) as a (-1)-sphere and not a (-1 )-disc, thus we never allow (1) as a simplex.

124

II. Hochster·Reisner theory for monomial ideals

Definition 3.1. Let 1: be a simplicial complex, and (1 a simple~ of 1:. Then we say that Ik E(1 is a sphere or a disc if lIkE (11 is homeom€)rphic to a sphere or a disc. If IkE(1 is not a sphere or a disc we say (1 is 81,ngular, or a si1UJularity of 1:. Sing 1: is the subcomplex of 1: generated by the singular simplices of 1:. Remark 3.2. If we have a simplex (1 E Sing 1:, this does not imply that IkE(1 fails to be a sphere or disc. Sing 1: is the complex generated by the singular simplices, thus all we can say is that (1 = (1' where Ik E(1' fails to be a sphere or disc. However, we always know that facets of Sing 1: have bad links. >

Definition 3.3. ,A simplicial complex 1: will be called a comhinatarial manifold-with-8, or simply a manifoU, if: a) 1: is equi-dimensional, and b) Sing 1: = 0. Definition 3.4. A comJnnatorial sphere (resp. comlnnatorial disc) will mean a combinatorial manifold-with-81: such that 11:1 is homeomorphic to a sphere (resp. a disc). We will write c-sphere, c-disc, for short notation. Lemma 3.5. Let 1: be an equi-dimensional simplicial complex. Then the ,/ollowzng conditions are equivalent: (i) 1: is a manifoU. (ii) For all 81,'mplices (1 E 1:, lkE(1 is a sphere or disc. (iii) Far all simplices (1 E 1:, IkE(1 is a c-sphere or c-disc. (iv) Far all 81,"mplice8 (1 E 1:, StarE(1 is a c-disc. Proof: (il ~ (ii) by definition of manifolds. (ii) ~ (iii): The implication (~) is trivial. The implication (=?) results from the fact that the link of any simplex in 1: is an iteration of links of vertices (see Lemma 1.10). (iii) ~ (iv): The implication (~) is easy to prove, since IkL'(1 is just IkstarEu(1, and StarL'(1 is a combinatorial manifold by hypothesis. To prove the implication (=?) recall that StarE (1 is just an iteration of simplicial cones over IkE(1. Thus it suffices to show that if 1: is a c-sphere or c-disc, then Conep(1:) is a c-disc. But this follows from the link formulas of Lemma 1.14, q.e.d.

Remark 3.6. According to our definitions, an equi-dill1ensional simplicial complex is a combinatorial manifold if and only if Sing 1: 0, i.e. the link of every simplex is homeomorphic to a sphere or disc. This is weaker than the usual definition, which would have every link PL-homeomorphic to the standard simplicial sphere or disc (it is enough to require this for links of vertices). Our definition is a hybrid of the topological and PL approaches which makes it easier to detect the singular simplices. . Proposition 3.7. Let 1: be a simplicial complex. Then if 1: is a manifold, 11:1 is a topological manifoU-u;ith-8. Proof: Let P be any point of 11:1. We have to show that P has an open neighborhood U in 11:1 homeomorphic to an open subset of the closed half-space Hll R" where d is the dimension of 1:. Now, P lies in the interior of some simplex (1 of 1:. Note that we may perform a modified barycentric subdivision of 1: (with P as the barycenter of !(1!) to

=

§ 3. The topological Cohen.Macaulay criterion of Schwartau

125

make P into a vertex in a new triangulation E' of lEI. Under such a barycentric subdivision we have IStarE 0'1· ~ IStarE.PI. Since E is by hypothesis a manifold, it follows from Lemma 3.5(iv) that IStarE,PI is homeomorphic to a disc (of dimension d). Thus the required neighborhood U above is provided by IStarE.PI, q.e.d. Now we want to analyze the singularities of cones and of stars. Proposition 3.8. Let E be an equi-dimensional simplicial complex. Then we have: (i) (s~'ngularities of cones). There are three p(xmoilities for the simplicial cone Conep(E): 1) a) E is a c-8phere or c-dz'sc and Conep(E) is a c-disc. b) E is a manifold other than a c-sphere or c-disc and Sing Conep(E) {Pl. 2) Sing E 9= 0 and Sing C<>nep(E) = Conep(Sing E) 9= 0. (ii) (singularities of stars). Let 0' be any szmplex of E. Then for any vertex v E 0', 0' E Sing E zJ and only if 0' E Sing StarEv. Proof: (i) This proof depends on the link formulas tx, {3, y of Lemma 1.14. (ii) Note that v EO'=? 0' E StarEv, thus the statement is meaningful. The proof follows by showing that ( *)

IkE 0' = IkStarL" 0' •

To show this, note that in any complex, the link of a simplex 0' is obtained by finding all simplices of the complex containing 0' and "pulling off" 0'. But since v E 0', {simplices of E containing O'} . {simplices of StarEv containing O'} and (*) is proven. We may now apply (*) to show that Sing E, Sing StarE v have the same facets; hence we have an equality of complexes Sing E Sing StarEv, q.e.d. Next we need to make some observations and collect known results on quasimanifolds. Definition 3.9. A simplicial complex E is a quasi-manifold if (i) E is equi-dimensional, and (ii) Sing E contains no codimension 1 simplex of E. Remark 3.10. This definition means that the link of every codimension 1 simplex is a O-sphere or O-disc; i.e. every codimension 1 simplex is contained in at most two facets. This constitutes only part of the usual definition of "pseudo-manifold" found in the literature. Definition 3.11. A simplicial complex E is called a regular non-quaai-mamYold (RNQM) if (i) E is equi-dimensional, and (ii) Sing E 9= 0 and is a union of codimension 1 simplices of E. Examples 3.12. (i) Any manifold is a quasi-manifold. (ii) The complex E of Example 1.15 is not a manifold, but is a quasi-manifold. (iii) The complex E of Example 1.16 is not a quasi-manifold, but is It regular non-qultsimanifold. (iv) Let E be an equi-dimensional simplicial complex. If lEI is a topological manifoldwith-o, then E is a quasi-manifold.

126

II. Hochster-Reisner theory for monomial ideall!

Proof: Suppose that E is not a quasi-manifold. But since E is equi-dimensional, 80 there must exist a simplex u E Sing E of codimension 1 in E. Let P be any point of lui. Then any meighborhood of P in iStarl:ul is a neighborhood of P in lEI. Therefore if we show that IStarl:ul is not a topological manifold at P then lEI is not a topological manifold at P, and the proof is finished. But u is contained in 3 facets of E, therefore IStarl:ul is a "fan". Separations theorems from topology guarantee that a fan cannot be a topological manifold. This does prove (iv), q.e.d.

We still need more facts on orientable simplicial complexes. Definition 3.13. Let u be a simplex in a simplicial complex E. Then u becomes an oriented simplex if we choose an arbitrary fixed ordering of the vertices. The equivalence class of even permutations of this fixed ordering is called the positively oriented szmplex +u; the equivalence class of odd permutations is the negatively oriented simplex -u. Notice that an orientation of u induces through the simplicial boundary operator an orientation on each boundary-face of u. Definition 3.14. Let E be a quasi-manifold. Then we say E is coherently oriented if the facets of E are oriented in such a way that any two facets meeting in a codimension 1 simplex induce opposite boundary orientations on that simplex. Definition 3.15. Let E be a quasi-manifold. Then we say that E is orientable if it is possible to orient E coherently; otherwise we say E is non--orientable. Remark 3.16. Note that if E is an orientable quasi-manifold, and if E' manifold of the same dimension, then E' must be orientable.

E is a quasi-

We now extend the definition of orientability to arbitrary simplicial complexes. Definition 3.17. Let E be a simplicial complex. Then we say that E is orientable if: a) E is equi-dimensional of dimension d. b) Every d-dimensional quasi-manifold E' c E is orientable. Examples 3.18. (i) Any triangulation of a cylinder is an orientable quasi-manifold of dimension 2. (ii) Any triangulation of a Mobius band is a non-orientable quasi-manifold of dimension 2. (iii) The quasi-manifold of Example 3.12(ii) is orientable. (iv) The RNQM of Example 3.12(iii) is orientable. Lemma 3.19. Let E' c E be equi-dimen8ionalsz'mplicial complexes of the same dimen81,·on. Then: E is orientable ~ E' is orientable. Proof: Immediate from the definition of orientability.

Corollary 3.20. Let E be a simplicial complex, anit v a vertex of E. Then: E is orientable ~ Starl:v is orientable.

Lemma 3.21. Let E be a szmplicial complex. Then: E is orientable

{?

Conep(E) is orientable.

§ 3. The topological Cohen-Macaulay criterion of Schwartau

127

Proof: As 'in the proof of Lemma 1.13, we need the 1-1 correspondence:

{facets of E) + (;~) • {facets of Conep(E)) . This correspondence preserves equi-dimensionality, and preserves coherent in a natural way; thus the Lemma 3.21 is proven. Corollary 3.22. Let l' be a

8~mplic£al

orie~tations

complex, and v a vertex of E. Then:

l' 'is orientable :::} IkI v 'is or£entable. Proof: IkIv is the base of the simplicial cone StarIv. We now apply Corollary 3.20 and

Lemma 3.21. Corollary 3.23. Let l' be a s£mplicial complex and then IkI

(1

(1

any ~mplex of E. If l' 'is orientable

is orientable.

Proof: As in Lemma 1.10, any link IkI (1 is just an iterated link of vertices. Now apply

Corollary 3.22, q.e.d. It is not too difficult to show that equi-dimensional simplicial complexes of codimension 0 or 1 are all c-spheres or c-discs. Therefore it was Schwartau's idea to investigate simplicial complexes of codimension 2. All his results depend on the following decomposition theorem:

Theorem 3.24 (Schwartau [1], Theorem 165). Let d > 1, l' an equ£-d£mensional d-dimensional ~mplicial complex of codt'mens~on 2. Then there exist subcomplexes S, St, G = S n St of l' such that: (i) S =!= 0, St =!= 0, and l' S u St. (ii) S 'is a c-sphere or c-d'i8c of d~menstOn d. (iii) St 'is an equi-dimensional simplicial cone of d£mens~on d and of cod£mension < 2. (iv) G is conta1ned in the base of the cone St. (v) G is a nonvoid union of (d-1)- and (d

2)-~mplices.

By considering orientable simplicial complexes of codimension 2 Schwartau obtained the following result (see Schwartau [1], Theorem 167): Theorem 3.26. Let l' be an orientable quasi-manifold, either:

s~mplicial

complex of

cod~mension

2. Then if l' 'is a

a) E £8 a manifold, or b) Sing E has a facet of codtmension 2 in E.

The key in analyzing the singularities of codimension 2 complexes is the following interesting lemma proved by Schwartau [1], Lemma 166: Lemma 3.26. Let X, Y be two-dimensional simplicial ocmplexes, each of which 'is a manifold or an RNQ~!, joined along two disjoint line segments. Then if cod~m (X u Y) 2, X u Y must be a M 6bius band. We can now sharpen Theorem 3.25 and we get the topological Cohen-Macaulay criterion of Schwartau [1], Theorems 172, 173, 174 and 175.

128

II. Hoohster-Reisner theory for monomial ideals

Theorem 3.27 (Schwartau's topological Cohen-Macaulay criterion). Let .E be an equidimensional orientable 8implicial complex of codimension 2 and d~"mension d 2. Then the following conditions are equ~valent: (i) E is Oohen-Macaulay. (ii) Sing E is equi-dimensional of dimension d - 1 or i8 0. (iii) E i8 a manifold. (iv) E is a regular non-quasi-manifold. This is Schwartau's beautiful Cohen-Macaulay criterion. Example 3.28. The complex of Example 1.15 is not Cohen.Macaulay, but the complex of Example 1.16 is Cohen-Macaulay. This is now a direct consequence of Theorem 3.27. In the following observations we show how to put Schwarlau's topological CohenMacaulay criterion into practice. Let E be an equi-dimensional simplicial complex of codimension 2 and dimension d. Then by definition E is a subcomplex of L1 d+2, the standard (d + 2)-simplex. It follows that each facet of E contains all the vertices of L1 d+2 save two. Thus we may denote each facet of E by two parameters, namely by the vertices it does not contain. If we now draw vertices to indicate the vertices of L1 d+2 , each facet of E may be represented by an edge connecting two vertices. This is the socalled associated graph of E. Example 3.29. Consider the ideal I (xo, Xl) () (X2' xa) in S = K[xo, ... , xa]. Then I is a square-free monomial ideal of height 2. Therefore the Reisner complex (see § 1) has codimension 2. We have a decomposition of Er into its facets given by Er

E(31•. :&,)

U

E(31,,31.)'

By the definition of Reisner complexes, of S except Xo,

Xl'

is the simplex given by all the variables o 1 Therefore in the graph for E[ this simplex is represented as - ; E(:&••z,)

3

similary, the simplex

-o

1

3

2

E(z •• z,)

2

is represented as _ . The total graph for

E[

is thus:

Notice that in Pi the ideal J defines two skew lines; by coincidence the graph is a simplicial model of this fact (also note that the graph of E[ in this case happens to coincide with E[ itself). Remark 3.30. In case that a monomial ideal I of S is not a square· free monomial ideal, we must take the polarization of J before obtaining the Reisner complex (see Definition 1.1). Now suppose dim E 2 and T is a simplex of codimension 2 in E. Since IkET is equi-dimensional of dimension 1 and since codim E 2 it follows that codim IkET 2 (linking reduces vertices by at least as much as dimension). Thus IkET must be a i-dimensional subcomplex of .1 3 , the standard 3-simplex. It follows that ikE T is either a 1-sphere, or a 1-disc, or

t I, or contains a sUbcomplex of the form --1-.

§ 3. The topological Cohen.M.acaulay criterion of Schwartau

129

In the last case (T, v) is a singular simplex of E containing T. Thus if T is a facet of Sing E this cannot occur. Alternately lk£T could also not be a sphere or dies. Thus we must have Ik£T =

a

c

b

d

I !.

Conversely, if T has such a link, T must be a facet of Sing E. For T is clearly in Sing E, so we only need to check that any codimension 1 simplex of E containing T is non-singular. But the link in E of any such simplex is simply the link of a vertex in Ik.!,'T. This is always a O-disc and we are done. Thus we have shown:

T ,is a facet of Sing E of codimension 2 in 1: § lk£ T

=

I I. a

c

b

d

Notice that T has such a link if and only if (T, a, b), (T, e, d) are facets of E and (T, a, c), (T, a, d), (T, b, c), (T, b, d) are not. Thus T is an isolated codimension 2 singularity of E if and only if the edges ab, cd appear in the associated graph of E and the edges ae, ad, be, bd do not. Notice that the four missing edges are the only possible edges which could connect the disjoint edges ab, cd to each other. Conversely, suppose there exist disjoint edges ab, cd in the graph which are not connected to each other by any other edge of the graph. The facets of E represented by ab, cd must then intersect along a codimension 2 simplex T of E (given by il d+2 - {a, b, e, d} ), and lk T must be precisely

a

c

b

d

I !; hence l' is a facet of Sing E of codimension 2

in E. Therefore_we have shown that there is a 1-1 correspondence: pairs of disjoint edges in the } {Facets of Sing of codimension 2 in E} ~-+ graph of E not connecteq. to • { each other by any other edge Therefore Sing E has no facets of codimension 2 in E if and only if given any two edges of the associated graph, either they are connected to each other or else there exists a third edge connected to both of them. Let E be an equi-dimensional simplicial complex of codimension 2, and write G for the associated graph of E. Consider the dual graph G*: each edge of G becomes a vertex in G*, and two vertices of G* are connected with an edge if and only if the two corresponding edges of G meet at a vertex. Recall that the diameter of a graph is the largest number of edges needed to connect any two given vertices of the graph. In this context, we hav~ proven:

Theorem 3.31. Let E be an eqm'-dimensional simplicial complex of codimens~on a880eiated graph, and G* the dual graph of G. Then:

2, G the

assume) ( dim £';;:: 1

1. Sing E

conta~"ns

no facet8 of codimension 1 in E {::::

g

G eontainAJ no triangle8,

assume)

( dim£;;::2

2. Sing E contain8 no facet8 of codimension 2 in 1: {::::=g G* na8 diameter 9 Buchsbaum Rings

2.

130

II. Hochster-Reisner theory for monomial ideals

Corollary 3.32. Let I be an equi-dimensional orz'entable simplwi.al complex 0/ codimenstOn 2.; G the a8soci.ated graph, and G* the dual graph. Then: A) I is Cohen-Macaulay ~ G* has di.ameter 2. B) I Us a c-sphere or a c-disc ~ G* has di.ameter 2 and G contains no tri.angles. Proof: If dim I = 0 then I is always Cohen-Macaulay (see Example 2.14(ii)), and the result follows. If dim I 1, I must be a subcomplex of Ll 3 • I is Cohen-Macaulay if and only if I is connected (see Example 2.14(iii)), and again the result follows directly. Thus we may assume dim I 2, whence both parts of Theorem 3.31 apply. If codim I < 2, the corollary is a trivial consequence of Theorem 3.31 by use of our observations after Corollary 3.23. Thus we may assume codim I = 2. But then we finish the proof as follows: A) :::}: Theorem 3.31 and Theorem 3.27. A) ~: Theorem 3.31, Theorem 3.27 and an application of the following fact which improves Theorem 3.25: Let I be an equi-dimensional orientable simplicial complex of codimension 2. Th~n either I is a manifold or else Sing I is a non-empty union of codimension 1 and codimension 2 simplices of I. B) :::}: Theorem 3.31. . B) ~: Theorem 3.31 and the just mentioned fact. We also note that if I is a manifold then I is a c-sphere or c-disc, q.e.d.

Now let 1 c S = K[xo, ... , x n ] be a monomial ideal of height 2; we relate the above theory to the Cohen-Macaulay property for S/l. First note that Sll cannot be CohenMacaulay unless 1 is equi-dimensional and unmixed; that is, every associated prime of 1 has height 2. It follows that the Reisner complex I associated to 1 is equi-dimensional and of codimension :;; 2. In fact by these results, each facet of I is given by a Reisner complex I(Xa,Xb) c LlN for (xu. Xb) an associated prime of Polar 1 c SIN} = k[xo, "', XN]' By the definition of Reisner complexes, X a , Xb are the two vertices of LlN not contained in the facet I(x•. x.); hence this facet correspond to the edge :-----1 in the graph G which we associate to I. In short, G (or G*) may be obtained directly from the primary decomposition of Polar 1. Thus we now refer to G* as "the graph or the primary decomposition of Polar 1". In addition, we now call 1 orienlable or a qua8't"-mani/old, etc., if the Reisner complex I of 1 has these properties. It follows immediately from Corollary 3.32A); CoroUary 3.33. Let 1 c S be an orientable monomi.al Ukal 0/ hei,ght 2, and let G* be the graph 0/ the primary decomposition 0/ the pol{lrization 0/ 1. Then: S/l is Coken-Macaulay zj and only tj (i) I i8 equi-dimensional and unmixed, and (ii) G* has di.ameter 2.

Example 3.34. Let 1 c S be the square-free monomial ideal (Xo, Xl) n (Xl' X2) n ... r (Xk-l, Xk)' It is not difficult to see that 1 is an orientable quasi-manifold. Thus Corollary 3.33 applies. The conclusion is: S/l is Cohen-Macaulay if and only if k 3.

§ 3. The topological Cohen-Macaulay criterion of /3chwartau

131

We now apply Schwartau's theory to answer the following question: Consider the ideal

Which exponents a, b, c, d 0) define arithmetically Cohen-Macaulay curves in P~. Examining the dual graph G* of the Reisner complex of I we get the following general result: Example 3.35. Let G* be the graph associated to Polar 1. Then G* has diameter if and only if:

+c

b

Case 2. a, b, c > 0, d = O;a

+c

Case 1. a, b, c, d

>

0: a

+ d + e, for e b

+

2

-1,0, 1.

1.

Case 3a. a, b > 0, c = dO: always. Case 4. a

>

>

0, b d = 0: never. 0, b = c = dO; always.

Case 3b. a, c

Proof: Left to the reader (see also Schwartau [11, Theorem 186).

Now we recall the original motivation for this section; Consider the following ideals in S: and

We will see from liaison addition in Chapter III that S/1 is always Cohen-Macaulay, and S/I' is always Buchsbaum. Example 3.35 provides also' an explanation for this phenomenon independent of liaison addition. By Example 3.35 the rings Sj]' are not Cohen-Macaulay since the 'associated Reisner complex E has isolated codimension 2 singularities (i.e. Sing E has facets of codimension 2 in E). In addition it follows that the Reisner complex for the ideal]' has precisely n isolated codimension 2 singularities. In order to show that S/1' is a Buchsbaum ring with invariant n we have to use another method. Applying Schwartau's liaison addition from Chapter III we will prove that the curves in P~ defined by l' have liaison invariant 11" (up to shift); that is, 8jl' is always Buchsbaum. Furthermore, by constructing the minimal graded free resolution of the ideal from Example 3.35 we can sharpen the result of Example 3.35 (see Schwartau [1], Chap. 3) ;

!

Proposition 3.36. The ideal (xo, xll a n (Xl> x 2)b Il (X2' xa)C n (xa, xo)t! defines an arithmetically Cohen-Macaulay curve in Case 1. a, b, c, d Case 3a.

Pk if and only if:

+ d + e, for e = 0; a + c < b + 1.

> 0, d = a, b > 0, c = d = a, c > 0, b d

Case 2. a, b, c Case 3b.

> 0; a + c

b

0: alwaY8. 0; never.

Ca8e 4. a> 0, b = c ='d = 0; always. 9*

-1,0,1.

'132

II. Hochster-Reisner theory for monomial ideals

§ 4.

Further applications to algebraic topology and combinatorics

In this paragraph we will investigate properties of Buchsbaum complexes of simplicial complexes. For example, following P. Schenzel [1], we will examine a term which can be interpreted as measuring the error when we no longer have the Cohen-Macaulay case of simplicial complexes. We note that we actually do not need Schenzel's approach for these applications to combinatorics (see, for example, Corollary 4.7' and Lemma 4.14'). The first topic of this pltragraph will 15e the characterization of Buchsbaum modules using dualizing complexes. For this we will follow an idea initiated by R. Kiehl [1]. P. Schenzel [1] obtained some generalizations of Kiehl's approach by improving the locally Cohen-Macaulay criterion of Reisner of Theorem 2.4. We will first of all review the relevant commutative algebra. Let M': ... -'>- Mk -'>- Mk+1 -'>- ... be a complex of A-modules, By ,8M', resp, "M', we denote the truncated complex ,., -'>-

jJfk

lV8 -

1 -'>-

MH2)

-'>-

-'>- , •• -'>-

Im(2If$<-1

-'>-

MB)

-'>-

0

resp.

o If r

<

Im(M'+l

-'>-

lVr+1

..•

-'>- ~Jfk -'>- .,'

s we have Hk(,BM') ,

~

=

{OHk(M')

for k for 'I'

or k ~ s, k < s, r

<

where <;(M') ,,(,sM'), Now M'[tJ will denote the complex M' shifted t places to the left with changed sign ofthe boundary map, that is, (M'[t]) = lV,,+I and dM,[t] 1)' dM " n E Z. To state the first main result we will use some notation from Chapter 0, § 3, The following theorem was first published by P. Schenzel [1].

Theorem 4.1. Let M de1wte a Noetherian A-module with d:= dimAM > 0, 'J'hen the following conditions are equivalent: (i) M is a Buchsbaum module, (ii) -rdl1rm(M) is quasi-isomorphic t() a complex of k-vector spaces, (iii) -rdl1rm(M) ~ c"(M), where C'(M) is a complex of k-vector spaces with CI(M)

= { ~!n(lJf)

for 0 i otherwzse

< d,

and trivial boundary homomorphisms. If A possesses a dual£z£ng c()mplex lA' then the above conditions are equivalent to: (iv) '-d HOffiA(M, I A) is quas£-isomorphic to a complex of k-vector spaces.

Before we will embark on the proof, we will show a lemma which is of interest in its own right.

Lemma 4.2. Let A be a local ring possessing a dual£z£ng complex lA' Let P' be a bounded complex of finitely generated free A-modules such that pi = 0 for £ < O. Assume that the cohomology modules Hi(F" ® M), i E Z, are A-modules of finite length for a Noetherian A-module M. If Ld.HomA(M, I A) 'is quasi-isomorphic to a complex of k-vector spaces, then ,d(F" @ M) is quasi-'isomorphic to a complex of k-vector spaces.

§ 4, Further applications

133

Proof: By the definition of the dualizing complex we get HomA(M, l~)

0

for ~' < -dim M,

Therefore KM := H-d(HomA(M, lA'» is the first non-vanishing cohomology module by virtue of the Local Duality Theorem, Hence we have the following short exact sequence of complexes

0-* KM[d]

-*

HomA(M, lA)

-*

LdHomA(M, fA)

-*

0,

Applying the derived functor 11 Hom(F', ) and HomA( ,E) we get

0-* HomA(11 Hom(F", '-dHomA(M, lA)' E») -*

HomA(11 Hom(F', KM[d]), E)

-*

-*

I1rm(F'

M)

0

by virtue of the Local Duality Theorem. Here E denotes the injective hull of the residue field k. Now we remark

and

HomA(11 Hom(F', KM[d]),

E) "-' HomA(HomA(F', KM[d]), E).

From this it follows that

(HomA(11 Hom(F", KM[d]), E»)i

=

0

for all i

< d,

Therefore, the short exact sequence induces a quasi-isomorphism Td(F'

HomA('-dHomA(M, fA)'

E») '" Tdl1rm(F'

M).

Because all the cohomology modules Hi(F' ® M) are modules of finite length, it follows that I1rm(F' ® M) = F" M, If T_aHomA(M, fA) is quasi-isomorphic to a complex of k-vector spaces, the same is true for

F' ® HomA(T_aHomA(M, fA'>,

E).

Thus, our statement follows from the quasi-isomorphism given in (*).

Proof of Theorem 4.1. First of all we remark that a complex X' is quasi-isomorphic to a complex of k-vector spaccs if and only if X' A is quasi-isomorphic to a compiex of k-vector spaces. Furthermore, a Noetherian A-module M is a Buchsbaum module if and only if M®AA is a Buchsbaum module over A, see Lemma 11.13. That is, without loss of generality we can assume A = A and M iI, where ~ denotes the m-adic completion. That means, we can assume A to be a quotient of a regular local ring R by the Cohen Structure Theorem, compare for example Matsumura [1]. J"et fA be the dualizing complex. Then we have

Therefore, the conditions (ii) and (iv) are equivalent, Also, a complex of k-vector spaces s quasi-isomorphic to its cohomology complex. Hence, (ii) § (iii), Now, we will show (i) => (iii). To this end we make use of the following:

134

II. 'Hochster-Reisner theory for monomial ideals

Proposition 4.3. Let

r: K" -+ D

be a homomorphi8m of complexes of A-modules su,ch that

(a) K' is a complex of Jc..vector spaces, and

(b) HV): H'(K") -+ H'(D), i E Z, is a surjective homomorphism, Then the complex. L' is quaIJi-isomorphic to a complex of k-vector spaces,

Bi.

Proof: We denote by Bk., the image of the homomorphism Kl-l -+ Ki resp. V-I -+ V, and we denote by Zk" Z~, the kernel of the homomorphism Ki -+ Ki+l resp, V-+ V+l. Then we have the following commutative diagram with exact rows

)

Hi~K')----):'()

where the homomorphisms Bk. -+ BL Z~, -+ ZL and Hi(K') -+ Hi(£") are induced by r. Because K" is a complex of k-vector spaces, the canonical homomorphism Zk -+ Hi(K') splits. Furthermore, Hi(j'): Hi(K') -+ Hi(D) is a surjective homomorphism of k-vector spaces, i.e., it splits. That is, we get a homomorphism Hi(L') -+ £i. In fact, it is a homomorphism of the cohomology complex H'(L') of L' to L', which induces isomorphisms on the cohomology modules. This proves Proposition 4.3, Next we prove (i) =? (iii) of Theorem 4.1. By Theorem 1.2.15, for a Buchsbaum module M, the canonical homomorphism of co~plexes TtlK'(mj M) -+ TdK"'(m; M)

induces surjective homomorphisms on the cohomology modules. Because K'(m; M) is a complex of k-vector spaces it follows that rtlKOO(m; M) is quasi-isomorphic to a complex Erm(M) in the derived of k-vector spaces, see Proposition 4.3. Because KOO(m; M) category, we have that T-dE HomA(M, I A) is quasi-isomorphic to a complex of k-v~ctor spaces by virtue of the Local Duality Theorem, Theorem 0.3.4. For the proof of Theorem 4.1 it remains to show (iv) =? (i). To this end we consider the Koszul complex K'(q; M) of M with respect to an arbitrary parameter ideal q (Xl' .. ,' Xd) of M. By the definition we have K"(qj M) = K'(q; A) ®AM,

where K'(q; A) is a complex of finitely generated free A-modules such that the cohomology modules Hi(qj M) of K'(q; M) are A-modules of finite length, compare Chapter 0, § 1 for properties of Koszul complexes, By virtue of our Lemma 4.2 it follows from (iv) that TdK'(q; M} is quasi-isomorphic to a complex of k-vector spaces. In pa,rticular we have mHd-1(q; M) = 0

§ 4. Further applications

for an arbitrary parameter ideal q of M. We set q' following short exact sequence 0.-+ Hd-2(q'; M)@AAjxdA

.-+

H·H(q; M)

.-+

(Xl' ..• ,

Xd-l)'

135

Then we have the

q'M :Xdjq' Jl .-+ 0, ~

compare Chapter 0, Lemma 1.6. Therefore we get m· (q'M :Xd) Buchsbaum module by virtue of Proposition 1.1.10.

q'111', i.e., .Jl is a

The following corollaries indicate, in our opinion, how substantial Theorem 4.1 is. First, we recall that an ideal a of a regular local ring R is called perfect, if Rja is a CohenMacaulay ring. That is equivalent to the vanishing of Hi(E Hom(Rja, R») ~ Ext~(R/a, R)

for all i

n. For Buchsbaum rings a corresponding result is valid.

dim R - dim Rja

Corollary 4.4. The larol ring RIa is a Buchsbaum ring ?/ and only i/ TnE Hom(Rja, R) is to a complex 0/ k-vector spaces, where k denotes the residue field 0/ R.

qu.a~i-ilJomorphic

The proof follows from Theorem 4.1 because E Hom(Rja, R) is up to a shift isomorphic to the (normalized) dualizing complex of Ria. If M is a Buchsbaum module, then it follows that T-d

HomA(ilf, fA)::::::: Hom".(C·(M), k»)

by the Local Duality Theorem and Matlis duality. Corollary 4.5. Let M be a Buchsbaum module. Let F' be a bounded complex 0/ finitely generated free A-module8 8uch that Fi 0 lor i < 0 and F" @Ak has trivial boundary map8. A8sume that Hi(F' @M), i E Z, are modules o//inite length. Then we get

m· Hi(F'@M)

=

0

and i

diIDt Hi(F' @M)

= 1: rank FH dim". Hfn(M)

i

lor 0

< dim M.

Proof: First we notice that m· Hi(F'@M) 0, i < dim M, follows from Lemma 4.2. Then the quasi-isomorphism (*) given in the proof of Lemma 4.2 implies Hi(F'@M)

= H'(F'

C'Ull») ,

i

<

dim M.

Because F'@k has trivial boundary homomorphisms it follows that i

dim". Hi(F @ C'(M»)

= 1: rank Fi-v dim". Hfn(M) ,

i

<

dim Jl.

0=0

Next we will apply Corollary 4.5 to the Koszul complex K' (q; M) of M with respect to a parameter ideal q = (Xl' ... , XII), d = dim M, of 1.1I. Corollary 4.6. Let M be a Buchsbaum module, and let q = (Xl> ••• , Xd), d arbitrary parameter ideal of M. Then we get

mHi(q; M) and

0

dim M, be an

136

II. Hochster-Reisner theory for monomial ideals

By results of Auslander-Buchsbaum [1] and Serre [2] we know that d

lA(M/qM)

eo(q; M)

1:

l)i-lIA(Hi(q; M»)

.=1

for an arbitrary parameter ideal q (Xl' ... , Xd), d dim M, of a Noetherian A-module M. That is, in the case of a Buchsbaum module M we have

which is another proof of Proposition I.2.6. Also Theorem 4.1 provides a Koszul complex characterization of Buchsbaum modules. Corollary 4.7. Let M denote a Noetherian A-module, d = dim M. Then the jollowing conditions are eqmvalent: (i) M iY a Buchsbaum module, (ii) mHd-l(q; M) 0 jor every parameter ideal q = (Xl> •.. , Xd) 01 M, (iii) mHr(q; M) 0 lor every parameter ideal q 01 M and aU 0 r < dim M, and (iv) lor every parameter ideal q 01 M, .dK(q; M) iY quas~'-isomorphic to a complex oj k-vector spaces.

Prool: First we note that we can assume A = .A and M = k without loss of generality. That is, we can assume that A possesses a dualizing complex. Then, (i) =:;, (iv) follows by Lemma 4.2. The implications (iv) =:;, (iii) =:;, (ii) are trivial. Finally, (ii) =:;, (i) is proven by the same way as the statement (ii) (i) of Theorem 4.1. The analogy of the previous corollary to the Koszul complex characterization of Cohen-Macaulay modules is obvious. Furthermore, we want to mention here that Suzuki [3] also gave a characterization of a Buchsbaum module by using the Koszul complex generated by a syst€m of parameters of the module and the standard properties of systems of parameters of Buchsbaum modules. To this end he obtained a new statement concerning the cycles and the boundaries of the Koszul complex. In view of our remark given in the preface that the theory of derived categories is not needed for our applications to algebraic topology and combinatorics we want to give a new and simple proof of Corollary 4.7. This approach will enable us to give an elementary proof of Lemma 4.14 below. It is precisely this Lemma 4.14 which does supply the key for applications to combinatorics (see, for example, the proof of Theorem 4.19 below).

Corollary 4.7'. Let M be a Noetherian A-module oj dimen8'ion d > 1. Then the jollowing conditions are equivalent: (i) M is a Buchsbaum module. (ii) m· Hd-l(Xl> ... , Xd; M) = 0 lor all systems 01 parameters Xl' ... , Xd oj M. (iii) m· Hi(Xl' ••• , Xd; M) = 0 jar all systems 01 parameters Xl> ••• , Xd 01 M and all i< d. (iv) m· Hi(Xl, ••• , x,; M) o jar , all parts 01 systems 01 parameters Xl> ... , x" r S d, oj M and all i < r.

137

§ 4. Further applications

Proof; The implications (iv) ~ (iii) ~ (ii) are triviaL For the proof of (ii) ~ (i) we have that the top row of the commutative diagram of Lemma 0.1.6 does yield an epimorphism: Hd-l(X I , .•. , Xd; M)

-'i>

HO(Xd; Hd-I(Xl, ••. , Xd-l; .;}l))

c::-:: (Xl' •.. , Xd-l) . M :Xd!(Xh "', Xd-I) . M, Hence our assumption now provides that

m· ((Xl' "., Xd-I)' M:Xd/(X h

''',

Xd-I)' M)

O.

By applying Proposition 1.1.10 we therefore get the result (i). It remains to prove (i) ~ (iv). We will prove the following result: Let M, Xl> • '" Xr be as above. We denote by d' the differentiation of the Koszul complex K'(xl> •.. , Xr ; M). If 0 i < rand Yl> ••• , Ys, 8:::;; d - t', are elements of m such that Yt> ••• , Y., Xi" .,., Xi, is a part of a system of parameters of M for all 1 il < ... < ii r, then we show the following claim to be true:

Claim. Ker d i

()

(Yl1 .•. , Ys) Ki(Xl> ••• , Xr ; M) ~ 1m d'-I.

Having this one obtains the implication (i) ~ (iv) as follows: First, choose a EJj M/(xi" .• ,' Xi) , M)' -basis Yl> ••• , y, of m (see our Definition 1.1.7 and Pro-

( l-s;,j,< .. ·
1:::;; l < t. Then the claim implies

Yl • Ker d i ~ Ker d i n (Yl) . Ki(Xl' ••• , X r ; M)

1m d H

;

that is, Yl • Hi(Xb ..• , Xr ; M) = 0 for all l = 1, "., t. Hence we get condition (iv). Now we prove the claim: We will use induction on r. If r sider the case i = 0; Ker dO () (Yl' "., Ys) KO(x l ,

".,

1 we have only to con-

= (O:M (Xl' ... , Xr )) n (Yl> , .. , Ys)' = (O:Mm) () (YI' "" Ys) , M = 0 1m d- 1

Xr ; M)

M

(see Lemma 1.1.14). If 1 < r d we put K' := K'(XI' .'" Xr ; M)

and

L'

;=

K'(XI' , .. , X,-l; M)

with differentiation J;. Then Xi = D EJj V-I, and for the element (a, b) E Ki wit? a ED and bE D-I we have: di(a,b) = (di(a), (-l)i xra

+ di-l(b)).

If (a, b) E Ker dl () (Yh . '" Ys)' Xi then we have: (i) a E Kerd i n (Yh ... , Ys)' D and (ii) bE (Yl> ... , Ys) • LH, (_I)i x,a di - 1(b) O. If i < r 1 then (i) and the induction hypothesis imply that

+

a

di-I(a') with a' ED-I.

138

II. Hochster-Reisner theory for monomial ideals

If i = r 1 we have V-I = M and 1m dr- 2 equation in (ii) one obtains:

(xl' .'"

X r-l)

,M, Therefore with the

a E (1m dr- 2 :;\fXr) n (Yl> .. " Ys) • M

(tX l, .. "

x r- 1 ) M :xr) n (Yl, ... , Ys) . 1'1f

(Xl> ... , x r- 1) • ill: m)

n (Yl'

..• , Ys) . M

S (Xl> •.• , Xr-l) . M which by applying Lemma 1.1.14, equals 1m dr2 ; that is, a = dT - 2(a'). Hence we obtain for all i < r that a di - 1 (a'). Then

dH ((-l)'x ra'+b)

(-l)'xra

+ di-1(b)

0;

that is, (-1)' xra'

+ b E Ker di - I

n (Yl> ... ,

1m di - 2 •

y" x r )· Li-l

Thus we see that b = 1).-1 xral + di - 2(b') with b' E £1- 2 ; that is, (a, b) = di - 1 (a', b') i I E 1m d - • This proves the claim and therefore also CDrollary 4.7'. Next we are interested in the Buchsbaum property of the canonical module of a Buchsbaum ring. We recall the definition of the canonical module.

Definition 4.8. A Noetherian A-module KA is called a canonical module ?f A if KA@AA

HomA(Hi!t(A}, E),

as A-modules, where E

d = dim A,

E(k) denotes the injective hull of the residue field k of A.

The notion of the canonical (or dualizing) module was introduced by Herzog and Kunz in [1]. It is uniquely determinated if it exists. In case A possesses a dualizing complex lA' A has a canonical module. More precisely, KA '::: H-d{lA) '

d = dim A.

'This follows from the Local Duality Theorem, compare § 3, Chapter O.

Theorem 4.9. Let A denote a d-dimenswnal Buchsbaum ring which has a canonical module K A. Then KA is a Buchsbaum module with H:n(KA) "-' HomA(Ili!t-i .1(A), E)

lor 2

i

<

d.

Prool: Since all statements are preserved by passing to the completion A of A we can assume without loss of generality A complete. Now we apply the derived functor to the short exact sequence of complexes 0-)00 KA[d]

-)00

lA

-)00

T-alA

-)00

O.

By the Local Duality Theorem we get 0-)00 Hrm(KA[d]) -';. E

-)00

Hrm(LdlA)

-)00

O.

That means, the complex Hrm(LtllA) is isomorphic tD the mapping cone D£ Hrm(KA[d}) -)00 E. Because E is concentrated in degree zero we obtain

§ 4. Further applications .

139

by a simple calculation. Now, if A is a Buchsbaum.ring, r._d.IA, and therefore also the left complex in (**), is isomorphic to a complex of k-vector spaces. From (**) it follows that r dl1rm(KA ) is isomorphic to a complex of k-vector spaces. By our Theorem 4.1 this proves the first statement. Because l1rm(LdIA):::::'

r ... ,

Ld

the isomorphisms for the local cohomology modules follow from (**) by the Local Duality Theorem, q.e.d. Theorem 4.9 answers affirmatively a question posed by Goto and Shimoda in their paper [1], where the result is proven for the case dim A = 3. Naoyoshi Suzuki [1] was the first who asked this question in 1978 at the first symposium on commutative algebra in Japan. He also proved in his report at the symposium Theorem 4.9 in case dim A = 3. In his talk at the 4th symposium, which took place during the period 3-6 November 1982, N. Suzuki [6] gave an elementary proof of Theorem 4.9 by using extensively a lemma on Buchsbaum rings proven by S. Goto. We will give a brief sketch of Suzuki's proof. First we mention the following interesting lemma by Shiro Goto. Goto's Lemma. Let ~W be a Buchsbaum module for dimension d be a part of a system of parameter8 of M and Sll r

(yr', ... , y:') M: 1If m

E

E

••• ,8,

2 and let Yl' ... , Y" r :::;; d positive integers. Then

Y:~1-1 ..... y:~.-l(Yil' ... , Yi.) M:Mm).

k=O l~j,< ... <jk~r

Proof: Using an easy induction argument it is sufficient to prove the following statement: Let N be a Noetherian A-module and a A be an ideal so that NjaN is a Buchs-

baum module of dimension 2. Then for every parameter element a with respect to NjaN and all n > 0 we have (aN

+ anN):N m =

aN:Nm

+ all-1(aN +

aN):Nm).

Passing to M := NjaN we may assume without loss of generality that a

= O. Then

hence anM:m

=

(aIlM:m) n (a ll - 1M

=

O:m

=

+ O:m)

+ all-1(aIlM:m):an-l) O:m + afl-I(aM + O:m):m)

+ (all - M n (aIlM:m)) O:m + afl-1(aIJM:afl-1):m)

O:m

1

O:m

a,,-l(aM:m) ,

q.e.d. Now we sketch Suzuki's proof of Theorem 4.9 for modules: The canonical module KM of a Buchsbaum module M is also a Buchsbaum module. We set JJ'(.) = Hom...{H!n(.), E(k)). We may assume that A = A and we procede by induction on d dim M. Let d > 3 and additionally that depth M > 0 since H'fn(M) H~(MjH:h(M)). Let al> ... , ad be any system of parameters for K M • We have an exact sequence

o ~ K l /aKM ~ K{M/aM) ~ Dd-l(M) ~ 0, l,[

where a

(aI' .•• , ad)'

140

II. Hochster-Reisner theory for monomial ideals

Consider the long exact sequence of Koszul homology modules with respect to a' = {a2, ••• , ad},

EI(a'; K M,) -+ EI(a'; V) -+ KM/(a) KM -+ KM'/(a') K M, -+ V -+ 0

where M' = M/aM and V

D'H(M). If we have that the mapping

EI(a'; n) : EI(a'; K,w) -+ E 1(a'; V)

is a zero map, then the equality

holds. On the other hand, we have

and therefore, by the induction assumption, we can conclude that the difference lA(KM/(a) KM) - eo(a; K M)

does not depend on the chOIce of the system of parameters at> ... , ad for M; which is the definition of the Buchsbaumness of K M • Consider the direct system {Ei(a~, ... , a~; M), «1>~.n+l} with the limit E~(M) (see Chapter 0, § 1), where Ed-i(a; M) ~ Ei(HomA(K.(a; A), M))~ Ei(a; M).

Let an {a~, ... , a~} and a'n {a~, ... , a:}. A commutative diagram is induced by the exact sequence (see Lemma 0.1.5) 0 -+ aM -+ M ~ MjaM -+ O. Ed-I(a", M) - - - - - - - - - - -..... Ed-I(a"; M/)

II!

II!

E 1 (a";M)

1\b:'

H'f;;I(M)

Eo(a''';M')EBEj(a''';M')

1IP:'.

E'f;;I(M')

d

and

1: a/i

o.

(#)

j=2

ltsuffices to show that for any j 2, ... , d, Ii· A = 0. Let z E E'fn-I(M). Then there exists (u, v) E ZI(a"; M) c: Ko(a'fI, M) EB K1(a'fl; M) such that «1>l:c(lu, vI) = z. Note first that the cycle condition implies that a!u E (a~, ... , a~) M; hence uE UM(a;, ... ,a~).

We claim that h·«1>l:c,·E I (a",n) (Iu, vi) the homology class of a cycle c. Let Iii,

vi

(##)

O. Here we use the notation lei for

(lu, Ivl) := HI(a"; n) (Iu, vi) E Eo(a'''; M')

EI(a'''; M').

§ 4. Further applications

141

It is not too difficult to see that (/>~'(Iu,

vi) =

(I(az, .•• , ad)

ul, 1(1)

E Ho(a'fI+l; M/)

EB Hl(a'fl+l; M ' ).

Therefore it suffices to show that

with lui E Ho(a;+I, ... , a~+l; M/aM) M/(a, a~+l, ... , a~t1) M. By Goto's Lemma, from (it it) we obtain the following expression: 'U

= .E

ai~luI

I<;;{2 ••••• d}

with 'UJ E UM(a,; i E I) and aI set I of {a2' • ,., ad}

1. We must show that for any sub-

Ii . (/>M~l(l(az, , •. , ad) a~-lihi) If I 9= {2, .. " d}, there exists i

For I

~

O.

I, hence

= {2, ... , d} ,

Ii' (/>~,;1(I(a2' ... , ad)fI-l alaII)

Ii' (/>;'~1(I(a2' ... , ad)" uII)

= a/I' (/>;';1(I(a2' ... , ai' , .., ad)" aj-lull) , by the cycle condition (it),

= -Lad;' (/>~,;I(I(a2' ... , aj, ... , ad)" aj-luII) i+i -_

"I i ' 'PM' """+l{I(a2' •• ,' Ujai, A • "', ad )fI ain-lai11+1UI I) •

-,,;;.,

i+i

Since lai +lUI I = 0

in

M/(a~d,

... , a~+l) M',

we conclude in this case also that

It· (/>~~1(I(a2' .. ,' ad),,-l alaII)

0

as required, q.e.d. Our Theorem 4,9, proved by P. Schenzel in [1], is an extension of the main result of Kiehl's paper [1] to Buchsbaum rings. Using the criterion of Theorem 1.2.10, Kiehl showed that KA is a Buchsbaum module, if i_ttIA is isomorphic t.o a complex of kvector spaces, If A is a two-dimensional local ring admitting a canonical module K A , then K,d is a Cohen-Macaulay module. Therefore, the converse of Theorem 4.9 is not true in general, even if we assume that A has small dimension or that KA .is a CohenMacaulay module.

142

II. Hochster-Reisner theory for monomial ideals

Theorem 4.10. Let A denote a local riWJ haviWJ the canonical module K A • Assu1ne A saf:i8fies 8 2 , i.e.,

~that

depth All > min(2, dim All) for all p E Spec A. If KA is a B'IJJ)hsbaum module, then A is a B'IJJ)hsbaum riWJ. Proof: First without loss of generality we can assume A A, i.e., A possesses the dualizing complex I;. by virtue of the Cohen Structure Theorem. For dim A < 2 there is nothing left to prove. Therefore, we can assume dim A > 2. By Proposition 0.3.6, we have

Supp KA

= Spec A.

It follows from Theorem 4.9 that H~{KA)'-"'" HomA(H~-i+l(A), E)

for 2

i

<

d.

Since KA is a Buchsbaum module H:n(KA) is isomorphic to a k-vector space. Hence this is also true for H~-;+I(A) for 2 i < d. By this property and the assumption of 4.10 we get that H~(A) has finite length over Aim. Therefore we get that All is a CohenMacaulay ring for all prime ideals p m. Since

'*

dim A

+ dim All

dim Alp

for all prime ideals p, it follows by Proposition 16 ofthe Appendix, that T-diA is a complex whose cohomology modules are modules of finite length. Therefore

T-dI;,......, I1r m{T-dl jJ. By the isomorphism (**) given in the proof of Theorem 4.10 we get

(Tdl1rm(KA») [d

+ 1]'-"'" T=JIA "" LaIA,

where we have used the fact depth A

Ht(T_dI;') = 0,

i

2, i.e.,

= 0, -1.

Now, if KA is a Buchsbaum module, then the left complex in the above chain of isomorphisms is quasi-isomorphic to a complex of k-vector spaces. Therefore this is also true for LaI;' which proves our statement by Theorem 4.1, q.e.d. Proposition 4.11. If ~n addzlWn KA in Theorem 4.10 is a Oohen-Macaulay module, then A itself is a Oohen-MacauZay ring. . ' Proof: If KA is a Cohen-Macaulay module, then it follows from the proof of Theorem 4.10 that Hi(T-dI;') = 0 for all i E Z. By the Local Duality Theorem we have H:n(A) = 0 for all i,* dim A. Thus, A is a Cohen-Macaulay ring, q.e.d.

Having Theorem 4.1 we can now sharpen Reisner's local Cohen-Macaulay criterion of Theorem 2.4. Theorem 4.12. Let 1: be a finite connected simplieial complex with dim 1: = d - 1. Then . the folloun1uj conditions are equivalent: (i) (K[1:])p is a Oohen-Macaulay r~1uj for all pri1ne ideals P different from the irrelevant

ideal m.

§ 4. Further applications

(ii) For aU0 =t= a E E we have

143

1i j(lk,1;l1; K)

0 il i =t= dim Ik EI1. (iii) II X = lEI is the geometric realization 01 E, then Hj(X, X points p E X and all i =t= dim X. (iv) There is a canonical isomorphisms -r-dD~ ~

j

p; K)

=0

lor all

Lo(C.(E; K)* [1]),

where DE denotes the dualizing complex 01 K[EJ and C.(E; K) the reduced sZ'mplicial chain complex with coellicients in K, i.e. Cj(E; K) Z'8 the K-vector space 01 i-laces 01 E. (v) E is a Buchsbaum complex, i.e., K[EJ is a Buchsbaum ring. Prool: It follows from Theorem 2.4 that (i) § (ii) § (v). The equivalence of (ii) and (iii) is even an exercise in topology. For this we note that if a is non-empty and p is a point interior to a, we have the following isomorphisms by excision: a)

p;K),

where starE a denotes the closed star of a in E (see our Definition 1.8). ~ Hj(starE a, (bd a)

* (lk(a)), K)

by deformation retraction, where lk a denotes IkE a, denotes the boundary of a,

* is the join operation and bd a

::::::: H j-l( (bd a) * (lk a); K) by the long exact cohomology sequence, ~ Hj-dimu-1(lk a; K)

by the suspension isomorphism. This proves the equivalence of (ii) and (iii). (iv) =} (v): LaDE is isomorphic to a complex of K-vector spaces in the derived category. Therefore the assertion is immediate from Theorem 4.1 (see also the proof of Corollary 4.4). Hence it is enough to show (i) =} (iv) for proving the theorem. (i) =} (iv): We apply again some ideas of G. A. Reisner. Under the assumption (i) Reisner [1], Theorem 2 proved HomK(Ko(!!i; K[EJ) j K)~ C.(Ll; K) (1],

where Ko(!!i; K[E]) denotes the Oth graded piece of the Koszul complex K(;J2; K[E]) with respect to the elements ;J2 = {Xl' ... , xnl which generate the irrelevant ideal of K[ E] . From this we get K o(;J2j K[E])

~

C'(E, K) [-lJ.

Since K[E] is F-pure resp. has a presentation of relative graded F-pure type it follows from Hochst€r-Roberts [2], Theorem 1.1 and Theorem 4.8 that -rdKo(:r; K[EJ) ~-rdKij,

where Ko denotes the Oth graded piece of the complex K

lim-+ K(:r' ; K[E]). I

144

II. Hochster·Reisner theory for moiiomial ideals

Also we know that Tdj{~ ~

,dK in the derived category D(K[IJ)

since Hi(K) ~ lt~(K[I]) [H~(K[I])lo for 0 <::;; i < d by using Lemma 2.5 (ii). We now use ol?-r assumption (i) which implies the finiteness of the local cohomology modules lt~(K[I]), i =t= dim K[I] (see Proposition 1.3.4). Therefore we obtain

.aK ~ .a{C·(I, K) [-1]). The assertion (iv) now results by virtue of the Local Duality Theorem, q.e.d. By the Local Duality Theorem and Theorem 4.12 we get a result which relates the local cohomology modules of K[I] to the reduced simplicial homology of I with coefficients in K. Corollary 4.13. Let I be a Buchsbaum complex. Then there are z80morphisms

.Following P. Schenzel [1], 6.3, we next will present some further applications of the purity of the Frobenius in the case of the graded k-algebra k[I] if I is a Buchsbaum complex. This is of some interest for our investigations regarding the Upper Bound result for connected manifolds. Let N be a graded module and p E Z an integer. We denote again by N(p) the graded R-module whose underlying module is the same as that of N and whose grading is given by [N(p)]; = [N]p+h i E Z. The graded ring R is assumed to be a graded kalgebra which is equi-dimensional and such that R:p is a Cohen-Macaulay ring for all prime ideals \:l different from the irrelevant ideal m. Furthermore, we assume that R is F-pure resp. has a prescntation of relative graded F-pure type. By Theorem 4.1 we have that LdD~, d = dim R, is isomorphic to a complex of k-vector spaces. Next we will prove the graded counterpart to Corollary 4.7. Lemma 4.14. Let R be a a8 def%'ned. We denote by :!i = (Xl' ... , Xd) • R a parameter ideal of R oonsis#ng of forms of degree t. We 8et 2:8 = {Xl' •.. , x B }, 8 = 0, 1, ... , d. Then it follows for the Koszul complex K'(:!i.; R) that rK(:!i.;R) i8 isomorphic to a complex of k-vector spaces. In particular, for the cohomology modules we hat¥! H'(2:B; R) ::::: EB i=o

~.,( (r

i)

t),

0 <::;; r

<

s,

where

Proof: First of all we show the assertion for s = d. In this case the cohomology modules H'(:!i; R), 0 <::;; r < d, are modules of finite length. We have a short exact sequence

of complexes

§ 4. Further applications

I

145

where KR[d] denotes the canonical module KR shifted d places to the left. Applying the functors lJ Hom(K(f; R), ) and Hom( ,E) we obtain 0->- Hom(lJ Hom(K(;r; R), T~~), E) ->- Erm(K'(f; R)) ->- Hom(lJ 'Hom(K(f; R), KR[d]), E) ->- 0

by virtue of the Local Duality Theorem. Since (Hom(E Hom(K(;r; R), KR[dl),

for all i

<

E))i = 0

d, the exact sequence induces an isomorphism

Td Hom(lJ Hom(K(;r; R), LdD~), E) ""' TdlJrm(K(;r; R)) in the derived category of R. For the left complex we have rdK(f; R)®RHom('LdDR' E). By considering the structure of the dualizing complex it follows rdK(;r; R).®RC·(R)::::: rdK(f; R) where we have used ,TdlJrm(K(f;R))""' r'K(f;R) since TdK(;r;R) is a complex whose cohomology modules H'(;r; R), 0 r < d, are modules of finite length. By our assumption C'(R) is a. complex of k-vector spaces, therefore also K'(;r; R)®RC'(R), is isomorphic to a complex of k-vector spaces. This proves our first statement in the case s d. Taking into account that Ki(f;R)

=

R(t)(it)

and

K(;r;R)®RC'(R)

has trivial boundary homomorphisms, the formula for the cohomology modules follows 1 we have an exact seimmediately. For an arbitrary 0 s;; s s;; d and an integer quence for the cohomology modules of the Koszul complexes

o ->- Hr-l(f._l; R) ®RRfx;R ->- H'(;r8-l> x!; R). If s d and r < d, we saw that m annihilates the left modules for all i2. 1. By Nakayama's 1.emma it follows that m· H,-l(f._l; R) = 0, i.e., for 8 - 1 = d - 1 and, r < d the cohomology modules are of finite length. By an induction argument it follows that H'(f.; R), 0 r < s, are modules of finite length, in fact k-vector spaces. Re.peating the above arguments we have, as in the case s = d, that r K' (f.; R) is isomorphic to a complex of k-vector spaces. Thus, the formula for the cohomology modules can be derived here also in a similar manner, q.e.d.

Lemma 4.14 provides a key to applications in combinatorics by use of homological algebra in derived categories. We want to describe an elementary approach to these applications by giving a new and simple proof of 1.emma 4.14. Also our Lemma ,4.14' improves Lemma 4.14. Lemma 4.14'. a) Let M be a Buchsbaum module 0/ dimension d and let {Xl' •.. , Xd} be a system 0/ parameters 0/ M. Then we have (i)

H~(M/(Xh ... , x;) 1\f)

4> (EEl H~i(M)) for all i and j with i + j < d. I=(}

10 Buchsbaum Rings

({)

146

II. Hochster·Reisner theory for monomial ideals

(ii) H'(x I , b) Let R

••• ,

=

EB H:;I(M)) ( ({)

Xi; M) '"

EB R;

lor all i and j with i < j.

be a Noetherian graded ring with Ro

=

K a lield, m

R i • Let

i:2:0

M be a Noetherian graded R-module 01 dimension d. Let {Xl> ••• , xa} be a By8tem 01 homogeneOUB parameter8 01 M with ti := degree 01 Xi lor i = 1, ... , d. Then we have (i') !1:n(M/(XI' •.. , Xi)'

M)

lor all i and j with i

+j

(ii') IP(xv ... , xi; M) '"

(

lor all i and j with i

<

Prool: a) (i) is clear for j

0.....;.- M/(XI' .•. , .....;.- M/(x l ,

••• ,

EB

( 1=0

<

tp,))

!1;;i(M) (-tp1 - ...

l:>;p,< ...
d. !1;;-'(M) (tp,

1:S:P1<···
+ ... + tp,)

j.

O. If i

>

0 we have the following exact sequence:

• M : m -='4 M/(x I, ... , xj-I) . M Xi)' M.....;.- O.

Applying the local cohomology functors H:n(

) we get an exact sequence:

0.....;.- H:n(M/(XI, .•. , Xi-I) . M) .....;.- H:n(M/(x l , .....;.- H:;l(M /(Xl' .•. , Xj-I) • M : m)

(E)

••• ,

Xi) • M)

. . .;.- O.

Note that

H:;l(Mj(XI' ... , Xi-I)' M : m)~ H:;I(M/(XI' ••. , Xi)' M) and that all modules occuring in this sequence are annihilated by

H:n'( M / (xv ... , xi) . M )

m. Hence we get

i ( M j (Xl> ... , Xi-I) ·.J.ll ) EB Hm i+ l( M /(Xl' •.• , X,-2) . M) Hm

and an easy induction on j proves the result (i). (ii) l'his result is clear for j 1 since HO(XI; M)

If j

>

O:MXl~ H~(.lll).

1 we consider the following exact sequence (E') (see Lemma 0.1.6)

0.....;.- Hl(Xi; Hi-I(XV ... , xj-I; M)).....;.- Hi(Xl' ... , Xj; .J.ll)

(E')

.....;.-}]O{Xi; }]i(X1' ... , xj-I; M)) .....;.- O.

If i

<

j we have

Hl(X;; }];-I(Xl, ... , xj-I; M)) '" Hi-l(Xl> •.. , XI-I; M). If

t'

<

j

1 we obtain

}]O(x;; lJi(xl' ... , Xi-I; M) ""' Hi(Xl' •.. , X;-I; M) by Corollary 4.7'. Hence we have from (E') for i < j - 1 }]i(XI' ... ,

Xi; M) "" Hi-l(x v ... , Xi-I; M) EB

}]i(Xl' ..• , Xi-I;

M)

§ 4, Further applications

since these modules are 'annihilated by m (see Corollary 4,7'), If i =

147

1 - 1 we get

HO(xi; Hi-I(X I "", Xj-I; M)) ~ (Xl' "" Xi-I) , M :

m/(x I ,

"" Xj-I) , M

Hr;.,(M/(Xl> ,." Xj-I) • M)

and we therefore have: Hi-I(X I , "" Xi; M) "" Hi- 2 (xl> "" xj-I; M)

Hr;.,(M/(xl> .," XI-I)' .lll").

(i) and an easy induction on j prove (ii), b) The exact sequence (E) of part a) has the following form: 0->- M/(xl> "" xj-I) , M : m( - t i ).!4 M/(x 1 , ->-

M /(Xl> "" Xi) • M

->-

".,

xj-I) . M

0,

Therefore again we obtain (i/) by induction on 1. For (ii') we point out that the exact sequence (E/) has the following form:

o ->-IP(xi; IP-l(xl> ""

Xj-I; M)) (tj) ->-IP(xI , , .., Xi; M)

->- !fO(Xj; lli(xI' "" xj-I; M)) ->- 0,

Furthermore we have: llO( Xj; llH(x v .'" Xi-I; M)) ((XV"" Xi-I) • M : m/(x1,

""

XI-I) , M)(tl

+ .,' + tj- 1 )

+ .. , + tj-I)'

~ llr;.,(M/(Xl> "., Xj_l) , M) (tl

Hence our result (ii') follows by the same arguments as above, q.e,d,

Remark 4.15. It follows from Lemma 2.6 and Lemma 4,14 that ll~(R) ~ [Hi(;!i; R)]o.

0

i

< d.

For this it is not necessary to assume that if {Xl> ., "Xd} is a system of parameters consisting of forms of the same degree. Following the reasoning of the proof it is seen immediately that it is enough to assume for if = {Xl' , •• , xn} to be any set of forms m. Or put differently, it is not necessary contained in m and such that ROO;!iR to take a direct limit in order t{) comlJUte the local cohomology modules via Koszul complexes for those rings, This was proved by M. Hochster and J, L. Roberts in [2], Theorem 1.1(b), in case of an P-pure graded k-algebra over a perfect field k of prime characteristic p and in Hochster-Roberts [2], Theorem 4.8(b), in case R has a presentation of perfect graded P-pure type. Hence, in the case of the ground field of characteristic zero our result leads to a slight improvement of Hochster-Roberts [2], Theorem 4.8(b). For our considerations in the following t{)pics Lemma 4.14' has an important application with respect to quotients of certain ideals. It allows us to give an explicit description of certain Hilbert functions. 10*

148

II. Hochster-Reisner theory for monomial ideals

Corollary 4.16. Let R denote as belore a graded Ie-algebra. Let 2: = {Xl' ... , Xd} be a system 01 parameters consisting ollorms 01 degree t. Then, lor 1:C;; s:C;; d there are isomorphisms s ~-l

((Xl' .'"

where ri

= (s

~

XS-I)

R : Xs )/(xv , .. , XS-l) R '"'-'

1£"( -t't) ,

1) dimk[H~(R)]o'

Prool: For the cohomology modules of K08zul complexes there are the following short exact sequences 0----+ Hs-2(2:s-t; R) ----+ Hs-l(:fs; R) ----+ (2:s-tR : xs/2:s-tR) ((s -

1)

t)

0,

where it was used that llS-2(2:,H; R) is annihilated by m. Since this short exact sequence is a sequence of Ie-vector spaces it splits. By virtue of Lemma 4.14' our statement now follows, q.e.d. In point of fact, the graded rings considered in I..emma 4.14' are Buchsbaum rings by virtue of Theorem 2.7. ('A)ITesponding results hold for an arbitrary graded Buchsbaum ring without assuming the purity condition. However, for two reasons we restrict ourselves to this special case. Purity implies that the non-vanishing cohomology of T d!1rm(R) is concentrated in at most the zE,lro'th graded piece. This is no longer true for an arbitrary graded Buchsbaum ring. For our purposes here it is enough to consider pure rings for which the statements can be formulated in an easier manner. Let .1 denote a finite simplicial complex with the vertex set {Xl>"" x n}. Let /; be the number of i-dimensional faces of .1. Thus 10 nand 1-1 = I, since .1 has the unique I)-face 0. The vector I (/-1,/0' ... , Id-t), d dim .1 1, is called the I-vector of .1. Since 1e[.1] is a graded algebra, we can associate to it its Hilbert lunction H(m, k[.1]), that is H(m, 1e[.1]) = dimk[Ie[.1]jm, mE Z,

+

where [k[.1]jm denotes the k-vector space of all homogeneous forms of degree m in k[.1]. For m large, H(m, k[.1]) coincides with a polynomial, the Hilbert polynomial. R. P. Stanley [3], Proposition 3.1 showed that the I-vector describes the Hilbert function exactly.

Proposition 4.17. For the Hilbert lunction H(m, k[.1]) we have H(m, k[.1])

where

(~!)

=

=i:i:>

0 lor i

+

(m i

1 and

1),

C=~) =

Prool: I,et Xi denote the image of

Xi

m 2:

0,

1.

by the canonieal projection

k[Xh ... , xn] ----+1e[.1).

x:' .....

A k-basis of [k[.1]jm consists of all monomials x = X: such that deg x all = m and Supp (x) E .1, where the support Supp(x) is defined by

+ ... +

Supp(x)

=

{x;

i

a;

> O}.

8

at

§ 4. Further applications

If

(J

E .1 has exactly i

+

1 elements, then the number of monomials of degree m

+

2:: 0

(m . 1). Hence, the propo~ition is

whose support is contained in (J coincides with proved, q.e.d. In particular by Proposition 4.17 dim k[L1) = dim .1

149

~

1.

Next we will prove an estimate of the number of i-faces of certain simplicial complexes. To this end we define integers hi by (1 - T)d

E H(m, k[L1]) Tm = 11,0 + hIT + ... + hdTd. m2'O

It is easily seen that the degree of the polynomial on the left does not exceed d. The vector h = (11,0' • '" hd) is called the h-vector of .1. In fact, the form!!>] power series E H(m, k[L1]) Tm is the Poincare series of the graded k-algebra k[L1). By the Theorem m2'O

of Hilbert-Serre there is a polynomial/(T, R) such that F(T, R)

I(T, R)j(1

T)d

for the Poincare series F(T, R) of a d-dimensional graded k-algebra R. We recall that H(m, ) and F(T, ) are additive functions on short exact sequences. For our particular situation we remark: knowing the I-vector of .1 is equivalent to knowing its h-vector. Proposition 4.18. For the h-vector and I-vector 61 a linite simplicial complex the 10Uowing relations hold:

i~ (-1)0-' (~ -:) Ii-!

hv lor v

0, 1, ... , d, d

= dim .1

and

IH

ito (~ _:) hi

+ 1.

The prool follows by an easy calculation. We omit it. Now we prove one of the main results of our applications in algebraic topology and combinatorics. Theorem 4.1D. Let .1 denote a simplicia}, BuchBbaum complex with n vertices and we denotp d = dim .1 1. For the I-vector and the h-vector 01 .1 there are the lollowing bounds

+

IV-I

d) (:) (v

(V E.

,,-2

i~-l

1,

1) dimk H,(L1; k)

+1

and

h. (n - d+v v 1) where 0

v

(-I)"

(d) .=1:. V

v- 2

(-1)i dim k

Hi(L1;k),

1

d and k denotes an arbitrary lield.

Prool: By the previous Proposition 4.18 we have an exact expression of the Hilbert function or even of the Poincare series of k[L1].

150

II. Hochster-Reisner theory for monomial ideals

'Vhat now follows is another way to calculate the Poincare series by techniques from commutative algebra. To this end we consider the associated graded k-algebra R k[,1], where k denotes the fixed field. Without loss of generality we can assume k an infinite field. Otherwise we extend k to the field of rational functions k(t) in one {x}' ... , Xd) of variable. Then there exists a homogeneous system of parameters:li R consisting of forms of degree 1. Let x E R be homogeneous of degree one. Then .there is the following exact sequence of graded R-modules R(1) --+(R/xR) (1) --+0.

Because F'(T, ) is additive, it follows that (1 - T) F'(T, R) = F'(T, RlxR) - TF'(T, OR:X).

Iterating this argument d-times we obtain a-} (1 - T)d F'(T, R)

where Q;, 0 S i d

F'(T, RI;rR)

T(l - T)i F'(T, Q;),

1, denotes the quotients

((x}' ... , Xd_H)R: Xd-;/(X ll •.• , Xa-H)) R.

By virtue of Corollary 4.16 we get F'(T, Q;)

a-i-}

= ,1:

(d

1=0

It now follows from simple calculations that (1- T)d F'(T, R)

=

F(T,R/;JdR)

-1: (d)v (.EI(-1)v-i-Idimk[l!~(R)]0) T". ~=}

'=0

In particular, the last equality shows that F'(T, Rj;JdR)

flo

+ flIT + ... + flaTd

does not depend on the system of parameters ;r (consisting of forms of degree 1). We define ((Jo, flI' ... , fla)

fI

the fI-vector of the simplicial Buchsbaum complex ,1. By the definition of the h-vector it follows that

h. = g. where 0

v

(~) v

.:i (_1)V-i-l dimk[l!~(R)]o, 1

.=0

d. Using the second formula of Proposition 4.18 we get

.1: (d)v '=0

v-I

(V-l) . . dimk[l!:n(R)]o, ~

v 0, 1, ... , d, by some simple calculations. In point of fact, the fI-vector of ,1 is the Hilbert function of the O-dimensional graded k-algebra R/;JdR, i.e. fl.

dimk[R/;JdR]v,

0

<: v s:: d.

§ 4. Further applications

151

Therefore, gv is not larger than the number of all linearly independent forms of degree v in n - d variables. That is, gt

d~V-1),

(n

Since

E~

i~o

e i) V-

.

i~ v

g.

%

0

v

i)

(d (n V- i

d.

d+v V

1)

= (:),

the two inequalities follow by replacing the local cohomology modules by the reduced simplicial homology, see Corollary 4.13, q.e.d. We know (see Corollary 2. 12(i)) that the graded k-algebra k[A] is a Buchsbaum ring, if X = IAI is a connected manifold. This implies:

Corollary 4.20. 11 the geometric realization X manilold, then we have

Iv~(

v

for v

n

+

)_( d ) 1 v+ 1

IAI 01 a I~nite

simplicial complex A i8 a

};1 (.t +v 1)dimkfl;(A;k), .

;~-l

= 0, 1, ... , dim A, resp.

d+ v - 1) v

(-1)~

(d) v

~-2 E ;=-1

+

for v = 0, 1, ,.., dim A 1, where H,(A; k) denotes the reduced 8implicial colwmology 01 L1 with coeflicients tn an arbitrary lixed lield k.

R. P. Stanley [3] characterized those numerical functions H which are Hilbert functions of a certain graded k-algebra. (This result goes back to F. S. Macaulay.) This leads to a slight sharpening of Theorem 4.19. In the expression for the upper bound of the h-vector the sum ~-2

E

1)t dimk fli(A; k)

i~

1

is a partially reduced Euler-Poincare characteristic of A. It is not surprising that the homology of A appears in the expression for the upper bound of Iv. If the homology does not vanish, there are cycles which are not boundaries. That is, some faces are deleted in A. The sum

1:

2

;=-1

(~- 1) dimk fl;(A; k) z+ 1

in the bound of the lv's coincides with dimk Qd-v, i.e.,

f.-l

(nv)

If A is a Cohen-Macaulay complex, then the reduced simplicial homology modules fl .(A ; k), 0 < z· d 2, vanish. In this case we obtain the formulas proved by R. P. Stanley [1] and [8].

152

II. Hochster-Reisner theory for monomial ideals

The purpose of the sequel is to introduce some of the foregoing ideas to partially ordered sets, posets for short. I..et P be a finite poset with rank function Q. We say that P is a Cohen-Macaulag p08et, if the associated simplicial complex LI(P) is a CohenMacaulay complex. This concept originated with K. Baclawski [3]. In particular, he showed that the simplicial cohomology of LI(P) with coefficients in a field k coincides with a cohomology theory on the poset with coefficients in certain diagrams, compare K. BaClawski [3, 4, 5]. Related to the foregoing we say that P is a bouquet, if it has nonzero reduced simplicial cohomology at most in the highest possible dimension. For our purposes we need a more general notion than that of a Cohen-Macaulay poset. Definition and Proposition 4.21. Let P denote a finite p08et. Then the following conditions em P are equivalent:

Hj(X, X - p; k) = 0 for everg point p E X and everg i =1= dim X, where X denotes the geometric realization of LI(P), (ii) Everg open intervall (x, g) of P is a bouquet, except p08sible for (x, g) = ((;, i). (iii) k[LI(P)] is a Buchsbaum n"ng.

(i)

H P satisfies one of these equivalent conditions, we say P is a Buchsbaum p08et. Proof: The equivalence of (i) and (U) follows from results of J. Munkres [1], Lemma 2.2, and K. Baclawski [3], Proposition 3.3. That (i) and (iii) are equivalent was proved in our Theorem 4.12, q.e.d.

We remark that K. Baclawski [3, 5] considered this kind of posets under the name "almost Cohen-Macaulay". In his papers it is a useful concept for induction proofs. We prefer the name Buchsbaum poset, because k[LI(P)] is a Buchsbaum ring which has additional structure by virtue of our results. Examples of Buchsbaum posets are simplicial complexes LI such that ILII is a (connected) manifold, see Corollary 2.12. Next we will discuss some Buchsbaum preserving operations on posets. Let P and Q be finite posets. The order-dual of P, denoted by P*, is the poset obtained by reversing the order of P. The product of P and Q, denoted by P X Q, is the cartesian product of P and Q with its order defined by (x, g) :::;; (x', g') if and only if x:::;; x' and g g'. The ~"nterval p08et of P, denoted by Int(P), is the poset of closed intervals of P, ordered by inclusion, i.e., [x, g]

[x', g']

if and only if x' :::;; x and g < y'. This means, Int(P) is given by the induced order as a subset of p* X P. Next we recall some results of K. BacIawski [3], § 7. Theorem 4.22. Let P, Q denote Buchsbaum posets. Then P*, P xQ, and Int(P) are again Buchsbaum posets. Proof: H P is a Buchsbaum poset, then p* is a Buchsbaum poset too. Because LI(P) = LI(P*) and (x, y) is an open interval in P*. For the rest we refer to K. BacIawski [3], § 7, q.e.d.

Next we discuss some more terminology for posets. Let # denote the Mobius function of P defined by #; {(x, g) E P X

P I x:::;; g}

-)- Z

§ 4. Further applications

such that (a) ",(x, x) = I for x E P, (b) E ",(x, y) = 0 for all fixed pairs x

153

< z in P.

Let S c {I, ... , d} =: [dJ be a set consisting of ranks of P. The rank selected 8'Ubposet with respect to S is defined by Ps

I

{x E PI e(x) E S).

In addition, we denote by Oi.(P, S) the number of chains Xl < ... < Xs in P such thaf {e(XI), .'" e(xs)} = S. It follows from the definition, that Oi.(P, S) denotes the number ot maximal chains in P s . Now we define ,

=E

{J(P, S)

(_1)cardS\'P

Oi.(P, T),

'PcS

where the sum is taken over all subsets T of S. Conversely we have Oi.(P, S)

= E {J(P, T). 'PcS

The numbers Oi.(P, S) and {J(P, S) were introduced and examined by R. P. Stanley [4, 5] for various posets. Another related invariant of P is its zeta polynomial, see Stanley [5]. If m is a non-negative integer, define Z(P, m) to be the number of chains

oS

Xo

Xl

S

...

xm

i

in P.

Z(P, m) is a polynomial function of degree d

+ 1 of m.

It follows the existence of

constants eo, el> ... , ed and ho, hi> ... , hd such that Z(P,m)

d+l =.EeH

(m).

1=1

'/,

and

E Z(P, m) Tm

(1 - T)dH

hoT

+ hlTt + ... + hdTd+l,

m:
see Stanley [3]. In this reference it is also notet that ei

E

Oi.(P, S)

and

SeIdl

cardS=;

hi

= E

{J(P, S).

SeIdl

cardS=i

Let 1= (/-1> 10' ..., la-I) denote the I-vector of d(P), i.e., /; is equal to the number of ej. chains Xo < Xl < ... < Xi in P. It is clear that IH Theorem 4.23. Let P denote a Buchsbaum poset with dim P

lor v

=

+ 1 = d. Then we have

0, 1, •.. , d and an arbitrary lixed lield k.

Prool: By Philip Hall's Theorem, see Rota [lJ, Proposition 6, it follows that {J(P, S)

=

(_I)1+c8rd S ",(P s )

:== (_1)1+cafd S X(P S ) ,

154

II. Hochst~r·Reisner theory for monomial ideals

where i(Ps) denotes the reduced Euler characteristic of X(Ps)

PSI

i.e.,

= 1..: (-1)' dimkHi(Ps; k). '20

Now, by the results of K. Baclawski [3] on rank selected subposets, Theorem 6.5, we get

Hi(P S ; k)~ Hi(P; k)

for i

< card S - 1.

A corresponding result was proven implicitely by J. Munkres [lJ, Theorem 6.4. By virtue of the above formula for the hi we obtain hv

1..:

(_1)1+0 (

Se[dl cudS_

:t

(-I)' dimkHi(P; k)

i=-1

+ (_1).-1 dimkHv-1(Ps; k)). •

Thus, our statements follows, q.e.d. If we apply Theorem 4.23 to the case of a finite simplicial Buchsbaum complex L1, we obtain an exact description of the g-vector, introduced in the proof of Theorem 4.19, in terms of the reduced simplicial homology.

Corollary 4.24. Let L1 = L1(P) be the order complex of a Buch8baum poset P. Then we have for the g-vector of L1: gv

1..:

dim"HH(Ps; k),

Se[dl

card S=v

o :::;; V

d, where d = dim L1

+

L

,

.

Chapter III On liaison among curves in projective three space

The underlying philosophy of liaison is to examine the closed subschemes of Pi.: by introducing equivalence relations on them, see the beautiful paper of Peskine and Szpiro [3]. This allows one to study the closed subschemes indirectly by investigating the induced equivalence classes. The liaison equivalence relation orginates from the tenet that the simplest closed subschemes of Pi.: are the complete intersections. Two arbitrary closed subschemes are then equated if they bear an appropriate relationship to some complete intersection. The main subject of this chapter, but not exclusively, is liaison of curves in P1:. Roughly speaking, liaison considers two curves in P~ to be equivalent if their union is a complete intersection. That this idea for curves in p3 has interested algebraic geometers for over a century is attested by, for instance, Rohn and Berzolari [1]. In a fundamental paper, P. Rao [1] has shown that curves in P1: which are generic complete intersections and without isolated or imbedded points are classified up to liaison by a certain graded module of finite length over K[xo, Xl' X 2, xa]. This "liaison invariant" vanishes if and only if the curve is arithmetically Cohen-Macaulay. For instance, the liaison invariant of a line in P~ is 0, whereas the liaison invariant of two skew lines in P~ is K. In § 1 we show how famous classical results (see for example Apery [1,2], Gaeta [1J, Peskine and Szpiro [3] or Artin and Nagata [1]) may be generalized for arithmetical Buchsbaum curves in P~. We also point out a different formulation of the liaison invariant by using a certain dualizing complex. This allows us to study the liaison invariant in any codimension via the theory of derived funct.ors and categories. Rao's work raises the following question: Does there exist a geometric addition of curves in P~ corresponding to the direct sum of their liaison invariants? This is the liaison addition problem. Merely defining the sum of two curves to be their union will not suffice, as evidenced by the preceding examples of lines. In § 2 we exhibit a solution to the liaison addition problem by using results of Phil Schwartau's [1] magnificent thesis. We therefore find that not only there is a way to add curves in P~, but that an explicit procedure is possible: that is, equations for the added curve may be written • the equations of the curves being added. The addition procedure turns out down from to be quite simple, and in fact it admits a purely intrinsic formulation reminiscent of liaison itself. ' Our aim in § 3 is to study curves linked to lines in P~. Our results deal mainly, but not exclusively, with the case of a line lying on a nonsingular quadric surface. By using a very explicit description we may construct several examples' of non-CohenMacaulay and non-"Buchsbaum curves in P~. Coupling the
156

Ill. On liaison among curves in projective three space

R. Hartshorne on set-theoretic complete intersections in Pk and also comment further on a counter-example to Hartshorne's question given by Peskine and Szpiro [3]. In § 4 we apply the results of the third paragraph to the notion of "seH-linked" curves, recently studied by Rao [2], and we are able to give new examples of non-self-linked curves in Pk. It is still an open question to determine those liaison classes in Pk which' contain a self-linked curve. Liaison addition of § 2 provides one step tQward the solution of this problem by use of Phil Schwartau's [1] thesis.

§ 1.

On liaison among arithmetical Buchsbaum curves in P'

It seems that A. Cayley [1], p. 152, in 1847 was the first who posed the problem to describe the liaison class of a complete intersection. Nowadays it is well-known that a curve in p3 is in the liaison class of a complete intersection if and only if the curve is arithmetically CQhen-Macaulay. Furthermore, L. Gruson and C. Peskine [1] have given a complete classification of arithmetical Cohen-Macaulay curves in P3. Discussing some results from Bresinsky, Schenzel and Vogel [1] we will begin in this paragraph to investigate the next simple case, that is the liaison classes which are characterized by finite-dimen'sional vector spaces of dimension 1. By using the theory of Buchsbaum rings this means we will study liaison of arithmetical Buchsbaum curves in P3, see Peskine and Szpiro [3] and Rao [1]. Definition 1.1.l£t R be a Cohen-Macaulay local ring and let a and 0 be two ideals of R. The ideals a and 0 are said to be (algebraically) linked (written a,....., 0) if there is an R-sequence Xl' ••• , xn in a n 0 such that a = ((x]' ... , x,,)): 0, and 0 = ((Xl> ... , xn»): a. We will make use of the following equivalent definition: I£t a, 0 be two ideals of the local Gorenstein ring R. The ideals a and 0 are algebraically linked by a complete intersection! (Xl> •.• , Xg) can 0 if (i) a and 0 are ideals of pure height g, and (ii) a/!' R "-' HOIllR(Rlo, RI! . R) and o/! . R ~ HomR(R/a, RI! . R). We now consider the equivalence class of ideals generated by linkage, that is we consider the transitive hull of this relation. We sayan ideal b is in the hnkage cla88 of a if there are ideals Cl' ... , Cn such that We remark that the linkage classes of complete intersections are reasonably ~ell-under­ stood in codimension ::;; 3. I£t a be an ideal of the local ring R. If R is normal, then if a is an unmixed ideal of codimension 1 is in the linkage class of a complete intersection if and only if the associated element a in the class group of R is trivial. If R is regular, and a is an unmixed ideal of codimension 2, a is in the linkage class of a complete intersectipn if and only ifR/a is Cohen-Macaulay, see Peskine-Szpiro [3]. If R is regular and a is an ideal of codinlCnsion 3 such that Ria is Gorenstein, then a is in the linkage class of a complete intersection (see Watanabe (1]).

§ 1. On liaison among arithmetical Buchsbaum curves in pa

157

Moreover, a and 0 are linked (geometrically) by a complete intersection! := (Xl' ••• , if a and 0 have no components in common and a n o ! . R. Two subschemes of P" of codimension 2 are 8aid to be linked geometrically if their scheme theoretic union is the complete intersection of two hypersurfacea. For instance, the twisted quartic curve C in p3 given parametrically by (t4, t3 u, tu3 , u 4 ) and the union X of two skew lines in p3 are linked since we have for the defining ideals I(C) and I(X) = (XO, Xl) n (X2' xa) of C and X, respectively: Xg)

I(C) n I(X)

This liaison of C and X was discovered by G. Salmon [1], p. 40, already in 1848 and a little later in 1857 again by J. Steiner [1], p. 138. Nowadays we know that algebraic and geometric linkage generate the same equivalence relation, see Rao [1], Theorem 1.7. This equivalence relation generated by linkage is called liaison. Now let C c: Pi be a curve, that is, C is an one-dimensional subscheme of Pi over an algebraically closed field K, equidimensional, locally Cohen-Macaulay and a generic: complete intersection. A. P. Rao [1] studied the following invariant, due to R. Hartshorne: M(C) where 3c(v) is the twisted ideal sheaf of C. Then M(C) is a graded 8:= K[xo, Xl' X2' X3]module of finite length. Furthermore, M(C) is invariant up to duals and shifts in gradings, under liaison. It follows from Rao [1] that for each graded 8-module of finite length, there is a liaison equivalence class, and that the module determines that equivalence class. We recall that a curve C c: p3 with defining ideal I(C) is said to be arithmetically Cohen-Macaulay or arithmetical Buchsbaum if the local ring 8(z" .....,.)/I(C) =: A of the vertex of the affine cone over C is a Cohen-Macaulay or Buchsbaum local ring, reap. Since M(C) Htz•.... ,z.)(8jl(C)) = Htz., ....Z.).d(A) we sce that C is arithmetically Cohen~Macaulay if M(C) O. Moreover we get from Proposition 1.2.12 that C is 0 where m ~ (x o, Xl' X 2, Xa) . A; 'arithmetically Buchsbaum if and only if m· H~(A) that is, M(C) is a finite-dimensional vector space over AIm. Also it is known (see, e.g. Peskine-Szpiro [3]) that liaison respects the Cohen-Macaulay property. From local algebra we get the following more general result: Theorem 1.2. Let R be a local Goren.stein ring of·dimen.sion d 1 and with 1'lULxi1'lULl ideal m. 8upp08e that the ideal8 a, 0' c: Rare lz'nked. Then we have: (a) Ria tS a Buchsbaum ring if and only if Rio is a Buchsbaum ring. (b) In addiJion, assume that the local cohomology module8 H~(R/a) have finite length over R for all integers i 0, 1, ... , t - 1 where t dim Ria dim Rio. Then we get H!;1(R/o)

HomR(H~(Rla), E)

for all i

1, ... , t - 1,

where E is the injective hull of the residue field of R.

Remark. The focus of this paragraph on liaison among arithmetical Buchsbaum curves is on the statement of this theorem. In proving it the key is Theorem 11.4.1 by applying homological algebra in derived categories. In the preface we did mention that we will

158

III. On liaison among curves in projective three space

describe a different and up to now unpublished proof of Theorem 1.2. Our proof is elementary by virtue of the fact that it uses only some basic concepts from Chapter I. Proposition 1.28 below immediately opens the way to our approach (see the proof of Corollary 1.30). Proof of Theorem 1.2: First we prove the statement (a) of the theorem. For this we need a new invariant under liaison extending M(G) of a curve G. Looking at it by way of

commutative algebra this invariant is defined for an arbitrary ideal a of a local Gorenstein ring R. We will use the theory of derived functors and categories, see Chapter 0, § 3. If a and 0 are linked by a complete intersection r = (Xl' •.. , Xg) then we have for the canonical or dualizing module Ka of RIa:

Ka

Ext~(Rla, R)::::: HomR(Rla, RI(r)

ol! . R,

and Hence we have the following exact sequence:

o

--')0

KQ

--')0

R/r . R

--')0

Rib

--')0

o.

From this we get the sequence: 0--')0 KQ[ -g]

--')oil HomR(R/!. R, R)

--')0

Rlb[ -g]

--')0

0

by use of the isomorphism Rlr' R ~

Let

I~

11 HomR(R/!. R, R) [g].

be the dualizing complex of Ria, that is, I~ = HomR(R/a, E~)

where E~ is a minimal injective resolution of R over itself (see Chapter 0, § 3). Note that in the derived category the complex I~ is isomorphic to 11 HomR(R/a, R). Factoring out the first non-vanishing cohomology module KQ of I~ we get a short exact sequence:

o

--')0

Ka[ -g]

--')0

I~

--')0

J~

--')0

0

where J~ is up to a shift in grading the truncated dualizing complex of Ria. Using the canonical map

induced by the canonical epimorphism

we obtain the following commutative diagram of complexes with exact rows: 0--')0 Ka[ -g]

II

--')oil HomR(RI! . R, R)

t

ra

where fP is defined in the obvious manner.

--')0

R/b[ -g]

i~

--')0

0

§ 1. On liaison among arithmetical Buchsbaum curves in p3

159

By applying the derived functor 11 HOIDR( , R) and using the Local Duality Theorem we get the following commutative diagram:

0-11 HomR(Rlo, R) [g] - RI"{. . R -11 HomR(K u, R) [g] - 0

t

t~

o-

!1 HomR(J~, R)

II

_Ria

Now we will show that cp induces a quasi-isomorphism between Ji,[g] and 11 HomR[J~, R). Therefore we consider the induced homomorphism on the homology modules. Hence we get the commutative diagram with exact rows:

o_

HomR(J~,

and isomorphisms for

R) _ Ria

i> 1

Extuii(Ka, R):::: Extuii+l(Rlb, R)

II

Extuii(Ka' R):::: Ext~+1(J~, R)

By using ExtMRlo, R) :::: al! . R and the surjectivity of the mapping RI"{. . R _ Ria we get from the diagram: HomR(J~, R)

=

0

and thereforeExt~tl(Rlo, R)

ExtMJ~, R).

Hence cp induces a quasi-isomorphism: Ji,[g] :::: !1 HomR(J~, R).

Note that J~ is therefore a new invariant (up to duality and shifts in gradings) under liaison. Assume now that Ria is a Buchsbaum ring then J~ and also !1 HomR(J~, R) are quasi-isomorphic to complexes of k-vector spaces (see Theorem II.4.1). We apply again this statement and get that Rio is a Buchsbaum ring since J~ is invariant under liaison. Replacing a by b we obtain the converse. For the proof of Theorem 1.2 it remains to prove the statement about the local cohomology of Rfa and Rio. In order to do this we use an observation whieh is of some interest, in its own right. Before proving the statcment (b) of Theorem 1.2 we therefore collect some facts that follow from this observation. We also obtain the interesting Corollary 1.4. Lemma 1.3. Let R be a local Goren,~tein ring of dimengion d 1 and with 'llULximal ideal m. Suppose that the ideals a, 0 c:: R are linked. Then we have the exact sequence: 0_ Ria -

Ex~(K("

R) _ ExtuR+1(Rfb, R) _ 0

and canonical i~omorphisms for i

>

g

Ext~(Ka' R) "-' Extk+1(R/b, R),

d - dim Ria

160

III. On liaison among curves in projective three space

that i8, for the loca'[ cohomology moduJes we get:

o -711t;;l(Rjb) -711~l(Ka)

-7

HOllln(R/a, E) -70,

dim Ria and E i8 the injective hull of the residue field of R, and for i

where t


ll:n(K a) ~ ll;;;l(Rjb). Proof: Using local duality it suffices to prove this lemma for the "Ext" cohomology. We have again the exact sequence (see the proof of statement (a) of Theorem 1.2)

0-7 Ka

-7

RhR

-7

RIb

-7

O.

'*'

Since Extk(Rh: . R,R) Ofod gand Ext~(RI!' R, R) '::::- Rh· R we get the following sequence by applying the functor E HomR( ,R):

o

-7

Ext~(Rlb,

and isomorphisms for

~.

R)

>

-7

R/! . R

-7

Ex~(Kg,

R)

-7

Ext~+l(Rlb,

R) -70,

g

Extk(Kg, R)~ Ext~+1(Rlb, R). By use of

Ext~(Rlb,

R)

a/-,; . R

we obtain our assertion, q.e.d.

Corollary 1.4. Let 0:, b and R be a8 in Lemma 1.3. For an integer r ditions are equivalent:

2 the following con-

(i) Ria 8atisfie8 the Serre condition (ST); that i8 depth R.. min(r, dim R .. ) for aU 1-1 E Spec R. (ii) ll:n(Rlb) = 0 for all integer8 t - r < i < t where t dim Rib. Proof: The exact sequence

results in the following canonical isomorphisms ll~l--i(Ka)::::::: llm'(J~)

for all integers i > 1 by using local duality and applying the functor I1H~( ). Lemma 1.3 therefore provides the isomorphisms

for all integers 1 < i < t. It follows now from Schenzel [1], Theorem 3.2.2(iii) that Hmi(J~) o ~. < r if and only if (S,) holds for Ria.

=

0 for all integers

Proof of 8tatement (b) of Theorem 1.2: First, we note that we again have for all i> 1: H~l-j(Kg)::::::: H;/(J~).

From our assumption and the Local Duality Theorem (see 0.3.4) we get Hmi(J~)

=

H-i(J~)

for all z' ~ t - 1.

161

§ 1. On liaison among arithmetical Buchsbaum curves in p3

Hence we have for 2

i

t -

1

H~I-i(Ka) '" HomR(H;"(Rla),

since H-i(J~) '"'"' HomR(H;"(R/a), q.e.d.

E)

E) for

i

*' t.

Lemma 1.3 then yields our assertion,

Remark 1.5. (i) Using the fact that Ria is a Cohen-Macaulay ring if and only if J~ 0 we obtain another proof that liaison respects the Cohen-Macaulay property. (ii) Let 0 p3 be a curve with defining ideal 1(0). Applying local duality we get: M(O) '" l1M8jl(0)) ~ If Homs(Jj(C), 8) [-2].

Since Jj(C) is invariant under liaison we recover the invariance of M(O).

Examples 1.6. The simplest curve in p3 which belongs to the liaison class corresponding to a vector space of dimension 1 is the union of two skew lines in P3. Having this curve the specific data in the papers from the late 19th century render examples of arithmetical Buchsbaum curves with invariant i(O) = 1. For instance we get O~, 0:, ~, O~ from the paper of M. Noether [11 or Oiij, O~i, O~) o~g from the paper of K. Rohn [11, where O~ is a irreducible curve of degree d and of genus g. We have studied the curves O~ and O~ in our Introduction. We recall that we have found the resolutions of the curves We made the claim that the curves O~ have the following resolution if is arithmetically Buchsbaum:

0:.

0:

o -? 8(-6) - ? 8 4 (-5) -? 8(-2) ED 8 S( -4) -? 8 -? 8/1(0~) -? O. In order to construct this resolution we prove the following more general result. Lemma 1.7. Let 0 be a curve in ps which is lz"nked to two skew l~"nes 't"n p3 by two hypersurlaces 01 degree I and g, resp. Then we get the 10Uou>ing Iree resolution 01 8/1(0) where 8:= K[xo, ... , xs]:

o -? 8(-1- g) -? 8 4(-1- g + 1) -? 8(-/) ED 8(-g) ED 8 2(-1 -?

8

-?

8/1(0)

-?

g

+ 2)

O.

In proving this lemma we must use for a curve 0 the property of being ideally the intersection of d hypersurfaces. Therefore we give the following definition. Definition 1.8. A projective variety V c hypersurlaces if there exists a surjection

p~

is said to be ideaUy the

~"ntersection

01 d

II.

ED Opn(-a,) -?:Jv-?O i=1

for some integers a;, i = 1, ... , d. (Note that Opn is the structure sheaf of p .. and :J v is the ideal sheaf of V.) This definition is equivalent to saying that there are homogeneous elements 11, ... , I", in the defining ideal 1(0) of V such that l( V)/(/l> ... , la) is a K[xo, ... , x,,]-module of finite length, that is, or 1( V)L',=l 11 Buchsbaum Rings

(/1> ... , /11)1.>:,=1

for i

0, 1, ... , n.

162

III. On liaison among curves in projective three space

A reason for the importance of this concept is the fact that for curves in pa the property of being the zero scheme of a section of a rank-two bundle on p3 is connected to that of being ideally the intersection of three surfaces. Furthermore, the same argument as used by A. P. Rao in [1.], suitably refined, can be used to prove the following Proposition 1.9. II a curve 0 in p~ if; ideally the t"ntersect£on 01 the three surlace8 w£th delz"n£ng equations/ l = 12 = la 001 degrees I, g, h respect£vely, then we have:

that if;, we have up to duals and sMits zn gradzngs 1(0)/(/1,/2,/3) """ 1l!:"".r".r.,z.)(K[xo,

Xl'

X 2,

x 31/1(0)).

Prool 01 Lemma 1. 7: Let V denote the union of two skew lines in P3. Let 0 be any curve in pa which is linked to V. Knowing the resolution of V it follows from A. P. Rao [1] (see Remark 1.10) that 0 is ideally the intersection of three hypersurfaces. Our Theorem 1.2(a) implies that 0 is arithmetically Buchsbaum with an invariant ~'(O) = 1. Therefore we get from Proposition 1.9 that the defining ideal 1(0) is gen~rated by precisely four elements. From these facts we get the resolution of 0 following the argument from Rao [1], Theorem 2.5 (see our Lemma 1.14 below), q.e.d.

Remark 1.10. Another consequence of the property of being id.eally the intersection of three surfaces is that the homogeneous ideal 1(0) of a curve 0 is generated by precisely three elements if and only if 0 is ideally the intersection of three surfaces and 0 is arithmetically Cohen-Macaulay (non-complete intersection). Continuing in this vein, on discovers that curves in p3 are much more complicated if their homogeneous ideals are generated by precisely four elements. Related to this we want to prove the following theorem. Theorem 1.11. Let 0 c p3 be any curve. Tn.;, 10lloW£ng conditions are equt"valent: (i) 0 if; ar£thmetically Buchsbaum (non-Oohen-Macaulay) and 0 if; ideally tlu; t"nter8ection 01 three hyper8urlaces, say 11 12 = 13 = o. (ii) There are homogeneous elements 11,/2,/3, I. whwh provide a mint"mal base lor 1(0), and Xi ·14 E (/1,/2,/3) lor £ 0, 1,2, 3. Before proving this theorem, we need two lemmas. p(M) will denote the number of elements in a minimal basis of the module M over a local ring. Lemma 1.12. Let 0 c p~ be an arithmetically Buchsbaum curve W£th £nvarmnt £(0) Tlu;n we get lor tlu; delin£ng t"d~al 1(0) 010: p{I(O))

3· £(0)

1.

+ 1.

Lemma 1.13. In addition to the hypotlu;8i8 01 Lemma 1.12, suppose that 0 if; ideally the inter8ection 01 three surlace8. Tlu;n we have 1'(1(0)) = 4

and

£(0)

=

1.

In order to prove these two lemmas we apply the following striking result of A. P. Rao [1], Theorem (2.5).

§ 1. On liaiBOn among arithmetical Buchsbaum curves in p3

Lemma 1.14. Let 0 be a curve in P3. Let M(O) have a minimal free

Then 8(0)

K[xo,

Xl> X

163

re80lut~'an

z, xa]/I(O) has a minimal resolution 0/ the form

o ~ L, ~ La EB

r

m

1

1

e 8( -li) ~ EB 8( -e;) ~ 8 ~ 8/1(0) ~ 0,

Pro%/ Lemma 1.12: We set i

=

i(O), then we have M(O)

=

K(Pj)' Tensoring the

Koszul resolution of K we get the following minimal free resolution of the invariant M(O), say

o ~ L, ~ La ~ L z ~ Ll ~ L o ~ M(O) ~ 0 where L j are free 8-modnles of rank i·

(~), 0 < j

4. Applying Rao's Lemma 1.14

we conclude that a minimal resolution of S/I(O) is:

o ~ L, ~ La

r

m

EB 8( -li) ~ EB 8(-e;) ~ 8

~8/1(0) ~

0,

1

where m = p(I(O)). Therefore the Euler-Poincare characteristic of the vector spaces of this resolution implies q.e.d. Proof 0/ Lemma 1.13: It follows from Proposition 1.9 that p(I(O)) st3tement and Lemma 1.12 provide the assertion, q.e.d.

3

+ i(O).

This

Pro%/ 'l'heorem 1.11: First we prove the implication (i) =? (ii): Our Lemma 1.13 shows that i(O) = 1 and p(I(O)) = 4. Therefore we get from Proposition 1.9 that the S-module 1(0)/(/l> /Z, /a) has precisely one generator which provides one basis element, say It, of 1(0). Thus 1(0) = (/1' /2' /a, I,). Furthermore, we have (xo,

Xl> X z,

xa) . 1(0)/(/1' /2' /a)

=

0

since 0 is arithmetically Buchsbaum, that is, Xi . /, E (/1' /2' /3) for i 0, 1, 2, 3. (ii) =? (i) of Theorem 1.11: We have (Xo, Xl, X 2 , Xa) (/1' /2' /a) : ft. It follows that (/1' /2' /a) equals (/1' /2' /a, /,) up to a primary component belonging to (Xo, Xl' X 2 , xa), that is, 0 is ideally the intersection of the three hypersurfaces /1 = /2 /a = O. By assumption we have (xo, Xl' X 2 , X3) • 1(0)/(/1' /2' /a) O. Proposition 1.9 therefore shows that 0 is arithmetically Buchsbaum (non-Cohen-Macaulay). This concludes the proof of Theorem 1.11, q.e.d. The proof implies the following remark. Remark 1.10. Let 0 C p3 be a curve. Assume that 0 is ideally the intersection of three hypersurfaces, say /1 = /2 /a = 0, and p(I(O)) = 4. If 0 is arithmetically Buchsbaum, then there are homogeneous elements /1' /2' /a, /, such that 1(0) (ft, /2' /3, I,) and the first module of syzygies of (/1' /2' /a, I,) has a linear second syzygy.

o·

164

III. On liaison among curves in projective three space

Moreover, it is not too difficult to show the following fact by using the method of the proof of Lemma 1.12 and Local Duality; Let C c p3 be an arbitrary curve and let l

O-EEl 8(-d j ) ~ F2 -Fl -8 _8j1(C)_0 j=l

be a minimal free resolution of 8/1(C), where A, aI' ... , d l are suitable integers. Then C is an arithmetically Buchsbaum curve if and only if (/> is obtained up to an isomorphism of F2 by multiplication with the matrix Xo Xl X 2 Xa

0 0 0 0

. . . . . . . . . . . . . . . . .. 0 ... 0 )

~ ~ ~ ~ ~o Xl X2 Xa ~. ~

m= (

o

•••

~ • ~. ~.

• • • • ••

0 0 0 0 ............... 0 Xo Xl

X2

0 ... 0

Xa 0 ... 0

Hence p.(F2 ! 4A and therefore we see again that p.(1(C)) 3i + 1 if C is an arithmetically Buchsbaum curve with invariant i i(C) (= A). If C is an arithmetically Buchsbaum curve with p.(1(C)) = 4 then A 1 and = (xo, Xl' X 2, XaJ, that is, the second module of syzygies is generated by exactly one linear syzygy (see also Amasaki [1] and [2]). The following two examples shed some light on the case when the homogeneous ideal 1(C) of a curve C in p3 is generated by precisely four elements.

m

Examples 1.16. 1. Take the curve C given parametrically by (8', sOt2, 8t6, t7). It follows from Renschuch [2J, p. 324, that 1(C) (/1,/2,/3,/4) where

We get from our characterization of the monomial arithmetical Buchsbaum curves in p3 (see Theorem 1.20 below) that C is not arithmetically Buchsbaum. It is easy to see that C is ideally the intersection of the three surfaces 11 = 12 14 = O.Therefore this example shows that curves C in p3 with p.(1(C)) = 4 and the property of being ideally the intersection of three surfaces are not, in general, arithmetically Buchsbaum. 2. Here we give an example of a curve in p3 such that p.(1(C)} 4 and C is arithmetically non-Cohen-Macaulay-Buchsbaum but C is not ideally the intersection of three surfaces. The explicit description of such a curve, we believe, is not clear. The possibility of such a construction was suggested to us by David Eisenbud. It makes use of the theory of finite free resolutions. Let C be the curve in p3 with defining ideal 1(C)

(XIX2Xa(XoX2 - XIXa), x~xa(x~

x~), XiX2(XoXl

x 2xa), xa(x~ - xlx a) (XoX2

xi))

: (/1> 12, la.i4)'

An easy computation now shows that a minimal free resolution of 8/1(C) has the form: 0_8(-8)

m+ 8 4(-7) ~ 8 4(-5) r

8 _8/1(C)_0

~

1. On liaison among arithmetical Buchsbaum curves in pa

165

with

18 -

c~·o 2

X3 -

2

Xo

-XIX3

X22 0

x23

0

0 XIX3 - XOX2 -XIX2 -XOX3

x: )-

-X~

0

By applying Lemma 1.14 and by our Remark 1.15 we get from this explicit resolution that C is arithmetically Buchsbaum and C is not ideally the intersection of three surfaces. Remark 1.17. The preceding investigations show that the property of being ideally the intersection of three surfaces is useful in studying curves in P3. The result of Abhyankar [1] in A3, namely that every non-singular curve in A3 is ideally the intersection of three hypersurfaces, is not true in general for curves in P3. It was first pointed out by C. Peskine and L. Szpiro [3] that there exists a non-singular curve in f3 which is not ideally the intersection of three hypersurfaces. A. P. Rao [1] proved that every liaison class contains curves that are not ideally the intersection of three surfaces, and that there is a liaison class that does not contain any curve that is ideally the intersection of three surfaces. With regard to this we get the following corollary. Corollary 1.18. Every liaison equivalence class correspondiTI{J to a finite-dimensional vector space of dimension> 1 does not contain any curve that is ideally the ~ntersection of three ~rfaces, hence conta~ns no curves com~ng from sections of rank two bundles. Interestingly, we can also read this result as follows: Let C c: p3 be a curve that is ideally the intersection of three surfaces. Assume that (xo, Xl' X2, X3) annihilates the first local cohomology module .,.)(K[xo, ... , x3]/I(C)) then tMs vector space has a dimension :c:;: 1.

In., . . .

Proof: Lemma 1.12 and Lemma 1.13. With the use of Corollary 1.18 we can construct irreducible curves in p3 which are not ideally the intersection of three hypersurfaces by applying the methods of EvansGriffith [l] (see our Chapter V, § 4). We mention here still another explicit description of curves in p3 which lie in a liaison equivalence class corresponding to a finite-dimensional vector space of dimension > 1. The procedure to obtain them is based on an old geometric idea. For instance, we claim that there exist irreducible curves C!~5 in p3 of degree 42 and genus 145 which belong to the liaison class corresponding to a vector space of dimension 3. In order to prove this assertion we need to mention that K. Rohn [2] studied the .residual intersection of special classes of space curves lying on any surfaces of degree 4 already in 1897. From these specific data we get that the rational twisted cubic curve q, counted with multiplicity 6, is linked to irreducible curves C~~5 by two hypersurfaces of degree 4 and 15. Therefore our claim follows if we show that the local ring A := K[xo, ... , x 3](.,.......,).):J3

is a Buchsbaum ring with invariant i(A) = 3, where .):J is the defining prime ideal of cg. We will prove for this the following lemma.

166

III. On liaison among curves in proj

Lemma 1.19. Let lJ = (XOX2 x~, XoXa - XIX 2, XlXa xi) =: (/1> 12' la) be the delining prime ideal 01 eg. Put R K[xo, ... , xa]!.,,, ... ,.,,). Let n ~ 1 be an tnteger. Then RIlJ" is a (local) Buchsbaum ring q and only zj n 3 in which case the ~nvar'iant8 01 the Buchsbaum r~ngs are given by i(RIlJ) = 0, i(RIlJ 2 ) = 1, and i(RI'tJ 3 ) 3. Furthermore, lJ2 and lJa are the delining weals ul eg, counted with multiplicity 3 and 6, respectively.

Prool: First, we will show that RIlJ3 is a Buchsbaum ring with i(RIlJ3)

3. The same argument, suitably modified, can be used to prove the statements for RI'tJ 2 • Note that RIlJ is a Cohen-Macaulay ring. We look first at the ideallJ3 + (xi): Let U(lJ3, xi) denote the intersection of the primary ideals of R belonging to lJ3 (x~) of dimension 1. Using localizations of lJ3 (xi) at the prime ideals belonging to the radical of lJ3 (xi) it is not too difficult to show that

+

+

U(lJ3, xi)

=

+

(x~, x~, x~, X~Xh xoXi)

n (xi, xoX~

3XIX2Xi, X~X2X~ -

xoXixi

+ 2X~X2X3' XOXIX; -

2X~X2Xi,

Ii, I~, tr/2' lila, lill' li/2' 11M3)' Some nasty, but routine calculations show (xo, Xl'

X2'

Xa) . U(lJ3, x~)

(lJ3, xi).

+

Therefore it follows from Proposition 1.1.10 that RIpS (x~) is a Buchsbaum ring. Lifting x~ (see Propositi~n 1.2.19) then provides that RIpS is a Buchsbaum ring since lJ" is a primary ideal for all n 2 (this follows from Achilles-Schenzel-Vogel [1], Example (6.9)). Secondly, we calculate the invariant i(RjpS). We know that for every primary ideal q generated by a system of parameters of RIlJ 3 we have:

i(R/,p3)

L(Rlpa

+ q) -

eo(q, Rlp3)

(see Theorem 1.1.12). Take q (xo, xa) in RllJ3. Bezout's theorem provides that 18. It is not difficult to show that L(RI(lJ3, Xo, xa)) 21, i.e. i(RjlJ3) 3. Thirdly, we show that RllJ" is not a Buchsbaum ring if n 4. Let U(\:>", X2) denote the intersection of the primary ideals of R belonging to \:>" (Xi) of dimension 1. Using localization we get

eo(q, Rlp3}

> +

U(\:>", X2)

(X2' (xo, Xl)") n (X2' (xi, x 3)")

and

(\:>", X2)

= (X2' (xi, XoXa,

x\xs)").

2 be an integer and put n 2k. For 1:= xikx: E U(\:>", x 2) we see that 2k 1. Choosing g X~k12X~ E U(\:>", X2) we have XtY ~ (\:>", x 2). We get from Proposition 1.1.10 and Corollary 1.1.11 that RIp" is not a Buchsbaum ring for n > 4, q.e.d. Let k

xtf ~ (\:>", X2)' Let n

+

Finally, we want to give an illustration of some of our investigations by studying the class of monomial space curves in p~, that is, curves = p~ given parametrically by

e

where d

> b>

a

1 are integers with g.c.d. (d, b, a)

1.

§ 1. On liaison among arithmetical Buchsbaum curves in p3

167

We set 8 := K[xo, Xl' X2. x a], and let 1(0) 8 be the defining ideal of O. We set R:= 8/1(0). Let f-l(M) denote the number of elements in a injnimal basis of the module M over a local ring. We have the following theorem: Theorem 1.20. Let 0 be a monomial curve zn P~ over an algebraically closed field K. Then the follow£ng cond£tio1uJ are equ~'valent: (a) 0 is an arithmetically non-Oohen-Macaulay Buchsbaum curve.

(b) f-l(1(0)}

=

4 and 0 lies on a quadric. (xoXs - XIX2' x~x~n-l x~n+1, xoX~n - X~nX3' x~nTI - xin-lx~) for an integer n > 1. (d) 0 is g£ven parametrically by (8.m, S211+1t211 - 1, 8211-1t211+1, t411 ) for an ~nteger n

(c) 1(0)

=

1 such that a min£maZ fin£te resolU'#Qn of 8/1(0) over 8 has

(e) There is an znteger n

the followzng form:

o ~8(-2n

> 1.

0:1

3)

~

2)

S4(-2n

~

8(-2) EB

8 3 (-2n

- 1)

~

8

~811(0) ~O.

(f) EBH1(pa,3dv)},......X(-2n

1.

1) for an integer n

v

(g) 0 and the union of the 2 skew lz'nes

Xo

Xl

=

0

U

X2 = Xs

= 0 are Hnked.

We need some preliminary results for the proof of Theorem 1.20.

Remark 1.21. Assume 0 is an arbitrary monomial curve. Then the forms of the defining ideal1( 0) are of the following types: 1. There is exactly one form of the type

2. There are forms of the type G

=

x~'x~'

xi'x~',

minty;)

> 0,

Yl

<

<X

and

Yz

< fl·

min(6;)

> 0,

61

<

<X

and

62

< fl·

3. There are forms of type H

=

xg,~.

-

:ti'x:',

This follows easily by studying the parametric representation of 0, see also BresinskyRenschuch [1], Lemma 2. We note that in Bresinsky-Renschuch [1] there is given an algorithm for computing a base of the defining ideal of an arbitrary monomial curve in pa. Next we want to relate the perfectness of 0 to the number of generators f-l(1(0)}.

Lemma 1.22. Let 0 be a monomial curve £n Pic. Then 0 is arithmetically Oohen-Macaulay £f and only if f-l(1(0)} 3. Proof: First of all let us assume that 0 is arithmetically Cohen-Macaulay. We consider the homogeneous coordinate ring of O. This ring is a two-dimensional Cohen-Macaulay ring. In particular we have R R ez.) n Rez,l> since {xo, xa} is a homogeneous system of parameters of Rand R ez,) n Rez,)/R,...... Y:n(R) = O. Hence it follows that there is no quotient E R ez,) n R(Z,) which is not contained in R. In other words, there does not exist a form of type H contained in 1(0). Now assume that there are at least two different

168

III. On liaison among curves in projective three space

forms G = xCxg - xrx~ and G' = xg'xf - xr'x~' of the second type. Without loss of generality we may assume p > p' and q < q'. By multiplying with xg'-q resp. with xr;-p' we get

xrx;xg'-q -

xc-p'xr'x~'

E I(C).

Since I(C) is a prime ideal it follows that there exists a form of type H. But this contradicts the Cohen-Macaulay property of R. For the converse we remark that C is in particular ideally the intersection of three hypersurfaces. Applying Proposition 1.9 we see that M(C) = 0, that is, Cis arithmetically Cohen-Macaulay, q.e.d. According to Lemm3l 1'l.22 one may hope that there are similar results relating the Buchsbaum property to increased numbers of generators. The only result in this direction which is known to us is the following Corollary. Corollary 1.23. Let C be a monomial curve. Assume that ,u(I(C)) = 4. Then the minimal number of generators ofl!:n(R) is equal to 1, to.e. ,u(l!:n(R)) = 1. The converse does not hold. Proof: First of all two forms of I(C) are defined to be FI and F 2 , see Remark 1.21. Now assume there exist two forms of type G. Then the discussion given in the proof of Lemma 1.22 yields the existence of a form H E I(C) which contradicts ,u(I(C)) = 4. A similar discussion shows that the existence of two forms of type H also results in contradiction. Therefore there are exactly one form G and exactly one form H. Since R(z,) n R(zj R ~ l!:n(R) it now follows that l!:n(R) has exactly one generator. For an example which disproves the converse statement see Example 1.27(ii) below, q.e.d.

Lemma 1.2<1. If C is arithmetically non-Cohen-Macaulay and C lies on a quadric then the defining equation of thts quadnc is given by XoX3 - X 1X 2 = 0. Proof: Using the above-mentioned Remark 1.21 and the parametric representation of C we get the following possibilities for defining equations of the quadric:

(i)

XOX2 -

(iv)

X 1X 3 -

xi = xi =

0,

(ii)

XOX3 -

x~ =

0,

(v)

XOX3 -

X1X2

0,

=

(iii)

XoXa -

xi =

0,

0.

We will show that C is arithmetically Cohen-Macaulay if the defining equation of the quadric has the form (i), (ii), (iii) or (iv). There are several possible ways to prove this assertion. For instance, we get from Bresinsky-Renschuch [1] that ,u(I(C)) = 2 if we have cases (ii) and (iii). Having (i) or (iv) we obtain ,u(I(C)) < 3. Therefore Lemma 1.22 says that C is arithmetically Cohen-Macaulay. Another possibility is to apply the same methods used for proving Lemma 1.22 and the implication (a)::::;. (c) of Theorem 1.20. From this we get l!:n(Sjl(C)) = 0 if we have the cases (i), ... , (iv), qoeod. The proof of Lemma 1.24 implies the following interesting fact. Corollary 1.25. Let C be a monorm"al curve in P3. If C lies on a quadric cone then C is anOthmetically Cohen-Macaulay.

§ 1. On liaison among arithmetioal Buchsbaum curves in pa

169

Proof of Theorem 1.20: We will show the following implications:

The difficult part is to prove the implications (a) =';> (c) and (b) =';> (a). The other implications are more or less clear. (a) =';> (c): Since 0 is not arithmetically Cohen-Macaulay there exists at least one form H = xg.x~' - x1'x;' contained in a minimal generating set of 1(0), see the discussion in the proof of Lemma 1.22. Therefore the following element is a non-zero element of the first local cohomology module: ~ := x~'lxg.

=

x~'lxg. E R(z,)

II

R(z.,/R ~ liin(R).

Since R is a Buchsbaum ring Hin(R) is annihilated'by In particular it follows that

H

= xox~

-

XiX3

with (

ill,

i.e.,

Xi'

$ E R for i

= 0, 1,2,3.

1.

We choose ( minimal with respect to the property HE 1(0), Then it follows easily

Next we consider the quotients

This gives two forms

contained in 1(0). By comparing both

Xl,

x2-terms we get a form

contained in the prime ideal 1(0). By factoring out irrelevant terms we get a form of type H by an easy discussion. By the Buchsbaum property of R we see, as above, that it is of the following form :

H'

=

xoX{ - xi' X3

with ('

<

y.

But this contradicts the minimaIity of y. Therefore both forms in (*) have to be equal. It follows ( PI ( - 0:2 1 and Po 0:3 = 0, i.e. the form is given by XoX3 - X 1X 2 • Because XoX3 XIX2 E 1(0) it now follows that there is no other form of type G. On the other hand, there is also no other form of the type H contained in 1(0) because it would be equal to xoX; - X~X3 with iX > y by virtue of the Buchsbaum property of R. But 0: > ( is not possible because this reduces modulo Fl resp. F 2 • Therefore

170

III. On liaison among curves in projeotive three space

we get the following forms:

F2 =

X~+l -

xClxi,

H

xoX~

xixa

contained in I(C). Since FI resp. F2 is uniquely determined we get y = 2n. This proves _ (a) o? (e).

(c) o? (d), trivial. (d) o? (g): Applying Bezout's theorem we get that

I(C) n (xo, Xl) n (~, xa)

(XoX3 - XIX2' xoX~n - X~nX3)

for all n

1

since the degree of C is given by 4n, that is C and the skew lines Xo = Xl = 0 and X2 Xa = 0 are linked. (g) o? (e): From (g) follows (a) by applying Theorem 1.2(a}. Hence from (d) we obtain that' C and the union of the two skew lines are linked by two hypersurfaces of degree 2 and 2n + 1. Therefore Lemma 1.7 implies assertion (e). (e) o? (b): The assertion results immediately from the term S( -2) EEl sat -2n 1). (b) o? (a): Lemma 1.22 and Lemma 1.24 show that C lies on the quadric with defining equation XoXa - XIX2 = O. Hence our Example 1.27(iii} yields the elements lof a minimal base for I(C). From these elements (a) follows from Theorem 1.11. (c) o? (f): It is easily seen that C is ideally the intersection of the three hypersurfaces

Hence Proposition 1.9 proves claim (f) for the Rao invariant of C. (f) o? (a) is trivial since H:n(SjI(C)) is a vector space of dimension 1. This concludes the proof of Theorem 1.20, q.e.d. Remark 1.26. We note that S. Goto in [2] proves a Buchsbaum criterion for an arbitrary affine semigroup ring using the main results of Goto-Watanabe [2]. In fact, S. Goto . [2] computes the first local cohomology module H:n of such a Buchsbaum ring in terms of the underlying semigroup. But these methods do not provide a proof of Theorem 1.20. We want to conlude by studying three examples and by adding some remarks. Examples 1.27. (i) The following example shows the usefulness of Theorem 1.2 in projective space Pll with n > 4. Take the surface F in p5 with defining ideal a = (Xo, Xl, X2) n (Xl' X2, Xa) n (X2' Xa, X,) n (X3' X" Xli) n (X,; xs, Xo) n (Xs, Xo, Xl)'

By using Theorem 1.2 we will prove that F is arithmetically Buchsbaum which also follows from Proposition V.2.7 and Example V.2.9(ii). To accomplish this, let G be the surface in pli with defining i~eal b

(xo,

X 2,

x,) n (Xl>

X 3,

xs).

§ 1. On liaison among arithmetical Buchsbaum curves in

pa

171

I

Then F is linked to G since a n {I (XoXa, XIX" X2X.). If follows immediately from Proposition 1.2.25 that G is arithmetically Buchsbaum. Theorem 1.2 then finishes the proof of our claim. (ii) In order to prove Corollary 1.23 of Lemma 1.22 we investigate the following example. Take the curve 0 given parametrically by from Renschuch [2] that

(8 8,

87t, sSt·, tS). It follows immediately

that is; 1(0) is generated by precisely five elements. We get that !1fn(R) ~ (xi/xo, 1) RjR

where R

=

8/1(0).

Hence !1fn(R) is genemted by precisely one element. The computation of !1fn(R) is accomplished by applying the same methods used in the proof of Theorem 1.20(a) =} (c). (iii) In connection with Theorem 1.20(a) =} (c) and Lemma 1.24 we now consider the monomial curves lying on a non-singular quadric. We collect some properties of these curves. Their proof follows easily by applying methods of Renschuch [21 or of the proof of Theorem 1.20. Let 0 be a curve given parametrically by

Then we have the following properties: (a) The defining ideal 1(0) has the following minimal basis:

where

(b) 0 is ideally the intersection of the three hypersurfaces defined by = 0, :4 - x~x!:-a = 0, and XoXa - X 1X2 = O. (c) dimK[Hfn(R)]i

{

~i -

a

+ 1) (b -

1 - i)

otherwise.

b - a-I.

(d) ",(Hfn(R)) (e) In(Hfn(R))

=

~ ~ ~-ax~

=

e-; 1). +

At the end of this paragraph we finally come to the elementary proof of Theorem 1.2 as remarked after the statement of Theorem 1.2. Let M be a Noetherian A-module. As in the graded case (see Corollary 0.4.16) we can define a Noetherian A-module MX in the following way (A a local ring): Take a free A-module F and an epimorphism :n:: F -70- M such that Ker:n: <;;; m . F. Set ~

MX := Coker(Hom(M, A)

-70-

Hom(F, A))

Coker Hom(:n:, A).

172

Ill. On liaison among curves in projective three space

We note that F and hence MX are uniquely determined by M (up to an isomorphism). If M is free itself, MX = O. If Supp M n Ass A = 0 (which means that Hom(M, A) 0) we set MX Hom
Proposition 1.28. Let A be a local Gorenstein ring with d: the injective hull 01 the residue lield k Noetherian A-module. Then:

dim A 2. Let E denote Aim. Furthermore, let M denote a non-Iree

(i) II M is an equidz"men8z"onal locally Oohen-Macaulay module then the same is true lor MX. II dim M = d

lor all z' = 1, ... , d - 1, lor i

O.

(ii) II II! z"8 a d-dimen8ional Buchsbaum module nith depth M a (d-d2~men8ional) Buchsbaum module wiih I(MX) I(M).

>

0 then

~lfX

is also

Prool: (i) If dim M < d then MX is a free A-module (since Hom(M, A) 0) and there is nothing to prove. If dim M = d (i) follows by the same argument as in the proof of the graded version of this statement, see Corollary 0.4.16.

. (ii) Take a free A-module F and an epimorphism n: F ~ M with Ker n ~ m· F. Set U:= Ker n. Since M is a non-free A-module, U 9= 0 and since Ass U ~ Ass F Ass A we obtain that dim U = d. Let +> E Spec A be minimal. Then All is an injective All-module and therefore (MX)1l = HomAp(Ull' All) (HomA ( ,All) is an exact ll functor). But HomAll(Ull , All) 9= 0 for all +> E Ass U 9= 0, i.e. we have dim MX = d. An easy computation shows that depth Hom(M, A) > 2, i.e. we obtain from the exact sequence 0 ~ Hom(M, A) ~ Hom(F, A) ~ MX ~ 0:

>

Therefore depth MX 1. Also, there is an exact sequence 0 ~ MX ~ Hom(U, A) _ (U,.tt) _ Extl(M, A) _ O. Since by Corollary 0.3.5 Hom(ExtI(M, A), E) H'f;;l(M) we see that lA(Extl(M, A)) < =, especially is m· Extl(M, A) = O. Therefore we get (since depth Hom(U, A) 2):

>

Extl(M, A)

~(Extl(M, A)) "'" H:n(JlfX).

If now d =.2, MX is a Buchsbaum module by Proposition 1.2.12. We next use 2. If d 2 the proof is already complete. Let d 3. induction on d Take now an element x E m with dim MX Ix, lJx d - 1. We will show that MX Ix . ~lfX : (m) is a Buchsbaum module. Then by Proposition 1.2.23 MX is a Buchsbaum module. Let Ass A := {+>l> ••• , +>r} and assume x 1 U ••• U ~., x E+>8+1 n ... n +>, where 1 s r. If 8 < r, Ann MX t;t +>.+1 U ... U +>, (since dim AjxA Ann MX = dim MX jxM X = d - 1). Take an element y E +>1 n ... n +>. nAnn MX with y 8+1 U... u.\J, and set z:= x + y. Then zMX = xMX and z
+

§ 1. On liaison among arithmetical BuchsbauJll curves in p3

173

= Hom,,(N, AjxA) for any A-module N we obtain a commutative diagram with exact rows and columns

o t

o t 0-* Hom(M, A)

-*Hom(F, A)

t

t

0-* Hom(M, A)

-* Hom(F, A)

t

t

-* MX

t

0-* HomA/.'rA(M/xM,AjxA) -* Hom,,/x,,(FjxF, AjxA) -* (MjxM) X -* 0

t

t

o

Extl(M, A)

t

o From this we obtain an epimorphism #: MX -* (MjxM)X «MjxM)X is defined as an AjxA-module but is now considered as an A-module). Let C := Ker #. Then the above diagram gives exact sequences (1)

0 -* ~~fX -* C -* ExVUlf, A) -* 0 (snake-lemma),

(2)

0 -* C -* MX 4

-* O.

(MjxM)X

From (1) and (2) we obtain the following exact sequence since the composition MX -* C -* MX from (1) and (2) is just the map '-4fX ~-+- MX); (3)

0 -* Extl(M, A) -* MX jxMX -* (MjxMV -* O.

Since d

3 we have depth(MjxM)X Extl(M, A)

1, i.e.

Hi'n(Ext1(M, A))

H~(MX /xMX)

xMx : (m)/xMx .

Therefore (3) gives rise to an isomorphism 1l:fXjxll{X

;(m)~

(M/xM) X

•

By the induction hypothesis (Mix",,! is a Buchsbaum module over A and hence over A/xA of dimension d - 1 dim A/xA) we know that (M/xM)X is a Buchsbaum module over A/xA and hence over A. Therefore MX/xMx :(m) isa Buchsbaum module. It is I(MX) I(M) by (i), q.e.d. Corollary 1.29. Let A be a local Gorenstez'n ring with d : = dim A > 2. Let M be a ddimen8'ional Noethen:an A-module with dim Alp = d tor all p E Ass M which has no tree direct summand. Then M i8 a Buch8baum module it and only it MX is aBuchsbaum module. Proot: It is easy to show that by the assumptions made there is an isomorphism M ::,.. (MX)x. The statement follows from the above proposition, q.e.d.

The following corollary establishes Theorem 1.2. Corollary 1.30. Let A be a local Gorenstein ring ot dimension d > 2. Let a, b A be unmixed ideals without common components 8uch that a n b = 0, a g; b, b g; a. Then A/a is a Buchsbaum ring it and only it Alb i8 a Buch8baum ring. In this case I(A/a) I(A/b). Proot: Set M := A/a. Then MX

Alb and M

(MX)X

A/a, q.e.d.

17 -1

III. On liaison among curves in projective three space

§ 2.

On lia.ison addition and applications

In this paragraph we will solve the liaison addition problem by following P. Schwartau's [1] work. We ar~ able here to use this liaison addition to prove new theorems about liaison realization and self-linkage. Since the liaison procedure is explicit, we are able to solve a long-standing problem of liaison: to find explicit curve in Plc corresponding to the liaison invariant EB K for all n 1; that is we will construct arithmetical Buchs-

.

baum curves C in p3 with an invariant i(C) = n. Essentially the only well-understood case previously was the case n = 1, represented by the example of two skew lines above. Using this example and the liaison addition, we produce explicit examples for all n > 1. The ideal of the curve C with i(C) n will be the monomial ideal

note that this reduces to the skew lines when n = 1. It is then surprising to discover that a slight change in the exponents above can make all the liaison invariants vanish. For example all the monomial ideals

have liaison invariant 0; that is they define arithmetically Cohen-Macaulay curves in P~. Liaison addition in its most general form is an addition of resolutions of ideals. Of course one already knows that the direct sum of two such resolutions does provide a resolution, but not of an ideal. NEvertheless, our discovery is that for ideals of grade 2 the direct sum is very nearly the answer. The key result is the following 2. Choose Theorem 2.1. Let I, I' be homogeneous ideals 01 S = K[xo, ... , XII] 01 grade any hmnogeneouselementsl E I and /' E I' such that {t, j'} i8 an S-sequence (such a choice is possible). Then /'. I + I . I' i8 a hOmL>(leneous ideal 01 grade exactly 2, and there exist graded Iree resolutions F, F', G 01 I, I', /' . I + I· I' over S such that G

F (-deg j') EB F'( -deg I)

up to relations aruJ. generators.

Remark 2.2. The theorem is stated only for the case which interests us: for homogeneous ideals of S K[xo, ... , X,.l. However, the first proof, given by P. Schwartau, applies to any commutative Noetherian ring, as long as we make the extra assumption that pd [ and pd [' < 00. Another proof we shall give, due to David Buchsbaum, shows that even this assumption is unnecessary. In either proof, the grading of S plays no helpful role. The existence of I, /': We may obtain the first non-zero-divisor 1 by choosing any non-zero homogeneous element of I. We next need a homogeneous element /' E I' which is a non-zero-divisor on S/(/), which we try to obtain by advoiding the associated primes ~1' ••• , fl" of (/). If this is impossible, then for every degree d we get: 1',; c ~1 U ••• U ~II' Therefore I' c ~1 U ••• u ~,.; that is I' C fli for some i. This is a contra1. Therefore the desired /' exists. diction, since grade I' > 2 but grade fli

§2. On liail!On addition and applioa.tions

175

Prool 01 Theorem 2.1: David Buchsbaum has pointed out that once we known I' . I + I . l' is the ideal to add resolutions, a simple "coordinate-free" explanation is possible. Namely, use the standard exact sequence:

0-1" I nl· I' ~ 1" I EB I, I' -I" I + I· I' - 0 where 01 is defined by 8 H- (8, Now observe that since I, I' are relatively prime, I' . I n I . I' represent the ideal I' . I + I . I' as: coker ((f . f')

=

(f. f'). Thus we can

I' . I EB I· I')

where

f·/' H- (f '1', -I' f'). We now use this to write the resolution of I' . I + I· I' as a cokernel. Namely, the Comparison Theorem guarantees that the map IX (homogeneous of degree 0) lifts to a homogeneous chain map iX of resolutions. The sequence of cokernels will then form a graded complex G, ending in the ideal/, . I + I· I'. The resolution G will be exact if the li~ts can be chosen as inclusions, and will be free if the lifts are split inclusions. Under certain weak assumptions, this can be done as follows: Choose graded free resolutions F, F/, of I, I' over S: t

F: .. , -EB S(-Ck)~ 1

a

b

S(-bi)~EB S(-ai) ~ S,

I

I

r'

b'

a'

1

I

I

EB S( -4) ~. . EB S( -bj) ..!.4 EB S( -al;)

F/: ... _

<1>; ....

S.

We now demand that these resolutions select the elements 1', I of the hypothesis; that is, that the original resolution F of I selects I E I as the image of (0, ... ,0, 1) and that F' selects I' E I' as the image of (1, 0, ... ,0). Then from the cokernels one gets a graded free resolution G of the ideal/, . 1 + I . I', this can be done as follows: F(-deg

a

b

n ...

deg j')

EB8(-h l - deg/')

EB 8(

-a, -

deg f')

EB ~

F'( -deg /): ...

~

if

EB 8(

-c" - deg I) .-,. EB 8( -hj -- deg f) .-,. EB

+i

tI

I I

I I

I I

I I

I I

I I

I I

I

8(

-a;

deg f)

+I I I

I

. 0 ; S(-deg/-degf') ~ (1·/').-,.0

... .-,.0

But since the resolution of (/ . f') is so short, we see that G will be equal to F( -:-deg f') F'( -deg I) up to relations and generators, as claimed in the theorem. It now only remains to prove that /' . 1 + I . I' has grade exactly 2. But this follows from the inclusions; (f, f') . (1

q.e.d.

II

1') S; I' . I

+ I· l' S; (f, f') II I

n 1',

176

III. On liaison among curves in projective three space

The following principle fact is a direct consequence of the proof of Theorem 2.1. Corollary 2.3. For all i Ext~{f' . I

+j

2 we have: • 1', S)

S) (deg!') EB Ext~(1', S) (deg j),

and therejore jor all Ext~(S/j'

3:

+ j . I', S)

= Ext~(S/I, S) (deg!,) EB Ext~(S/I', S) (degj). Now we will show how to solve the liaison problem. As is known Rao has proven that curves 0 Pkare classified up to the liaison by the graded moduleEB Hl(Pk, dc(V)). v

In the following we intend to use Schwartau's alternate formulation of the liaison invariant by use of Ext~(S/I(O), S). This allows us to investigate the liaison invariant via the theory of free resolutions. Definition 2.4. Let 0 be a curve in Pi. Define as above JIf(O) to be the graded S-module EB Hl(p3, dC(V)) and let E(O) be the graded S-module Ext 3 (S/I(0), S), where 1(0) v

is the defining ideal of 0 and S

=

K[ XO,

Xl> X 2 ,

xa].

Using our notations and results from Chapter 0, § 4 (see the section on dualization) we will first show how Rao's result may be stated in terms of Schwartau's module E(O) rather than M(O). This follows immediately from Lemma 2.5. For any curve 0 c::: Pi, E(O)

((M(O)))V (4).

ThU8 E(O) determines the same liaison class oj modules as M(O). Additionally we have the jollowing transjorrnaUon law: Suppose, C, C' are curt'es Pi such that C and C' are algebraically ltnked by j, !'. Then: M(O') E(C')

~n

(M(C))v (4 - degj - deg!'), (E(C))v (deg j

+ deg j' + 4).

Prooj: First we note that the proof of Corollary 0.4.7 provides the following property: M(C)::::;. Hf"' .......AK[xo, .'" xaJ/I(C)).

Hence our duality theorem (see Theorem 0.4.14) yields the first assertion. The transformation law follows from Corollary 0.4.17, q.e.d. Now, recapitulating the liaison addition problem was the following: Is there an addition of closed subschemes in P} which induces an addition of their liaison classes? On might think that in the case of curves in Pi (without isolated or embedded points) this problem is already answered by Rao's theory of the liaison invariant. Each such curve determines a specific graded S-module of finite length. Thus the direct sum determines a unique algebraic liaison class which may be defined as the liaison sum. The reader will note, however, that this method is not only unsatisfactory but invalid.

§ 2. On liaison addition and applications

177

It is unsatisfactory because the addition has no description in terms of the eurves alone. It is invalid because of the following faet: The direct sum of graded S-modules does not induce a sum on liaison classes of graded S-modules. For example, E, E(l), and Homs(E, S) are all in the same liaison class, but E EB E, E(l) EB E, and Homs(E, S) EB E all determine different liaison classes. Therefore the direct sum of liaison invariants does not induce an addition in liaison classes of curves. So the Rao theory does not solve the problem for curves in p~, but instead complieates the issue by raising another question: Is there an addition of curves in p~ which induces the direct sum on their liaison invariants? On might at first try the scheme-theoretic union as a means of adding eurves, but the union of one line with a skew line shows that this idea is not very promising since scheme-theoretic union does not preserve the 0 class. The geometrie formulation of our liaison addition will reveal just why scheme-theoretic union is iJ?sufficicnt. Here now is Schwartau's solution of the liaiso~ addition problem: Definition 2.6. Let V, V' be closed subschemes in P1c of codimension 2 with defining ideal I(V) =: I and I(V') : l' (note that (xo, ... , xn) is not associated to I and 1'). Choose any hypersurfaces I containing V, I' containing V', such that (j, f') is a complete intersection in P1c. We then define VI I'V' to be the closed suhscheme of P1c defined by the homogeneous ideal j' . I + I . 1'.

+

Theorem 2.7. Let V, V' be closed suhschemes in P1c of codimension 2, and let VI deltned as above. Then: 1. (xo,"" xn) is not a!JsociaJed to I' . I I(VI+I'V') 2. VI

+ j'V' be

+ I . 1'; that is

1'·1+1'1'.

+ j'V' is agat"n 01 codimension 2 in P1c.

3. For all i

3 we have

Ext's{SfI( VI

+ I' V)) "-' Ext~(SII, S) (deg f')

Ext~(SII', S) (deg I).

+

4. VI I'V' is locally Cohen-Macaulay and equi-dimensional il and only il V, V' are locally Cohen-M.acaulay and equi-dz~mensional.

+

5. VI j'V' is arithmetically Cohen-Maeaulay il and only zj V, V' are arithmetically Cohen-Macaulay.

+

6. If I is a non-zero divisor rJWdulo l' and I' is a non-zero-divisor rJWdulo I, then VI I'V' = V u V' u (/, f') as closed sWschemes 01 P1c; that Z8, I' . I I . l' = I n l' n (/, f').

+

The last equatz'on is always true set-theoretically, even without the hypothesis 016. Corollary 2.8. Let C, C' be any two curves z"n P~ with delim"ng ideals I, l' and liaison z"nvariants E, E'. Choose any !Jurlaces I containing C, j' contat"nt"ng C', such that (/, j') is a complete z'ntersection curve in P~. Then 1. CI I'C' is a curve z"n p~, with liaison invariant E{deg f') EB E'(deg f) and delinz'ng I . 1'. ideal I' . I 2. II c, C' are without isolated or embedded poz"nts, the same is true 01 CI I'C'.

+

12

+

Buchsbaum Rings

+

178

III. On liaison among curves in projective three space

3. GI + I'G' 0 u G' u (f, f') set-theoretically. 4. II the surlace I does 'Mt contain any component 01 OJ, and I' does not contain any component 01 G, then GI + /,G 0 u G' u (f,j') scheme-theoretically; that is,

f' . I

+ I· l' = I

n l' n (f, 1').

Prool: Theorem 2.7 we only need to remind the reader that the property of "no imbedded points" is for curves equivalent to the "locally Cohen-Macaulay" property stated in the earlier theorem. Remark 2.9. We may always choose I, I' of the same degree d. Then the liaison addition procedure will give us a curve 01 + I'G' with liaison invariant E(d) E8 E'(d) ~ (E E') (d). Thus the added curve in this case lies in the liaison class associated to the module E E8 E'. However, for Buchsbaum curves we get the following statement: Corollary 2.10. GI + 1'0' is arithmetically Buchsbaum il and only il G, 0' are arithmetically Buchsbaum. (Note that I and I' have not necessarily the same degree.)

+

Prool 01 Theorem 2.7: 1. Assume that (xo, •.• , xn) is associated to I' . I I· 1'. By the Auslander-Buchsbaum theorem we have pd(S/1' . I + I· 1') n + 1; that is "1

(S//, . I

+ I· 1', S) =f: O.

Corollary 2.3 provides either

Ext~+I(S/I,

S) =f: 0

Ext~+l(S/I',

or

S) =f: O.

Hencc either pd S/I

n

+1

or

pd S/1'

n

+ 1;

that is (xo, •.• , xn) is associated to I or l' which contradicts our assumptions. This proves 1. 2. We are to show ht I(VI + /'V') = 2. But since S is Cohen-Macaulay it is enough to show grade of I(VI + I'V') 2. This fact follows from 1. and Theorem 2.1. 3. This follows from 1. and Corollary 2.3. 4. and 5.: Since V, V', and VI + I'V' all have codimension 2, the properties at issue in 4., 5. depend only OIl the vanishing or finite length of Extl( , ) (With suitable arguments) for 3 i::;; n. But then assertion 3. implies 4. and 5. 6. The reader will easily see that the hypothesis of 6. forces /' . I

+ I . l' =

I n l' n (f, f') .

Note that even without the hypothesis of 6. we always have this equation up to radical. This follows from the inclusions:

(I n 1') . (j, j') <;; f' . I

+ I . l' <;; I

n l' n (I, f'),

which always holds. This with our assertion 1. proves 6. and the theorem, q.e.d. Next we discuss some applications of liaison addition and some new examples of curves ill p~ defined by monomials. First of all, we need the so-called power formula.

179

§ 2. On liaison addition and applications

Theorem 2.11 (Power Formula). Let 0 be any curve in Pk with deft'mng ideal I and liaison tiwariant E. Ohoose any 8-sequence f, I' E I such that f, I' are homogeneous of the same degree d. Then

U, n" . I

deft1wS a curve tn

P1< with liaison invariant (ffi E)

(nd).

Proof: We use induetion on n, the ease n 0 being obvious. Therefore assume n> 0 and that the theorem is true for n - 1. Thus U, n,,-l ·l defines a curve 0( ..-1) with

n.

(4 E)

invariant ((n - 1) d). Notice that this ideal contains /", and that U", 8 is a complete intersection. Thus we may consider the liaison sum O("-l)/" 1'0. Corollary 2.8(1) then implies that this curve has defining ideal 1'((1, n"-l . l) + f" . I U, l, and liaison invariant

+

((4 E) ((n

l)d)hd)

E(nd)

(4 Ehnd)

n" .

E(nd) =

(Eel Ehnd) ,

q.e.d.

n.

Example 2.12. Let 0 be any complete intersection curve with defining ideal (/, Then 0 is arithmetically Cohen-Macaulay and therefore has liaison invariant O. It is clear that in the case of the trivial invariant the "same degree" hypothesis in the Power Formula is not needed. Thus we may apply the formula to the 8-sequence {f, f'} E (f,1';.8. We deduce that (f, f') (f, defines a curve in Pk of liaison invariant 0; that is an arijJlmetically Cohen-Macaulay curve. This is a special case of the well-known theorem that all powers of a complete intersection are perfect. Furthermore, let I be any perfect ideal in K[ X o, Xl' X 2 , xaJ. Then the Power Formula may be applied just as above to {f, f'}, an 8-sequence' in l, to conclude that the ideal I must be perfect.

n" . (/,

n"+l

(/, n .

Example 2.13. Let 0 be two skew lines with defining ideall = (xo, Xl) n (X2' xa). Then I has generators (XOX2' XoXa, XlX2, XlXa). Thus I contains the 8-sequence XOX2' XIXa, and we may apply the Power Formula. Note that 0 has liaison invariant [((4) by use of Remark 1.5 and Lemma 2.5. Therefore we conclude that for all n 1, (XoX2' x1Xa)·-1 • ((Xo, Xl) n (X2' xa»)

+

defines a curve in Pk of liaison invariant 1("(2 2n). This gives us the first explicit examples of curves in P1< corresponding to the liaison invariants K" for every n::::: 1; that is, we have equations defining arithmetically Buchsbaum curves 0 in Pk with Buchsbaum invariant i(O) n. Also we have the following. primary decomposition: (XoX2' xIXa),,-1 • ((xo, Xl) n (X2' xal) (Xo, Xl)" n (Xl> X2),,-1 n (X2' Xa)" n (Xa, XO) ..-l.

Example 2.14. Consider the curve 0 in I

=

(xo, Xl) n (Xl> X2) n (X2' Xa)

Pk with defining ideal (XoX2' XIX2' xlxa).

Note that 0 is arithmetically Cohen-Macaulay. This follows immediately, for example from Proposition 1.9. Once again we may apply the Power Formula with respect to the S-sequence (XoX2' XIX3). Hence we conclude that for all n 1 (XoX2' xIXa) .. -l . ((Xo, Xl) n (Xl' X2) n (X2' Xa») 12*

180

III. On liaison among curves in projective three space

defines a curve in p~ of liaison invariant 0; that is these curves are arithmetically Cohen-Macaulay. We have the following primary decomposition: (XoX2'

=

,

'

X I X 3),,-1 •

(XO,

Xl)"

((Xo,

Xl)

n

(Xl'

X2) n (X2' X3))

n (Xl> x 2)" n (X2' X3)" n (X3'

XO)"-l.

Remark 2.15. Each of the above examples presents curves in p~ defined by monomials. We want to point out that not every liaison class in p~ contains such a curve. This assertion is a consequence of S. Goto's and K. Watanabe's paper [2] on Z"-graded rings. The main point here is following: If C is a curve in p~ defined by monomials, then the quotient S/I(C) is a Z4-graded S-module in the sense of Goto-Watanabe. It follows that the liaison invariant Ext~(S/I(C), S) has also the structure of a Z4_ graded S-module (see p. 243 of the just mentioned paper). Such a structure is preserved by K-duals and shifts, so we conclude that every module in the liaison class of E(C) possesses a Z«-graded structure. But not every graded S-module of finite length has a Z«-graded structure, and therefore not every liaison class contains a curve defined by monomials. Another application of liaison is the so-called liaison realization. Following Schwartau [1] we will explain this phenomenon. No complete proofs will be given here. The point of view discussed in what follows was introduced in the fundamental paper by Rao [1]: There is a 1-1 correspondence between the liaison class of curves in p~ which are equidimensional without imbedded points and generic complete intersection and the liaison classes of graded S-modules of finite length. But this 1-1 correspondence of Rao does not produce a curve C c: p~ of liaison invariant E for each graded S-module E of finite length. It only gives a curve for each liaison class of modules; that is the specific result of Rao says that for any graded S-module E of finite length there exists a curve C c: p~ of liaison invariant E(v) for some shift of E. It is in fact not true that an arbitrary graded S-module of finite length can be realized as the liaison invariant of a curve in P~. Inde~d, we have the following lemma: Lemma 2.16. If A =1= 0 z"8 a graded S-module of fz"nz"te length, then for v ~ 0 z"t 'is z"mp088zOle to realz"ze A( -v) a8 the lun"80n z"nvarz"ant E(C) of a curve C c: P~. The proof of this lemma rests on the following Claim 2.17: Any non-zero liaison invariant E(C) "begins" in degree

=

-4.

After having established this claim it is easy to prove the lemma. Simply twist the given module A until it begins in degree -4. This would be impossible if A were to have infinitely many non-zero negative degrees, but this is precluded by finite length. Therefore high negative twists of A cannot be realized as liaison invariants. However Rao's result quarentees that some twist A(v) can be realized, and an examination of his proof shows that any twist A(v + 2n), n > 0 can be realized. Unfortunately the bound for v is not proved to be sharp, and there is no way in general to obtain information about the twists A(v 2n + 1), n> O. But this phenomenon may be clarified considerably through liaison addition:

+

Theorem 2.18 (Liaison Realization). If a graded S-module E can be realz"zed a8 the lz"az"8on znvariant of 80me curve C c: p~, then the same 'is true for each p08uive tw'ist E('YJ), 'YJ > 1.

§ 3. On curves linked to lines in p8 and applications

181

Prool: Choose any homogeneous element 1 E 1(0) such that (X;j, I) is a complete intersection. Now choose any curve 0' lying in the plane Xo = O. Since any plane curve is a complete intersection, 0' has liaison invariant O. Therefore by liaison addition the curvf2 01 x;jO' has liaison invariant

+

E("I)

+ O(deg/) =

E("I)'

A disadvantage in this method is that we can only produce curves without isolated points, whereas Roo produced non-singular curves.

§ 3.

On curves linked to lines in P' and applications

Consider the non-singular quadric surface XoXa XlX 2 = 0 in P3. Such a surface has two rulings, and we will examine "m by n" configurations L of lines where L consists of m lines from one ruling and n from the other. We will show: (i) The "m by n" configuration is arithmetically Cohen-Macaulay if and only if Im- nl< 1, (ii) the configuration is arithmetically Buchsbaum if and only if In - ml 2. Following Geramita-Maroscia-Vogel [1] we will outline an geometric approach to the content of (i) and (ii) by use of liaison. Therefore our proof is entirely different from the linear algebra proof of these results given by Geramita-Weibel [1]. Of course there are still different methods to study such configurations. By applying our method we are also able to produce equations defining connected curves in pa which are set-theoretically complete intersections and not arithmetically Buchsbaum. Therefore we get counter-examples to an even stronger formulation of a conjecture of R. Hartshorne [3] (Conjecture 5.17 on p. 126). Our simplest counter-example to Hartshorne's conjecture is the curve 0 c:::: p~ given by the following ideal 1(0) in K[xo, Xl' X 2 , xa]: 1(0) =

(XO,

Xl) n (X:!, Xa) n

(Xo -

X2,

Xl - Xa) n (xo - Xl,

X2 -

Xa).

o is a set-theoretic complete intersection of the quadratic Q and cubic K defined by the equation xoXa - XIX2 = 0 resp. (xo - Xl)

(X2 -

Xa) (Xo - Xl -

X2

+ Xa)

O.

The connected cpl"ve 0 is linked to the union of the following two skew lines: Xo

X2 =

0

and

Xl

Xa = O.

Hence 0 is not arithmetically Cohen-Macaulay but arithmetically Buchsbaum. The same method suitably refined, can be used to construct curves which are set-theoretically complete intersections and not arithmetically Buchsbaum. The key here is that these curves are linked to the union of h skew lines which lie on a non-singular quadric in p3 with h > 2. Also we are able to give the equations for these examples (see Proposition 3.3 and Remark 3.4, below). From the point of view of commutative algebra we have the following fact: Let F be an irreducible form in S = K[xo, Xl> X 2 , xa] and let V(F) c:::: p~ be the surface defined by F = O. We let R S/(F) be the homogeneous coordinate ring of

182

III. On liaison among curves in projective three space

V(F). If .\:l is a homogeneous prime ideal in S having height 2 and containing F then .\:l describes an irreducible cur\7e V(.\:l) on V(F). If V(.\:l) contains at least one point which is not a singular point on V(F), then the local ring. V(F) along the subvariety V(.\:l) is a discrete valuation ring. Consequently, for each integer n > 0, there is a unique .\:l-primary ideal of S which contains F and has length n. Our aim in this paragraph is to investigate the special situation in which .\:l describes a. line on V(F). First we need the following lemma.

Lemma 3.1. Let H be an irreduc~'ble form in S := K[xo, ••• , x r ] (K any algebraically closed field, r 3) which can be expressed as H

=

Xo'

F -

Xl'

a,

where F an a are forms 'l:n K[X2' ••. , x r ] of degree iX 1. Let A := K[xo, ... , xr]/(H). Set .\:l = (xu, Xl) • A. Then, the local ring of the variety V(H) along the sUbvariety V(.\:l) is a discrete valuation ring. Proof: It will suffice to show that there is a point on the linear subvariety V(.\:l) of V(H) which is a simple point on V(H). Suppose this is not the case, i.e. every point on V(.\:l) is a singular point of V(H). Consider the point (1, 0, ... , 0) P on V(H) and note. that it is not on V(.\:l). Let Q be a point ~m V(.\:l). The line connecting P and, Q meets P with multiplicity > iX and meets Q with multiplicity 2. Since iX + 2 > deg H, this line lies on V(H). Since this was true for all points of V(.\:l) we have that the hyperplane formed by these lines is conta.ined in V(H). This implies that V(H) is a hyperplane, which is a contradiction, q.e.d. We will say again that a homogeneous ideal Ie S := K[xo, ... , xr] is a CohenMacaulay (respectively, Buchsbaum) ideal if the local ring at the maximal homogeneous ideal of K[xo, ... , xr]/I is a Cohen-Macaulay (respectively, Buchsbaum) ring. In the following we will examine some properties of "multiple" lines which are on a non-singular quadric surface in P3.

Theorem 3.2. Let H xoF xla be as in Lemma 3.1. Let.\:l = (Xo, 'J,)-primary ideal in S of length n 2) which conta1,ns H. Then: (i)

q

Xl)

c S. Let q be a

(.\:In, H).

(ii) If, moreover, F E (X3) and a E (X2) then if q Z8 a Buchsbaum iileal we must have n 2 and H iXXoX3 PXIX2 with iX, P E K. (iii) If n = 2, r 3 and H iXXoX3 PXIX2, iX, P E K (tX • P =!= 0), then q is a Buchsbaum ideal with 1,nvariant 1. (iv) If r > 3 and n > 1, then q is not a Buchsbaum iileal.

Proof: From Lemma 3.1 we have that there is a unique .\:l-primary ideal in S of length n which contains H. Thus, to prove (i), it suffices to show that an ('J,)n, H) is a primary ideal of length n. Claim 1. an := (.\:lA, H) is .\:l-primary. We know of several ways to prove this claim. We will give an elementary proof following Geramita-Maroscia-Vogel [1].

§ 3. On curves linked to lines in

pa and applications

183

Since ROO(a.) p, it will be sufficient to show that an is unmixed. Suppose this is not the case and write an = 'I r. 'II ... n q., where 'I is p-primary and 'II, ... , 'I. denote the embedded primary components of an. Hence ROO(q;) : Pi ::J (xo, xd. Thus, there is a form A, no monomial of which is divisible by Xo or Xl> such that an : A i? an. Let BE K[xo, ... , x r] be a form such that B ~ an and B· A E a •. Then B E 'I ;::;;;; (xo, xd. We may consider B as an element in (K[X2' •.• , x r]) [xo, XI] and write B as a sum of forms in Bl + B2 + ... + B t where deg B; = i (as a form in xo, Xl)' Since this ring; B p" ;::;;;; a. ;::;;;; 'I we may assume t n t . Note that a.. may also be considered as a homogeneous ideal in (K[X2"'" x r ]) [xo, XIJ. Thus we may write

B· A

BlA

+ B2A + ... + BtA.

Since deg A 0 (in xo, Xl), deg BiA i. Therefore, B j • A E a. for 1:::;; i:::;; t. But, the only forms in a.. of degree < n (in xo. Xl) are the multiples of H. Thus H I BiA, 1 i t. Since H is irreducible and H { A we obtain that H I B i , 1 t, and so B E a•. This contraction establishes Claim 1.

Claim 2. an is p-primary of length n. Prool: Consider the following chain of length n of ideals: a" ~ (a",xg- l ) ~

...

~ (an.x~) ~ (Xo, XI)

= p.

It is easy to see that even after localizing at p, the inclusions in this chain remain proper. (One argues by degree in xo. Xl as in claim 1.) Thus an has length at least n. Using the well-known criterion from Zariski-Samuel [1J, Vol. 1, p. 237, Cor. 2, we see, after localizing at p, that this is a saturated chain of primary ideals and so a. has length n. This completes the proof of claim 2 and hence the proof of (i) of the theorem. Prool 01 (ii) : Suppose H has the required form and 'I is a Buchsbaum ideal. Then, since Xa is not a zero-divisor modulo 'I, ('I, xa) is also a Buchsbaum ideal (see Corollary 1.1.11). Let U(q, xa) denote the primary component of ('I, xa) belonging to (xo, Xl> xa) = Rad(q, X3)' By the special form of H we obtain that U(q, Xa) = (xg, Xl' Xa). By Proposition 1.1.10 we must then have ('I, xa).

(Xo, ... , Xr ) • U(q, Xa)

In particular we obtain that X I X 2 E ('I, Xa) = (p .., H, xa)' Hence simple degree considerations (and the special form of H) give that deg H 2 and that H <X • xoXa - {l. X I X 2• Also, we must have xi E ('I, xa) and this shows that n:::;; 2 since ('I, Xa) = (p", H, xa)' This completes the proof of (ii). Proolol (iii): It follows immediately from the assumptions that 'I

(x~, x~,

XoXI,

txXoXa -

{lX 1X2)'

The assertion (iii) now results since the subvariety defined by 'I is linked to two skew lines given by the following equations: (<xxoXa -

{lX l X 2, XoXl)

=

(xo,

X2)

n

(Xl>

x 3 ) n q.

Assertion (i) and Bezout's Theorem provide this equality. Hence Theorem 1.2 completes the proof of (iii) .•

184

III. On liaison among curves in projective three space

, Proof of (iv): This is clear from (iii) since in a local Buchsbaum ring every localization at a non-maximal prime ideal must be Cohen-Macaulay, see Corollary 1.1.11. This concludes the proof of the theorem.

We now apply Theorem 3.2 to the study of a special class of reducible curves lying on a non-singular quadratic surface in P8. These observations also yield the above-mentioned applications to set-theoretic intersections of algebraic subvarieties in n-space (see, for example, Kunz's book [1] or the report in Stiickrad and Vogel [6]) . . Let K be an algebraically closed field of arbitrary characteristic. I...et V be an algebraic subvariety in n-space; that is V is a subvariety of affine space Alc or projective space Plc. Then we have the following classical problem which is a major open problem in algebraic geometry today: What is the smallest number s of equations defining V1 We denote byara V this number s. Looking at it from the point of view of commutative algebra we then get the following fact: The radical of the defining ideall(V) of V is the radical of an ideal generated by ara V (homogeneous) polynomials. Let d be the dimension of the algebraic subvariety V in n-space. Then we have always the following bounds:

n- d

ara V
We also say that V is a set-theore#c intersection of ara V hypersurfaces. If n - d = am V then V is called a set-theoretic complete intersection. We recall that V is called an (idealtheoretic) complete intersection if I(V) is generated by n - d polynomials. Nowadays, ~e have the following major question in this area: Classical problem. Is every (connected) curve in 3·space the set·theoretic intersection of two hypersurfaces1 There are some recent results in Al which pertain to this problem. Nothing of real interest is known about general results on set-theoretic intersections of curves in Pl where K is a field of characteristic zero. Considering the cone over a curve in Plleads R. Hartshorne [3], p. 126, to the following conjecture in local algebra: , Let A be a regular local ring containing a field of characteristic zero. Let B be a local ring which is a finitely generated flat A-module. Then the reduced local ring Bred of B is also a flat A-module. One important special case of this conjecture is the following question: Let 0 be a curve in P~ over a field K of characteristic zero. Is then 0 arithmetically CohenMacaulay if 0 is a set-theoretic complete intersection? As it turns out this is not the case, however. The first counter-example was given by C. Peskine and L. Szpiro in [3]. Their curve is not irreducible but arithmetically Buchsbaum (see our Remark 3.5, below). As R. Hartshorne pointed out to us there are irreducible curves of genus 5 and degree 8 in P~ which are set-theoretic complete intersection and also not arithmetically Cohen-Macaulay. We note that these curves are again arithmetically Buchsbaum because their liaison invariants are given by £2 (see Rao [2]). Motivated by the preceding we investigate the following stronger formulation of I Hartshorne's question: Let 0 be a curve in P~ over a field K of characteristic O. Is 0 \ then arithmetically Buchsbaum if 0 is a set-theoretic complete intersection?

§ 3. On curves linked to lines in p3 and applications

185

However, this also is not true. The assertion results immediately from the following proposition (see our Remark 3.4) . • Proposition 3.3. Let C = Cm •• be a reducible curve lying .on a non8ingular quadric Q 1,n pa, where C m•• is of the form

Cm.•

(Ll

U ...

u Lm) u (L~ u ...

U

L~)

where the Lb 1 mare line8 in one rulinfl of Q and the £j, 1 1 11, are line8 in the other ruling of Q. AS8Ume m n. Then (a) C is always a set-theoretic complete intersection, and a complete inter8ection when n m. (b) C ill arithmetically Cohen-Macaulay if and only if n m 1. (c) C is arithmetically Buchsbaum if and only if n - m 2. Note the following basic facts of the ruled surface Q in pa: First, Q is equal to the Segre embedding of pl X pl in pa, for a suitable choice of coordinates. Therefore Q contains two families of lines {L;l, {Li} , each parametrized by t E pl, with the properties: if L t =1= Lit, then L t n LfI. 0; if L; =1= L~, then L; n L~ 0, and for alit, u L t n L; = one point; that. is, the quadric surface in pa is a nded surface in two different ways. m, 1::;; i < n) Proof: (a) Let IIi.; denote the plane containing the lines Li and Lj (I Then, it is easy to check that the curve C = Cm•• is the set-theoretic complete intersection of the quadric Q and the reducible surface, i'ay F, of degree n given by:

F

Ill • I Ilz.2 ..... Ilm•.,.lIm.(m+l) ...•. Ilm.(n-m)'

By Bezout's Theorem, when m = n, C;".m is a complete intersection of F and Q. (b) We first show that if C is arithmetically Cohen-Macaulay then n - m::;; 1. In fact, if n m h 2, Cm." is linked to the union of h skew lines lying on Q. This follows immediately from assertion (a) since C •.• is a complete intersection. But the union of h skew lines on Q is linked to a multiple line lying on Q whose defining ideal is preci- ' sely the primary ideal q = (pll, Q) we studied in Theorem 3.2. This assertion results 2 it follows from immediately from the proof of (a) and Theorem 3.2(i). Since h theorem 3.2(ii) and (iii) that q is not a Cohen-Macaulay ideal. Since we have the liaison of Cm." and the curve with defining ideal q the conclusion follows from Remark 1.5(i). 'Ve now assume that n - m 1. If n = m then C•.• is a complete intersection by (a) and therefore is clearly arithmetically CohEn-Macaulay. If n m 1 we obtain from (a) that the ideal of the configuration is linked to the ideal (XO, Xl) and thus again is Cohen-Macaulay by liaison (see Remark 1.5(i)). (c) This is proven exactly as (b) by again using Theorem 3.2 and of course Theorem 1.2. This concludes the proof of Proposition 3.3, q.e.d. Remark 3.4. Note that the configurations of lines in Proposition 3.3 for which n - m > i give an infinite family of counter-examples to the above-mentioned conjecture of Hart~ shorne. In particular when n - m > 2, none of these examples is arithmetically Buchsbaum. The simplest example m = 1, n = 3 was examined at the beginning of this l section. The equations follow immediately from the proof of Proposition 3.3.

186

III. On liaison among curves in projective three space

Remark 3.S. We now discuss the previously mentioned counter· example given by Peskine and Szpiro [3J, 1.7: They exhibited a connected curve of degree 6 in Pk which is a set-theoretic complete intersection and not arithmetically Cohen-Macaulay. We will show that this curve is arithmetically Buchsbaum. More precisely, their example is the union of the irreducible quintic, say 0, given parametrically by (S5, trit, st', t5 ) and the line, say L, defined by (xo, xd. Thifl quintic has the following prime ideal, say,,:

Notice that the sextic 0 u L lies on the qnadric surface Q = XoXs - XIX2 = O. It is linked to a multiple line lying on Q. The liaison is given by the following equation:

Bezout's theorem and Theorem 3.2(i) imply this equality. Therefore we can apply our observations. By Theorem 3.2(i) and (iii), (x~, xi, ~X3' X 1X 2 ) is an (X2, .rs)-primary ideal of length 2 and therefore is a Buchsbaum nonCohen-Macaulay ideal. Hence, by Theorem 1.2, the sextic 0 u L is not arithmetically Cohen-Macaulay but arithmetically Buchsbaum. By using another liaison we will see that 0 u L is a set-theoretic complete intersection. The point here is that the quintic 0 is linked to anotlaer multiple line lying again on Q. The liaison is defined by the following equations: XoXs -

(XoX3

XI X 2,

xf - X~X2)

= "

n ((xo, x 1)S,

XoXs -

X1X2)'

Bezout's theorem and Theorem 3.2(i) establish this equality. Therefore the property that the sextic 0 u L is a set-theoretic complete intersection follows immediately.

, \

Another application of our observations about multiple lines lying on a quadric surface in p3 is given in the following section on self-linked curves. We want to conclude this section with some work done by Migliore in [1 J and [2J. Let L 1 , ••• , L", be a set of mutually skew lines in ps and let L;, .. " L~ be another such set (we may assume m n). We assume also that Li meets Lj in a point, for all i, j. We shall call the configuration in p3 formed by the union of these lines a Om.,,-configuration. Note that if m 3 then Om, .. lies on a smooth quadric surface. Hence for most choices of m and n our Proposition 3.3 tells us when the curve Om." is arithmetically Cohen-Macaulay and when it is arithmetically Buchsbaum, leaving open only the case O2 .". Since the Buchsbaum property of a space curve is reflected in the Hartshorne-Rao module in a simple way (see Corollary 0.4.7), the new techniques of J. Migliore [1, 2J can be used to complete Proposition 3.3. First, the results of Proposition 3.3 and Geramita-Weibel [1] are summarized in the following theorem. T.heorem 3.6. (a) A 0 0•1 i8 arithmetically Oohen-Macaulay, a 0 0 •2 is arithmetically Buch8baum but not

Oohen-Macaulay, and a 0 0 ,,, is not arithmetically Buchsbaum lor n:?:: 3. (b) A 01.1 and a 01.2 are arithmetically Oohen-Macattlay, a 0 1.3 is arithmetically Buchsbaum but not Oohen-Macaulay, and a 0 1 ." is not arithmetically Bucksbaum lor n 4.

§ 3. On curves linked to lines in p3 and applications

187

(c) If a C m •n lies on a smooth q'}1adric surface then it is arithrru?-tically Cohen-Macaulay if

and only if n m S 1, and it is arithmetically Buchsbaum if and only if n 'In S 2. (d) A C2 • 3 and a Cu are arithrru?-tically Cohen-Macaulay if they do not lie on a quadri<; surface. Proof: Proposition 3.3 provides (c). Note that this proof is entirely different from the linear algebra proof given by Geramita-Weibel [1], § to. By using local cohomology it is not too difficult to show (a), (b) and (d). A more geometric approach to the content of (d) is given in the following proof: Consider a Cu-configuration not lying on a quadric surface. 1,et QI denote the unique nonsinglllar quadric surface containing L1> L2 and L~, L~, L~ and let Q2 denote the ' unique non-singular quadric surface containing L l , L2 and L~. L~ and L~. Select any point P E L~ such that P ~ L1> P ~ L2 and let Ll denote the line on Ql which meets L~ at P. Also, Jilick any point Q E L~ such that Q ~ L l , Q ~ L2 and let L2 denote the line on Q2 which meets L~ at Q. Clearly LI =f: L2 since we already have foul' lines (Ll' L 2, L~, L~) in the intersection of QI and Q2' Let Ill> respectively Il2' denote the tangent planes to Ql at P and to Q2 at Q. Clearly, III =f: Il2 since L~ ~ III and L4 ~ Tl2 and L~

n L~

O.

Claim. The C2 •r configuration above is linked to a Cohen-Macaulay curve T, by the cubic surfaces F = QIIl2 and G = QPl and therefore such a C2,4-configuratio~ is arithmetically Cohen-Macaulay. Proof: We have

and it remains to identify L with (Ill n Il2)' First observe that L meets LI and L2 and also meets L~ and L~, hence L is distinct from the other eight lines of (*). Let 6 = Ll U L2 U L. We have only two possibilities: (1) LI n L2 =f: 0, in which case 6 is a plane curve and hence Cohen-Macaulay; (2) Ll n L2 0, in which case 6 is still Cohen-Macaulay since L, Ll> L2 form a C1,2 which must lie on some non-singular quadric surface and'the result now follows from Proposition 3.3(b), q.e.d. Thus the Cohen-Macaulay and Buchsbaum properties of Cm.n-configurations are fully understood, expect for the case of C2.7. for n::::: 5 if the configuration does not lie on a quadric surface. Following Migliore [2] we now answer this question. 'l'heorem 3.7. Consider C2.,,-configurations not lytng on a quadric surface. (a) A C2,5 is ariihrru?-tically Buchsbaum but not Cohen-Macaulay.

(b) A C2,6 is ariihrru?-tically Buchsbaum but not Cohen-Macaulay provided it does not lie on a cubic 8'1trface. A C2,6 on a C~tbic surface is not arithrru?-tically Buchsbaum. (c) A C2 ." is not arithrnetically Buchsbaum for n::::: 7.

We want to describe the basic idea of Migliore's proof since this proof requires ,\' new technique: namely to examine the module structure of the Hartshorne-Rao module in some detail.

".

188

III. On liaison among curves in projective three space

Put S:= K[xo, Xl, X 2 , xa] for an algebraically dosed field K. Given a graded Smodule ~lf EB M" of finite length, the action of Sl HO(PS,O(I») between two nEZ

consecutive components (Le. tP,,: SI -+ Hom(M", ~?Jfn+l») gives rise to a degeneracy locus, which can be thought of as lying in PSI (P3)*. Namely, let V".r P{L E Sl I dim tP,,(L) r} and let V" V .... where 8 = max {r I V,,:r c: (PS)*) (or else V" 0). These loci are isomorphism invariants and are preserved under duals and shifts. Let C be a curve of ps. For the Hartshorne-Roo module these loci are related to the geometry of C itself. The main philosophy that emerges from Migliore [1] is that the degeneracy locus generally corresponds to those planes H in ps which meet C non-generically, either containing a component of C or having C n H impose an unusually small number of conditions on some plane curves on H. By applying these ideas Migliore [1] also derives some necessary conditions for M(C) to have components in negative degrees (see Lemma 3.8 below). The main point to be made here is that although the dimensions of the various components of M(C) are relatively easy to compute, the module structure requires much more careful study. The object of the following is to examine this structure in some detail: Let M = EB M" be a graded S-module o! finite length, and let SI HO(PS, 0(1»). "EZ

The S-module structure of M is given by the collection of vector space homomorphisms:

tP.: S} -;. HomK(M", .If'+l)' Since tPn is trivial if either ~lf" or ..tlfn-l-l is zero, assume that this is not the case for some choice of 11. If we choose bases for .M" and for M"+l' and if L = tXoXo + tXIX} + tX2X2' + tXaXa E Sl' then tP" can be viewed as a (dim M,,+}) X (dim M"l matrix An whose entries are linear polynomials in the lXi' Also, we must understand how the S-multiplication

tP,,: SI -;. HomK(M ,,(C), ~lf"+l(C») is induced for M

M(C). Let L E Sl' Then L gives a map of sheaves

3c(n)..::.!:.-'/< 3c(n

+ 1)

by usual multiplication, and this induces the homomorphism tP,,(L) on the first cohomology. Observe that the map xL is injective. A natural way to underRtand tP,,(L) is then to find the cokernel of the map xL and study the associated long exact cohomology sequence. We will use this idea by considering the following exact sequences: We have the exact sequence O -+ 3c(n)

xL --'/<

3 0 (n

+ 1) -;. 3 HLnCIHL(n + 1) -+ 0

since the cokernel of the map X L comes by restricting to the hyperplane H L defined by L = O. Taking cohomology, one gets

o -+HO(PS,

3c(n») ~ HO(PS, 3 0(n

-;.HO(HL' 3H~CIHL(n

+ 1»)

+ 1») -;.M,,(C)

.(LJ~ ,If''+l(C) -;. ...

§~. On curves linked to lines in p8 and applications

189

Now, recall that a curve 0 in pa is arithmetically Buchsbaum if and only if the Hartshorne-Rao module M(O) EEl HHPk, 3c(n») as a graded S-module is annihilated flEZ

by the irrelevant ideal (xo, Xl' X 2 , x 3 ) of S (see Corollary 0.40.7). Equivalently, 0 is arithmetically Buchsbaum if and only if the maps fP,.(L): Mn(O) --+ Mn+l(O)

are zero for all n E Z and all L E 81> thus our techniques are applicable to prove Theorem 3.7.

Applying these techniques we can draw some simple conclusions about, curves 0 whose Hartshorne-Roo modules M(O) are shifted sufficiently far to the left. Specifically we want to say something when M(O) has components in negative degree. We will state without proof the following striking lemma of Migliore, Proposition 2.11 in [1]: Lemma 3.S. Swppose n < -1. Then dim Mn(C)

< dim Mn+l(O).

In some sense this statement is as good as we can hope to obtain since there are examples where dim Mo(O) = dim MI(O). For example, let 0 be the disjoint union of 0 otherwise. a conic and a line. Then dim Mo(O) = dim M1(0) = 1 and dim Mn(O) An interesting generalization of this is, to let 0 be the disjoint union of a line and a 1 for 0 n t plane curve of degree t + 1. Then calculations show that dim Mn(O) and dim M .. (O) = 0 otherwise. Proof of Theorem 3.7. We shall make frequent use of the following exact sequence: 0->- 3 c(n) --+ Op.(n) --+ Oc(n) --+ O.

Furthermore, we will apply the following basic fact of our O2 • ,,-configuration : hO(Oc •. n(d») = (n

+ 2) (d + 1)

2n

for d> 1.

To see this, note that no more than two lines in the 02.n-configuration meet at any point, and that. the restriction of HO(Oc •..
100

III. On liaison among ourves in projeotive three spaoe

Proof of (a): Note first that dim M 1(C 2 • 5 ) 0, dim M 2 (C2 • 5 ) = 1, and diln M a(C 2 • 5 ) = hO( d 0,..(3») 2. We will show that dim M r (C2,S) = 0 for r > 3. First suppose that dim M a(C2 •s) 1. Then hO(dC",(3») 3. Note that even in the "worst" case, when L l , L 2 , L~, L~, L; and L~ all lie on a quadric surface Q, we still have hO(dC,,.(3») = 2. (Any cubic containing C2,5 is the union of Q with a plane containing L~.) Hence if hO(dc,.,(3») 3 we must have two cubics containing C2 ,5 and meeting properly. These link C2,5 to a curve Cf. By Lemma 2.5 we get dim M a(C2 •s) f dim llf_l(C ). But since dim Mo(C') 1, Lemma 3.8 shows that this is impossible. Thus dim Ma(Cz.s) 0. In order to show that dim M r (C2 • S ) for r 4 we link C2,5 to a curve C' via a cubic and a quartic surface and apply the same reasoning as above. Therefore M(C2.5) is non-zero in exactly one component, so C2 •5 must be Buchsbaum.

°

Proolol (b): If the C2,6 does not lie on a cubic surface then we have dim M1(Cul = 0, dim M 2(C2 • 6 ) = 2, dim M a(C 2 •6 ) = 0, and dim M 4 (CU ) = hO{d c,..(4») - 7. If there exist two quartic surfaces containing C2 •6 and meeting properly then they link C2 •6 to a curve C' with dim M r(C2,6) = dim .iW4_ r(C'). In particular dim M1(C') 0, and = dim Mo(C') = dim M_l(C') ..• (since deg C' = 8 and not every hyperplane meets C' in eight collinear points). Therefore M(C2,6) is non-zero in just one component and C2,6 is therefore Buchsbaum. If all quartics containing C2 • 6 have a common component then the condition hO(dc,,.(4») > 7 forces (,\.6 to have the property that eeven of the eight lines, say Lh L 2 , L~, "', L~, lie on a quadric surface Q. (Note that any quartic surface containing L~, ... , L~, must contain Q.l But then C 2 • 6 lies on a pencil of (reducible) cubic surfaces, contradicting our hypothesis and completing the first part of (b). For the second part of (b) we first consider the case just mentioned. Then dim M 2(C2 •6 ) 2, dim M a(C 2•6 ) = hO(dc,..(3») 2 and we have the above-mentioned exact sequence

°

For a general hyperplane H]" C2 •6 n HL consists of seven co-conical points and one additional point. Then hO(HL' dC"tnH ,,(3») = 2, so tP2 (L) cannot be the zero map. Therefore C2,6 is not Buchsbaum. The only remaining case is when 02.6 lies on a unique cubic surface S. We first claim that S must be irreducible. Clearly S cannot be the union of three planes. If S is the unio:i\ of a quadric Q and a plane H then Q must contain exactly five of the lines Lj. But then Q also contains Ll and L 2 , therefore as above H (and hence S) is not unique. Therefore S is irreducible. h°(:Jc,,,(3») 1 and by (*) we only need to show that there Now dim M a(C 2 •6 ) exists ap HL for which hO(Ih, dC",n llL(3») < 3. Equivalently, we will show that not every hyperplane section of C2,6 lies on a ,cQnic (since clearly the general hyperplane section does not have five collinear points). In Chapter 4 of Griffiths-Harris [1] th~re is a description of all cubic surfaces and the lines they contain. ]'rom it one can check that the only irreducible cubic surface 8 in pa which contains a C2,6 is the projection from p4 of a Steiner surface. Hence ,8 is to be counted twice along Ll (say) and it contains infinitely many lines, each meeting Ll and L 2 • Among these lines are L~, ... , L~, and no four of them can lie on the same quadric surface. Next let Ql be the quadric containing L~, L~ and L~ and let

§ 3. On curves linked to lincs in pa an applications

191

Q2 be the quadric containing L~, L~ and L~. Then Ql () Q2 consists of L 1, La and two other lines (or a double line). A general hyperplane HL meets L~, L~, L~, Ll and L2 in a unique conic, namely H L () Ql' Similarly, H L meets L~, L;', L~, Ll and L2 in a unique conic HL () Q2' Since QI Q2' HL () O2•6 cannot lie on a conic which finishes the proof of (b).

*'

Proal 01 (c): We have dim 1'\12(02 ... ) n - 4 and dim 1'\13(02 ... ) = 2n - 12 + hO(J o•.• (3»). By (*) we have to show that hO(HL' ,JO•.•nHL(3») < n 4 + hO(J o .,.(3») for some Ih. Some thought shows that this is true since n > 7, q.e.d. Remark 3.9. ,We know that the technique of Liaison Addition introduced in the second paragraph of this chapter is useful in constructing examples of Buchsbaum curves. For example, one can form a 02,6 not lying on a cubic surface by "adding" two pairs of skew lines. Similarly, one can construct a Buchsbaum configuration of lines 0 for which 1'\1(0) is one-dimensional in two cQnsecutive components. We conclude this parapraph with two examples by considering again multiple lines. First let it be &aid that a totally different approach to Theorem 3.2 for double lines can be found in Geramita-.Maroscia-Vogel [2]. Example 3.10. We would like to construct in this example a double line on a nonsingular cubic surface in p3 which is not arithmetically Buchsbaum. Consider G xoX~ - XIX~ and note that by Theorem 3.2 q = (x~, x~, XoXl' G) is (xo, xI)-primary of length 2 and is not arithmetically Buchsbaum. If we letF xr X~2 then F E q. If the characteristic of the base field is 3 then F + 0 describes a non-singular cubic surface and therefore q describes a double line on this surface which is not arithmetically Buchsbaum. One may also show that q is linked, by F and 0, to the monomial curve with generic point (8 7, 8 5t2, 8t 6, 1'). This curve has been shown not to be arithmetically Buchsbaum (see Theorem 1.20). Examples 3.11. In this example we show that Theorem 3.2(ii) does not extend to lines lying on a nonsingular quadric hypersurfaces in pit (n > 3) . .More precisely, consider the following ideal in K[ xo, Xl, ••• , X4]

Then a defines a curve in P4(K) and it is easy to check that a

.):I () q

where .):I is the defining prime ideal of the monomial curve, 0, (8 5, sat2 , 8 2ta, 8t4 , t5 ) and q is an {Xl' x2, xal-primary ideal of length Q = xi - xoXa + xi - X 2X 4 which is non-singular for char K arithmetically Cohen-.Macaulay since the defining prime ideal .):I of

*'

with generic point 3. Now a contains 2. Note that 0 is 0 is given by

It follows by liaison (see Remark 1.5(i)) that q is also arithmetically Cohen-Macaulay. Note that here we may only say that q describes "a" 3-fold line on the quadric hypersurface Q but that "the" 3-fold line on a nonsingular quadric surface in pa is not even arithmetically Buchsbaum.

j

192

Ill. On lia.ison among curves in projective three space

§ 4.

On self-linked curves in pi

Let 0 be a curve of P8. Following Roo [2J, Schwartau [1J and Geramita-Maroscia-Vogel [1] we wish to study the geometric situation where twice 0 is the intersection of two hypersurfaces, i.e., where there exist two hypersurfaces which meet exactly on 0 and which make contact with each other with multiplicity two, along 0 (see Corollary 4.9 below). We wiIllook at the phenomena algebraically using the techniques of liaison. o will be called self-linked or linked to itself jf we can find a complete intersection X containing 0 such that 0 is residual to itself in X, i.e.

where de is the ideal sheaf of O. The simplest non-trivial example of self-linkage is the case of the twisted cubic curve in pa, which, even though not a complete 'intersection, is set-theoretically a complete intersection by means of a quadric and a cubic which touch along O. Catanese [1] ipresents a method of Gallarati where, starting from this example, more complicat€d examples are' created by preserving self-linkage under liaison. This method thus creates many examp~es of self-linked curves which are arithmetically Cohen-Macaulay. The other commonly known example of self-linkage is the example of two Kummer surfaces touching along a smooth curve of degree 8 and genus 5. This example was also discovered by M. Noether [1] in 1883 and has been recently treated by W. Barth [1] in a study of the Mumford-Horrocks bundle on pt, It is a curve that is arithmetically Buchsbaum with Buchsbaum invariant 2. One question that arises immediately is to find those liaison equivalence classes of curves in p3 which contain a self-linked curve. By studying liaison we see that an immediate necessary condition for a curve o to be self-linked is that its liaison invariant be self-dual up to grading. Since the liaison invariant characterizes liaison classes, this is a restrictive condition on the liaison !)lass in order for it to contain a self-linked curve. As an example, the liaison class in ps of two skew lines, contains no self-linked curve in case the characteristic of the ground field is not equal two. Nowadays, it is still an open question to determine those liaison classes in p3 which contain a self-linked curve. Liaison addition from § 2 of this chapter provides one step toward the solution of this problem, in the form of the following theorem and its corollary. First we recall a definition. Definition 4.1. Let 0 be a curve in Pi.:, K an 'al~ebraically closed field. 0 is said to be self-linked if there is a complete intersection V containing 0 such that de

Ann{dc/dvl

where de is the ideal sheaf of O. Now, if 0 is defined by the ideal 1(0) in S = K[xo, Xl> X2, xa] and V is defined by the ideal (F, G) c S, where F and G are forms of 1(0), then it is easy to see that this definition is equivalent to (F, G):I(O)

1(0).

So, in particular, if 0 is self-linked then 1(0)2 $;; (F, G) $;; 1(0) and hence 0 is a settheoretic complete intersection. Of course, the converse is not true in general.

§ 4. On self-linked curves in p3

193

Theorem 4.2. Suppose two curves C, C' c: p~ are algebraically h'nked (see Definition 1.1) by /, /'. Then the curve C/ /,C (see Definition 2.6) is sell-linked by /2, /,2.

+

J

Proof: Since C and C' are algebraically linked by /, /' we get: a) /, /' E I(C) n I(C'), b) (/, /') is a complete intersection, c) (/,/,):I(C) = 1(C'), d) (/,/,):1(C') = 1(C). Now a) and b) guarantee that C/ + /,C' makes sense. This new curve has defining ideal /' ·I(C) + /. I(C') by Corollary 2.8. We are to show it is self-linked via the complete intersection /2, j'2. Since 12, /,2 E/'· I(C) + /. 1(C') we have to prove:

+ / . I(C')) =

(12, j'2): (I' . 1(C)

/'. 1(C)

+ / . 1(C').

We show this property in two steps: 1. ;:;;;): We must show that for all i, i E 1(C), i', l' E I(C') we obtain: (/' . i

+ /. i') (/' . i + /. f') E (12, /,2).

Sinee i· l' and i' . i are in (/, /') by either c) or d) the assertion follows immediately. 2. ~ : Suppose 'If' E (/2,/'2) : (I' . I(C) + /. 1(C')). We have to show 'If' E/'· 1(C) + /. I(C'). We have 'If' • /' ./ E (12, j'2). But then the relative primeness of /, /' implies: 'If'

·f E (12, /,2):/

Hence 'If' E (/, /,2):/, 'If'

(/,/'2).

(/, /'). Set

= r ./ + l! . /'

for some r, l! E S.

We finish by showing that r E 1(0') and l! E I(C). The fact that

r ./ + l! . /'

E (/,12) :(t' . 1(0)

+ /. I(C'))

implies that for all elements i E I(C), (r ./

+ l! . /') ./'. i

E (12, /,2).

Hence r ./. /' . i E (/2, /,2). The relative primeness of /, /' again provides: r . i E (/, f'). Therefore we obtain r E (/,/,}:I(C) 1(C') by c). Similary, by using d) above we can show l! E 1(0) as desired, q.e.d. Remark 4.3. It is not too difficult to show that this theorem has the following converse: I,et C, C' be two curves in p3K and (/, f') a complete intersection containing both C and C'. Assume that the curve C/ + !'C' is self-linked by 12, /,2 then 0 and C' are algebraieally linked by /, /'. Hence the liaison addition yields a characterization of the property that two curves are algebraically linked via self-linkage. We now concentrate on the promised corollary to Theorem 4.2. 0 or Corollary 4.4. Let E be a graded S-module 0/ /~:nt'te length. Then lor I' liai,<Jon class associated to the module E EB EV(2v) contains a sell-linked curve.

l'

<{' 0 the

Proof: By the theory of liaison realization (see Theorem 2.18), some twist E(ll) ean be realized as the liaison invariant E(C) of a curve C c: P~. Since E('rj) is again of finite length it is possible to construct an algebraic link of C and a new curve 0' via 13 Buchsbaum Rings

194

III. On liaisOn among curves in projective three space

any homogeneous S-sequence (/' f') ~ 1(C) where 0' is defined by the homogeneous ideal quotient (f, f') :1(0). The equivalent Definition 1.1 shows that 0 and 0' are algebraically linked. Then by Theorem 4.2, we know that 01 + 1'0' is a seH-linked curve in P1. But this curve has liaison invariant by Corollary 2.8 given by

EB E(O') (deg I) = E(rJ) (deg f') EB [E(rJ)" (deg 1+ deg I' + 4)] (deg f)

E(O) (deg f')

by the transformation law of Lemma 2.5 E(rJ

+ deg f') EB EO( -rJ + 2 deg I + deg I' + 4).

Now, this module determines the same liaison class as the module

EB EV(2 rleg 1 2rJ + 4) '1!:= deg I rJ + 2. E

:E

EB E"(2v)

where We now point out that in choosing the homogeneous S-sequence (I, f') . S ~ 1(0), the first element I may be taken as any non-zero homogeneous element of 1(0). In particular we may replace I by Xo • I, x~ • I, ~~ . I, ... etc. Thus we obtain self-linked curves as above for all v O. To obtain the desired curves for v 0 simply replaue E by E" in the argument above. We then obtain self-linked curves in the liaison class of E"EB E(2v) for v 0; but these are the same liaison classes as those of the k-duals E EB E"( -2v), which finishes the proof, q.e.d.

<

Remark 4.6. In characteristic 2 this result has been proven (independently) .by Rao [2] using some transversality theorems of S. Kleiman. :Following Schwartau [1] , advantages of our method are, that it applies to any characteristic, and that explicit bounds for v may be given as follows below, since we have produced specific bounds for v during the course of the proof. ]!'or "v ~ 0" it suffices that min{deg 1- rJ

v

+ 2 I E(r,) is realized by a curve on the surface II.

If the module E itseH can be realized as the liaison invariant of a curve in P1:, we may write a bound for v as follows: v

defl

+ 2,

where I is the least degree surface in P1: on which E may be realized. For "v 0" it is enough to have

<

v

-min{deg 1- rJ

+ 2 I E«rJ) is realized by a curve on the surface II.

If E" itself may be realized, we may write a bound as follows: v~

-(degl

+ 2),

where I is the least degree surface in P1: on which E" may be realized. We now show that a full converse to Corollary 4.4 is impossible. We will see that in characteristic 2 the liaison class of arithmetically Buchsbaum cur· ves with Buchsbaum invariant 1 (i.e. the liaison class of two skew lines in P1:) contains a self-linked curve. In fact, the following theorem gives the curve as (~, x~, XoXl' xoxa - X 1X 2 ) with Schwartau's liaison invariant K(4). At any rate, we 6.ave now demonstrated as

§ 4. On self·linked curves in p3

195

promised that Coronary 4.4 has no full converse. Hence we come back to our observations of § 3 of this chapter.

Theorem 4.6. Let C. denote the curve

~n

P=:C defined by the ideal

1J)here ~ = (xo, xd, Q = XoXa X 1X2 and n (i) If n > 2 then C" is not self·linked, (ii) C2 is self-lznked if and only if char K

2. Then; 2.

Proof: We shall give,the proof in three steps. Step 1: = (~2", ~"Q, Q2) is ~-primary for aU n? 2. To prove this we use an argument similar to the one we developed in the proof of Theorem 3.2 (see Claim 1 of that proof). More precisely, if is not primary then we can find a linear form L = (XX 2 (Jxa((X,{J E K, ((X,{J) (0,0)) such that a;:(L) ~ a•. Let H be a form in K[xo, Xl> X 2 , xa] such that H ~ and HL E a;. Now consider H as a polynomial in K([X2' X3]) [xo, xd and write H = HI + ... + H t where deg H j (in xo, Xl) is i. Since H ~ and ~2" ~ we may assume that t 2n -.: 1. Thus we have

a;

+

+

a;

a;

a;,

a;

and since deg L 0 (in xo, xIl, deg HjL i. Since a; is still a homogeneous ideal, when considered in (K[x 2 , xa]) [xo, Xl], it follows that HiL E a;, 1 ~ i ~ t. Hence we obtain: If i= I,H I LEa 2 =;,H I =0;

if 2

if n

<

i~

n, HjLfI E a!=;,Q2j Hi;

i

t, H;L E a! =;, Q i Hi and therefore Hi

=

QH'i

where deg H'; n (in xo, Xl) and therefore Hi E ~"Q. Thus, HE a;, which is a contradiction. Step 2: is a ~-primary ideal of length 3n, for all n > 1. It is enough to check that in the localization K[xo, ... , xaJ/) we have the following saturated chain of primary ideals:

a;

a;

(a;,

x~n-l)

c: (a~, X~·-l, x~n-2) c: ...

c: (a;, ... , x~, X~-lXl) (a~, ... , x~-I, x~-2xtl

(a;,

~.-l,

... , xZ)

(a;, ... , x~, X~-lXl' X~-l) (a;, ... , x8-1, X~-2Xl' x~-2)

c: (a;, ... , x:;-l, X8-2Xl> x8-2, x~-IXI)

(a;, ... , x8-2, X~-IXl' Xij-3) c: ...

c: (a;, ... , x~, XoXI) c: (a;, ... , x~,

of length precisely (n

+ 1) + 2(n -

1)

(a;, ... , x~)

XoXl>

+1=

x o) c: (a;, ... , x~,

XOXl>

xo, Xl),

3n.

We are now ready to prove the theorem. Suppose en is self-linked (n > 2). Then we must have two forms F, G in that (F, G): an = an and consequently a; ~ (F, G) a". 13*

an' such

196

III. On liaison among curves in projective three space

It is clear from Step 1 that o~ c (F, G). Let a = deg F, fl = deg G. Then from Step 2 we must have a • fl < 3n. We may as well assume a::;; fl and since (F, G) ~ On that a > 2, fl > 2. We distinguish two cases:

(i) n> 2: If a> 2 and a::;; fl we must have ex < nand fl < n. But then Q I F and Q I G which is a contradiction. If a = 2 then F = Q and therefore (F, G) is a ~-primary ideal of length 2fl containing Q. By Theorem 3.2 we obtain that (F, G) = (Q, G) = (~2P, Q) which is impossible since (~2P, Q) is never a complete intersection. (ii) n = 2: We first show that if char K =1= 2 then O2 is not self-linked. Suppose O2 is self-linked by two forms F, G. Then, in order to have o~ c (F, G) c (~2, Q) we must have deg F = deg G = 2. (This follows since, by Step 2, o~ has length 6 and we know (~2, Q) has length 2. We also know that neither O2 nor o~ is itself a complete intersection.) We distinguish two cases: (a) Either F or Gis nonsingular.

In this case, (assuming F is nonsingular), we have that (F, G) is a primary ideal of length 4 whose associated prime defines a line on the nonsingular quadric surface F = O. But from Theorem 3.2 we see that this ideal is not a complete intersection. Thus this case cannot occur. (b) Both F and G are singular. It is easy to see that the general quadric in O2, ax~

+ flxr + yXoxl + b(xox3 -

(a, fl, y, 15 E K),

XIX2)

is singular if and only if 15 = O. Thus both F and G have this form. Since both F and G cannot have a common factor we are reduced to consider two cases:

+ yxox, x~ + flxi,

(i) F = x~

G = xi

+ y'XOXI>

(ii) F = G = XOX I • In either case, we must have Q2 E (F, G). In case (i) this gives that

+ y'x~ + 2X2X3) E (F, G). yX5 + y'xi + 2X2X3 (char K =1= 2) ~ (xo, XI), xoxl(yx5

But

and thereforp XoXl E (F, G). Thus xi) in this case and hence 2XOXIX2X3 E (x~, xi) which is a contradiction. In case (ii) if we use the fact that Q2 E (F, G) we find that (x~x5 xixi) E (F, G) and therefore xi(flx5 x~) E (F, G). Since flx5 xi ~ (xo, XI) we obtain xi E (F, G) from which (F, G) = (x~, XOXI), a contradiction. Thus, if char K =1= 2, O2 is not self-linked. To see that O2 is self-linked in characteristic 2 it will suffice to establish the following: (F, G)

=

(x~,

+

Claim. (x~, xi): (~2, Q)

=

+

(~2, XOX3

+

+ XIX2) for any field K.

Proof: We have (x~, xn: (~2, Q)

=

[(x~, xi) :xoxd n [(x~. xi): Q]

= (xo, XI) n [(x~, xi): (XOX3 -- XIX2)] = (x~, xi): (XO.1:"3 -

XIX2).

§ 4. On self· linked curves in p3

+

197

+

Clearly (~2, XoXa ;(IX2) <;; (xg, x~): (xoxa '--- X1X2). Now (~2, x oXa X1X2) is an (xo, Xl)primary ideal of length 2 (see Theorem 3.2) and therefore if this inclusion of primary ideals were strict wc would have (X~, x~): (xoxa -

X l X2 )

= (xo, Xl)'

This is clearly false and the inclusion is an equality. This concludes the proof of Theorem 4.6. Remark 4.7. The statement (ii) in Theorem 4.6 was also proved by Rao [2] (Example 10) and by Schwartau [1] (Corollary 7.7). The extra information we obtained in Theorem 3.2 allows our method of proof to proceed in a more direct fashi!Jn. Rao [2] shows that from (ii) of Theorem 4.6 it follows that the liaison class of arithmetical Buchsbaum curves with invariant 1 contains no self-linked curves, when char K =1= 2. Therefore we want to look at an example from this liaison class which shows that self-linkage' is not preserved under liaison, when char K = 2. Example 4.8. The curve O2 is linked to the rational nonsingular quartic curve in p~ for arbitrary characteristic of K. The liaison in this case is given by the following surfaces:

since the rational nonsingular quartic is given parametrically by (s4, sat, sta, t4 ).

NQw, this quartic is not self-linked in any characteristic. If it were self-linked, it would be a set-theoretic complete intersection, one of the surfaces being the (unique) quadric containing it by Corollary 4.9 below. But it is well-known that the quartic is never a set-theoretic complete intersection of thli quadric with any surface which contains the quartic. This fact is true in any characteristic (see, e.g., Roloff-Sttickrad [1], Lemma 1). Hence this quartic is an arithmetically Buchsbaum curve which shows us that when char K = 2 self-linkage is not preserved under liaison. It is possible to give more general example of this phenomenon which show that self-linkage is not preserved under liaison in any characteristic (see Geramita-MarosciaVogel [1], Example 2.4). We have used the following corollary: Corollary 4.9. Let 0 be any t'rreducible curve in pa. Assume that 0 is self-linked by surfaces F and G. Then deg F . deg G

=

tu'O

2 . deg 0

.

~

Proof: Let ~ be the defining prime ideal of O. If S = K[xo, ... , xa], then SiJ is a twodimensional regular local ring and therefore ~2. S is never a complete intersection. Since 0 is self-linked by F and G we obtain: ~2 ~

(F, G) c ~.

Since ~2. SiJ =1= (F, G) . SiJ and ~2. SiJ is a ~. SiJ-primary ideal of length 3, we get that (F, G) is a ~-primary ideal of length 2; that is, deg F . deg G = deg V = 2 . deg 0, where V is the curve with defining ideal (F, G), q.e.d.

198

III. On liaison among curves in projective three space

We conclude this chapter with the following remark: Remark 4.10. In 1983, Juan C. Migliore [1J has given a elassification of sets of skew lines up to liaison. It is shown that if 0 and 0' consist of t:::::: 3 and i' skew lines, respectively, then their linkage properties depend primarily on whether or not they lie on a smooth quadric surface. If 0 lies on a quadric surface then 0 is linked to 0' if and only if t = t' and 0' also lies on the same quadric; that is, it is shown how the quadric fully determines which sets of skew lines are in the liaison class of O. If 0 does not lie on a quadric then it is not evenly linked to any other set of skew lines. If 0 is in general position then it is not oddly linked to any set of skew lines, but in special cases it can be oddly linked to at most one other set. Hence it is shown that if 0 does not lie on a quadric surface then it is essentially unique with respect to liaison. Furthermore, there is a classification of "double lines" up to liaison. More precisely, given two schemes 0 and 0' of degree 2, each supported on a line, a necessary and sufficient set of conditions is given for 0' to be in the liaison class of 0 and a description how they can be linked. The reader is referred to Migliore's thesis for more details (see also Migliore [2J, [3]).

·

-

Chapter IV Bees modules and associated graded modules of a Buchsbaum module

Hironaka [l], in his paper on desingularization of algebraic varieties over a field of characteristic 0, in order to deal with singular points develops the algebraic apparatus of the associated graded ring and introduces the theory of normal flatness. Such an approach necessitates a deep investigation of blowing-up and monoidal transforma00

tions. Let R be a Noetherian ring, q c R an ideal. type. !ence Proj Proj (EB qn)

C~Oqn)

is a finite type

EB qn is a graded R-algebra of finite

proje~ti~e

(Spec R)-scheme. We will set

= BlqR = the blowing-up of R with center q. We recall the terminology that

n-O

BIqR is called a quadratic transformation of Spec R if q is a maximal ideal in R. If q is a prime ideal, 'BlqR is called a monoidal transformation of Spec R. Let f be the projection BlqR -+ Spec R. The 'scheme-theoretic inverse image f-l(Spec R(q) in BlqR (i.e., the fibre above the subscheme defined by q) is isomorphic as R(q-scheme to Proj ( EB q"(qn+l). Then the n~O

result on the resolution of singularities of algebraic varieties can be stated as follows: Let K be a field of characteristic O. If X is an algebraic K-scheme, say reduced and irreducible, then there exists an algebraic subscheme D of X such that (i) the set of points of D is exactly the singular locus of X, and (ii) if (/: X -+ X is the monoidal transformation of X with center D, then singular.

X

is non-

In Hironaka's proof of this statement, the notion of normal flatness plays an important role. We say that X is normally flat along D if the stalk of grb(X) at every point of D is a free OD ... -module for all non-negative integers p. This chapter now has its origin in an effort to extend Hironaka's observations to a general situation due to G. Faltings [1] and M. Brodmann [1]: In an abstract manner of speaking, consider an affine scheme (X, Ox) and a point p E X where p is an isolated singularity. Let X be pure-dimensional of dimension d. Take d elements fl' ... , fa E Ox(X) defining p set-theoretically, that is, VUl"'" fa) = {pl. Then we want to investigate the blowing-up Blq(X) of X with center Spec 0 x(X)(q, where q := Ul' ... , fa). That is, we do not investigate the blowing-up BI(p)(X) at p but we consider the blowing-up at an "infinitesimal neighbourhood", say p, of p. The following two facts are well-known (see M. Brodmann [1]): (i) Blq(X) = Proj ( EB qn), where n;;,O

EB qn is the Rees algebra with respect to q.

,,;;'0

200

IV. Rees modules and associated graded modules

(ii) There is the following commutative diagram; Bl(p)(X)

Proj ( EB

.,.;;,0

t'P

Bl(p)(X) = Proj ( EB ,,;;'0

m;) ~ X

tid q,,)

~X

Assume now that p is "large enough". Then by results from Faltings [11, Bl(p)(X) is Cohen-Macaulay. This observation leads to the theory on the l}facaulayfication of X; that is, we only want to obtain from Hironaka's desingularization process that X is (locally) Cohen-Macaulay. The first global statements were obtained by G. Faltings [1]. Using Brodmann's [1] arithmetical approach we have to investigate the affine cone over the blowing-up BIq(X), that is the Rees algebra EB qll, and the conormal cone n:?O

defined by the form ring (associated graded ring)

EB qlljqll+l.

Therefore, taking this

ll~O

notion locally, we can work throughout in the theory of local rings. Such an approach involves a deep study of Rees rings and form rings with respect to an ideal which is not the maximal ideal of a local ring. Hence our main object in the following is the study of the stability of the Rees rings and form rings of parameter ideals with respect to the Buchsbaum property (see, for example, Theorem 3.3). These investigations and the main results of this chapter include and improve the observations of M. Brodmann, S. Goto, Y. Shimoda and N. V. Trung. Now, let A denote a local ring with maximal ideal m. In the sequel we will use the following notation: ' Let q be an ideal of A. We set

EB qll'j'll C

A[ T},

where '1' is an indeterminate.

nc.O

Clearly, Rq(A) is a graded A-algebra (containing A and contained in A[T)) offinite type over A, i.e. Rq(A) is Noetherian and [Rq(A)],. qllTII for all n>- 0 and [Rq(A)]1I = 0 for n < O. (Note that for a graded ring R and a graded R-module M we denote by [M]" the Abelian group of homogeneous elements of degree n in M. In our situation [M]" is even an [R}o-module, i.e. an A-module.) Rq(A) possesses only one homogeneous maximal ideal, which is the ideal

mEB

qllTII.

Furthermore, we put Gq(A) := Rq(A )jqRq(A) which is again a Noetherian graded A-algebra (with induced grading). Rq(A) is called the Ree8 algebra and Gq(A) is said to be the a880ciated graded algebra (or form ring) of A with respect to q. Let M now be an A-module. There is no difficulty to define in a similar way as above the Rees module Rq(M) and the associated graded module Gq(M) of M with respect to q (see § 1). Clearly, Rq(M) is a graded Rq(A)-module and Gq(M) is a graded Gq(A)module. If M is a Noetherian A-module then it is easy to see that Rq(M) and Gq(M) are Noetherian modules. We also will make use of the following definitions in the sequel: We say that Rq(M) or Gq(M) is a Cohen-Macaulay (Buchsbaum) module if Rq(M)ilJl or Gq(M)ilJl has this property. :Furthermore, we say that Rq(M) or Gq(M) is a locally

§ 1. Some preliminary results

201

Oohen-Macaulay module if Rq{M}(\ll) or Gq(M}(,lll are Cohen-Macaulay modules for all homogeneous primes \13 =F roc of Rq(A) or Gq(A).

The aim of this chapter is to discuss the relationship between the Buchsbaum property of a Noetherian A-module M and the Buchsbaum ((,.-ohen-Macaulay, locally CohenMacaulay) property of Rq(.M) and Gq(M). Since this is possible in a very satisfactory way for all parameter ideals q of M (see Theorem 2.1, Theorem 2.10, Theorem 3.2 and, above all, Theorem 3.3) we restrict ourselves in § 2 and § 3 to this case.

§ 1.

Some preliminary results

Here in the first paragraph we collect (and prove) some basic results concerning Rees rings and Rees modules and associated graded rings and modules. Let M always denote a Noetherian A-module although many of the following facts remain true if M is not Noetherian. Let q A be an ideaL We set

EB q"MT" c

('I' a:{l indeterminate)

M[T]

,,:'0

and

Gq(1lf) := Rq(M)jqRq(M) ~

EB q"Mjq!l+lM. ,,:'0

Both, Rq(M) and Gq(M) are modules over Rq(A). (Clearly, Gq(M) is a Gq(A)-module, but we always consider it as an Rq(A)-module using the natural epimorphism Rq(A) -+ Gq(A).)

Rq(1lf) is called the Rees module and Gq(M) the associated graded module of llf with ' respect to q. Obviously, Rq(M) and Gq(M) are graded Rq(A)-modnles (Noetherian if M is Noetherian) with 0 for n < 0, [Rq(M)]" = { qIMT" for n O. Identifying for simplicity the modules [Gq(M)],.

0 { qfJMjqfJ+IM

qIlMT"/q"~lMT!I+1

for n for n

and q"M/q,,+lM we have

< 0, O.

If IJI c Rq(A) is a (not necessary homogeneous) ideal then we set for each i EN;

[1JI]j:= (forms of degree iin IJI)

=

IJI n [Rq(A)],

IJI n qiTi,

It is clear that IJIh := EB [IJIJj is a homogeneous ideal of Rq(A}. It is the biggest homo;«0

geneons ideal contained in IJI. If IJI is a prime ideal then IJIh is prime as well and [IJIJo [lJIh]o is a prime- ideal in A. (If \13 E Proj Rq(A) = Blq(A) and if I: B~ A -+ Spec A is the projection mentioned above then 1(\13) = [\13]0') Conversely, if ~ is a prime ideal in A, then ~ EB (~ n qi) Ti and ~* ;= ~EB EB qiTi j~o

i>O

are homogeneous prime ideals in Rq(A}. As in the local case we denote by SnpPRqIAIRq(M) or, if no confusion is possible by Snpp Rq(M), the set 01 all primes \13 01 Rq(A) with Rq(M)$ =+= 0 and by Ass Rq!A)Rq(1lf) the set 01 all associated pnmes 01 Rq(M}. Since Rq(M) is a graded Rq(A )-modnle, Ass Rq(M) consists only of homogeneous primes (see Matsumura [1], Prop. (lOB), p. 62). Also all

202

IV. Rees modules and assooiated graded modules

minimal elements of Supp Rq{M) belong to Ass Rq{M). Clearly the same is true for Gq{M). Our first aim is to calculate Supp Rq(M) and Supp Gq(M). First we note that for every homogcneo~s prime ideal $ of Rq(A) with [$ h qJ' we get by Lemma 1.2.27: Rq{M)'l3~

R q(M)i'J3}

(for the notation see the remarks made in conjunction with I..emma 1.2.26). Similar to Lemma 1.2.27 we can prove the following Lemma 1.1. Let.\) E Spec A '" V{q). Then (i) Rq(A).,o A[XJc ~A.,[X]e· (X a variable), where e := .\)A[XJ + X . A(X] and e' := .\)A.,[X]

+X

. A.,[X].

(ii) For any A-module M we have

Rq{M).,o ~ M[XJe:::::= M.,[XJc" Prool: It is sufficient to prove (i) since the !lame a~guments work for (ii). The isomorphism A[XJe:::::= A.,[XJc· is obvious. We construct an isomorphism g: A[X]e -+ Rq(A).,o in the ,same way as in the proof of Lemma 1.2.27. We choose an element x E q "'.\) and define g(X) := x· T. We leave the remainder of the proof to the reader, q.e.d.

Corollary 1.2. Rq(M).,o i8 a Cohen-Macaulay module il and only il M., is a Cohen-Macaulay module. We note that this is not true for the Buchsbaum property. Let $ c Rq(A) be a prime ideal. It is easy to see that Rq(M)\U =1= 0 (or R q(M)('J3) =1= 0 if $ is homogeneous) implies [$]0 E Supp M. H qT ~ $ then $ is homogeneous and has the form $ .\)* with.\) := [$Jo. Then R q{M)('J3) ~ M~ =1= 0 if .\.1 E Supp M. The same is true for Gq(M): H Gq(M)'J3 =1= 0 (or Gq{M)('J3) =1= 0 if $ is homogeneous) then [$]0 E Supp M n V(q). H $ = .\)*, .\.1 E Supp M n' V(q) we obtain Gq(M)'J3:::::= (MlqM)~ =1= O. Collecting all these facts we get for every homogeneous prime ideal $ c Rq(A): R q(M)'J3 =1= 0

if and only if Rq(M)($) =1= 0,

Gq(M)$ =1= 0

if and only if Gq(M)(\U) =1= O.

Furthermore, if $ is a homogeneous prime ideal of Rq(A) with $ tt {.\)* ; .\) E V(q)}, Lemma 1.2.27, 1.2.26 and Lemma 1.1 show that Rq(M)(\Ul (Gq(M)($)) is Cohen-Macaulay if and only if R q(M)'J3 (Gq(M)$) is Cohen-Macaulay. We prove some further lemmata: Lemma 1.3. Let $ c Rq(A) be a homogene()'IM prime ideal. (i) 11 [$]; ~ qiTi lor some i';2 1 then we have lor x E qi: 11 xTi E $ then x E $. The converse is true il q [$]0' (ii) AS8ume that [$]0 E Supp M and [$Jl ~ qT. Then we have lor every p E Ill: (Rq(M) (p))('J3) = (qPRq(M))('J3) = xPRq(M)($)

with xT tt [$h.

lor all x E q

§ 1. Some preliminary results

203

Every such x is a non-zero d£v1.8or with respecl to Rq(M}(~) and we have an exact seque~

o -+ Rq(~lf}(lfll -=-.. Rq(M)( 'Ill -+ Gq(~lf}('Il) -+ O. Proof: (i) Let xT' E ~. We choose an element y E qi with yT' ~ [~]i' Then x(yT'} = y(xT}' E ~, hence x E ~. 0. Choose y E qi"[~]o. Let q <;;;; [~]o and assume x E ~. Then q'" [~]o 9= ~Hor all Then y ~ ~ and (xT') y x(yT'} E ~, hence xT' E ~. (ii) Let pEN. Since in (Rq(M) (P})!\lll only elements of degree ooccur as numerator, we see that (Rq(Ml (p})(\ll) (qPRq(M})(\lll'

yTp Let x E q, xT ~ [~h. Then we have for every y E qP; Y x P • -'- in R(A }('Il)' TherexPTP fore (qPRq(M))(\lll = xPRq(M)(\ll)' Now a straightforward calculation shows that x is a non-zero divisor of Rq(M}(jlll' If we localize the exact sequence

o -+ qRq(M} -+ Rq(M) -+ Gq(M) -+ 0 with ~ and use (qRq(M) )(\ll) q.e.d. -Corollary 104. Let ~ 1:>* (t'.e, [~]i

~

xRq(M)(\ll) '" Rq(M}('Il) we get the desired exact sequence,

be as ~n Lemma 1.3 but aS8ume that 1:> := [~)o ;;t2 q. Then either qlTI for all £> 1) or ~ = ~ (£.e. [~]i (1:> n ql) Ti for all i> 0).

=

Proof: If [~]i qiT' for some i 1 then (qT)' = qiT' ~,hence qT ~ and therefore [~]i qiT' for alIt'> 1. If [~]i ~ qiTI for all i 1 then we get by Lemma 1.3: If x E 1:> n ql then x E ~ and therefore xTI E [~]i' If xT' E [~]i then x E ~ and x E qi, i.e. x E 1:> n q', q.e.d.

Corollary 1.5. Let ~ Rq(A) be a prime ideal of the form ~ ~ or ~ = 1:>* with 1:> E Spec A. Let N be an A -module and a8sume that 91 1.8 a graded Rq(A }-module which 1.8 a (graded) submodule of N[T) with (91)0 = N. Then 91(\ll) '"'-' N~. Proof: We consider the natural homomorphism (of A~-modules)

f; N~ -+ 91(\ll) = [91]0 and all pEA" 1:> = [Rq(A)]o " [~]o. P Let pT' E 91(\ll) with v E [91]; <;;;; NT', pT' E [Rq(A)]i "[~]i' Then v = nT' with

which is defined by f

v

.

(!!:..) = !!:.. for all n E N p

n E Nand pEA" 1:> by our assumption (if ~ if

~= If

1:>* then i

!!:.. E Ker / p

0), Hence

p

p E q' " (1:> n qi) C A "1:>,

is surjective.

(n E N, pEA" 1:>) then there is an i E N and a qT' E Rq(A) "

(q E qil with qTin = 0, i.e. qn

this implies !!:..

~ f (!!:..) and / pT< p

= ~ then

0, i.e. Ker /

= 0. But by the same argument as before = 0 and / is an isomorphism, q .e.d.

~

q E A "1:> and

Lemma 1.6. Let ~, :0. c Rq(A) be homogeneous prime ideals with ~ :0. and eitkef' [:O.h ~ qT or [~h qT. Then there is a prime ideal bE Spec Rq(Al£]. with Rq(A)(\lll '" (Rq(A}(£].»)u and there/ore Rq(M}('Il) '::::: (Rq(M}(£].»)\).

204

IV. Rees modules and associated graded modules

Proof: If [\13h = qT then [Qh = qT and \13 = pi, Q P: with Pi> P2 E Spec A. Then Rq(A )(\)3) A p" Rq(A )(0) = A p, by Corollary 1.5 and our statement follows by a standard argument of local algebra. Let [ell ~ qT. Choose x E q with x'l' ~ O. Then xT ~ >,\5. Let e denote the maximal ideal of R q(A)(\)3). Then the natural homomorphism f: Rq(A )(,c) --+ Rq(A )(Ill) induces a homomorphism

g: (Rq(A )(C))/-'(e) --+ R q(A)(\)3). aT' Let -;;;: E Rq(A)(\)3) (a, p E qt, pTj ~ ~.

=g

\13).

Then

pTj

-.~. ~

E Rq(A )(0)

"'"

fl(e) and

aT' ~

(a,!,' (~Ti )-1), i.e. g is surjective. x''l'' x'T'

Let

iX

E Ker g. We may assume without loss of generality that ex E Rq{A )(D)' Write

aT- . h . T' "C"'I S· aTj O·ill R q(A )(\)3)t herelsalE . . N and apEql,. iX - . Wit a,qEq',q • \flY. illce-. qT' qT'. pTJ ~ \13 with apTi+1 = pTiaTi = 0, i.e. ap = O. But l~Tl. E Rq(A)(O) "'" rl(e) and . ~~ pTJ aT' -~ . 0, i.e. iX = O. Hence g is injective and therefore an isomorphism, q.e.d. xiT' pT'

Lemma 1.7. Let \13 Rq(A) be a (not necessarily homogeneOUB) prime ideal. Then 0 if aM only if there i8 apE Supp M with P <;;;;; \13.

'*'

'*'

RlLl1~),¥

*'

Proof: Assume P g::: \13 for all P E Supp M. We may assume that M 0, i.e. Supp ill 0. Let {PI' ..., P.,} denote the set of minimal elements of Supp Jl:l. Then there are lil E Pi "'" \l3 for ~. = 1, ... , s and, consequently, Iii' .•.• lis E (PI n ... n P8) "'" \13. Since for all i = 1, ... , s the coefficients of 'YJi (considered as a polynomial in T) lie in Pi> all coefficients of lit ..•.. lI. lie in lJl n ... n P•. Therefore there is a hEN such that all cDefficients of rl ('YJl' ••• '))8)" lie in Ann 1'1:1. Hence 'YJ • Rq(M) 0, i.e. Rq(M)\j3 = 0 since 1] ~ \13. Assume there is a lJ E Supp"Y with P <;;;;; \13. Then there is abE Spec Rq(A)\)3 with (Rq(M)\j3)u ~ Rq(M)p ~ Rq(~l:l)~ :::.:: Jl:l;

'*' 0

(see Lemma 1.2.27 and Corollary i.5). Hence Rq(M)'Il

Corollary I.S. Let qC

\~

'*' 0, q.e.d.

be as before. Then Gq(M)\j3 =1= 0 if aM only z/ RQ(Jl:l)\j3 =1= 0 and

[\13]0'

'

Proof: The "only-if-part" follows by the remarks made above. Assume that Rq(M)'Il 0 and [\13]0 E V(q). Then there is a lJ E Supp M with P \13. If we replace \13 by \13h, the biggest homogeneous ideal contained in \13, then p <;;;;; \l3h' Therefore Rq(M)\j3. 0 and [\13h]O [\13]0 E V(q). If Gq(M)'Il. 0 then Gq(M)\)l 0 since there is abE Spec Rq(A)\Il with (Gq(ll:l)'Il)U:::':: Gq(~l:l)'Il •. Therefore we may assume without loss of generality that \13 is homogeneous and it is now sufficient to show that Gq(M)('Il) O. If ~ [\13]; then Gq (1l:l)('ll) ~ (MjqM)['llJ. oF 0 since [\13]0 E V(q) n Supp Jl:l Supp ll:ljqM. If \13 [\13]t then [\13]1 ~ qT. Now take the exact sequence of Lemma 1.3(ii). The element x used there belongs to q c [1.l3]0, hence x is not a unit in Rq(A)('ll)' Therefore Gq(ll:l)(\)3) :::.:: Rq(M)('ll)/xRq(Jl:l)(\)3) 0, q.e.d.

'*'

'*'

'*'

'*'

'*'

'*'

'*'

§ 1. Some preliminary results

205

Proposition 1.9. dimA M if q ~ 'P for all'P E Supp M UJ'lth dim A/'P { 1 + dimA M otherwise,

dimRq(A)Rq{M)

dim Rq(AP q(M)

=

dimM,

dimAM.

Proof: Let lJ E Supp M. Then ~ E Supp Rq(M) by Lemma 1.7 and Rq{A )/~ '" Rq(A/'P) is true. Therefore we get by Lemma 1.7: dim Rq(M) = sup {dim Rq(A/lJ) IlJ E Supp M}. Thus for the first statement it is sufficient to prove: Let A. be an integral domain. Then we have for every ideal q A: jf q 0, dim A { 1 + dim A' if q 9= O.

If q = 0, Rq(A) = A (with the trivial grading) and we are done in this case. If q 9= 0, write q = (ai, ••. , at)A with t:2: 1. We use induction on t. If t = 1 there is an isomorphism A[ X] ,...., Rq(A)

(X a variable),

which is given by X H> alT. This finishes the proof in this case since dim A[X] = 1 + dim A (c.f. Matsumura [1], Theorem 22, p. 83) . . Let t:2: 2 and set q' (al> . '" at-t) A. Then there is an epimorphism (X a variable)

which is given by X H> atT. Now Rq'(A) and Rq(A) are integral domains ami dim Rq,(A) = 1 + dim A by our induction hypothesis. Also alX at(atT) E Ker n, hence Ker n 9= 0 and we get dim RiA)

=

dim Rq,(A) [X]/Ker n

=

dim Rq,(A) = 1 + dim A.

dim Rq,(A) [X] - 1

°

NtJ4t let 'Po ~ lJI ~ '" ~ 'Pd, d dim A be a chain of prime ideals in A. (Then 'Po = and 'Pd m.) Since q 4= 0, 0 ~ 0* ~ 'P! ~ ... ~ 'PLI ~ m* is a chain of prime ideals in Rq(A), Le. dim Rq(A) 1 + dim A and our first statement is therefore proven. Now let us calculate dim Gq(M). If dim Rq(M) dim M we deduce from the following two epimorphisms (of Rq(A)modules) Rq(M) -+ Gq(M) -+ M/qM (underlining an A-module N means consider N as an Rq(A )-module by tl}e canonical epimorphism Rq(A) -+ A where N is equipped with the trivial grading) : dimAM = dimAM/qM =dimM and we are done in this case.

dimRq!A) MjqM:;; dilllRq!A) Gq(M) < dil1l Rq (A) Rq(M)

206

IV. Roos modules and associated graded modules

. Assume now dim Rq(M) = 1 + dim M. Then there is a prime ideal .\.l E Supp M with dim A/.\.l dim M and q g; .\.l. From the proof of the first statement we know that

{\l3 E Supp Rq(Ml I dim Rq(A)/\l3 = 1 + dim M} If> I .\.l E Supp M, dim A/.\.l = dim M, q g; .\.l} • Now choose an element a E q with a q.\.l for all .\.l E Supp M, with dim A/.\.l = dim M, q g; i>. Then dim Rq(llfljaRq(M) = dim M and hence the canonical epimorphism Rq(M)/aRq(J.lf) -r Gq(M) shows that dim Gq (M) dim M. Thus we have to prove that dim Gq(M) > dim M. Let V denote the intersection of all those primary submodules of M belonging to prime ideals .\.l in Supp.J.H with dim A/.\.l dim M and q g; .\.l. Then by the natural epimorphism Gq(M) -r Gq(M /V) we have dim Gq(M) > dim Gq(M /V) and since dim MjV dim M we may assume without loss of generality that dim Aj.\.l = dim M, q g; .\.l for each prime ideal .\.l in Ass M. Now the set )8 := {o

10 ideal in A, 0

~ q

and there is an n E N with

o"-lil{

q"M}

is not empty (q2 E )8) and contains a unique maximal element, say qQ. (If 0 1, V2 E )8, q"J.lf, b~-lM c q"'M, then (VI + V2)""J-} M c q""'M.) Clearly, qo ~ q since q"M =F 0 for all n E N. Choose an a E q with a q qQ, a q.\.l for all y E Ass M (c.f. Matsumura [1], (1.B), p. 2). Then by the same argument as above dim Rq(M)jaRq(M) = dim M and we have a natural epimorphism Rq(M)jaRq(M) -r Gq(M). By Lemma 1.7 \l3 is minimal in Supp Rq(M) if and only if \l3 = f> for some minimal .\.l E Supp M. Hence all minimal elements of Supp Rq(M) have the same dimension. By Krull's Hauptidealsatz the same is true for Rq(M)jaRq(M). If aT E (1, for all 0. E Supp Rq(M)jaRq(M) with dim Rq(A )/0. = dim M then there is an integer n E N with (aT)fI(Rq(M)jaRq(M)) = O. This implies aflM ~ aqflM or a,J-}M c qflM + O:Ma q"M since O:Ma 0 by our assumption. But this is impossible since a q qo. Hence there is a prime ideal 0. E Supp Rq(M)/aRq(M) with dim Rq(Al/o. dim M and aT q o.. If bE q then beaT) a(b'l') E 0., hence bE 0. and therefore q c [0.]0' i.e. 0. E Supp Gq(M) by Corollary 1.8. This shows dim Gq(M) dim Rq(A)jo. dim M, q.e.d. b~-IM ~

Corollary 1.10.

dim Rq(M) = dim Rq( ..H)m

ami,

dim Gq(M) = dim Gq(M)m.

The exact sequence of Lemma 1.3(ii) has an interesting consequence: Let \l3 c Rq(A) be a homogeneous prime ideal with [\l3h =F qT and assume Gq(M)('ll1 =F O. Then Gq(.M)('ll1 is a Cohen-Macaulay module if and only if the same is true for Rq(M)(\ll)' We say Rq(M)nlll an9 Gq(M)('ll1 are simultane()U,8ly Cohen-Macaulay. Per definitionem, the zero module is Cohen-Macaulay. Hence the assumption Gq(M)('ll1 =1= 0 is necessary for this statement. The last lemma of this paragraph shows that we can omit this assumption if we allow \l3 to vary in the following way. Lemma 1.11. Rq(M)(\ll1 is Cohen-Macaulay for all \l3 E Proj Rq(A) '" (.\.l* l.\.l E V(q)} if ami, only if Gq(M)('ll1 is Cohen-ltfacaulay for all \l3 E Proj Rq(A) '" (i>* l.\.l E V(q)}. Proof: The "only-if-part" follows immediately from the above remark. Assume that Gq(M)('ll1 is Cohen-Macaulay for all

\l3 E P(q) := Proj Rq(A) '" l.\.l* l.\.l E V(q)}.

" § 2. The Buchsbaum property

207

0, Rq(M)(\lll is Cohen-Macaulay. Let Gq(M)(\ll) = 0 and = ~* or ~ = fl with ~ E Supp M " V(q) and Rq(M)(~) Mfl by Corollary 1.4 and Corollary 1.5. Take a minimal prime ideal :0 of Rq(A) containing qRq(A) + fl. If all these :0 are of the form 0 = b* with b E V(q), then [rad(qRq(A) + fl)h qT, i.e. there is an integer n with q"T" = [qRq(A) + fll" = (q"+1 + (q" n ~») Ttl. But this is impossible. Hence we can find a 0 E P(qJ with [0]0 E V(q) and fi O. By Lemma 1.7 and Corollary 1.8 Gq(M)(£.l9= 0 and this implies that Rq(M)(£.) is Cohen-Macaulay. Now [Olt ~ qT and I.€mma 1.6 guarantees the existence of a prime ideal b E Spec Rq(A)(\lll with Mfl ~ Rq(M)(\lll = (Rq(M)(\ll))u and this is a Cohen-Macaulay module, q.e.d. Let

~

E P(q). If Gq(M)(\ll)

Rq(M)(~) 9= O. Then [~]o ~ V(q) by Corollary 1.8, i.e. ~

O. Then Rq(M) is locally Oohen-Maca111ay£f and Corollary 1.12. Assume pimA M /qM only l/ Gq(M) is lorAIlly Oohen-Maca111ay.

Remark. The property "Cohen-Macaulay" in the statement of Lemma 1.11 can be replaced by every other local property which is stable under localizations, which is preserved if we factor a module having this property by a non-zero divisor and which is "discovered" in such factor modules where we have some possibilities of choosing "admissible" non-zero divisors. In particular we have Corollary 1.13. Rq(M)(\lll is a BU£hsbaum module for all ~ E Proj Rq(A) " if and only if Gq(M)(\lll is a BU£hsbaum module for all ~ E Proj Rq(A) "

{~* 1 ~ E V(q)} {~* I ~ E V(q)}.

Proof: The only difficulty is to conclude from the Buchsbaum property of Gq(M)(\ll)' where Gq(M)(\lll9= 0, the Buchsbaum property of Rq(M)(\ll)' But this is possible using the exact sequence of Lemma \.3(ii) where the element x must be contained in the square of the maximal ideal of Rq(A)('\l!) (c.f. Proposition 1.2.19). :For this replace x by x 2 if necessary, q.e.d.

§ 2.

The Buchsbaum property of Bees modules and associated graded modules

The aim of this paragraph is to show that the Rees module Rq(M) and the associated graded module Gq{M) is Buchsbaum whenever M is Buchsbaum and q is a parameter ideal of M. To th~s end we calculate the local cohomology modules Him(Rq(M»), i:::;;; dim M, and HIDl(Gq{M»), i < dim M. . Theorem 2.1. Let M be a Noetherian A-module. M is a Buchsbaum module if and only if Gq(M) is a Buchsbaum module for all parameter ideaifJ q of M. Moreover in this su·uation we have and I(G q{M») = I(M). Proof: Let d := dim Jf. If d 0 there is nothing to prove. I.et d > O. If Gq{llf) is a Buchsbaum module for every parameter ideal q of M, then we obtain for any parameter ideal q of ill: Gq{M)/q'l'Gq(M) M/qM, i.e. q'l'Gq{A) is a parameter ideal of Gq(M).

208

IV. Rees modules and associated graded modules \

Assume q

(Xl' •.. , xa)A. But q'

;=

(Xl' ... , xa~d A. Then

[(xIT, ... , xa~lT) Gq(M) :Oq(M) xaTJo

= (q2M + q'M):~ Xd/qlJf =

(x~M

+ q'M):Xd/qM

XdM

+ (q'M:xa)jqM.

Since Gq(M) is a Buchsbaum module, this last module is annihilated by m, i.e. (xaM + (q'M:xa)) (xaM + q'M):m. If we replace Xa by x~, n 1, then for sufficiently large n q'M:x~

q'M:(m)

and, using Krull's Intersection Theorem, we obtain

q'M:xa c;;;;; q'M:(m)

n x~M + (q'M:x~) c n (x~M + q'M):m ,,>0

n>O

q'M:m. Therefore M is a Buchsbaum module by Proposition I.1.10(i)'. Now let M be a Buchsbaum module and q a parameter ideal of M. Assume qM = (Xl> ... , xd) M, d dim M > O. We prove inductively on d: (a) [H~(Gq(M))lll = for all n > - i and all i and

° °

[H~(Gq(M))l..

for all n

<

-%.

and all i

<

d.

(b) For all i there is a natural (A-)monomorphism

p~'d; [Him(Gq(M))]-i

H:n(M)

which is an isomorphism for (c) The canonical maps

~.

< d.

A~: HI(IDli Gq(M)) -Him(Gq(M))

are surjective for all i < d. Our theorem will follow by Theorem 1.2.15. Let depth M > O. Then [0 :Oq(MIIDlJ..

q"M n (q"'1 1Jf:M m) n (q,,+2M: M q)!q,,+lM

q"+lM/qn+1M =

°

by Lemma 1.1.15. Hence depth Gq(M) > 0. Now let M be an arbitrary Buchsbaum (c.£. Lemma 1.1.14) we have a monomorphism module. Since (O:Mm) n qM H?n(M) = O:M m - M /qM which extends to an exact sequence of graded Rq(A )-modules

°

O-O:Mm _Gq(M) _Gq(M) _0,

where M:= M/O:Mm. Since depth M > '0, we obtain a:n isomorphism HMGq(M)) III which gives rise to an isomorphism

~ H~(O:MIll) =

°

Clearly, p~ is natural (i.e. if there is a Buchsbaum module N of (positive) dimension such that q is again a parameter ideal of N and a map t: 1U _ N then there is a com-

§ 2. The Buchsbaum property

209

mutative diagram o

[H~(Gq(M))lo--~~-+ H~(M)

t ~ [H~(Gq(N))lo ~~ ~ H~(N) where the vertical maps are induced by I). Furthermore

HO(Wt; Gq(M)) =

O:Gq(M)Wt~ O:H'in(M)Wt

=

H~(M) = HMGq(M))

and A.~ is an isomorphism. Therefore (a), (b) and (c) are correct for i We note that for depth M > 0 we have an exact sequence

(E):O

--+

Gq(M) (-1) "',T ~ Gq(M)

--+

Gq(Mjx1M)

--+

= O.

O.

(It is only necessary to show that x1T is a non-zero divisor on Gq(M). But this is easy to see by virtue of Lemma 1.1.15.) If d = 1 and depth M = 1, we therefore have an exact sequence

o --+ HM Gq(MjXIM~)) --+ HM Gq(M)) (-1) "',T -+ HM Gq(M)) --+ 0, where H~(Gq(MjXIM)) ~ Mjx1M. Hence [llMGq(Jf))]n CY [H~(Gq(M))ln+lfor all n> 0 and this shows that [H~(Gq(M))ln = 0 for all n:2': 0 since HMGq(M)) is Artinian. We also have an isomorphism [Hk(Gq(M))l-l ~ Mjx1M . . If depth M = 0, H~(Gq(M))~ H!n(Gq(M)) with M := MjH~(M) and this proves (a). Furthermore, there is a natural isomorphism [H~(Gq(M))l-l ~ MjXIM

and a natural monomorphism

MjXIM

--+

H:n(M) ~ H:n(M)

coming from the exact sequence

0--+ M ~ M

--+

MjXIM

--+

O.

Thus (b) is proven and we are finished for d = 1. Let d > 1. If we have proven (a), (b) and (c) for Buchsbaum modules of dimension d and positive depth, we get in the case depth M = 0: Let M:= MjH~(M). Since H~(Gq(M))~ H~(Gq(M)) for all i> 0 we obtain (a). If we set f.l it := f.l~ for i > 0, (b) holds for all i. To verify (c) we again consider the exact sequence

o --+ H~(M) --+ Gq(M) --+ Gq(M) --+ O. From this we obtain for 0

Hi(Wt; ~(M))

--+ !!i(Wt;

1 o 14 Buchsbaum Rings


d commutative diagrams with exact rows

Gq(M)) ~ Hi(Wt; Gq(M))

1A!, --+

H~(Gq(M))

lA~ --+ H~(Gq(M))

--+

O.

210

IV. Reeli\ modules and associated graded modules

lb

is surjective. Since by (a) [Hk(Gq(M))].. 0 for all n =1= -i, the surjectivity of will follow if :n;i is surjective in degree -i. Hence we show: The monomorphism O:Mm --'.>- Gq(M) induces for all i~ d monomorphisms

[lfi(WC; O:Mm)]-i+l

--'.>-

lk

[lft(WC; Gq(M))]-HI'

Choose a minimal generating set {aI' .,., a,l of m. Then {aI' ,." a" x1T, •.. , XdT} is a minimal generating set of WC. Consider the Koszul complexes K'(WC; O:Mm) and K(WC; Gq(M)) and denote by e~,,·'t. , 1 il < .. ' < jq d, 1 il < ... < l"..-q ~ t, it· .. ·p-v O q~ p free generators of KP(WC; Rq(A)) of degree -q, where the upper indices are related to the elements xIT, ..•, xaT and the lower indices to aI, .•. , at, If d' is the differentiation in K'(WC; Gq(M)), we have to show for all d: p-

[1m dH]_i+l n [K'(WC; O:M m)]-i+l = 0

(d' acts trivially on K'(WC; O:Mm)). Let x E [KH(WC; Gq (1I:f))}-i+l with d'-I(x) E [Ki(WC; O:Mm).!-I+I' Write

:I:

x=

el ' ...j.-lp;. ...;.-l

1:S;;j.< ••• <j._I;S;d

wh~re pi".,i.-,

E [Gq(M)Jo

d·-I(x)

MjqM and where

:I:

=

e!, .. ·i·-'(a/pi,... i.-,)

I;S;I:S;;I

l:S;it< .. ·<j,_,<;;a

-

:I:

eil "';' (

l:s;;,< ... <j,:S;;d

t

h=l

d·-I(x) E [K'(WC; O:Mm)]-i+l implies

1. a,pi, ...i.- E O:M m for all 1 ~ 1

i1 < ... <

ji-l

d, 1

1~ t and

i

2,

:I: h=l

1::;; il

< ... < ii-I

d with pi, ... i,-,

Xpm it ... j .-, E

:I:

mj, .. ·j·-l

mod qM then 2. means

xqM + q2M

qE!i, ... j,-,)

for all p E {1, ... , d) " {il> ... ,1;-1)

q' :=

el), Set (p fixed)

(Xl> •• " X p -1> Xp+l, ••• , Xd)

A,

Then i.e. pj, ... i,-, E qM:mjqM.

Therefore we have for all 1 = 1, ... , t: aIPj,···j,-, 0, i.e. di-I(x) = 0 and this proves (c) for this case. :Finally it remains to consider the case depth M > O. If we apply the functors Hk to the exact sequence (E) we obtain (using the induction hypothesis) for all i isomorphisms [Hk(Gq(M))] ..

~~ [Hk(Gq(M))] ..+l

for all n> - i

§ 2. The Buchsbaum property

and for all i

211

< d (at least) monomorphisms

[H~(Gq(M)lln

z,T.

[Hk(Gq(M))]n+1

for all n

<

1.

-i -

Hence [H~(Gq(M)}ln 0 for all n ~ - i since H~(Gq(M)) is Artini~n. Now the map is the zero map. We construct ,uk inductively on i, i::;; d. If f.l'iil is constructed, i::;; d, then we have a commutative diagram with exact rows

,u~

1,-'.

""M/SIM

H!;-l(.JljXlM)

Hence there is a unique map f.l~: [H~(Gq(M)}l-1 -7- H:n(M) making this diagram com· mutativ. Clearly, ,u~ is a natural map. It is a monomorphism and an isomorphism whenever i < d since tp' is surjective and ,u~/~,M is an isomorphism in this situation. This proves (b). Now for i < d, we consider the commutative diagram with exact rows: o -7- [H~(Gq(M)ll-H -7- [H~(Gq(M))l-i -)- [H~(GqOll/xIM))l-i

11'~

11'~"'M

o It .shows that [H~(Gq(M))l-i-l = 0, hence [H~(Gq(M))ln = 0 for all n < - i by the above monomorphisms. Therefore (a) is proven. Now, from the exact sequence (E) result for each i < d commutative diagrams with exact rows

0-7- HH(IDl.; Gq(M)) -7- HH(IDl.; Gq(M/XI1lf)) -)- HI(IDl.; Gq(M))

1Aif'

1Ati~IM

1A~

Since ;'~I~lM is surjective by the induction hypothesis for i

;.k is surjective for all i < d and (c) is proven, q.e.d.

<

d ;'k(-l) and hence

Corollary 2.2. Let ill be a Buchsbaum module 01 dtmension d > 0 and 01 parameters 01 M. Set q (XI> ... , Xd) A. Then lor all integers n, rl>

where q"'M

14*

... , rd

Xa a system

>0

Milm< O.

Prool: It is sufficient to prove q'

Xl' ••• ,

. Fix r 1,

••• , rd

and denote for notational convenience: M /(X~I, ... , x~·) M,

212

IV. Rees modules and associated graded modules

Consider the canonical epimorphism ;11:: Gq(M)Jq'Gq(M) I(Gq (M)/q'Gq(11f))

eo(q'; Gq(M»)

-lo-

Gq(M').We get

+ I(Gq(M») + I(M)

rl

• •..•

rdeO( (xIT, ... , xdT); Gq(M»)

r1

• ••••

rd(l(Gq(M)/qTGq(M» - I(Gq(M»)

r l ' ....

rd(I(M/qM)

I(M»)

+ I(M)

+ I(M)

+ I(M) eo(~l, ... , x~,); M) + I(M) = I(M') = I(Gq(M'») , r1 •

•••• r~o(q;

M)

i.e. n is an isomorphism. Therefore q"M/W"

+ q"+1M)

[Gq(M)Jq'Gq(M)],,~

[Gq(M')J"

+ (X~l, ••• , ~.) M)J( qn+1 M + (X~', ••. , x~,) M) qIlM/
{qll M

hence V"

<;;;;;;

U"

V" =

+ qll+1 M for all n > O. This yields (U" + qll+lM) n (X~l, ••• , xd.) M Un + V,,+1 <;;;;;; U" + q''''.2M

and so on. Hence

Vn

<;;;;;;

n (Un + qmM)

U"

m>"

by Krull's Intersection Theorem, q.e.d. Next we investigate the Rees module Rq(M) where q is a parameter ideal of M and M is a Buchsbaum module. First we have: Proposition 2.3. Let M be a Buch8baum module and q a parameter ideal of M. Assume that d:= dim M > O. Then (i) ll~(Rq(M») = 0 m H~(M).

1, ... , d we have

(ii) For all i

[ll~(Rq(M»)l" '" { ~:;;-l(M) (iii) ilR 'll~(Rq(M»)

0

for all i

for for

n 0 and n -i+2 n

-i+ 1, 1.

d.

Proof: Before embarking on the proof we note that "parameter ideal of ill" means: There is an ideal q in A and d elements Xl> ••• , Xd such that qM = (Xl"'" Xd) M. This does not imply that q = (Xl> ••• , Xd) A although one can assume this in most cases.

Let Xl, ••• , shows that

Xd

be elements of m such that qM

ll~(Rq(M»)~EB(qIlMnH~(M»)TII

(Xl' ••• , Xd)

H~(M)

0

M. An easy calculation

m

,,~o

since q"M n H~(M) = q"M n (O:Mm) proves (i).

0 for all n> 0 by Lemma 1.1.14. This

§ 2. The Buchsbaum property

213

Let M := M /0 :M m. Then the exact sequences

0-+ O:Mm -+ M

-+

M

-+ 0

and

0

-+ O:Mm -+

Rq(M)

-+

Rq(M)

-+

0

give rise for i > 0 to isomorphisms H~(M) c:::: H~(M)'

and

llim(Rq(M)) '" llim(R(M))

and we can assume without loss of generality that depth M exact sequences

>

O. Now we have two

(E l ): 0 -+ Gq(M) (-1) ~ Rq(M)JxlRq(M).!.... Rq(M JxlM) -+ 0, (E2 ): 0 -+ Rq(M)

Rq(M) -+ R q(M)/x 1R q(11f)

We use induction on d. H d

=

1, R q(M)/x 1R q(M)

-+ O.

Gq(M) and (E2 ) and Theorem

2,1 imply:

o = ll~(Gq(M)) = llMRq(M)/x1Rq(M))

O:!!~(Rq(Ml) Xl'

hence ll?m(Rq(M)) 0 (every element of ll?m(Rq(M)) is annihilated by some power of *1)' Therefore we are finished with the proof in this case. Let d> 1. By the proof of Theorem 2.1 (statement (a)), we have for all i and all n - i + 1: [lliDi1(Gq(M)) 1)].. = O. Therefore (EI ) induces (with the induction hypothesis) for all i < d exact sequences

0-+ llim(Gq(M)) (:-1) ~ llim(Rq(M)/xIRq(M)) ~ llim(Rq(M/xIM))

-+

0 '

Qnd t~is implies [llim(Rq(M)/xIRq(M))] .. = 0 for all n 0 and n::;; -t', Furthermore, llIDi(Rq(M)JxIRq(M)) = 0 by this exact sequence, since the left term is concentrated in degree _to + 1 and all elements of the right-hand term are of degree > - i + 2. From (E 2 ) result for all d exact sequences

m'

lliil(Rq(M)JxIRq(M)) -+llim(Rq(M)) ~ llim(Rq(M)). As we will see later, xlllim(Rq(M)) But first of all we see that for n

~ :[lLim(Rq(Ml)]nXl

=

o and this implies (iii). o and n::;;

[0 :4(Rq(Ml)XI]"

-i

=0

+ 1:

(i::;; d),

and this implies [llim(Rq(M))] .. = 0 for n 0 and n - t ' + 1. We note (d. Sharp [1], Theorem 4.3) that for all t'we have llim(M) '" Him(M). Now we consider the exact sequences --(E3) 0 -+ qRq(M) (-1) -+ Rq(M) -+ M -+ 0,

(E,l 0

-+

Rq(M)

Since [llim(Rq(M))] ..

=0

0-+ H;;I(M)

-+

qRq(M) -+ Gq(M) -+ O.

for n > 0, we get from (Ea) exact sequences

-+

llim(qRq(M)) (-1)

-+ llim(Rq(M)) -+ 0

which yield isomorphisms

[llim(qRq(M))]-1 '" H;;l(M) , [Hk(qRq(M))] .. '" [llim(Rq(M))]rt+l

for all n::;; -2.

(*)

214

IV. Rees modules and associated graded modules

In particular [ll~{qRq{M))]" = 0 for n 0 and n < -i. Therefore (E,) together with Theorem 2.1 implies exact sequences (1 i ~ d):

o ~ ll~l(Gq(M)) ~ llim{qRq(M)) ~ llim(Rq(M)) ~ 0 and we obtain for all n;;:;:' - i + 2:

Together with the isomorphisms (*) we obtain (ii). Since Xl E Rq(A) is an element of degree zero and since Xln:;;l(M) 0 for all i~ d, we have xlllim(Rq{M)) 0 for all i d which concludes our proof, q.e.d. To prove our next main result we need some technical lemmata which we will now prove. We note that some of them are of independent interest (e.g., Lemma 2.4, Lemma 2.6 or Lemma 2.8). First of all we have an easy consequence of Proposition 1.1.9:

Lemma 2.4. Let M be a Noetherian A-module 01 dimension d and let a, b c:: A be idea18 with dimA MjaM = dimA MlbM = O. Assume we m,:e given an M-basis al> ••• , a, 01 a (d. Definition 1.1.7). Then there is an M-basis bi> ... , bs 01 b hamng the lollowiTl.fl eroperty: For all integerlJ p, q with 0 P t, 0 ~ q ~ s, p + q = d and all 1 i l < ... < ip ~ t, 1 il < ... < i q ~ s the elements ai" ... , ail" bit, ... , bi• lorm a system 01 parameters 01 M. Prool: Choose a common M/(a", ... , ai,') M-basis of b for all 0 < ... < z~ ~ d (see Proposition 1.1.9), q.e.d.

p ~ d and all 1

il

Next we have

Lemma 2.6. Let M be a Buchsbaum module 01 dimension d > 2 and let Xl> ••• , Xd be a system 01 parameters 01 M. II z E m is an element such that Xl, ••• , Xi-I, Z, Xi+l, ... , Xd is again a system 01 parameters 01 M lor all i 1, ... , d then we have lor all p, q with 1

p
(Xl' ... , Xp) (XP+l' ... , Xq) M:MZ) ~ O:Mm

+ (Xl' ... , Xp) ~M.

Proal: We use induction on p. If p = 1 and mE (XI(X2, ... , Xq) M):z then there is an m' E (X2, ... , Xq) M with zm xlm', hence m' E (zM:x l ) n (x z, ... , Xq) M ~ (zM:m) n (z, X2, ... , Xd) M = zM by Lemma 1.1.14. We select an element m" of M with m' = zm" and obtain zm = xlzm", hence m E xiM O:z = O:m xiM. Now let p > 2. Then we obtain using the induction hypothesis

+

+

(Xl> ... , xp) (xP+l' ... , Xq) M) : z =

(Xl' .. " Xp) (Xp+b ••. , Xq) M:z) n (xpM

+ (Xl' ... , Xp-l)

~ (Xl> ... , Xp-I) M

1- (xp(xP+l' ... , Xq) M:z))

~ (Xl> .•. , Xp-ItM

+ O:m + XpM) n (xpM:m + (Xl' ... , Xp-I) 1lf)

=

q.e.d.

+ (Xl> .•. , Xp_I) (xP+l' ... , Xq) M) :z)

O:m

+ (Xl> ... , xp) M,

n (xpM:m

M)

§ 2. The Buchsbaum property

Lemma 2.6. Let M be a B'UCMbaum module and of parameters ofllf. Then the homlYmorphism

Xl> ••• , Xi>

215

1 < i d a part of a system

Hi(m; (Xl> ••• , Xi) M).....,.. Hi(m; M) induced by the emhedding (Xl> ••• , Xi) M c M is the zero homomorphism whenever i

d - i.

Proof: Let i = 1 and denote by gi: Hi(m; xIM) .....,.. Hi(m; M) the map induced by the embedding x l M c:: M. For; d - 1 let I' E JJi(m; xIM). Choose a cocycle c E Ki(m; xIM) representing y. Then the image c of c in Ki(m; M) has the form c XIC' with c' E Ki(m; M). An easy calculation shows that c' is also a cocycle (choose an M-basis of m, use O:MXl = O:Mm and apply Lemma 1.1.14). Therefore x 1c' is a coboundary (since xIHi(m; M) = 0), i.e. gi(y) = O. Now gi 0 is equivalent to the injectivity of the map Hi(m; M) .....,.. Hi(m; M/XIM) obtained from the epimorphism M _ M /XIM by using the long exact cohomology sequence which results from the short exact sequence 0.....,.. XIM.....,.. M _ M/XIM .....,.. O. Repeating this argument again and again we get a chain of monomorphisms for ; ::;;d z' Hi(m; M).....,.. Hi(m; M/XIM).....,.. ... .....,.. HI{m; M/(x I ,

... ,

Xi)

M),

i.e. a monomorphism Hf(m; M) .....,..Hi{m; Mj(xI> ... , X, )M) induced by the epimorphism M - llf/(X1> ••• , Xi) M. This proves the lemma, q.e.d.

Lemma 2.7. Let M be a BucMbaum module with depth M > 0 and let a c:: A be a'ilc Uleal with dim,{ M/aM = O. ConsUler the double complex (K"', d~', d;) defined by Kp,q = KP( a; K'l(m; M)) and differentiations dr,ll: KM .....,.. Kp+I,q, dr,q: Kp·q .....,.. Kp.q+I given by the differentiatwn of the underl~·'ng Koszul complexes. Then we have for all p, q 0 with p

>

+ q::;; d: Ker df'q n Ker dr·1l elm dr,ll- i

•

Proof: We denote by (K~, d~) the Koszul complex K"(a; ... ) with differentiation d~ and by (K~, d~) the Koszul complex K"(m; ... ) with differentiation d~. For l 1,2 let e~? ..ii be free generators of K{, 1::;; il < ... < ~'j t/) where tl rankA/ma/ma and t2 := rank'{/mm/m 2• Assume that aI>'''' at, and bI> ... , b't are M-bases of a, resp. m, satisfying the statement of Lemma 2.4. Let K" := Tot K" with differentiation d', i.e. we have

EB

Kt =

KM

p+<1=1

and for

~

E KM, P

di(o.;)

+q

df,q(o.;)

i

+ (-1)P d:,q(a).

By H~, H~, H' we denote the cohomology of these complexes. Note that a free basis of Kp.q is given by the formal products e~:,\, ei;'~,j.'

1

il

< ... <

ip < t 1 , 1:::;;;1

In addition the map K' _ K'(al> ... , a", bI , e.tI,1...).'p •

e~2)

,... ·1., m

~

e·.,...... . m p.Il+I' .. ·I.+I'

is an isomorphism of complexes.

... ,

< ... <;q::;;

b,,; M) given by

(m E 1lf)

~.

216

IV. Rees modules and associated graded modules

Now choose x E a such that al,"" aI, and bl , ••• , bl. are a/!tain (MjxM)-bases of a and m (this is possible since dimA MlaM = 0). Let ~ E Ker di,q n Ker d~·q, p q::;; d. If p = 0 or q = 0 there is nothing to shpw since Ker di,q = 0 or Ker d~,q = 0 in this case (depth M > 0). Thus we assume p > 1, q > 1. Since xH~ = xH; = 0 for all i (x E a), there are lX E Kp-l.q and fJ E Kp·q-l with

+

=

d~-I·q(lX)

=

d~·q-l(fJ)

x~.

Now d~+l,q-ldi,q-l(fJ) = di,qd~,q-l(fJ) = d~,qdi-l,q(lX) = 0, i.e. there is a 1'1 E Kp+l·q-2 with di·q-l(xfJ) = d~+I,q-2(YI)' Again we have d~+2·q-2di+l.q-2(Yl) = di+ l ,q- l di+l. q- 2(yIl = 0, i.e. there is a 1'2 E Kp+2.q-3 with di+ l ,q-2(XYl) = d~+2·q-3(Y2)' and so on. If we put 1'0:= fJ, we find elements Yi E Kp+i.q-i-l, i> 1, with d~+i,q-;-i-l(Yi) = di+i-1,q-'(XYi_l)' If we put y~:= lX, similar calculations show that there are Yi E Kp-i-l.q+i, i> 1, with d~-i-l,q+i(xyi) = d~-i-2,q+i+l(Y~+1)' We define I'

:= i;::o ..r xP+q-i-I(( -1)" Yi + (-1)'~ yi) E Kp+q-l,

i 1),

(P - i)

+ +

p . .IS easy '2 ' t:i:= 2 . (Note Yi, .I ri = O'f' I t> _ P q. )Th en It ( to verify that dp+q-l(y) = O. Now XHp+q-l = 0, i.e. there is a <5 E Kp+q-2 with xy = dP+ q - 2 (<5). Write

h were

t:j :=

<5

= ..r

+

<5,., <5,. E Kr,s.

'+8~p+q-2

But this implies d~,q-Idi-I,q-I( (-1)" <5 _ I,q_l) P

+

= d~,q-I( (-1)" di-l,q-I(<5P-I.q-l) (-1)<'+1' d~,q-2(<5p.q--2)) p q = d~·q-l( (-1)" . x • x + - l ( -I)', . fJ) = xp+qd~·q-l(fJ) = xp+q+l~,

i.e. there is an element Let

1]

=

E Kp-l.q-l with xp+q+l;

where the sum is taken over all ii, 1 ::;; jl < ... < jq::;; t2 and

"'J

d~·q-ld~-I,q-I(1]).

ip. jl' ... , jq with 1

< i l < ... < ip::;; tl ,

where the indices are defined similarly. Then ..

xp+q+lm~'''·~· '1."'1'

=..r ..r (-l)"+vai b· p

q

u

.

•

.

Il~''''!v''''~

Ir'1"·'U ... 1p

E (ai'1 ... , ai l' ) (b·, ... , b.) M. 11 1'1

u=l"=1

By Lemma 2.5 and the choice of x and the ai's and b;'s,

rrJ., ...~. E (ai, '1 .• ""

••• , aj )

1

Consider for all 1 ::;; i l m·'1""1'.:=

M.

11

< ... < ""

~

l~j,< ...
ip ::;; tl the element

e\2) ·m~' .. ·i. E Kq = Kq(m' M). 11···1(1

'1""1'

2

'

§ 2. The- Buchsbaum property

217 '

~ E Kerd~,q implies that mi,... ,,, E Ker d~ n Kq(m; (ai" ... , ai,) M). But the embedding (ail' •.. , ai,) M c: M induces the zero homomorphism HJ( m; (ai" "', ai) M) - HI(m; M) by Lemma 2.6 for all; d - p. Therefore there are mil .. J, E Kq-l(m; M) (since' q d - p) with mi, ...'" = df-l(m~, ... i,,), i.e. ~

= dP,q-l 2:

(

"

~

l~jl< .••
e~1) • m/. .) 11<,,'. "I.h'l'

M - 1 E 1m d2 '

q.e.d. Lemma 2.8. Lei M be a Buduibaum module and q a parameter ideal of M (for further reference to thu lemma we note here that thu does not imply that q u generateq by dim M element8). If i, ; 0, then d;: Kf(q; qfM) _ K;+l(q; q1M)

maps into KJ+l(q; ql+ 1M) and therefore we can define complexes L~(M):

0 _KO(q; q-lIM) _Kl(q; q-1I+1M) _ ... ,

where h E Z and and qllM:= M for h< o. We denote the differentiativn by o~. Then

H'(L~(M»)"'{ HOI(q;M)

for i h.

Proof: It is clear that we only have to show Hi(L~(M») = 0 for all i> h, Let M M/O:m, The map M _ M induces an epimorphism of complexes L~(M) _ L~(M) which is an isomorphism for all degrees i> h (since qrM qrM for r > 0), Therefore Hj(L~(M») H'(L~(M») for t' > h and we may assume that depth M > 0 if dim M > O. We note that there is nothing to prove if dim M = 0, since, per definitionem, qrM 0 for all r > 0 in this case. We use induction on a := dim M. For a = 0 we are finished. Let d > 0 and (without loss of generality) depth M > O. Choose x E q such that dim M /xM a - 1 and q is again a parameter ideal of M/xM. Then there is an exact sequence of complexes (set M' := M/xM)

o _ L~+l(M) -

L~(M)

-

L~(M')

_ 0,

where for each i the map L1+1(M) _ L~(M) is obtained by multiplication with x (the exactness follows from xM n qrM = xqr-lM ~ qr-iM for all r E Z, see Corollary 2.2). The resulting long exact cohomology sequence gives for each i > h an isomorphism H'(L~(M») '" H'(L;_l(M»)

(use induction on i - h 1 and Hi(L~+l(M»)""'" H((L~(M») for i> h above exact sequence and the induction hypothesis on a). II we set h i - 1, we get an exact sequence

+1

by the

On the other hand, the exact sequence 0 _ M ~ M _ M' _ 0 induces an exact sequence 0 _ HI-l(q; M) _ H'-l(q; M') _ H'(q; M) _ 0, where HI(q; M('» = Hi(Lj(M('») for all j. Calculating the lengths of the modules occuring in (E), we

218

IV. Rees modules and associated graded modules

·obtain l(Hi(L;~l(M))):S;; l(H'(L;(M))) -l(Hi-l(L;_l(M')))

+ l(Ht-l(L;_l(M))) =

0,

i.e. q.e.d .

.Lemma 2.9. Let M be a B'UCksbaum module and Xl, ••• , Xd, d = dim M a system of parameters of M. Set q := (Xl' ••• , Xa) A. Then: (i) The embedding O:M me Rq(M) ind'UCe8 for all i:S;; d + 1 and all q < 0 monomorphiBms [H'(im; O:M m)]q -?- [Hi(im; Rq(M))]q.

(ii) Let depth M > O. Then the embedding XaM:M me Rq(M)/xaTRq(M) (induced by xaM :M m c M) results for all d in monomorphisms Hi(im; XaM:Mm) -+ Hi(im; Rq(M)/xaTRq(M)). Proof: For all p > 0, max (0, p - t):S;; h:S;; min (p, d), where t rnnkA/mm/m2 and all i h ••• , i p _ h , 11, ... 1h with 1 i l < ... < ip _ h t, 1 il < ... < h. diet be g~nerators of the (free Rq(A)-)module KP(im; Rq(A)) with deg = -h. Let d' denote the differentiation. O· and all 0 p t d if Note that [Kp(im; N)]q =l= 0 for any A-module N and only if max (0, p - t) -q min (p, d). To prove (i) we have to show that for all p d, q < 0

e!:::t.

et·t"

+

[KP+l(im; O:m)]q n [1m dP]q = 0

(note that d' acts trivially on K'(im; O:m)). We choose an M-basis al> ... , at of m. Let -~ =

P

I:

.

.•.

I:

•

e~'''·l.''. m~''''~'

•

h= -'1 lS.,< ... <'p+.+tSI

'1*··l'p"'''

'1*"'P-"

Tq+h

ISj,< ... <j_.sa

E [Kp(im; Rq(M))]q dP{~)

with I

il

(m{::J:__ E qq+l&M)

E [KP+l(im; O:m)]q, i.e. we have for all h = -q, ... , min (p

< ... < ip+H':S;; t,

I

il < ... < ilt

for for But q:S;;

since p

+ 1, d),

d:

h> -q, h= -q.

1, hence we get for h = -q:

+1+q

p

d (note that m~,.. .i·..·j-. = 0), i.e. dP(~) = 0 and this proves (i). '1"· 1'+1+"

219

§ 2. The Buchsbaum property

To prove (ii) let us denote by K' the Koszul complex K('iJR:; Rq(M)jxdTRq(M)) , with differentiation d' and by it the Koszul complex K('iJR:; xaM:m) with differentiation (j'. We have to show for all p d and all q with 0:::;:; q pI: [Jm dlHJ_ q n [KPJ- q ~ [1m d P- 1 ]_q

([Kq]" = 0 for n

<

-p and n> 0).

Let

( - - denotes the residue class modulo XdTRq(M)) , m~""~'

'l""p-It-l

E qli-qM, with dP- 1(E) , E [KP] -q •

First of all we prove inductively on

For 8 for 1

= 8

1

8

1 there is nothing to prove (put

E(l)

<

p

8,

P - q, that there are elements

:= E). Assume we have determined E(8)

q. Then

P-8+1

E

1=1

for alII Let

and

< ... <

i1

i 8- 1 :::;:; t, 1

il

< ... <

jP-I<+1 :::;:;

d.

using the notation of I..emma 2.8 11·

.:=

r'l""'~l

"

~

1$;'<"'<;p_.$a

e.. 1l!8lj,: .. jp-. E Kp-8(q; qP-8- QM') 11H·lp-.r-Jl..• t'_1

Then (jP-'(Pi, ... i._,) = o. By using Lemma 2.8 we find elements II~

.

f"""l· .. t "_l

=

"

~

1$].< ... <;.-.-.$4

e.. 1I),.. ·i.-.-, E LP-S-l(M') ,.···lp-,-lf""f,l ... i ,_l q

with P;, ... i H OP-.-l("" .. .i._,) (since p 8 > q). 1 E qP-.-q-lM Let m'J..· .. t.-·be elements with p';, ...;.-.... 'J,<*I,,-1 ;1 .. .i,,-1 q:::;:; h P 8 - 1 we put

m'i,i 1H.1.-:1 ..·t.-I-l mod xdM. For for h

<

q, h > p - 8)

for q

h

for h

=p

<

p -

8 -

-

8 -

1.

1,

220

IV, Rees modules and associated graded modules

Then an easy calculation shows that, we have with

dl'-I(~(8+1»)

Using

dl'-t(~(8»)

=

dl'-I(~) and we are done. we may assume without loss of generality that

~(I'-q)

where m{:j~.!'._, E M = [Rq(M)jxdTRq(MHg· dl'-I(~) E [KI'J-q means that

for all i lo and

"

with 1::;;

ip_q-l> i l " ' " iq+l

'0

< .,. <

il

ip-q-I

< t,

1::;; il

< ,.. <

iq+I

d

for all iI, .. ,' il'_q, it•. ," i q with 1 it < ... < ip-q t, 1::;; il < ... < i q d. Set M MjxdM:m and for mE M let m'denote the residue class of m in M. We consider the double complex K(q; K'(m; M)) introduced in Lemma 2.7, with a = q and M instead of M. We set ;;:= 'J

e\l) .• e~2).

'\'

~.

•

1::;;"<"'<',,_0_.::;;1

11"'}Q

m~· .. ·i.

.l .. ·.rQ-I

'1 ... 1. 1I - q-l

E Kq(q·' KP-q-I(m',

M)).

l::;;j,< .. ,
Then 'ii E Ker d!,P-q-l n Ker d~,p-q-I in the notation of Lemma 2.7. But then 'i'j E 1m d~'P-q-2 by Lemma 2.7 (q + (p q - 1) = p - 1 d - 1 dim M), i.e, there are m), .. ,i. E M with 'l····p-Il-*

p-q-l

m~'···t.

• 1···."....(t-l

E

=

(-1)1-1 aj

f

m:i,.-jq .

.1···t' ....P-q-l

1= I

for all 1 ::;; il

< '" <

mF ..~.

It ....''_q_l

ip-q-l::;;

:= m~' .. ·t.

.l •••• p-fl-l

t, 1 ::;; i l < , .. < jq::;; d, i.e.,

-

p-q-l

'\'

~

1=1

(_1)1-1 a·Jl m:i, .. j. . E XdM: m '1 •.• t, ... I;-ll-1

for 'all 1 Let

Now we are able to prove: Theorem 2.10. Let M be a Buchsbaum module 01 positive di'lneMian. Then Rq(M) i8 a BucMbaum module for aU parameter ideals q of M.

§ Z. The Buchsba.um property

221

Prool: Let q be a parameter ideal of M and Xl, ••• , X a, d:= dim M, a system of parameters of M with qM = (Xl' .•. , Xd) M. Let q' := (Xl' •.. , Xli) A. Then Rq(M) = Rq,(M} is a Buchsbaum module over Rq(A) if and only if it is a Buchsbaum module over Rq,(A), i.e. we may assume that q is generated by a system of parameters of M. We use induction on d. If d = 1 or d 2 then by Proposition 2.3 Hfm{Rq(M)) = 0 for i =1= 0, d + 1 (= dim Rq(M)) and IDCH~{Rq(M)) O. Hence Rq(M) is a Buchsbaum module by Proposition 1.2.12. Let d > 3 and let M := M /0: m. Consider the exact sequence 0-+ O:m -+ Rq(M) -+ Rq(M) -+ O.

Using Lemma 2.9{i} and the fact that Him(O:m) 0 for i > 0, we get for all 0 < i-::;' d and all p < 0 commutative diagrams with exact rows (the vertical maps are the canonicalones) 0-+ [H'(IDC; O:m)]p -+ [Hi{IDC; Rq(M))]p -+ [H'{IDC; Rq(M))]p -+ 0

t

0-+

o

trIll"

tlAt]p

[Hb{Rq(M))]p -+0.

-+ [Him{Rq(M))]p

If Rq(M) is a Buchsbaum module, then p.']p is surjective for all 1 i-::;' d by Theorem 1.2.15, hence [Ai]p is surjective for all 1 t' -::;, d and all p < O. But then Af is surjective since [Hb{Rq(M))]p 0 for p > 0 by Proposition 2.3. Since IDCHMRq(M)) = 0 (Proposition 2.3),10 is surjective too and Rq{M) is a Buchsbaum module by Theorem 1.2.15. Therefore we can assume that depth M > O. Set M:= M/xaM:m. Then by the induction hypothesis, RlM) is a Buchsbaum module, i.e. the canonical maps Xi: Hi{IDC; Rq(M)) -+ Hfm(Rq{M))

are surjective for all i d by Theorem 1.2.15. Now the kernel of the natural epimorphism Rq(M) -+ Rq(M) contains xaTRq(M), i.e. we have an epimorphism n: Rq(M)jxlITRq(M) -+ Rq{M).

By Lemma 1.1.14 we have for all p

> 0 with q'

[Ker n]p = (qPM n (XdM:m)) TP = (Xdq'P 1M

+ (q'PM

n (XaM:m))) Tp

(Xdq'P-l M) TP = 0

( - means here: residue class modulo xaTRq(M)) and for p [Ker n]o

0 we see that

XdM:m,

i.e. we obtain an exact sequence 0-+ XdM :m -+ Rq{M)/xaTRq(M) -+ Rq(M) -+ O.

From

Sharp [1],4.3., it follows

readily that

H~(xdM:m) '" H!m(xaM:m), I.e.

[Him(XdM:m)Jp = 0 for p =1= O. But [H!m(Rq(M))]p = 0 for p 0 (Proposition 2.3). By using this and Lemma 2.9{ii) we get for all i < d a commutative diagram with

222

IV. Rees modules and associated graded modules

exact rows (the vertical homomorphisms are again the canonical ones):

o -'>-IP(m; XdM :m) -'>-lli(m; Rq(M)jXdTRq(M)) -'>-ll'(m; Rq(1f1.)) -'>- 0

rfl

t·'.

o -'>-llim(xdM :m)

-'>-

llim(Rq(M)jxdTRq(M))

~1' -'>-

llim(Rq(M))

-'>-

0

We know that l' is surjective for i < d and we now show that A~ also is surjective for i < d. Then Ai is surjective for i < d, i.e. Rq(M)jxdTRq(M) is a Buchsbaum module. But then Rq(M) is a Buchsbaum module by Proposition 1.2.19, since xaT is a non-zero divisor with respect to Rq(M) and xaTllim(Rq(M)) = 0 for all i < d by Pr~position 2.3. Since dimR (A)XdM:m = dimA(xdM:m) dim M, the surjectivity of A~ for all q --i < d is equivalent to the Buchsbaum property of XdM :m (over Rq(A)), cJ. Theurem 1.2.15. But XdM:m is a Buchsbaum module over Rq(A) if and only if XdM:m is a Buchsbaum module over A (by virtue of the canonical epimorphism Rq(A) -'>- A). Consider the exact sequence 0 -'>- M -'>- xaM: m -'>- (Xd1ll: m)/xdM -'>- 0, where the first map is given bym t-+- xd,m. Now H:n{(XdM:m)/XdM) = 0 (sincedimA(xd1ll:m)/xd,M = 0) fur all i> 0, i.e. we have epimorphisms H:n(M) -'>- Hi(Xd1ll :m) for all 0 0). Therefore we get commutative diagrams (HO(XdM:m) Hi(m; M)

H'(m; xtlM :m)

-'>-

1 1.1

z.M:m

-'>-

H:n(xtl M :m )

(where the vertical homomorphisms are the canonical ones). But the surjectivity , of A~ for i < d implies the surjectivity of A!.M:m for i < d, Le. xdM:misa Buchsbaum' module, q.e.d. . Finally, we will give a characterizati.on of the Cohen-Macaulay pr.operty .of Rq(M) where q runs thr.ough all parameter ideals .of M. We have Theorem 2.11. Let M be a Noetherian A-module 0/ dimension d 2. Let S denote the set of non-zero divisors ~n A with respect to M. The following conditions are equivalent: (i) M is a Buchsbaum nwdule with H:n(M) = 0 for aU i =F 1, d. (ii) RQ(M) is a Cohen-Macaulay nwdule for all parameter ideals q of M. (iii) There is a d-dimensional Noetherian Cohen-Macaulay A-nwdule N with M ~ N <;; M s and m . N c M. Proof: (i) =} (ii) is an immediate c.onsequence .of Pr.opositi.on 2.3. (ii) =} (i): SinceH~(M) lll~(Rq(M))lo 0, we have depth M > O. By The.orem 3.3, which we will prove in the next secti.on, we get that III is a Buchsbaum m.odule. By using again Pr.oPosition 2.3 we find H:n(M) = 0 f.or all i =F 1, d. (i) =} (iii): Since H~(M) = 0, we have an exact sequence

o -'>- M -'>- HO(M) -'>- H:n(M) -'>- 0, where HO(M)

~

HomA(m"; M) (cJ. Lemma 0.1.8 for the notation). Since H:n(M)

n

is N.oetherian HO(M) is Noetherian as well and m . H;\(M)

=0

implies m . HO(M)

M.

§ 3. Blowing-up characterization of Buchsbaum modules

223

Also by Lemma 0.1.8 there is a natural embedding HO(M) ~ Ms (Ass M consists only of 1Il:inimal primes) and we have H~(HO{M)) = HMHO(M)) 0 and H:n(HO{M)) ~ H~(M) 0 for 2 ~. < d, i.e. HO{M) N is a C{)hen-Macaulay module. (iii) =? (i); We consider the exact sequence 0 ->- M ->-N ->-N/M ->- O. Since m(N/M) = 0, dimA(NIM) 0, i.e. H~(N/M) 0 for i> 0 and H~(N/M) NIM. Therefore the long exact cohomology sequence gives isomorphisms H~(M)::: H~(N) for 2, H~(M) = 0 (since H~(N) = 0) and an exact sequence 0 ->- N/M ->- H:n(llf) ->- H:n(N) ->- O. But N is a Cohen-Macaulay module and therefore H~(M) 0 for all i =1= 1, d and mH:n(M) "" m(N/M) 0, q.e.d.

§ 3.

Blowing-up characterization of Buchsbaum modules

Troughout this paragraph M denotes a non-zero Noetherian A-module. First of all we need the following

Lemma 3.1. Let q be a parameter ideal with respect to M. Assume that q is generated by a system 01 parameters 01 M. Let ~ be a prz~me ideal 01 Rq(A). Then (i) mRq(A) is a pnme ideal in Rq(A).

(ii) Rq(M)~ =1= 0 il and only il [~]o E Supp M. (iii) GqUW)~ =1= 0 2/ and only il [~]o = m. Prool: (i): There is an n EN with mlt(M/qM) = O. Therefore mnRq(M) qRq(M) <;:;::; mRq(M), hence dim Rq(llf)/mRq(M) dim Gq(M) dim M. Let A:= A/Ann M, if qA, iff:= mA. Then if is a parameter ideal of A and we have an epimorphism :rr:: Rq{A)/mRq(A) ->-Rii(A)/iffRij(A). If q = (Xl> ... , Xd) A (d dim M = dim A) then there are additional natural epimorphisms (Xl> .•. , Xd in-

determinates) : 'P: (A/m) [Xl' ... , X d] ->-Rq(A)/mRq(A) ,

defined by X j

1--+

xjT and

p: (A/iff) [Xl' ... , X,d

->-Rij(A)/ffiRij(A)

defined in the same way. Clearly, A/iff "" Aim and we have a commutative diagram (Aim) [Xl' ... , X d ].!..;. Rq(A)/mRq{A)

-"I

~"

(A/ffi) [Xl' ... , X d ].4 Rq(A)/iffRq(A).

Now (A/iff) [Xl' ... ) X d] is an integral domain of dimension d. As we have seen in the beginning Rii(A)/ffiRij(A) is a (graded) ring of dimension d. Hence p must be an isomorphism and therefore 'P is an isomorphism, i.e. Rq(A)/mRq(A) is an integral domain. This proves (i). (ii) and (iii): The "only-if-parts" are clear. By Lemma 1.7 and Corollary 1.8 it is enough to prove: [~]o E Supp M =? Rq(M)$ =1= O.

224

IV. Rees,modules and associated graded modules

Lsing the same argument as in the proof of Corollary 1.8 we may assume without loss of generality that ~ is homogeneous. It is sufficient to prove Rq{M)($) =+= O. If [~]o ~ V(q) then Rq(M)(,Xl) ~ M nll1 • =+= 0 by Corollary 1.4 and Corollary 1.5. If [~]o E V(q) then [~]o E V(q) n Supp M = {m}, i.e. [~)o = m. Take a prime ideall" E Supp M with dim All" dim M, Then l" E Ass M and we have a monomorphism All" --+ M which induces a monomorphism Rq(A/~) -'; Rq(M). Note that q is a parameter ideal of All", Hence it is enough to prove Rq(A/~)('$) =+= O. If RQ(A/~ )'$) 0, there is an 11, E N and apE q" such that pTn ~ ~ but pT"(A/~) = 0, i.e. p E f" Now write q = (Xl> , .. , Xd) A (d = dim M), take indeterminates Xl' .. " Xd and an element I E A[Xl' ... , X d), homogeneous of degree 11, with p = I(Xl> ... , Xd)' Let { denote the image of I in (AM [Xl' ... , X d]. Then {(Xl> ••• , xd) 0 and Theorem 21 of Zariski-Samuel [1], p. 292, shows, that all coefficients of { belong to the maximal ideal of A/~, i.e. all coefficients of I belong to m. But this means p I(xl> "', Xd) E mq" hence pT" E mq"T" = [~]o [Rq(A)). ~ ~, a contradiction. This concludes our proof, q.e.d. Now we can prove:

Theorem 3.2. Let M be a Noetherian A-module 0/ positive dimen8ion. Tken the lollowing statements are equtvalent: (i) RQ(M) is a locally Ooken-Macaulay rrwdule lor all parameter ideals q 01 M. (ii) Gq(M) is a locally Oohen-Macaulay module lor all parameter ideals q 01 M. (iii) M/H~(M) is a Buchsbaum rrwdule. Prool: The equivalence of (i) and (ii) is a consequence of Corollary 1.12. To prove the equivalence of (ii) and (iii) we first note that we' can assume without loss of generality that A is complete and that depth M > O. This is clear for (iii) (c.£. Lemma 1.1.13). On the other hand, Gq(M) ~ Gq(M) for all parameter ideaJs q of M (as Rq(A )-modules, hence as Rq(..4)-modules by using the canonical map Rq(A) -'; Rq (..4) of graded rings). Conversely, for a parameter ideal b of M (in ..4) there is a parameter ideal q of M (in A) with bnM/b n+1 M'"'-' qnM:jqnHM, i.e. G.(M) Gq(M) and therefore we can assume that A is complete. Furthermore, let n: denote the canonical epimorphism Gq(M) -'; Gq(MIH~(M)). Since for 11, 0 Ker[n:]" = qflM n (q... l 11f + H~(M))/qfl+lM

0

for all sufficiently large 11" there is an integer t > 0 with roll Ker n: = O. Therefore Gq(M)(\ll) '"'-' Gq(M/H~(llf)k.jl) for all ~ E Proj Rq(A) and we can assume that depth M

>R

.

First we prove the implication (iii) =? (ii). But this is a consequence of Theorem 2.1, since Gq(M)\JJl is a Buchsbaum module, hence Gq(M)'l3 is a Cohen-Macaulay-module for all ~ E Proj Rq(A) and all parameter ideals q of M (d. Corollary 1.1.11). Therefore Gq(M)(\ll) is a Cohen-Macaulay-module for all ~ E Proj Rq(A) by the remarks made in

§1. Now we prove the implication (ii) =? (iii). We use induction on d:= dim M. If d 1, there is nothing to prove. Let d > 2 and Xl> ••• , Xd a system of parameters of M. Set q (Xl' ... , Xd) A. If ~ E Proj Rq(A) with Gq(M)(\ll) =+= 0 then [~]o = m by Lemma 3.1 and therefore the prime ideal mRq(A) (Lemma 3.1) is contained in ~. Hence mRq(A)

§.3. Blowing-up characterization of Buchsbaum modules

225

.is the only minimal element in Supp Gq(M). Since depth Gq(M)(\ll) > 0 for every $ E Proj Rq(A) with mRq(A) $ 9Jl, Ass Gq(M) c:::;; {mRq(A), 9JC). Since xlT tt mRq(A), O:G q(M) Xl T is annihilated by some power of ,9Jl, i.e. there is a c > 0 with (q"M :M Xl) n qeM q"-IM for all n > c. Hence by Krull's Intersection Theorem qe(O:i\fXl) c:::;; (O:MX1 ) n qeM = () (q"M:MX1) n qeM = () q"-lM O. Therefore O:MXl c:::;; O:Mqe

**

n>c

n>c

0 since depth M > 0 and Xl is a non-zero divisor with respect to M. By the Lemma of Artin-Rees (c.f. Matsumura [1J (H.E), p.70) there is an integer r> owith q"M:MXl = q"-'(q'M:MXl ) c qeMforalln> r +c. This implies for n> r + c: ,, q"M:MXl (q"M:M Xl) n qeM = q - lM. =

We let M':= M/xlM. The canonical epimorphism Gq(M) -;.. GqUIf') factors through Gq(M)/xlTGq(M), Le. we have an epimorphism n: Gq(Ml/xITGq(M) -;.. Gq(M').

Let q' :=

(X2,"" Xd)

Ker[n]"

A. Then for all n > r

+ c:

(q"M n (xIM'+ q'fl+lM))/(qfl+lM (q/ ..+1M

+ (q"M n x

For all n < r + ewe have mt Ker[n].. annihilates Ker n, i.e. (Gq(M)/xlTGq(M))('ll)

F 'lf))/(q',,+lM

+ x1q'HM)

+ xlq"-lM)

O.

0 for sufficiently large t. Therefore a power of 9Jl

Gq(M')('tJ1

for all $ E Proj Rq(A).

Hence Gq(M') is a locally Cohen-Macaulay module. Assume we have proved (iii) in case a = 2. Then for a> 3 we know that M/x1M:M(m) (M/XIM)/H~(MlxIM) = M'/~(M') is a Buchsbaum module. Therefore M is a Buchsbaum module by Corollary 1.2.24, since A is complete and hence an epimorphic image of a local Gorenstein ring. Thus it remains to show (iii) in case a = 2. Let Xl' X2 be an arbitrary system of parameters of M. Since O:MX l = 0 as demonstrated above, Xl' X2 is a filter-regular sequence of M, see Definition 1 of the Appendix. Then by Proposition 16 of the Appendix, l(H~(M)) < 00. Therefore we can find an X E m with dim MjxM = 1 and xHfn(M) = O. Take y E m with dim M/(x, y) M = 0 and let q := (x, y) A. We set $ mRq(A) + xTRq(A). If Yisan indeterminate, we have an epimorphismip: (A/m) [Y] -)- Rq(A)/$ defined by Y ~ yT. Now $ is not 9Jl-primary, hence dim Rq(A )/$ 1. But (A/m) [YJ is a one-dimensional integral domain and therefore ip is an isomorphism. Hence $ E Proj Rq(A). Let

(I:=

xT yT

E Rq(A)('tJI' The natural embedding A c Rq(A) (of rings) induces a

homomorphism of rings

Let

aT"

E Rq(A)('tJ), where a, p E q", pT" tt $, i.e. p tt mq" + xq"-l. We write " aix'y,,-i, P = E bix'y"-' (ai' bi E A, i 0, ... , n). Then bo tt m and we have

ft =

"pT"

a

E

;=0

;=0

ft =

ao

-

bo

15 Buchsbaum Rings

+ {1ft ,

with

ft'

IV. Rees modules and assooiated graded modules

226

This shows that;' is surjective. Replacing A by 1Y we obtain by similar calculations an epimorphism of A-modules ,>

.

'P: lrI

-i>-

Rq(M)('J,HIgRq(M)(ill}

(we consider Rq(M)(ill)/gRq(M)(ill) via;' as an A-module). By Lemma 1. 7 a prime ideal ~ is minimal in Supp Rq(M) if and only if ~ = p, where l:.J is minimal in Supp M. Let l:.J be minimal in Supp M. Then dim AIl:.J = 2 and we have a chain of primes in Rq(A): pc mRq(A) c ~ c IDl (l:.J n qi c;;; mqi since q is a parameter ideal of A/+" compare with the argument used in the proof of Lemma 3.1). Hence dim Rq{M)(ill) = 2. Also it is easy to see that g ~ ,pRq(M)(ill)' This shows that g is a parameter element of Rq(M)(ill) and therefore Rq(M)('i)JgRq(M)(I'f,} is a one-dimensional Cohen-Macaulay module. Hence MJKer 1j! is a Cohen-Macaulay module. Claim: Ker'P = xM :y. xm', then 'P(m) = gm' 0, i.e. xM:y Ker 'P' Since O:MY = 0 . If mE X1Y:y or ym we have an exact sequence 0 -i>- M ~ 111 -i>- MJyM -i>- O. By applying the local cohomology functors we get a monomorphism (H~(M) = O}: H?n(M/yM) -i>- Hin(M). Now H~(MJyM) = (yM:(m»)lyjYJ. Since xHin(M) = 0, x(yM:(m») C;;; y-,W, i.e. yM:(m) C;;; Y1W:X C;;; yM :x 2 C;;; ... ylW: (m). Therefore yM:x = yM:x 2

••• =

yM:(m).

Let m E Kef 'P' Then there are n E H, m' E q" M, p E q" with pT" 4. ~ such t4at 'T" m g ~ in Rq(M)(\lll' Hence there are a i E H and a q E qf, qiT ~ ~ with

pTfI qTl(mypTfl+l xm'Tfl+l) 0, i.e. q(myp - xm') 0 (in M). Write p = xp' IXY", q = xq' + pyl with p' E qfl-l, q'E qH. Since p, q ~ ~, IX, P ~ m. Furthermore (x, q) A = (x, yl) A, i.e. x, q is a system of parameters of M and we have O:Mq O. Hence 0= myp - xm' = IXmyfl+l - xm" with m" m' - myp' E q"M. We write m"

+

II

xtyflimi' If x"ym~.

n>

0, x,,+lmn E yM, hence mn E yM:X"+l = Yll1:X. Let xfl+lm.

Then we obtain

y(lXmy" - x

n-l E x i y ..-i-lmPI)

0

'=0 with mJlI:=

mi

for 0

i< n

2, m~~l:= m n-l +~. Since O:MY

0, we get

1&-1

my

x

E x'yl&-i-lm11)

O.

ieaO

Repeating this procedure, we obtain after n steps my-xm~'J

0,

i.e.

IXmExlll:y.

Therefore m E xM: y (IX is a unit in A) and this proves our claim. Since M/xM:y is a Cohen-Macaulay module, X1W:y is unmixed in M, i.e. xM:y xM : (m). Therefore yH?n(MJxM) = y(xM : (m)JxM) O.

§ 3. Blowing.up characterization of Buchsbaum modules

227

Since xH:n(M) = 0, the exact sequence 0 --+ M ~ M --+ M/xM --+ o results in an isomorphism H~(M/xM)::::::. H:n(M) and therefore yH:nUI:1) = 0, where y is an arbitrary element of m with dimA M/(x, y) M O. If we choose an (Mjdl}basis Yh ... , Yt of m (d. Definition 1.1.7 and Proposition 1.1.9) it follows that YiHin(M) = 0 for all i = 1, ..., I, i.e. mHtn(M) = 0 and M is a Buchsbaum module by Proposition 1.1.12, q.e.d. In the last theorem of this paragraph we collect the main results of this chapter. Theorem 3.3. For a Noetherian A.module M of positive dimension with depth M the following coruiitions are equivalent: (i)

11{ is a Buchsbaum module.

(ii)

Gq(lll) is a Buchsbaum module for all parameter ideals q of M.

>0

(ii)' GqCJ1) is a locally Cohen-Macaulay module for all parameter ideals q of M. (iii) Rq(.M) is a Buchsbaum module for all parameter

~"deal8

q of M.

(iii)' Rq(M) is a locally Cohen--i.'lfacaulay module for all parameter ideals q of M. Proof: The equivalente of (i) and (ii) is the content of Theorem 2.1 and the equivalence of (i), (ii)' and (iii)' was proven previously in Theorem 3.2. The implications (ii) ~ (HY and (iii) ~ (iii)' are clear by the remarks of § 1. Finally, the implication (i) ~ (iii) , follows from Theorem 2.10, q.e.d.

Remark 3.4. Without the assumption depth ill> 0 (but dim M the following implications: (i)

¢>

(ii)

.~

(iii)

¢>

(iii)'

,u,

,ij, (ii)'

2) we only have

The following example shows that none of the implications (ii) ~ (ii)', (ii) ~ (iii) and (iii) (iii)' is gencrally reversible. Example 3.5. Let M be a Noetherian A-module of dimension 2 and with mHMM) We prove for every parameter ideal q of M: (i) !.!im(Rq(M») (ii)

0 for i

o.

= 1,2.

[!.!~(Rq(M»)]p {~~(M) n qPM) TP

for p for p

<

0, O.

°

Proof: Let M M/H~(M). Then depth Ii> and mH:n(Ii) = mH:n(M) = 0, i.e. M is a Buchsbaum module by Proposition 1.2.12. The~fore !.!im(Rq(M») = 0 for i 0,1,2 by Proposition 2.3 for all parameter ideals q of M (of M). The ker2:el of the natural epimorphism Rq(M) --+ Rq(M) must be !.!~(Rq(M») (since depth Rq(M) > 0). An easy calculation shows that

for p for p and this proves (ii). 15*

<

0, 0

228

IV. Rees modules and associated graded modules

If we apply

!Fm( ) to the exact sequence

o -+H~(Rq(M)) -+Rq(M) -+Rq(M) -+ 0, we find Hk(Rq(M)) ~ Hk(Rq(M)) 0 for i = 1,2 and this proves (i). Now let us consider the following two cases: (a) ~(M) = O. Then Wl.H~(Rq(M)) 0 by (ii) and Rq(M) is a Buchsbaum module for all parameter ideals q of M by Proposition 1.2.12 although M need not be a Buchsbaum module (see, e.g., Example 1.2.5). (b) mH~(M) =+= O. Then IDlH~(Rq(M)) =+= 0 and Rq(M) is not a Buchsbaum module but a locally Cohen-Macaulay module for all parameter ideals q of M. (a) shows that (iii) =} (ii) and (ii)' =} (ii) cannot be true in general and (b) shows that (iii)' =} (iii) is not true in general.

Chapter V Further applications and examples

§ 1.

A Buchsbaum criterion for affine semigroup rings

We have seen that the ring

K[84,

83 •

t,

8 •

ea, t4 J

is a non-Cohen-Macaulay Buchsbaum ring. In fact it is a special case of an affine semigroup ring. More general, we will prove in the following a Buchsbaum criterion for affine semigtoup rings. To accomplish this K will be a field, S a finitely generated (additive) submonoid of Nil, and L the subgroup of H := Z" generated by S. We set drank L. Let K[H) denote the group algebra of Hover K and let x a denote the image of a E H in K[H]. For every subset VofH we define K[V]:= 1: K ·xa. BEV

The ring K[S) of course coincides with the monoid algebra of S over K and may be considered as an H-graded subring of K[H]. For this and more technical results concerned the subject, compare the article [2J of S. Goto and K. Watanabe. Also we define

S :=

{a ELI t . a E S for some integer t

> O}.

S is called the normalizaLwn of S. It is known that K[S) coincidcs with the normalization of the ring K[S), compare M. Hochster [1], § 1. For simplicity we assume that S = L n N". In general, S is isomorphic to a monoid of this form, compare M. Hochster [1), § 2. Next we define additional notation. For i 1, ... , n let

Fi

{(aI' ... , a.) E S I aj

O}

and

S,

Let L j denote the subgroup of L generated by Fit 1 ~ t' + j. Set

S - Fj• n. We assume Li

+L

j

for

Then S is a finitely generated submonoid of S containing S (compare S. Goto and K. Watanabe [2], Proof of Lemma 3.3.8). S. Goto and K. Watanabe showed in [2], 3.3.3, that K[S] is a d-dimensional Cohen-Macaulay ring. Unfortunately, this result is not correct in general. A counterexample was given by Ngo Viet Trung and Le Tuan Hoa [1]. Moreover, this paper also contains a corrected version of the result of Goto and Watanabe. Especially, as a consequence of this new criterion, one sees that the property of K[S] being Cohen-Macaulay is dependent upon the characteristic of K. Therefore we will suppose the Cohen-Macaulay property of K[ S] for the following statement.

230

V. Further applications and examples

Theorem 1.1 (S. Goto 12]). Assume that KIS] is a Oohen-Macaulay ring. Let m be the unique H-qraded maximal ideal of K[S], i.e. m = K[S '" {On, and suppose that d := rank L 2. Then the following conditions are equivalent: (i) A:= K[S]m is a Buchsbaum n"ng. (ii) m· H~'A(A) = 0 for all i 0, ... , d 1. (iii) (S'" {O}) + S ~ S. In this case H~.A(A) 0 for all ~. =j= 1, d and leA)

= (d -

1)· :j:j:(S '" S),

where :j:j:(S '" S) denotes the numher of elements of S '" S. Proof: (iii)::::} (i): Set B := K[S]m' Then A <;;;:;; B <;;;:;; Q(A) and B is an intermediate ring such that B is a d-dimensional Cohen-Macaulay ring which satisfies the condition (iii) of Theorem 1V.2.11. Thus A is a Buchsbaum ring. The last assertion also follows from Theorem 1V.2.11, since by Proposition 1.2.6 . leA)

d-l(d ~ ,2: 1=0

1) .lA(H~'A(A»)

= (d

t

= (d - 1) . :j:j:(S '" S).

The implication (i)::::} (ii) is clear (see Corollary 1.2.4). (ii) ::::} (iii): By applying the functor H~( ) to the exact sequence

o ~K[S] ~K[S] ~KrS]/K[S] ~O we obtain H~(K[S]IK[S]) H~l(K[SD for every i = 0, ... , d - 2. Notice that SUPPK[s](K[S]/K[SD {m}. Assume this not to be true and let r:= dimK[s]K[S]/K[S]. Then 0< r d - 2 (compare Goto-Watanabe [2], Proof of 3.3.3) and (by our last result) H'm~i
is 8. non-Noetherian A-module (see Chap. 0, § 1, 3.). But this contradicts the fact that all H~'A(A), i 0, ... , d 1 are finite-dimensional vector spaces over A/m . A by our assumption. Therefore H~(K[S])'-"'" K[S]/K[S] which is annihilated by m. This S <;;;:;; S and we implies m· K[S] <;;;:;; K[S]. This is of course equivalent to (S'" {OJ) have completed the proof, q.e.d.

+

Corollary 1.2. Under the same hypotheses as in Theorem 1.1 suppose that (S'" to}) S c S. Then A := K[S]m is a Buchsbaum ring with leA) (d 1) • :j:j:(S '" S) and H~.A(A) 0 for all i =j= 1, d.

+

Proof: It suffices to show that S = S. First notice that m· K[S] K[S] and we have lK[S](K[S]/K[S])

<

<;;;:;;

K[S] since m· K[S]

00.

Thus the assertion follows from the fact that K[S] and K[S] are Cohen-Macaulay rings of dimension d. For the proof of the Cohen-Macaulayness of K[ S] see M. Hochster, [1], Theorem 1, q.e.d.

§ 1. A Buchsbaum criterion for affine semigroup rings

231

Next we want to discuss some examples: Examples 1.3. Let

1',

d be integers with d

2 and

l'

1 and let

d

T := {(aI' ••• , ad) E Nd

I E (11

0 mod r} .

;=1 Ii

For a subset 1 of {(aI' ... , ad) E Ne

I E aj

r} we set 8 := T "1. Then 8 is a finitely

1=1

generated sub monoid of Nd with T (8 " {O})

+Tc

Sand

8.

Thus, by Corollary 1.2, A := K[S]m is a d-dimensional Buchsbaum ring with 1(A) = (d - 1).:j:j:1 and H:n.A(A) 0 for~' 1, d. In particular, the famous ring K[8 4, 8 3 • t, 8' tS , f4] '" K[xo, Xl' X 2, X3J/~ is a special case of this example (take l' 4, d 2 and 1 (2,2». In his fundamental paper [2] W. Grobner showed that the ideal +'l is a speeial member of a large class of imperfect ideals. The result shows that projections of Veronesian ideals (perfect ideals) lead us to certain degenerations (imperfect ideals). In view of this phenomenon W. Grobner posed the problem to classify simple projections of Veronesian ideals, compare [2], p. 263, In the sequel we give a complete solution of this problem. In particular, we show that the local rings at the vertices of affine cones over almost all projections of Veronesian varieties are Buchsbaum rings: Let Md,m be the semigroup generated by 0 and all elements (ao, .. " ad) E Na+l with ao ad = mover Nd+l. For an arbitrary field K we denote by R d•m K[Md,m] the corresponding affine

'*'

+ ... +

selnigroUp ring. Rd,m is the coordinate ring of the image of Pi- in P~, N

(m ~ d) _ 1,

+ .,. +

by the Veronesian embedding. H (i) (io, .. ,' i d ), io id m, we denote by M~:~ the subsemigroup of Md,m generated by all (ao, ... , ad) E M d.m with (ao, ... , ad)

'*' The (i). affine semigroup ring R~'". is then the coordinate ring of the projection of the

Veronesian variety from infinity to the hyperplane X(i) we can assume

O. After changing the variables

For short we write (0) for (m,O, ... ,0) and (1) for (m we assume d > 1 and m > 2.

1, 1,0, ... ,0). In the following

Theorem 1.1. Let (i) properties : (a) depth R~:'".

"* (0), (1).

Then the affine semi(fl'oup ring R~~ has the follOOJing

= 1.

(b) R~.'". is a Buchsbaum ring wz'th 1(R~'".)

d.

0/ X := V(b~'".) c p~-l (b~~" is the de/in~1t.g ideal 0/ the projectWn Veronesian variety in P~-l) we have

(c) For the cohomology

0/ the

r-

=

232

V. Further applications and examples

I

n

dimK 0 Hr(x, e?x(n))

HO(X, e?x(n))

do.m. [R(i) ]

Hd(X, e?x(n))

l
-1

0

0

0

(-I)d. (n.:+d)

0

1

1

0

0

1

(m; d) _ 1

(m ;d)

0

0

(n.: +d)

(n.: +d)

0

0

>2

Proof: All statements of the theorem will follow from Example 1.3 by setting = M do•m '" {(io, ... , ia)}. This equality follows from simple calculations, q.e.d.

M~~~

In Theorem 1.4 only the cases (i) =l= (0), (1) are handled. Now we add some results concerning the case (i) (0), (i) = (1).

Theorem 1.5. (a) The al/ine semigroup rings R~~~,

Ri:!" and R~~4 are Oohen-Macaulay rings.

(b) For the local cohomology of R

R~~!,.,

m

3, d

2 we get:

dimK 0

n

[H~(R)].

S

-1

0

0

0

0

;;:::1

[H~(R)].

[Hi"n(R)]m 2

< rS

d

[H~I(R)].

0

(-I)d.

1

0

0

0

0

0

(e) For the local cohomology of R := R~~~, d

(n.: + d)

3 we have:

dimK 0

n

lHi"n(R)]., 0
< -1

< 3


d

[H~+I(R)].

lHi:t(R)].

[lfm[R]., 3

0

-n

0

( -1)do .

0

0

0

0

1

1

0

0

0

0

e.

nd+ d)

Proof: (a) According to Theorem 1 of M. Hochster [1] it is enough to show that the semigroup M~~!,. is normal, compare Corollary 1.2. In fact, this is easy to check. For the remaining two cases the proof follows readily.

§ 1. A Buchsbaum criterion for affine semigroup rings

233

, (b) We consider the inclusion map R~~~ --+ Rd,m' Let N be its cocernel. By an easy computation we obtain for the Hilbert function if n

=

if n

> 0.

0,

Hence for N it follows that H(n,N) =

for n = 0, for n > 0.

{~

The next step is to show that N is a one-dimensional Cohen-Macaulay module over But this follows because eo(m, N) 1. Since Rd ,.. is also a Cohen-Macaulay module over R~~~, we get !!.:n(R) !!.:n(N) and !!.~t(R) = 0 for r =F 2, d + 1. Now, comparing the Hilbert function and the Hilbert-Samuel function of R~~~ we get

R~~.

if n S 0, if n > 0.·

Thus, the statement has been proven. (c) The proof of this statement is obtained similarly. We remark only, that the cokernel N of the embedding R~J 4 Rd,2 is a two-dimensional C{)hen-Macaulay module over R~~~. Furthermore,

q.e.d. We also can consider multiple projections of Veronesian ideals. Here the situation is more complicated. We illustrate this by two examples. Examples 1.6. We consider R} := R~:~·Ij)) with (i) (4,2), (i) (3,3) and R z := R~~~)'(/)) with (k) = (4,2), (l) = (2,4). Then we get for the local cohomology: dimK 0

n 1 0 1 2

>3

(Rl]n

(R 2 ]n

[!!.~.(Rl)]n

[!f.:n.(Rz)]n

(!!.~(Ri)]n' i

0 1 5 12 6'n

°5

0 0 2 1

° °°

-(6· n 0 0 0 0

1

+1

13

6· n

+1

°

0 2

1,2

+ 1)

Hence, R2 is a Buchsbaum ring while Rl is not a Buchsbaum ring. Furthermore, the new results of Bresinsky [2] and Trung [8], [11] open the way to some deep lying applications of Goto's general Buchsbaum criterion in terms of semigroup ideals.

'234·

V. Further applications and examples

Some examples related to problems of Hironaka and Seidenberg First we need to mel!tion a question posed by Hironaka in a discussion (at the Uni~ versity of Halle in 1974), who asked whether one can construct normal (and factorial) non-Cohen-Macaulay Buchsbaum singularities. By using Segre products of graded Cohen-Macaulay modules we have constructed normal (local) Buchsbaum rings A for every given dimension d 3 and depth t 2 such that d> t (see Example 1.4.14). We now give another example: Let F be an arithmetically normal irregular surface. Let A be the local ring of the vertex of the affine cone over F. Then A is a normal non-Cohen-Macaulay ring. We claim that A is a Buchsbaum ring if Hl(F, Op(p)} 0 for all p =F O. Our assertion follows immediately from the following corollary of Proposition 2.4 below:

Corollary 2.1. Let F be an arithmetically normal surface in projective space Pic over an arbitrary algebraically closed f£eld K. Let A be the local n'ng of the vertex of the affine cone over F. If Hl(F, 0 F(P)) =F 0 for precisely one p E Z, then A is a non-Oohen-Macaulay Bu:chsbaum n·ng. We will give some examples of such surfaces. Example 2.2. Take any non-singular irregular surface F, any ample line bundle :l on F, and embed F in Pic by a sufficiently high power :legl. Then F is arithmetically normal and we know that Hl(F, 0F(P)) =F 0 only for P ~ O. Corollary 2.1 shows that the local ring of the vertex of the affine cone over F is a normal (non-CohenMacaulay) Buchsbaum ring. Example 2.3. G. Horrocks and D. Mumford [1] have constructed an indecomposable ra.nk 2 vector bundle 3' on P := P~ with the following exact sequence: 0-»- Op( -5) -')- 3'( -5) -;. Op -;. 01' -;. 0,

where Visan non-singular abelian surface. 3'is such that H2(3'(-5)}:::: K2, H2(3'(p- 5)) = 0 for P =F 0, Hl(3'(p - 5)) 0 for p 6 and P:::: -6. Therefore, if we embed P in P(T(Op(6))) = Pj?9

by a Veronese morphism, we get an embedding of V in Pj?9 which makes V arithmetically normal and such that Hl( V, 0 v(p)) = 0 if and only if p =F O. Let A be the local ring of the vertex of the affine cone over V c Pj?9. Then A is normal non-CohenMacaulay ring and Corollary 2.1 shows again that A is a Buchsbaum ring. By proving our Corollary 2.1 we relate these questions to some global questions by considering affine cones over projective varieties in projective spaces. By studying the vertices of these cones we can give a proof of Corollary 2.1. In order to do this we make use of local cohomology, for which we obtain the following vanishing theorem: Proposition 2.4. Let V be a subvariety all of whose irreducible components are of dimension d;::: 2 in projective 8pace'Pic over an arbitrary algebraically closed field K. Let (A, m) be the local n'ng of the vertex of the affine cone over V. Let t : = depth A > 2 and H~(A) = 0 for all i =F t, d + 1. Tlum we have: (i) A i8 a Bu:chsbaum n'ng with invariant I(A) =

G)

if and only

if HH( V, Ov(p)) =F 0

for precisely one p E Z and if for this p we have dim HH(V, Ov(p))

=

1.

§ 2. Some examples related to problems of Hironaka and Seidenberg

235

(ii) A is a Bw;hsbaum tum-Oohen-Macaulay ring and there exists an integer p E Z such that

I(A)

e)

=

dim Ht-1(V, Oy(p))

if and only if

Ht-l(V, Oy(j)) Proof: For d

>

=

0

for all

i =l= p.

1 there are isomorphisms (of A-modules)

EB Hi(V, Oy(p)) '" H~l(A) pe%

(see Proposition 0.2.3 and the arguments used in the proof of Corollary 0.4.7). If A is a Buchsbaum ring of Krull dimension d 1, then we have by Proposition 1.2.6

+

By hypothesis we obtain

I(A)

e)

dimAlm(H:n(A)).

This invariant for the Buchsbaum ring A, the above isomorphism, and Proposition 1.2.12 yield our assertions (i) and (ii), q.e.d. Proof of Oorollary 2.1. This assertion follows easily from Proposition 2.4(ii).

For the construction of factorial non-Cohen-Macaulay-Buchsbaum singularities we will apply the theory of abelian varieties. First we will show that an abelian variety with a very ample invertible sheaf defines always a Buchbaum ring. To be more precise: Let k denote any field and X an abelian k-variety with g dim X. Let:t be a very ample invertible sheaf on X. Then we consider the graded k-algebra

R :=

EB HO(X, :t®tI) ftE:%

with the naturally induced ring structure and

Rn := HO(X, :t®n)

for every n E Z.

In fact R is a finitely generated, non-negatively graded k-algebra. We put m =

EB Rn. n:21

Following Schenzel [1], 5.4, we will prove the following theorem. The key is our Proposition 1.3.10 and Mumford's [3] observations regarding abelian varieties.

>

dim X 1. Let :t be a very ample Theorem 2.0, Let X be an abelian variety with g inverHbleaheafon X. Then Rm is a Bw;h8baum nng with dim Rm g 1, depth Rm 2, and dimk H:n(Rm) =

I(Rm)

(i

g

1),2:;;.

= ( 2g ) g-1

g.

g, i.e.

+

236

v.' Further applications and examples

Proof: By using the results of D. Mumford [3], Section 16, we get the vanishing of

the cohomology

Hi(X, ,1'®") = 0

for 0


dim X and all n

'* o.

From this follows [g;;l(R)]n ~

H'(X, ,1'®n)

0

for all n

'*

0 and 0


dim X.

Also we have g~(R) 0 for i = 0, 1. It follows immediately from Proposition 1.3.10 that Rm is a (g + I)-dimensional Buchsbaum ring. Moreover we get from Mumford [3], Section 13, Corollary 2, that for n = and

°

°

dimX dimt Hi(X, ,1'®")

(~).

Therefore the results follow for the local cohomology of R m, the depth of Rm and for I(Rm), q.e.d.

In particular, let 0 be a smooth irreducible curve over a field k with genus g 1. Then the Jacobian variety J c of 0 is a g-dimensional abelian variety (see, for example, Mumford's [4] lectures on curves and their Jacobians). Example 2.6. Let k be a field of characteristic O. Let k(t) be the field of rational functions in one variable t over k. S. Mori [1] has constructed smooth and geometrically irreducible hyperelliptic curves 0 of genus glover k(t) such that for the Jacobian variety J of 0 and its theta divisor 8 the completion k of R

E& HO(J,

8®")

nEZ

with respect to m

E& HO(J,

8 0 ") is a factorial domain. Hence we get non-Cohen-

n~l

Macaulay factorial Buchsbaum rings Rm for g>- 2 with dim Rm = g and I(Rm) =

+ 1, depth Rm

2

(g 2g 1) g.

By using the Theorem 5.8 of Freitag-Kiehl [IJ, R. Kiehl [1] has constructed other examples of non-CD hen-Macaulay factorial Buchsbaum rings of depth 3 and, for example, of dimension 60. Moreover, there are non-Cohen-Macaulay factorial rings which are not Buchsbaum rings. This fact was mentioned by R. Kiehl [2] in a private letter: " ... Mit Hilfe Ihres Theorems 2 (see our Proposition 2.4) kann Herr Freitag ein Beispiel eines faktoriellen Ringes konstruieren, der nicht zur Klasse der BuchsbaumRinge gehOrt, niimlich: Der Kegelspitzenring des graduierten Ringes aller Modulformen zur vollen Siegelschen Modulgruppe geniigend hohen Grades ist faktoriell (see Freitag [1], Theorem 3.6), aber nicht Buchsbaum. Es ist aber nicht klar, ob die Komplettierung dieses Ringes faktoriell ist, dazu mii.l3te zumindest die Tiefe gro.l3er oder gleich 3 sein (s. Bericht von Lipman in den AMS)."

,§ 2. Some examples related to problems of Hironaka and Seidenberg

237

Today, the deep lying reason for this observation follows from results of Tsuchihashi' [1] and Ishida [1,2]. Other examples of factorial rings which are not Buchsbaum rings follow from Mori [1], Example 2.4. Also, Bertin's [1) four-dimensional non-Cohen-Macaulay factorial domain is not a Buchsbaum ring. Next we discuss p.gain Seidenberg's problem. A. Seidenberg [2], p. 620, has considered the difference between the dimension and depth of homogeneous polynomial ideals taking this difference as a measure of the deviation from the Cohen-Macaulay property. Our Example 1.4.14 shows that this measure does not describe the non-Co hen-Macaulay property well. In the following we describe another such class of projective varieties by given their defining equations. For this we examine our ideals of type (r, d) studied in Definition 1.2.13 and Lemma 1.2.14. We get the following class of projective varieties: Proposition 2.7. Let d, r be arbitrary integers with 1 S r < d. Then there exist projective varietie8 V c p~a-l over an arbitrary field k such that the local ring A 0/ the vertex 0/ the a/fine cone over V is a Buchsbaum ring with dim A

d

depth A

and

r.

Here A is given by the qu()tz'ent Adta where Ad = k[xl> .•. , X2d](z ......Z••l and a is the /ollounng ideal 0/ type (r, d) i/ r is odd: a

= (XIXd+l> X2Xd+2, ••• , Xd-IX2d-I, Xa, xi/ii,Xi,X; • •• , •• Xi,_,Xi) n =:

where

to

Xi

(XIXa+I, ••• , Xa-lXZ!l-l> Xu, Xi,Xi,Xi • ••••• Xi)

a1

n

az

is obtained by applying the permutatz'on

and the system 0/ indices (ib i 2, ... , ir) ru·ns through all r-tuples with 0 i,. < d. 1/ r is even, then a is given by Xi

< ... <

a

<

il

< iz

(XIXd+I, ••• , xd-lX2d-l> Xu, Xi,Xi,Xi • ••••• Xi)

n

(XIXd+I' ••• , Xd-lX2d-l, Xd, Xi Xi I Xi 3 •.•.• f 1

Xi r ).

Remark 2.8. This class of projective varieties provides another interesting application to canonical modules as given by Yoichi Aoyama in [11, Theorem 1. Example 2.9. (i) The ideal of type (3,5):

a

(x:;, XIX6' X2X7' XaXg, X4X9, X1XaX7, X 1X.X 7' X1X4Xg, X 2X 4X S )

n

(XIO' X I X 6 ' X 2X 7' XaXg, X 4X 9' X 2X 6 XS' X2X6X9, XaX6Xg, XaX7Xg).

(ii) The ideal of type (2, d) :

a

(xl> ••• , xa)

n

n

(X2' ••• , Xa+I)

(Xd+2' ... , X2a, xd

n ... n

n .•. n

(Xd+!' ••• , X2d)

(X2d' Xl> " ' , Xd-l)'

238

V. Further applications and examples

Prool 01 PrQ]losition 2.7: Lemma 1.2.14 shows that we only have to prove that the ideal a c Ad is of type (r, d). Let r 3 be odd. By Definition 1.2.13 we have to show that A d/a1 is a Cohen-Macaulay ring. It is not difficult to show that the elements Xl + Xd+l' X2 + Xd+2, ••• , Xd-I + X 2d-l> X2d define an Ad/aI-sequence of the d-dimensional ring Ad/a l • Hence Ad/a l is Cohen-Macaulay. If r is even, the proof is analogous, q.e.d.

§ 3.

On Buchsbaum rings obtained by glueing

We know already from the Introduction how a Buchsbaum ring is obtained by glueing, see Example 3 (3c). Following Shiro Goto [41 we now describe this procedure. Originally the concept of abstract glueing was introduced by C. Traverso [1] in order to clarify the structure of semi-normal rings. We also will study a key example involving this construction originally introduced by S. Goto [4] and G. Tamone [1]. In order to state our main result we need some definitions. Let S be a Cohen-Macaulay (not necessary local) ring with dim Sn = dim S for every maximal ideal (1 of S. Let R be a suhring of S and assume that S is module-finite over R. Hence R is again a Noetherian ring. For every prime ideal ~ (resp. $) of R (resp. S) we denote by k(~) (reap. k($» the field Rl)/~Rl) (reap. S'll/$S'll)' Let ~ be a prime ideal of R. We put W~~) =

{$ E SpecS' $ nR =

~}.

Notice that W(~) is a finite subset of Spec S. For every $ E W(~) let i'll: k(~) -7> k($) be the canonical monomorphism. We denote by 1($) the value of I at $ in k($) for IE 8 and $ E W(~). Then we have the following definition by C. Traverso [1]. Definition. We R' =

put~'

=

n $

\llEW(l)

and

{I E S , 3 C E k(~) such that 1($)

i'll(c) for every $ E W(~)}.

Then R' is a subring of S containing R and ~' is a prime ideal of R'. We call R' the glueing over ~. For an arbitrary local ring A we denote by emb (A) (resp. e()(A)) the embedding dimension of A (resp. the multiplicity of A relative to the maximal ideal of

4).

Now we are prepared to state the main result of this paragraph. Theorem 3.1. With the stated hypothesis let d dimRl)' Then A:= R~ is a BuChsbaum local ring 01 dimension d. Further, .mppose d> 0 and let ill denote the maximal ideal 01 A. Then we have:· (1) H:n(A) = 0 lor i,* 1, d. (2) I(A) = (d - 1) . [ 1: [k($): k(~)] - 1]. 'llEW(l) (3) emb (A) 1: emb (8m), [k($): k(~)]. PEW(l) (4) e()(A) 1: eo(8'1l)' [k($) : k(~)]. PEW(l.J) II d 2, the Cohen-..lfacaulaylication 01 A coincides with S'll" i.e., .l S'll"

§ 3. On Buchsbaum rings obtained by glueing

239

We will illustrate this theorem with the previously mentioned example from the Introduction. Let k be a field .with char k :+ 2 and 8 = k[x, y] a polynomial ring. LetR={jE8if(-1,O)=f(l,O)} and ~={fE8If(-1,O)=f{1,O) OJ. Then R is a subring of 8 and 8 is module-finite over R. Moreover ~ is a maximal ideal of R with W{~) = (~, C) where ~ (x 1, y) and C (x - 1, y). In this situation the glueing il' over ~ coincides with the ring R it.self, and the two points \~ and n of Spec 8 are glued together into a single point ~ of Spec R via the morphism Spec 8 --+ Spec R. Therefore we get that A = R:p is a Buchsbaum (local) domain of dimension 2 with J{A) 1. Clearly emb (A) = 4 and eo{A) 2.

+

In order to prove the theorem we need some lemmas.

Lemma 3.2. With the same hypothesis as stated in the theorem let B 8:p.. Then we have (1) ~' is also an ideal of 8 and m mB c: A. (2) ~' is a unique prime ~"deal of R' such that ~' n R = ~, and therefore dim A = d. (3) k{~)

=

(4) Max B

(5) dim 8 SS

k(~'), where k{tJ') denotes the /z"eld R~'/~'R~,.

=

{~B

I~

E W{~)}.

d for every

~

E

W(~).

Proof:

(1) This is trivial since lJ'

= n

~ by definition. ilSEW(:p) (2) It is again a trivial matter to verify the first assertion. The second one follows from the first, since A = ~, and d = dim R:p by definition.

(3) This follows immediately from (2). (4) Let ~ be an element of W(~). Then ~ n R'

~'by (2) since (~ n R') n R = ~. Hence ~B is a maximal ideal of B = 8:p.. Every maximal ideal n of B of course may be expressed as n ~B for some ~ E W(~) since ~' n R =~, which again follows by (2).

(5) Let ~ be an element of W{lJ) and ehoose a maximal ideal C of 8 containing ~ such that dim 8/~ = dim Sc.l~8e. Then 8:;:;. is a Cohen-Macaulay local ring with

dim 8 0

= dim S by the initial aSBumption on 8. Hence we have

dim 8'l!

dim 8 e - dim So/~80

= dim 8

and thus we see that dim 8ilS does not depend on

dim 8/ 1V> ~.

dim R - dim R/~,

Also dim B

B is module-finite over A. Thus we conclude by (4) that dim 8<,p ~ E W(~),

d by (2), since d for every

q.e.d.

Corollary 3.3. Let Q(A) denote the total quotient rz"ng of A and suppose that d depth A > 0 and Q(A) B.

>

O. Then

Proof: First note that B 8:p. is a Cohen-Macaulay ring as is 8 by the initial assumption on 8. Hence the A-module B is Cohen-Macaulay and of maximal dimension d, because dim Bn = d for every maximal ideal n of B by Lemma 3.2 (see (4) and (5). In particular depthA B > as d > O.

°

240

V. Further applications and examples

Let a be an element of m and suppose that a is B-regular. Then clearly a is A-regular and so we have that depth A > O. Also mB A by Lemma 3.2(1). Thus Be A[a- l ] and therefore we have the inclusion Be Q(A) as required.

Lemma 3.4•. (1) m = n n. nEMaxB

(2) lA(B/mi) =

1:

IS\ll(S\ll/~iS'.ll)' [k(~):k(fl)] jor every integer

i> O.

\llEW(\l)

Prooj: (1) Since

m = fl'B by m

(2) Let

(1) of Lemma 3.2, we have

n ~). B = \llEW(lJ) n ~B (\llEW(lJ)

n

n.

neMaxB

o be an integer. It then follows from (1) that mi = n ni and B/mi = EB B/ni nEMaxB

nEMaxB

by virtue of the Chinese Remainder Theorem. Hence lA(B/mi}

= 1:

lB(B/ni ). [B/n :A/mJ

neMaxB

and therefore recalling A/m Lemma 3.2 that

k(fl) by (3) of Lemma 3.2, we conclude by (4) of

as claimed.

= d by Lemma 3.2(2}. Note that if d::; 1, then A is a Cohen-Macaulay ring (see Corollary 3.3). Hence A is a Buchsbaum local ring with I(A) 0 in this case. Now suppose that d 2. Then, since B is a CohenMacaulay ring with dim Bn d for every maximal ideal n of B by (4) and (5) of Lemma 3.2, A is a Buchsbaum local ring with Cohen-Macaulayfication B and with Hk(A} (O) for i 4= 1, d (see '(1) of Lemma 3.2, Corollary 3.3 and Theorem IV.2.11. Moreover, by Proposition 1.2.6, we see that

Prooj oj Theorem 3.1: We know that dim A

I(A)

=

(d

1) dim Aim B/A

(since BIA ~ H:n(A»).

Also dimA/m

Bfm = 1:

[k(~} :k(fl)]

\l.\EW(lJ)

by Lemma 3.4. Hence we have the required equation I(A}

=

(d

1) . {

1: '.llEW(\l)

[k(~) :k(fl)]

1}.

Now consider the assertions (3) and (4). Suppose that d Then

>

0 and let

o be an integer.

§ 3. On Buchsbaum rings obtained by glueing

241

by Lemma 3.4, and therefore we get

= .E

lA(A/m')

!llEW(j:»)

lS!ll(S!llj~'S!ll)' [k(~):k(~)] -IA(BJA).

This implies the required equation

.E

eo(A)

!ll€W(j:»)

eo(S!ll)' [k(~) :k(~)]

since dim A dim S!ll = d Lemma 3.4 we see that emb (A)

> 0 for every

lA(m/m2)

E W(~) (see Lemma 3.2(5)). Again by

IA(B/m2) -IA(B/m)

.E

(lS'll(Slll/~2S'll) - IS!ll(S'll/~S'.l5»)' [k(~) :k(~)]

.E

emb (S!ll)'

'llEW(j:») =

=

~

\l!EW(j:»)

[k(~):k(~)].

This completes the proof of the theorem, q.e.d. Following S. Goto [4J and G. Tamone [1] we next will examine a key example: Let Kjk be a field extension of fields and S a finitely generated K-algebra of dimension 8. Suppose that S is a Cohen-Macaulay ring with dim SJ~ = 8 for every minimal prime ideal ~ of S. Let W be a non-empty. finite subset of Spec S and assume that dim Slll = d for every ~ E W. We put ~

n~.

!llEW

Then, by virtue of the Normalization Lemma, one may find elements of S such that S is module-finite over K[xl> X2' ••• , x s] and

Xl> X2, •• •

,x'.

Now let Then R is a subring of Sand S is module-finite over R. Moreover fl is a prime ideal of R with W(~)

and

W

R/~

k[Xl' X2, ••• , X.-d]'

It now follows that the glueing R' over ~ coincides with the ring R itself (see Tamone

[1], Theorem 1.3, for the details). Thus, by applying the above theorem to this situation, we obtain immediately the following results: (1) A = Rj:) is a Buchsbaum local ring of dimension d and with H:n(A) = 0 for i 9= 1, d. (Here m denotes the maximal ideal of A.) (2) Suppose that d> 0 and set d(~) = [S!ll/~S'll:k(xl> ... , X.-d)] for every ~ E W(~}. Then l(A) = (d -

emb (A)

I) . {

.E $€W(j:»

16 Buchsbaum Rings

.E

'llEW(j:»

d(~)

emb (S'll)'

1},

d(~),

242

V. Further applications and examples

and eo(A)

E

eo(SIl!)' d(~).

Il!EW{;,)

If d 2 and # W > 2, then A of course is not a Cohen-Macaulay ring. (3) Suppose d > 2. Then the Cohen-Macaulayfication of A coincides with Sm, i.e.,

A

SIl!'

For further examples see S. Goto [4]. We conclude by studying the following well-known example: Let k be a field and P k[ Xl>"" X d , Y1 , ••. , Yd ], d 2, a formal power serie sring. Let A P/'p n q and B = P/'p ffi P/q, where'p (Xl' "', X d ) P and q (YlJ ... , Y d ) P. Then A of course is a Buchsbaum ring. One may deduce this fact by using the above Theorem 3.1, since A is obtained from B by glueing over the maximal ideaL Clearly A B in this case, A coincides with the normalization of A and A is seminormaL We have obviously eo(A) = 2,

§ 4.

emb (A)

=

and

2d,

d - 1.

J(A)

Construction of Buchsbaum rings with given local cohomology

By using results found in the papers of Evans-Griffith [1] and Goto [3] we prove the following theorem. Theorem 4.1. Let d > 0 and ho, hI, ... , hd- l 2 0 be integer8. Then there is a Buchsbaum hi for local ring A with maximal ideal m 8uch that dim A = d and dim Aim H~(A} every 0 iSd-l. Moreover, if ho 0 (re8p. d 2 and ho hI = 0), then A may also be taken to be an integral domain (re8p. a normal domaz·n). Remark 4.2. In case d 2 and ho = 0 the method of construction of these examples is due to E. G. Evans Jr. and P. A. Griffith [1]. The Buchsbaum property for their results follows immediately by Lemma 1.3.10. Additional, the case ho =F 0 was studied by S. Goto [3] and the examples are obtained by idealization. First we will consider the ease d 2 and ho O. Suppose that A is a regular local ring of dimension d lution of A/m: o

~]?d+2 ~...

1\ ~]?o

We set Za:= im(J?8+l ~]?s} for 0

A 8

~A/m ~

d.

Lemma 4.3. (1) Z. i8 a Buch8baum A-module with dimA Zs (2) H!n(Z.}

~ {Ao/m

(3) The int'ariant J(Z.)

for i 8 + 1, for all i =F 8 + 1, d

+

1).

d 1 ( 8+

O.

+ 2.

d

+ 2.

+ 2. We take a

minimal reso-

§ 4. Construction of Buchsbaum rings with given local cohomology

243

Proof: The assertions (1) and (3) follow immediately from (2) by Propositions 1.2.6 and 1.2.12. For the claim (2), consider the exact sequence for O:S:; 8 d:

o -+Z. -+ F8 -+Z.-l -+ 0 where we set Z-l A/m. Then, after applying the functor H:n( ) to these sequences, the result follows by induction on 8, q.e.d.

Z1h•l ,

We now set M.:=

i.e. the direct sum of h. copies of

Z.,

for every 1:S:; 8

d-l

d

M •. Note that we may assume M =F O. In fact if M

1, and we set M:= 8=1

0, then h. = 0 for all 0 8:S:; d - 1 and every regular local ring of dimension d satisfies all the conditions required in our theorem.

Lemma 4.4. (1) M is a Buchsbaum A-module with dim" M = d (2) H:n(M)

{ (A/m)(hHl (0)

(3) The invari,ant I(M)

for i for i

= d-1 1: h.· 8=1

=

2, ... , d = 0, 1 and d

(d 1) 1 . s

+ 2.

+ 1.

(4) M is a non-free and reflexive A-module.

(5) Mp is a free A 1,module for every f' E Spec A '" m. Proof: The assertion (2) follows from Lemma 4.3. Clearly dim" M d + 2. Since we have the exact sequence 0 -+ Z. -+ F. -+ F S- 1 for all 8 = 1, ... , d - 1 we get that M is a reflexive A-module as is Z,. Now M is not free since depth" M:S:; d < dim" M = d 2. The assertion (5) is trivial since (Z.)p is a free Ap-module for every f' E Spec A '" {m} and for every 1 :S:;8 d - 1. Now we prove the assertions (1) and (3). Let q be a parameter ideal of A. Since dimA M = dim A, q is also a parameter ideal of Z. for every 8 = 1, ''', d - 1. Also since we have

+

a-I

1,,(.Llf/qM)

1: h. ·lA(Z./q • Z.)

8=1

and d-l

eo(q, M)

= 1: h• . eo(q, Z.) 9=1

we get for the difference=. d-l

IA(M/qM) - eo(q, M)

=

1: h•. (lA(Z./q. Z.) -

eo(q, Z.»)

8=1

a-I =1:h•. I(Z.), 8=1

a-I

1: h.·

8=1

byJ~emma4.3

(d+1)., 8

+1

thus the difference does not depend on q. Therefore we have proven the assertions (1) and (3). (Note that (1) also follows from Theorem 1.2.15, q.e.d.) 16*

244

V. Furtlulr applications and examples

Since M is a non-free and reflexive A-module we get the following lemma from the results of M. Auslander [1], Theorem B and Proposition 4.4:

Lemma 4.5. There is an exact 8equence O_F_M_I_O oj A-module8 8uch that F is a jree A-module and I i8 an unmixed ideal oj A oj height 2.

, In Lemma 4.5 we set X = A/I and ffi = mjI. Then the following lemma proves our theorem in case d 2 and ho O.

Lemma 4.6. X is a Buchsbaum local ring with dim X = d and dimiliii H~(X) = hi d - 1. .

tor OS;;

Proof: (ilearly dim X = d and ~(X) = 0 since I is unmixed. ConEider the following two exact sequences (exact by Lemma 4.5):. O_F_M _I _0

and

0_1 _A

-X _0.

By applying the functor 11:n( ) we therefore get: 11;;I(M)""'" 11;;1(1)

H~(X)

and

11;;1(1)

for every

i< d -

1.

Thus we see by Lemma 4.4 that Hk(x) ,....., H:n(X) '"" H;;I(M) '"" (Xjffi)(h,l

for all i

1, ... , d - 1.

Now we prove that X is a Buchsbaum local ring. For this purpose it suffices to show that X is a Buchsbaum A-module. From the above two exact sequences we obtain the following commutative diagrams of A-modules for every i d - 1: Ext~+I(A/m, M)

1

i+l

'I'M

H;;l(M)

-=+ Ext~d(A/m, I) ;+1

1 '1'[

Ext~(Alm,

X) -=+ Ext~+I(A/m, I)

I'I'~

1+1

1

'1'[

...

11;;1(1)

l1:n(X)

where the vertical maps are canonical homomorphisms. Since M is a Buchsbllum A-module by Lemma 4.4 we have that 9'if 1 is surjective (see Corollary 1.2.16) and therefore 9'~1 is also surjective. Hence we get that 9'~ is a surjection. This property implies that A is a Buchsbaum A-module by Theorem 1.2.10, q.e.d.

Remark 4.7. In case A is the localization of a polynomial ring k[Xl' ... , Xd+2] (k an infinite field) by the irrelevant maximal ideal (Xl>"" xm), Evans and Griffith [1] - hav-e shown that I can be taken to be a prime ideal. Moreover they have proved that A/I may also be taken to be a normal domain if hI = O. The assertion of our theorem is trivial in caEe d = 1 and

11,0

O.

We now will consider the case ho > O. The examples are then obtained by idealization. We need som", lemmas in order to describe the method of this construction. In the following we denote by A any local ring with maximal ideal m. Let V be a vector space over Aim of dimension t and let B := A X V denote the idealization of V over A; that is the underlying additive group of B coincides with that of the direct sum

§ 5. Some examples of Segre products

245

of A and V and the multiplication in B is defined by (a, x) . (b, y) = (ab, ay + bx). Note that B is again a Noetherian local ring of dimension d dim A and with maximal ideal e = m X V. Let p: B ~ A be the canonical projection, i.e. ~(a, x») = a for (a, x) E B. The'n clearly ker p {OJ X V and, applying the functor H~( ) to the exact sequence 0 -,. {OJ X V ~ B A ~ 0, we get the following lemma: Lemma 4.8. H~(B)

=

H:n(A) for i =l= 0 and H~(B)

Hg,(A) X V.

Lemma 4.9. B i8 a Buchsbaum ring if and anly if A i8 a Buchsbaum ring. In this case we have J(B) = I(A) t.

+

Proof: Let 0 be a parameter idea1.of B. We set q = p(O). Then q is also a parameter ideal of A since A/q is a homomorphic image of BIO via p. For every integer 8> 0 there is an exact sequence

o ~ {OJ X V ~B/o·4 AJq8 ~ 0, which implies lB(BJ08) = lA(A/q8) + t, in particular co(O, B) = eo(q, A). Hence we get that IB(BjO) - eo(O, B) lA(A/q) - €o(q, 4) + ti that is B is Buchsbaum ring if A is a Buchsbaum ring. Conversely assume that B is a Buchsbaum ringandleta1'" .,ad be a system of parameters for A. We set q = (al> .•. , ad) A and :0. = (/1' "', fa) B where fi = (a" 0) for 1 d. Then clearly :0. is a parameter ideal of Band p(O) = q. Therefore we again obtain that lA(Ajq) - eo(q, A) = J(B) t, that is A is also a Buchsbaum ring, q.e.d. 1 we take A to Now we prove the assertion of our theorem in case ho > O. If d be a Cohen-Macaulay ring of dimension 1. If d:::::: 2 we take a Buchsbaum ring A with dim A = d and dim Aim H:n(A) = hi for i = 1, ... , d 1, and Hg,(A) = O. This is possible by Lemma 4.6. Let V be an ho-dimensional vector space over A/m. Take the idealization B = A X Vof V over A. Then Lemma 4.8 and 4.9 imply that the local ring B has allthe properties required in our theorem for the case ho > O. This completes the proof of the theorem, q.e.d.

§ o.

Some examples of Segre products

1. In view of Corollary 1.4.15 we want to investigate Segre products of Buchsbaum varieties in p". Proposition 5.1. Let V ~ P~ and W ~ P'k (K any field) be loeally Buchsbaum varieties. In general, S(V X W), the Segre embeddz'ng of V X W z'n pl(lIl+n+m, will not be a locally Buchsbaum variety.

Before proving this by giving an example we describe an algorithm which enables one to calculate the defining equations of the Segre embedding of the product of two varieties using 'the equations of these varieties. To this end let V ~ P'k, W P~ be projective varieties given by homogeneous ideals a ~ Rl K[Xo, ... , X,,], resp. 0 R2 := K[ Yo, ... , Yml. Assume as given the generators of a and o. Our aim is to obtain generators for the defining ideal c of S( V X W) pl(m+n+lIl.

'

246

V. Further applications and examples

Let R:= a(R1, R z) K[Xo' Yo, .•. , X"' Y m], B:= K[Too, ... , T"m], Til variables. Then R is an epimorphic image of B given by the mapping Til 1-+ Xi' Y j • The kernel of this epimorphism is the prime ideal fl generated by Tij . TT8 Tis' Trj, 0 i < r :s;; n, 0 j < 8 m. Also there is an ideal c' c: R such that a(RI/n. Rz/b)

where c is the required ideal.

Rlc'::::- Bic

By Cartan-Eilenberg [1], II, Prop. 4.3(c), we obtain an exact sequence

i.e. c'

=

a(a, R 2 )

+ a(Rl' b). Therefore

{I, Yt·· .... Ybm ! I a generating form of n, io + ... + jm u {X~· ..... X;:.

I}

II} a generating form of b,

io

deg I}

+ ... + itt = deg g}

will be a set of generators of c/. Now c will be generated by the generators of the ideal p and the inverse images of the generators of c' (in B). (See also the remarks following Definition 0.2.5.) Now we are able to give our example which proves Proposition 5.1:

Pi be the surface given parametrically by

Example 0.2. Let V

{(tg, tgt]', totItz, tOt2(tz

to), t~(t2 -

to)}.

It was first studied by R. Hartshorne [1]. It is not difficult to verify that the ideal Pv of V is pv

XzX a, X OX IX 2

(XIX, XOX3X'

XoX~

-

XoX~

+ XiXa, XoX2 X a -

XoX2X'

XIX;,

+ X~) c: K[Xo, ... , X,] =: R

(see Renschuch [2], p. 334). First we show that V is a locally Buchsbaum variety. We know (see Hartshorne [1]) that sing V {(I, 0, 0, 0, O)}. Therefore we only need to verify the Buchsbaum. property of the local ring A of Vat the origin of Pi. Since dim A 2 and depth A 1 (A is an integral domain), we can apply Proposition 1.2.12. In order to use the propOsition we have to find one non-zero x E rn~ with rnA • U(x. A) x· A. Choose x We obtain

U(X; . A)

(Xi, X,, XOX2

+ XtXa) . A

n (Xi, Xa - X 4 , XO(X 1 - X 2 )

+ XIXa)' A

(X;, X~)·A

+ (X 4 , XoXz + X1Xa)· (Xa -

X" Xo(Xt - X 2 )

+ XIXa)' A

(X~,Xn·A

and therefore we see that rnA . U (Xi' A) variety.

<::::::;

Xi . A; that is V is a locally Buchsbaum

§ 5. Some examples of Segre products

247

Now . let X:= S(V X Pk) PlJc. By the previous algorithm the defining ideal 4lx of X is given by (we set Tij X 2i+;,O i< 4, 0 j 1): ~x

(XOX3 - X 1X 2 ' XoXs - XIX., XOX, X2X,

XaX., X 2X,

XaX6' X 2X 9

X,X g - X,Xs, X6X9 - X 7X S, X 2X S

X1X S, XoX9

X1X S,

XaXs, X,X,

X,Xs,

X4X~,

-

X2X 9

-

X,X 7 ,

XaXg - X,X7' X oX 2 X, - XoX:

+ X 2 X S , XoX2X, -

+X

XoX,X,

+ X~X7'

XIX;

+ X~X7'

XoXaXs

XoX~

XoX,X s

XoXfiXS -

+ X 2 X 6 X 7' XoX,Xg + X2X~, X 1X 5 X, - X IX 5X 9 + X3X~, XoX; + X~, XOX6X9 - XoXsXg + X:X 7 ,

XOX7X9

XoX~

XoX,X7 -

XoX,X g

2

X aX 7 , X 1X 3X 5

+ X2X~, XoX,X 7 -

+ X6X~, X IX X 9 7

XoX,Xg

XIX~

+ X~).

Let B be the local ring of X at the line X 2 = ... X g ,= O. We claim that B is not a Cohen-Macaulay ring. Then by Corollary 1.1.11 the local ring of X at the origin Xl X 2 = ... Xg = 0 cannot be a Buchsbaum ring. It is now easy to see that rad X 2 • B (X2' X 3 , X 4 , X,) . B and this last ideal is a prime ideal in B, since (~x, X 2 , X a, X" X,) is a prime ideal in K[X o, .. " XgJ. ((~x, X 2 , X a, X 4 , Xs) = (j)w, X 2 , X a, X" Xs) where j)w c K(Xo, Xl> X s, X 7 , X s, XgJ may be considered as the ideal of the Segre embedding W:= S(C X P~) of the product of the plane curve C given by ZoZlZ2 - ZoZ~ + Z~ = 0 with P~ in Pk. But j)w is indeed a prime idea!.) Now X,(X 6 - Xs) E X 2 • B. Therefore X 2 • B must have an embedded component and B is not a Cohen-Macaulay ring.

Remark 5.3. By studying a primary decomposition of

X~.

B we get from Proposition

1.2.12 that B is a Buchsbaum ring.

2. We now give examples of two arithmetically Buchsbaum varieties (non CohenMacaulay) such that their Segre product is again an arithmetically Buchsbaum variety. Let R := K[Xo, ... , X3]/~ (K a field), where j):=

(XoXa - X I X 2 ' X~ - X~X2' X~ - XIX~, XoXi - XiX a)

is the well-known ideal of Macaulay, and set m (Xo, Xl> X 2 , Xa) . R. We have that R is a graded Buchsbaum ring, see the Introduction, Example 3 (3b). First we investigate the local cohomology modules of R. By using the exact sequence

R-R/X2·R-O we obtain an isomorphism (i) lJ.~(R/X2· R) lJ.:"(R) r'oJ

1)

and an exact sequence (ii) 0 - lJ.:n(R) - lJ.:"(R/X 2 • R) - lJ.~(R) (-1) - lJ.~(R) - O.

248

V. Further applications and examples

Since (V, X 2 )

(Xo, Xl' X 2 ) n (Xi, X 2 , X 3 ) n (Xo, Xi, X 2 , X:) we see that

ll~(RjX2' R)

K(-2)

(generated by the coset of X1X a)

and this gives llin (R) :::::: II (-1). Since llin(RjX 2 • R) llir.{RjU(X 2 • R») and since RjU(X 2 • R) is isomorphic to a quotient of a polynomial ring by an ideal generated by monomials it is not difficult to calculate the Hilbert function and the Hilbert polynomial of this Cohen-Macaulay algebra. Hence we get by Lemma 14.3 and Corollary 1.4.4 [llin(RjX 2 • R)].

[llir.{RJU(X 2 , R))]"

[llin(RjX 2 , R)h

X.

=

0

for all n

2

and Since [llin(R)h ~ X the exact sequence (ii) yields for each n

1 isomorphisms

Cll~(R)]'-l:::::: [ll~(R)],..

Hence [ll~(R)]q = 0 for all q 0, i.e. e(ll~(R») = -1. Now let S be another graded X-algebra of dimension 2 with ll~(S)

0,

[llMS)]"

=

0

for all p =l= 1

and

e(ll~(S»)

<

0

(n := maximal ideal of S). Then S is a Buchsbaum algebra as well. We define T a(R, S) and get from our RUnneth relations (Proposition 0.2.10): ll~T(rl')~ llI(T)

a(lll(R), llO(S») tB a(llO(R), lll(S»)

0,

since e(lll(R») < 0, e(lll(S») < 0, a(llO(R»);;::: 0 and a(llO(S») O. Since [R],. ~ [llO(R)]., [S] .. :::::: [llO(S)]. for all n =p 1 the same must be true for T, i.e. we have CllinT(T)]" 0 for all n =p 1. Therefore T is a Buchsbaum algebra by Corollary 1.3.6. For instance, we can have S = R or S X[Xo, Xl' X 2 , XaJj(Xo, Xl) n (X2' Xa) since e(ll~(S») < 0 (see the following Example 3).

3. In Chapter 1, § 2, we gave an example (Example 2.5) of a local non-Buchsbaum ring A, of depth 0, dimension 2, m . H:n(A) = 0 for i = 0, 1. (See also Corollary 1.2.4). By using Segre products we will give now an example of a graded module M with m 'lltn(M) = 0 for i < dim M, depth M > 0 which is not a Buchsbaum module. Example 5.4. Let R X[XI' X 2 , X a, X 4 ], S:= X[Y l , Y 2 ] (Xl> .. " X 4 , YI , Y 2 indeterminates, X afield) and a (Xl' X 2 ) n (Xa, X 4 ) c R. We set M a(Rja( -2),8). It is a finitely generated a(R, S)-module. Let i.J := u(mR' ms) (the maximal ideal of a(R, S»). Since depth Rja( -2) depth Rja = 1, depth S = 2, Corollary 0.1.12 implies depth M 1. Clearly, dim M = 3. We know (see Proposition 1.2.25) that Rja is a Buchsbaum module with llinR(Rla) ~ !I and the exact sequence in the proof of Lemma 1.2.14 give rise to an isomorphism ll~plla) ~ ll~R(Rj(Xh X 2 »)tB ll~f«Rj(X3' X.»). Therefore e(ll~R(Rla») -2 and [ll~R(R/a)]-2 ~ X tB X X2). Therefore by our Runneth relations (Proposition 0.2.10) ll~(M)

lll(M) ~ a(lll(R/a) (-2), llO(S») tB a(llO(Rja) (-2), lll(S»)

K2

0

K2

§ 5. Some examples of Segre products

249 \

and coker(a(R/a(-2), B) -IlO(a(R/a( -2), S)))

m(M) '" cOker(M -IlO(M)) C'.:

a(Il:n,,(R/a) (-2), B)

K S( -2).

Hence depth M = 1 and n '1l~(M) 0 for all i < 3 = dim M. We claim that M is not a Buchsbaum module. Let Xi:= X;moda. Then n is generated by the elements Xi @ Y j with 1
= xi

Y l Y2 Yj

Xial

YIY2Yi=8I(XI@Y2Yl)E8I·M,

Y I Y 2 Yj

X2 a 2

Y I Y2 Yi

=

82(X2

C2

YIY;) E 82' M

and so on. Thus E (81,82 )' M: (xi Yi, ..., Y;) C (81,82)' M :(n). If M were a Buchsbaum module this would imply (Xl Y I )· m E (81,82 ) . M since (S1> 8 2)'1If:(n) = (S1>82)·M:n, i.e. there are elements m I ,m2 E'M with Xl Y~Y2 = 8 I m I + 82~' Comparing degrees it is easy to see that without loss of generality m I = 1 @ Uv ~ = 1 U2 with Ul, U2 E [B]2' This yields

m

Xl

x;

Yi Y 2

=

al

UIY l

+a

2

U2 Y 2'

But this contradicts the linear independence of the images of the elements Xl' aI' a 2 in [Rla(-2)]a = [R/ah, i.e. M is not a Buchsbaum module. Note that for this example the assumptions of Proposition 13.10 are not fulfilled.

4. Now we will give an example of a graded Buchsbaum module not satisfying the assumptions of Proposition 1.3.10. It shows that the criterion is not a necessary condition. Example 6.6. Let R:= K[XI' X s], B: K[Y" Y2' Y a, Y4] (Xl> X 2 , X a, Y I , ... , Y" indeterminates, K a field). Choose 0 =t= IE R, deg 1= 5, 0 =t= g E·B, deg g 2. Let M:= RII· R(2), N Big· S, P Cf(M, N) (as a Cf(R, B)-module) and m := maximal ideal of a(R, B). Now dim M 2, dim N 3, a(M) -2, a(N) = O,r(M) = 1, r(N) = -1, i.e. the assumptions of Theorem 1.4.6 are fulfilled and P is a Buchsbaum module. By Proposition 1.2.10 1l~(P) 0 precisely for i = 2, 3,4. Also (ll~(P)]n 0 for all n =t= and [1l;,,(P)]m 0 for all m =t= -2. Therefore the assumptions of Proposition 13.10 are not fulfilled since in this case (2 + 0) - (3 - 2) = 1.

°

6. We now give an example of two graded injective modules such that their Segre product is not an injective module (see Lemma 0.2.9). Example 6.S. Let R' K[X, Y] (X, Y indeterminates, K a field) and 1:= Q jective enveloppe of Rand J injective enveloppe of II = R}(X, Y) R.

in·

250

V. Further applications and examples

We know that Q

{:

k'

q homogeneous in R, q

* o} is the (homogeneous) field

of quotients of R. By Corollary 0.4. 10 and Theorem 0.4.14 we know that J "-' !.!;,,(R) (-2). We claim that Ext!/R.RlK, a(I, J») =f: O. . a(R, R) is generated (as a K-algebra) by the elements Zl := X X, Z2 X Y, Za := Y @ X, Z4 := Y Y E [a(R, R)h. Let m denote the maximal ideal of a(R, R). We choose four elements VI> ••• , V4 in [Q]o with V 1V4 - V2Va =f: O. Now e(!.!;,,(R)) = -2, i.e. e(J) 0, [J]o~K and [J]-l K(JJK. Since [a(1,J)]o .:::: [Q]o®KK [Q]o, by Zi f-l. Vi 1 a a(R,R)-homomorphism j: m -'? a(1, J) of degree -1 is given (notice that [a(1, J)}p 0 for all p > 0). We suppose Ext!IR.Rl(K, a(1, J)) = 0 and we trJ"to obtain a contradiction. We have an epimorphism a(1, J).:::: Hom"(R.R)(a(R, Rl, a(1, J))

-'?

HomaIR.Rl(m, a(1, J)),

i.e. there is an x E [a(1, J)J-I with j(z;) ZiX for i = 1, ... ,4. By studying the local cohomology module !.!;n(R) we see that [JJ-l "-' K (JJ K possesses a basis el> e2 with Xel Ye2 = 1, Xe2 = Ye l 0 (in K:::: [J]o). Using this we can write:

Then ZIX = XW 1 ® I, VI

= XWI>

V2

Xw2 ;

~x = Va

=

XW2 Yw l , V4

1, ZaX YWI 1, Z4X = YW 2 YW2 and we have in Q: V 1V 4

1. Consequently V 2Va = 0, a con-

tradiction. ..

This example also shows that the homomorphisms i" of our generalized Kiinneth relations (Proposition O.~.1O(i)) are not generally isomorphisms. Namely, for Ml = I, M2 J on the left-hand side of this formula we always get zero for all n > O. But the term on the right-hand side is not equal to zero (take for example n 1, N} N2 = K). 6. :lfinally, we give an example which shows that the sufficient criterion for Buchsbaum modules given by Theorem 1.2.10 is in general not necessary. Example 5.7. Let K be a field and Xl"'" X 2n , n R:= K[Xl> ... , X 2n ], n:= (Xl' ... , X 2n )· R, F Xi

3 indeterminates and set

+ .,. + X~". Let

By Lemma 1.2.14, Ria is a Buchsbaum-module over R with dim Ria == n, depth Ria 1. Therefore M:= Ria F·R is a Buchsbaum module over R with dim lYf = n - 1, depthM O. Let A R/F·R, m:=n·A, x,:=XimodF, i 1, ... ,2n. Then Xl> .. ,' X 2n is a minimal basis of m, Since F. M 0, M is also a Buchsbaum module over A (see, e.g. Corollary 1.1.11 and Lemma 1.1.6). If the canonical maps

+

tpk-: Ext~(K, M)

-'?

H~(M)

were surjective for all i 0, .. " n - 2, we then would obtain from the proof of Theorem 1.2,10, that the homomorphisms

k

Ext~(K, O:Mm)

-'?

Ext~(K, ..."tI)

(induced by O:.I{m eM)

§ 5, Some examples of Segre produets

n - 1. Let now

are injective for all i ". -+

251

An. J.4. A '" ~+ A

-+

be a minimal free resolution of K ker fh = im Y2 is generated by the

K

e:)

-+

0

=

2n and

-Xi>

0, .",0)

Aim over A, Obviously, we have n 1 syzygies 81i := (0, ".,0,

for all i, i with 1 i < i 2n (Xj is in the ith place and syzygy 8 := (Xl> "" x2n)' These syzygies form a minimal basis, i.e. we have n2

-Xi

Xj,

0, .",0,

in the jth place) and the

(2n) + 1. By applying the

n functors RomA( ,O:M m), RomA( ,M) to the above free resolution we obtain complexes (all !!quares are commutative): 0-+ O:.}[m ---.. (O:Mm)'" -+ (O:.I[m)'" -+ ... g'

-"-+ ...

The boundary homomorphisms of the above complex are zero, i.e. Ext~(K, O:Mm) (O:J[m)"'. Let Z:= kerg 2, B img1 • Then the following diagram with exact rows, is commutative:

if,

1 Z

... -*

Ext~(K, ill)

-+

O.

+ ". +

Let m := gl(Xl' ... , x,,, 0, ... , 0) E B (Xi is the coset of Xi in M). Set y := xi x; 0, then m (0, ... , 0, y) (gl is obtained by scalare multiplication with the syzygies 8tl and 8). Since Xi' Y for all i = 1, .'" 2n, {' E O:.l{m, i.e. m E (O:J[m)'" n B. Therefore Mm) 0, i.e. 12 is not injective. By using Lemma I.2.14 we see that H:nUIf) = 0 for al]z' 9= 0, 1, n 1. Since n - 1:::::: 2 and since 10 is even an isomorphism, we obtain that tplt is not surjective. If we consider .Ilf as a ring, an easy computation shows that also in this case also the canonical homomorphism

'*'

°

Extlt(K, is not surjective.

~'lf) -+

HMM)

(IJ maximal ideal of M)

Appendix On generalizations of Buchsbaum modules

It is not difficult to see that there are various possibilities to generalize the concept of Buchsbaum ring and module. A first step in this direction is our Proposition 1.2.1 where we have characterized those modules M for which mH~(M) = 0 for all i < dim M (they are not Buchsbaum modules, in general, see, e.g., Example 1.2.5 and Example V.5.4). From this point of view, we want to give in this Appendix an answer to the following two questions: 1. Which are the modules M (always Noetherian over the local ring A) such that 1, are N~therian modules (and hence modules of H~(M), i = 0, ... , dim M finite length)? (S€e Proposition 13, Th€orem 14, I.emma 15 and Proposition 16.) 2. With the hypothesis of 1.), what can we say about parameter ideals q of M for which

where dim lJf

>

0,

compare Proposition 1.2.6? ,(See Lemma 15, Theorem and Definition 14 and Theorem 20 as it relates to Definition 19.) Let A always denote a local ring with maximal ideal m. Let M be a Noetherian A-module. If M is a Buchsbaum module, then for any part Xl' ••• , Xr of a system of parameters of M and any prime ideal ~ E Supp lJf "'- {m} containing Xl, ••. , Xr the images of Xl> ••• , Xr in Av form an Mv-sequence (see Chapter 0, § 2 for the definition and Corollary 1.1.11 ). By virtue ofthis relation we generali ze the notion ofa weak M -sequence in the following manner. For this we recall first that for a submodule N of M: {m E J.lf I mil. m ~ N for some n EN}.

N:M(m)

Definition 1. Let Xl> ••• , Xr be a sequence of elements contain€d in m. It is called a filter-regular (f-regular) sequence with respect to lJf if for all i = 1, ... , r

It is immediately seen that a sequence of elements

and only if

x\ ... , Xr 1

1

Xl> ••• ,

x~

is an f-regular sequence if

in Av forms an Mv-sequence for all ~ E Supp M "'- {m} containing

~-,

On generalizations of Buc~bau:ln modules

253

In particular an M-sequence or a weak M-sequence is an f-regular sequence. First we note the following easily provable facts:

Xl ••••• X r •

Lemma 2. (i)

A sequence 01 elemenl8 Xl, " ' , Xr is I-regular with respect to M I-regular unzh respect to Mjll'/n(M).

(ii) II Xl> ••• , Xr is I-regular '/Lith respect to M, then lor any n y E m" such that Xl> ••• , X" Y is I-regular with respect to M. (iii) II Xl . . . . , Xr is I-regular with re&pect to ill then dim Mj(xl> ... , xr )· M = sup{dim M

if

and ooly

> 1 there

if

it is

is an element

1', O} •

.Prool: (i) is trivially true. For (ii) it suffices to show that m contains an f-regular element y with respect to M. If H'/n(M) M, choose y E mil arbitrarily. If H'/n(M) =l= M H'/n(MjH'/n(M») 0 implies depth M/ll?,.(M) > fI, hence there is an M/H'/nUll)-regular element y E mil and (ii) follows from (i). For the proof of (iii) we note that the case l' 0 is trivial. Thus let l' > O. If dim M 0, there is nothing to show. If dim 1ll > 0 then Xl is M/H'/n(M)-regular. Now Ass Mlll'/n(M) = Ass M" {m} shows that Xl lies outside of all minimal primes of M. Let M' Mlxl' M. Then dim M' = dim M 1 and we get by an easy induction argument:

dim M/(XI,' .. , x r )· M

= dim M'j(X2' ... , xr ), M = sup{dim M - 1',0) ,

= sup{dim M'

(1' - 1), OJ

q.e.d. Let M be a Noetherian A-module of dimension d > 1. According to Lemma 2(iii) each f-regular sequence Xl, ••• , Xd with respect to M is a system of parameters with respect to M. The converse is not true, in general.

>-

Proposition 3. For a Noetherian A-module M of dimen8ian d 1 the followirv; corulitioos are equivalent: (i). Each system 01 parameters Xl, ... , Xd 01 M i8 an I-regular sequence withre&pect to M. (ii) Each part Xl> ••• , Xr 01 a system 01 parameters 01 M is unmixed up to an m-primary component, i.e. lor all P E AssAM/(xl , ... , xr ), M" {m} we have dim Alp

=d

- r.

(iii) For any P E Supp M" {m} we have dim M

=

depthAlJ Mp

+ dim A/p.

(iv) For all p E Supp M " {m} dim 1ll

=

dimApM.,

+ dim Alp

and

di~lJMlJ

=

depthAlJMlJ'

+

(v) Supp M is catenarian (i.e. htAP htAq htA1qP/q lor all primes q pol Supp M), equidimensional (i.e. dim ~M = dim Alp lor all minimal primes P 01 Supp M) and MlJ is a Cohen-Macaulay module lor all P E Supp ~M "{m). Prool: (i) =? (ii): Suppose there exists a part of a system of parameters Xl> ... , Xr such that there is apE AssAM/-,;' M with 0 < dim AlP < d - l' (here -,;;= (Xl' ... , x r )· A}. Then we choose yEP such that Xl> ••• , X" Y is again a part of a system of parameters.

254

Appendix

By our assumption (i) we have

r.. M~:MYIr.· M ~ r.. M:Mm/r. . iff and Ass M/r.' M n Supp A/y. A

~

Ass M/r.·.M n {m}

~

{m}.

This is a contradiction since l' E Ass Mlr.· iII n Supp A/y. A. (ii) =? (iii): Let l' E Supp M '" {mi. We choose a natural number r maximal with respect to the property that there are elements Xl> ••• , X T in p forming a part of a system of parameters of M. By the maximality of r we get l' E AssAM/(xh ... , x r )· M. By the unmixed ness condition it follows that for all i = 1, ... , r

=

(Xl' ••• , Xi-I)' MlJ:MlJX;

(Xl> ... , Xi-l)'

M lJ ,

i.e. r depthAlJMlJ' But l' is minimal in Supp M/(x!> ... , x r )· M, i.e. d~lJ.M·lJ' Therefore r depthAlJ~:lflJ = dimAlJMlJ' We also have dim A/p dim ill - r and the conclusion is therefore proven. It is now immediate that (iv) and (v) are equivalent formulations of (iii). (iii) =? (ii): Suppose (ii) is not true. Then there is a part Xl, " ' , X T of a system of parameters, r minimal, such that the assumption is not true. Take r. := (Xl' ••• , x r )· A and .p E AssAM/r.' M with 0 < dim A/p < dim Mr. Hence r> dimAlJMlJ' By the minimalityofr we get that

=

Xl, ... , X

1

T

1

is an MlJ-sequence of maximallength. ThusdepthA M» lJ

r, which is a contradiction. (ii) =? (i) is trivial and the proposition is thus proven, q.e.d.

Next we prove the following interesting fact.

Lemma 4. Let 1ff be a Noetherian A-module. II

Xl> ... , X T is a part 01 a system 01 parameters with respect to 1ff then there is an I-regUlar 8equence YI> ..• , Yr with respect to M such that

(Xl' ... , x r ) • A

=

(Yl' .. " YT) • A.

Prool: We use induction on r. The case r 0 is trivial, therefore we assume r 1 and (XI> ••• , X'_l) . A (Yl' ... , Yr-I!' A, where Yl>"" Yr-l denotes a suitable f-regular se-

, quence. Now we choose Yr E (Xl' ... , X r) . A '" m . (Xl' ... , X T ) • A such that Yr ~ P for all p E AssAM/(Yl, ... , Yr-tl· M '" {mi. From this we get the desited result, q.e.d. Next we will show two technical results which will be useful to prove a relation between our f-regular sequences and reduced systems of parameters defined by M. Auslander and D. Buchsbaum in [1]. Proposition o. Let M be a Noetherian A-module arui ditions are sati8fied: (i) X It q for all q E ASSAM "-... {m}.

(ii) Let.p be minimal in AsSAAj(q' .p E AssAM/x· M.

+ X· A) for

SOlrle

X

E m. Supp08e the following con-

q' E AssAM'" {mi. Then we have

On generalizations of Buchsbaum modules

255

Proof: Let N Hg.(M) O:M(q') ~ M. Since SuppNjxN = SuppN n V(xA) V(q') n V(xA) V(q' + xA) and t' is minimal in V(q' + xA) it follows that t' E Ass NjxN. Since O:M/NX 0, the embedding N ill indu,ces a monomorphism N/(xN .....,... "lfjxM. Therefore t' E Ass MjxM, q.e.d. Proposition 6. Let J.lf be a Noetherian A-module with depth M there exists an integer n 8uch that depth MjxflM O.

H?n(M) =t= O. The short exact sequence

Proof: Let N

o

O. Suppose x E n1. Then

(N

+ x .. · M)/x'"

M .....,... Mix'" M

implies depthAMjxfl. M 0, since xn. M n N and (N + xn . M)/x" . M ~ Nix" . ll! n N q.e.d.

=

xn(N :MX") = 0 for sufficiently large n

N,

1 the following conditt'ons Theorem 7. For a Noetherian A-module M with d:= dim M are equivalent: (i) Each system 0/ parameters 0/ M is an I-regular sequence with respect to llf. (ii) Each system 0/ parameters 0/ M t's red·uced, i.e./or each s,l/stem Xl> ... , Xd 0/ parameters we have Xi ~ t' lor all t' E ASSAMj(xI' ... , Xi-I)' M. with dim Ait' dim M t'jor all i 1, ... , d. (iii) For each system 0/ parameters Xl, ... , Xd 0/ M we have

lA(Mj{xl> ... , Xd)' M) - eo(xl> ... , Xd)' A, M) lA(Xl> "', Xd-l)' M :MXd!(XI, ... , xd-d· M). (iv) Each part 0/ a system 0/ parameters 0/ M const'sting to m-primary compon.ents.

0/ d -

2 elements is unmixed up

Proof: (i) =;, (ii) is clear by (ii) of Proposition 3. (ii) {? (iii) is Corollary 4.8 of Auslander-Buchsbaum [1]. (ii) =;, (iv) is trivial, thus it remains to prove (iv) =;, (i). Without loss of generality we may assume d 3. According to Proposition 3 we have to show that a part Xl, • '" X T of a system of parameters of 1vl is unmixed up to mprimary components. The statement is true for the cases r d 2, d - 1. Therefore let us assume 0 r < d - 3. Then X 2 , "., X T is a part of a system of parameters of the (d 1)-dimensional module MjxlM. The condition (iv) remains valid for .M/xl· M by passing to Mjx l · M, hence an easy induction argument yields that (Xl>' •• , xTf· M is unmixed up to an m-primary component. To complete the induction step we have to show the assertion for r 0, i.e. for M itself. Assume there exists a t' E AssA ~M with 1 < dim Alt' < d. Then we choose a parameter X for M such that x ~ q for all q E AssA M "{m}. Next we choose q' E AssAAI(t' + x·A) with dimAlq' = dimAI(t' + x·A) dim Alt' - 1. By Proposition 5 we have q' E ASSAM/x ·1~{ with 0 < dim Ajq' < d -1 which contradicts the induction hypothesis for Mix, M. Furthermore if there exists an t' E ASSAM with dim Alt' 1, we choose a parameter X for M in t'. Then we have t" AI' E AssApMp and depthApM~ = O. For n 0 it follows that depthApMp/x'" .Jlp 0 and therefore t' E AssA Mjx'"M, which contradicts the induction hypothesis for Mjxfl.M. Hence the inductive proof is complete, q.e.d.

256

Appendix

Condition (iii) of the theorem is connected to our theory of Buchsbaum modules, more precisely, l..t( (Xl' .•. , Xli-I) • M:M Xd/(Xl> ••• , Xd-l) • M) is independent of the choice of Xl> ••• , Xd if M is a Buchsbaum module. Lemma 8. Let M be a Noetherian A-module with d:= dim M L Let ~ denote the m-adw completion. (i) If each system of parameters of M is an f-regular sequence of M, then the same holds for M. (ii) If A is a quotient of a Oohen-Macaulay ring the converse of (i) is true. Proof: Let Xl> ... , Xr denote a part of a system of parameters of M. Then it is also a part of a system of parameters of M. Since

(Xl' _..

J

Xr-l) . M: tlXr) n M

=

(Xl' .. " Xr- l ) . M:M Xr

the result (i) follows immediately. For the proof of (ii) let ~ E Spec A '" {m} be a prime ideal and i' := ~ n A. Since A is a quotient of a C{}hen-Macaulay ring the fibre ring k(i') ® A\ll of the canonical homomorphism

A41~A\ll is 'a C{}hen-Macaulay ring. (We may assume that A is a Cohen-Macaulay ring and apply Corollary (21.C) of Matsumura [1].) Since dimjlBMlB

= dimA 41 M 41 + dim k(i')

AlB'

resp. depthjs,uMlB = depthA41 M 41

+ depth k(i') ® A\ll'

it follows that dimjlBM!Il - depthjlBM!Il = dimA41Ml) - depthAl>Ml)' By our assumptions we get that SuppjM'" {m} is a Cohen-Macaulay set. Since A is catenarian, by Proposition 3(v) it is enough to show that SuppjM is equidimensional. Suppose ~ E SuppjM is minimaL Then we get by AssjM.=

U

AssjA/i"

A

that ~ E AssjA/i" A for i' ~ n A, which is minimal in Supp M. Because A is a quotient of a Cohen-Macaulay ring, we have dim A/~ = dim Ali' . A for all ~ E AssjA/i"..4 by Nagata [1], Theorem (34.9). Since dim Ali' = dim M the equidimensionality follows, q.e.d. Remark 9. Without additional assumptions on A the assertion (ii) in Lemma 8 does not remain valid. Ferrand and Raynaud (1] and Nagata [1] have constructed twodimensional local integral domains R such that 11, admits a one-dimensional associated prime ideal. R is a local ring for which every system of parameters forms an f-regular sequence but. 11, is not unmixed up to an m-primary component. Therefore it does not satisfy the equivalent conditions of Theorem 7.

On generalizations of Buchsbaum modules

257

A generalization of Buchsbaum modules is given by modules satisfying one of the four conditions of Theorem 7. We now give a co homological characterization of these generalized Buchsbaum modules (see our Proposition 16). For this we first state another generalization of the notion of a weak "W-sequence. Definition 10. Let M be a ~oetherian A-module with d dim M > 0 and q an m-primary ideal. A system of elements Xl> ••• , Xr is called a q-weak M-sequence if

q . ((Xl> ... , Xi-I) . M:M Xi) for all i

=

S;;

(Xv ... ,

Xi-I) .

M

1, ... , r.

Remark 11. Clearly, any q-weak M-sequence is an f-regular sequence. If there is an m-primary ideal q such that every system of parameters is a q-weak M-sequence then every system of parameters is therefore an f-regular sequence with respect to M. But the converse is not true in general. (For an f-regular sequence of M there exists of course an m-primary ideal q' such that this sequence is a q' -weak M-sequence.) As a consequence of the following it will be true, that the converse holds if A is it "good" local ring. Lemma 12. Let M be a NoetlU3rian A-module with d := dim M > 0 and q an m-priTlUliry ideal. If there exists a.system of parameters Xl> ... , Xd of M contained ~n m·q such that Xl' ••• , Xd is a q-weak M-sequence then

q . H:n(M)

=

0

for all i,* d.

In particular, H:n(M) is a finitely generated A-module for all i,* d.

The proof of this lemma is obtained word by word from the proof of the implication (i) ::} (iii) of Proposition 1.2.1. Therefore we omit it. It also is a stronger version of one implication of the following proposition. Proposition 13. Let M be a Noetherian A-module with d:= dim M > 0 and q an m-primary ideal. Then the following statements are equivalent: (i) There is a system of parameters Xl> ••• , Xd of ~W contained in q2 which is a q-weak JJ -sequence. (ii) Each system of parameters of M contained z'n q2 z's a q-weak M-sequence. (iii) q. H:nv.n = 0 for all z' dim M.

'*

The proof follows by using again exactly the same arguments as in the proof of Proposition 1.2.1. Thus we again omit it. Next we characterize those A-modules 1~f for which all local cohomology modules H:n(M),i dim M are finitely generated, i.e. of finite length.

'*

'fheorem 14. Let M denote a Noetherian A-module with d:= dim M > O. Then the follou'1'ng conditions are equivalent: (i) lA(H:n(M)) < 00 for all i 0, ... , d 1. (ii) There is an m-priTlUliry ideal q such that every system of parameters of M is a q-weak M~-sequence. 17

Buchsbaum Rings

258

Appendix

(iii) There is an m-primary ideal I and a system 01 parameters x~, ... , x~ is a I-weak M-sequence lor all n > 1. (iv) There is an integer t such that

Xl> ••• , Xd

01 M such that

lA(M/!.M) - eo(!,M):S;; t lor all parameter ideals! 01 M. (v) There is an integer s and a system 01 parameters

Xl> ••• , Xd

01 M such that lor all n > 1

lA(M/(x~, ... , x~). M) - eo((x~, ... , x~). A, M):S;; s.

Prool: (ii) =9 (i) and (iii) =9 (i) are consequences of Proposition'13 since in each case there is an m-primary ideal I) and a system of parameters in 1)2 which is a I)-weak M-s€quence. To show (i) =9 (iv) we prove the following more general lemma : Lemma 15. Let M be a Noetherian A-module with d:= dim M ideal ~ 01 M we then have:

>

O. For every parameter

1)

(d -:lA(H~(M)). 1 (ii) IllA(H~(M)) < ex:) lor all j = 0, ... , d - 1 then there is an m-prima/ry ideal q Buch , that equality holds in (i) lor all parameter ideals ! ~ q. (i) lA(Mj!. M) - eo(!, M):S;;

~El

1=0

Prool: If lA(!!~(M)) = ex:) for some j, 0 < j < d, then (i) is trivially true. Therefore assume lA(H{.(M)) < ex:) for all j = 0, ... , d - 1. Then there is, an m-primary ideal I) such that I) . H k(M) = 0 for all j < d. We show by induction on d that (i) holds with equality if ! ~ 1)2"-'. If d = 1 then ! = X • A with x E m and O:MX ~ O:M(m) If x E I), O:MX

=

O:MI)

=

H'/n(M).

= H'/n(M). This gives (see Lemma 0.1.3(vi)!)

lA(Mjx. M) - eo(!, M) with equality if x E I). Let pow d > 2 and! sequences

= lA(O:MX):S;; lA(H'/n(M))

= (Xl' ••• , Xci)' A

be a parameter ideal of M. We have two exact

0-+ O:MXI -+ M 14 MjO:MX l -+ 0

and 0-+ MjO:MX l ~ M -+ Mjx l · M -+ O.

By Lemma I.2.2 O:MX l ~ H'/n(M) with equality if Xl E I). Therefore H~(O:MXI) = 0 for all j > 0 and H~(p) is an isomorphism. Also Hk(i). H~(p) = H~(i. p) = H~(XI) = Xl' From the second exact sequence we obtain for all j > 1 (by using the isomorphisms H~(P)) (E j ): H{;;l(M) -+ H{;;l(MjXl . M) -+ H~(M).

Thus for j

<

d we have

1)2.H{;;l(MjXI·

M)

lA(H{;;l(MjXl . M)):S;; lA(H{;;l(M))

=0

and

+ lA(H~(M)).

On generalizations of Buchsbaum modules

If ); ~ 1)2'-" Xl • H~(M) = 0 for all j rise to short exact sequences

o

-l>

H1;;I(M)

-l>

< d and

H1;;l(MjXl' M)

-l>

259

therefore the exact sequences (Ej ) give

M~(M)

-l>

O.

\,

Hence ip this case lA(H~~l(M!Xl' M») = lA(H~-I(M») + lA(H~(M»). Now by the induction hypothesis (applied to M':= M!Xl' M) we get for t:= (X2' ••• , xa)' A, by using Lemma 0.1.3(vi) (note eo(!', O:MXl) = 0 by I..emma 0.1.4): eo(~',

lA(M!);. M) - eo(!, M) = lA(M'!t .M') -

:<;; di;2

M')

(d . 2) lA(H~(M'»)

j=O

(with equality if);'

1

~ tJ2

4

•')

,

:to e~ 2) (lA(H~(M») + lA(H{;I(M»)) 2

(with equality if !

~ tJ2--

1 )

q.e.d. We continue the proof of Theorem 14. (iv) =? (v) is trivial. (v) =? (iii): Let nl> ... , na > 0 denote integers and set D(nl' ... , na) := lA( M /(X~l,

•.• , x;J") , M) - eo( (X~', , .. , Xd

4) •

A, .If),

First we prove that for integers ml' ... , md with mj > nj for i = 1, , .. , d: D(nl> .. ,' nd)

D(~, ... , md)'

Since D(nlJ ..• , nd) does not depend on the order of the elements in the system of parameters it is sufficient to prove D(1, .. " 1, n) D(1, .,.,1, n + 1) for all n > 1. For i = 1, .. ,' d - 1 we write M j := (Xl> ... , Xi-I)' M:x1!(XI> ... , Xi-I)' M, Then by Lemma 0.1.3(i), (v) and (vi) we get D(1, .. ,,1, n

+ 1) -

D(1, ... , 1, n)

d-l

a-I

,J: eO(xi+1, •••• Xd-l> x;+1)· A, MI) - ,J: eo(x;;:l, ..., Xd-l, x d)· A, Mi)

!'~I

.=1

+ lA(XI> ... , Xd-I)' M :Xd+1!(XI , ••• , Xa- 1)' M) - lA(XI> , .. , Xd-l) . M :Xd!(Xl, ... , Xd-l)' M) a-I

J: eo(x/+1, ,."

Xd-!> xa), A,

M)

.=1

+ lA( (Xl' ..., XII-I) • M: X;+1j(Xh ... , Xd-l) , M :x~) I..et now n > 0 be an integer and let 1

O.

i:<;; d. For all m > n we get by Lemma 0.1.3(vi):

lA((X~, ''', xi_I' x41' ... , x:r) . M :xr;(x~, ... , xi_I, xl"+l> ... , x:I')' M) D(n, ... J n, m, ... , m)

17*

D(m; ..., m)

8

260

Appendix

by our assumption. Therefore

o

\

or equivalently, (x~, ... , Xf-l' x~V

••• , X

d )' M:xi

~ (x~, ... , Xi __ l' X1"+l' ... ,

X

d)· M:m s .'

By applying Krull's Intersection Theorem (the iiltersection is over all m

n) we obtain

i.e. x~, ... , x~ is an m8 -weak M-sequence for all n 1 and (ii) is proven. (iv) =} (ii) follows from the validity of (v) =} (iii), q.e.d. . Finally we will now give an answer to the problem mentioned in Remark 11.

Proposition 16. Let A be an epimorphic image of a local Cohen-Macaulay ri1UJ and let M be a Noetherian A-module with d:= dim At 1. Then the following conditions are equivalent: (i)

lA(H;"(M»)

<

00

'*' d.

for all i

(ii) Every system of parameters of M is an f-regular sequence with respect to M.

(iii) There is an m-primary ideal q such that every system of parameters of M is a q-weak M -sequence. (iv) M;> is a Cohen-l1!acaulay module (over A;» for all l:> E Supp M "-.... {m} and dim 1W = dim All:> for all mtllimal primes l:> in Supp M. Proof: The equivalence of (i) and (iii) is clear by Theorem 14. The equivalence of (ii) and (iv) follows from Proposition 3(i) § (v), since Spec A and hence Supp M is catenarian by the assumption on A. Remark 11 yields the implication (iii) =} (ii). Finally, , we prove the implication (ii) =} (i). Let B denote a local Gorenstein ring with maximal ideal e such that the m-adic completion A of A is a quotient of B. Since A is an epimorphic image of a Cohen-Macaulay ring, (ii) holds if we replace M by M = M ®A A and A by A and then by B (consider M as a B-module) by I.emma 8. Also (see, e.g. . i'" Sharp [2], Theorem 4.3) we have for all i that H~(M) Hm(M) (as B-modules) and i A. A Hm(M)~ H~(M) ®AA and this results in

lB(H~(M») = lB(H~(Ah)

lj(H~(.LW»)

lA(H~(M»).

Therefore we can assume without loss of generality that A is a local Gorenstein ring. Let n:= dim A. Let l:> E Supp M with dim All:> > O. Assume there is an integer j with n - d < j:::;; nand l:> E Supp Ext~(M, A), l:> ~ Supp Ext~(M, A) for all i > j. (Note that E:x:t~(M, A) = 0 for i < n - d and i> n.) Then by local duality (see Corollary 0.3.5): H;:;:lm{A/Pl-'(M;» r-v HomA;>(Ext~;>(M;>, Ap), I):::::: HOffiA;>(Ext~(M, A)p, I)

{

'*'

0 for i > j, 0 for i j,

where 1 denotes the injective hull of the residue field Ap/l:> Ap of Ap. Therefore n - dim All:>

j

depth Mp

dim Mp

d - dim All:>,

On generalizations of Buchsbaum modules

- 261

since llfp is a Cohen-Macaulay module by the equivalence of (ii) and (iv). Hence i . n - d, i.e. Supp Ext~(M, A) ~ {m} for all i> n d, and this means that Ext~(M, A) is of finite length for all i> n - d. Therefore we obtain again by local duality, H:n(M) ~ HomA(Ext~-i(M, A), E) (E denotes the injective hull of Aim). is of finite length for all i < d, q.e.d. Reexamining our Lemma 15 we are lead to investigate y~t another type of systems of parameters in a Noetherian A-module M. These observations go back originally to :.vI. Brodmann [2] and N. V. Trung [6] and [10]. Following Trnng [10], we prove

Theorem and Definition 17. Let M denote a Noetherian A-module of dimension d > O. Let Xl, •• " Xa be a system of parameters of M and let q := (Xl> ~ •• , xa) A. The following conditions are equivalent: (i)

lA(M/(xi, "" x~) M) - e (x~, ... , x~ iM)

(ii) IA(lll/(x~I" •. , x~·)

IA(M/(xl, . '" xa) M)

e(x~', . ,., x~'IM)

lW) -

for all

= IAM/(xl, ... , Xa) M) - e(Xh .,., xi/1M)

(iii) (IA(H:n(M))

00

IA(M/(xl> .. "

Xd)

for all i < d and)

M) - e(x1> ... , xalM)

d-I

.J.:

1=0

(iv) xr', ,.,' (v) X~',

x~·

, •• , X~d

is a q-weak

e(xl' ... , xalM),

~lf-sequence

for ail n 1 ,

11 1, ••• ,

nd

1,

(d -. 1) l.t(H~(M)). . '/.,

... , lid

E {I, 2}.

is a q-weak M-sequence for all n1> ,.,' nd> 1.

(vi) qH:n(M/(xl, ... , Xi) M)

=

0 for all i,

i

0 with i

+j

< d.

A system of parameters Xl>"" Xd of M fulfilling these equivalent conditions is / called a standard system of parameters of lll.

Proof: As in the proof of Theorem 14((v) =? (iii)) we let

Clearly this number does not depend on the order of the elements Xl> , •• , Xd and for. 1:;; ml nl, .,., 1:;; md < n'; we have by the proof of Theorem 14: D(mb .'" md) ::;; D(nl' ... , nd)' Hence Theorem 14 and Lemma 15 prove the equivalence of (ii) and (iii). The implid, cation (ii) =? (i) is trivial. Now we prove (i) =? (ii) by induction on N := J.: n; If N d there is nothing to show, Let N > d and without loss of generality assume lId = max{nh "" nd)' If nd = 2, we then have by (i):

D(1, .. ,,1):;; D(nl' , .. , nd)

D(2, .",2) = D(l, ... , 1)

which finishes the proof in this case, If nd

D(nl, ... , nil_I> nd - 1)

=

3, we have by the induction hypothesis'

D(l, ... ,1) =.D(nl' , .. , nd-l, 1).

,262

Appendix

Since nd - 1 2, 'Ye find by using the proof of the implication (v) =;. (iii) of Theorem 14 (for simplicity we set here Yi := x7' for i = 1'00', d - 1, n := nd):

o=

D(nJ> 00:' nd-l, nd -

1) - D(nJ> "0' ntl-l, nd

2)

tl-l

=.E e(Yi+l> .•. , Yd-h xdl(Yl' ... , Yi-l) M :Yi!(Yh .0., Yi-l) M) '=1

hence

e(Yi+l' 00', Yd-h xdl(Yl' 0", Yi-l) M :Yi/(Yh •.. , Yi-l) M)

0

for ,i = 1, ... , d - 1 and (Yl' ... , Yd-l) M :X;-l

(Yl' .," Ytl-l) M :X;-2.

Therefore (Yl' ... , Yd-l) M :x; = (Yl' ... , Ytl-Il M: X;-l and this implies D(nl' .;., nd) - D(I, ... ,1) = D(nJ> .", ntl) - D(nh ... , nd-I> ntl

1) = O.

Now the implication (v) =;. (iv) is trivial and (iv) =;. (vi) follows by Proposition 13 «i) ~ (iii)), ,¥ext we prove (vi) =;. (iii). Clearly, l(H!n(M») < 00 for i < d (let j 0). We prove by induction on m, 1 m < d:

Eo (m ~ 1) l(H!n(M/(Xl, .. ,' Xtl_m) M)) 1

l(M/qM)

eo(q; M) =

(note: eo(q; M) = e(xl' , •• , xdIM), c.f. Lemma 0.1.3), First by Lemma 0.2.2 and Theorem 7 we find D(I, .. " 1) = l(M/qM) - eo(qj M) = l(q'M:Xd/q'M),

where q':=(xh""xd-l)A. Since xtlH~(M!q'M) xd(q'M:(m)/q'M) =0, we have q)M:(m) = q'M:Xd and therefore D(I, ..., 1) = l(H~(M!q'M») and we are done for m = 1. Let 1 < m S d. Write q. := (Xl> ... , Xi) A for 0 < i < d. Since for 1 SiS d we have dimA qi-lM :Xi/qi-lM = 0 the exact sequence

0.....". qa-mM:xd-m+1/qa-mM - M/qd-mM.....". Mjqd-mM:Xd-m+l- 0 induces for all i

> 1 isomorphisms

H!n(M/qd-m M )::::: H!n(M /qd-m M :Xd-m+l) '

If 'we apply the functors H!n( ) to the exact sequence 0.....". Mjqd-mM:Xd-m+l-4 M/qd-mM.....". M/qd-m+lM - 0,

where g is obtained by multiplication by Xd-m+h the composition of the above isomorphisms with the maps induced by g in the resulting long exact cohomology sequence is just multiplication with Xd-m+l (on H!n(M/qd-mM» which is zero for m - 1 by our assumption. Since xd-m+1H~(Mjqd-mM) = 0, we get qd-mM:Xd-m+1 = qd_mM:(m), , hence H~(Mjqd_",M :Xd-m+1) O. Therefore the long exact cohomology sequence splits for all i < m - 1 into exact sequences

o . . .". H!n(M/qd-mM) -

H!n(M/qd-m+1M) .....". H:;l(M/qd-mM} .....".0.

On generalizations of Buchsbaum modules

263

where we have replaced H:;l(Mjqd_mM:Xd_m+1) by H:;l(Mjqd-mM) using the above isomorphism. This yields with the induction hypothesis

:E (m ~ 2) l(Hln(M/qd-m+1M ») 2

D(I, ... , 1) =

.=0

~

= i~2 (m" t'

2) [l(Hln(Mjqd-mM») + l(H:;l\Mjqd_m M »)] = ~1 (m i 1) l(Hln(Mjqd_m M »). (iii) follows for m = d. Next we prove (ii) =} (vi). li (ii) holds then (iii) is also true. 1, consider the embedding We use induction on d, li d

O:MXl c O:M(m)

= Irin(M).

Since by (iii) l(O:x}) = l(MjXIM) e(xIIM) 1(~(M»), O:x I = ~(M) and this implies xI~(M) = O. Let d > 1. Since every system of parameters is f-regular (Theorem 7), we obtain with M := M /x~M, n > 1, q' := (X2' "., xt./) A:

l(M j(x;', ••. , x;,) M) - e(x;', "" xd"IM) = D(n, n 2, ... , nd)

D(n, 1, ... , 1)

I(Mjq'M) - eo(q'; M).

Hence 0 = q'H:n(M j(X2' •.• , Xi) M) q'Hln(Mjx~, X2, ... , Xi) M) for all i> 0, i 1 with i + i < d and it remains to consider the case i 0 (set n = 1). The proof of the previous implication (vi) (iii) shows that there is a.n epimorphism for all i> 0 and all n '? 1:

H:n(Mjx1 M ) ~O:H::I(M}X~. Since every element of H:;l(M) is annihilated by some power of ,Xl> we get q'H:;l(M) = 0 for all 0 S; i < d - 1. Since (ii) does not depead on the order of the elements Xl> ••• , Xd, we find that qH:n(M) "0 for all 0 < i < d. On the other hand H~(M) = 0: x~ for sufficiently large n and the exact sequence

where the first map is induced by multiplication by 0): (since H~(MjO:x~)

x~,

gives rise to a monomorphism

H~(M) ~H~(Mjx~M).

This shows that q'~(M) 0, hence q~(M) 0 and this concludes the proof of the statement. Finally we prove (vi) =} (v). Let n l , ••• , nd 1. Since (vi) is equivalent to (ii) and (ii) does not depend on the order of the elements Xl> ••• , Xd, we can assume without loss of generality that 1 nl nd' li nl > 2, (v) follows from Proposition 13. li there is a j, 1 j d with nl ... = nj = 1, then we are finished if i d. to

264

Appendix

<

d, xit+i, ... , X~d is a system of parameters of M/(x v .. " Xj) M contained in Xd)2A, i.e. Xl> ... , xi> xr,+i, "', X~d is a q-weak M-sequence by Proposition 13 and '(vi), q.e.d.

If j

(Xj+l' ... ,

This theorem and Lemma 15 imply Corollary 18. Let M be a Noetherian A-module 0/ positive dimension. There is a standard system 0/ parameters 0/ M i/ and only i/l(H~&l'lf)) < 00 lor all i <. dim M. Moreover, in this situation there is an m-primary ideal q 0/ A such that every system 0/ parameters 0/ M conta~ned in q is a standard system 0/ parameters.

The corollary gives rise to the following definition: Definition 19. Let M be a Noetherian A-module of dimension d> O. An m-primary ideal q of A is called a standard ideal with respect to M if every system of parameters of M contained in q is a standard system of parameters of M. Note that M is a Buchsbaum module if and only if m is a standard ideal with respect to M. From this point of view our next (and last) theorem is not only a generalization of the important Theorem 1.2.15 but it may also be considered as an alternative way of proving this main result (without using the "Ext-functors" and injective resolutions). We note that statement (ii), or better, the equivalence of (i) and (ii) essentially goes back to S. Goto. Theorem 20. Let 111 be a Noetherian A -module 0/ dimension d > 0 and let a c A be an ideal which is m-primary. Assume we are given an M-basis Xl' ••• , X, o/a. The /oll~ng statements are equivalent: (i) a is a standard ideal with respect to M. (ii) The canonical maps

l1- :Hi(a; M) ~ Hk(llf) are surjective lor all i < d. (iii) XI" ... , Xi. iIJ a standard system 0/ parameters 0/ M lor all systems ill ... , id 0/ integers with 1 ~ i l < ... < id t. (iv) Every system 0/ parameters 0/ M conta~ned zn a is an a-weak M-scquence. Before proving this theorem we need some lemmata. But first of all we state: Corollary 21. Let Xl> .. " Xd be a system 0/ parameters 0/ the Noetherian A-module M and assume that d := dim M > O. Then Xl"'" Xd iIJ a standard system 0/ parameters 0/ M if and only i/ q = (Xl' ••• , Xd) A + Ann M is a standard ideal 0/ M.

Now we prove: Lemma 22. Let M be a Noetherian A-modu,le with d:= dim M > 0 and lA(Hk(M)) < 00 lor all i < d. Let Xl> ••• , x" r 1 be a part 0/ a system 0/ parameters 0/ M and let q := (Xl> ... , xr ) A. Then the maps

Hk(M) ~ H~(M) are ilJomorphilJms/or all i

(znduced by q ~ m, t:.f. Chapter 0, § 1, 3.)

< r.

I,. , 265

On generalizations of Buchsbaum modules

Proof: For i = 0, ... , r, M; ponents. Hence we have for i H~(M)

=

O:M(m)

=

(Xl' •.. , Xi)

M is unmixed in M up to m-primary

COlll-

0: O:M(q)

=

Hg(llf).

We have for all i> 0 a commutative diagram (c.f. Lemma O.1.5(ii)) H~UW) =+H~(MjH~(llf))

t

t

H~(M) =+H~(MIH~(M))

Therefore we can assume that depth 111 > 0. Then x := Xr is a non-zero divisor with (Xl' ... , X r - l ) A. Then for all n 1, q' is a part of a system respect to M. Set q' of parameters of M/x'flM and l(H~(M/xflM)) l(H~(M)) + l(H:'; I(M)) < 00 for all t' < d 1. Consider the following commutative diagram O-+M

M -+ M/xfI.M -+ 0

~II

~

~x

o -+M --+ M %1'1+1

-+MjX'fl+IM -+ O.

By applying H~(M) (resp. H~(M)) and forming the direct limits of the corresponding long cxact cohomology sequences we get for all i > 0 isomorphisms (since the liinit of the direct system H~(llf) H~(M) is zero):

-=+ ...

lim H~-I(.M /xn M) =+ H~(M)

(the same is true if we replace q by m).

n

Hence we find for i


(use Corollary 0.1.7)

H~Uff)~ ~H:';~I(M/xflM) n

H~(M),

~ Il~~~l(MIx" M) II

q.e.d. Lemma 23. Let M be a Noetherian A-module of dimension d > 0 with lA(H~UJ{)) <'00 for all i < d. Let U be an m-primary ideal of A and assume that t is an integer with 1 t~ d 8'lJCh that for all r = 0, ... , t - 1 U[(Yl' ... , Yr) M:M(m)] £; (Yl> ... , y,) llf for all parts Yl> ••• , Yr of 8YsterruJ of parameter8 of M contained in u. Let Xl> •.• , X, be a part of a system of parameter8 of M contained in u. Then for all nl> .•. , nt 1 we have:

>

(x~'+1, ... , xr'+I) M:MX~"' .... x;"

=

t

(Xl' •.• ,

X,) llf

+ E (Xl> .. ·,:t b

•.. ,

X,) M:Mtf)·

i=l

Proof: It suffices to prove "£;". We use induction on t. For t 1 we have (n:= nIl: = xIM + O:x~ xIM + O:(m) = Xlllf + 0:0. I.et t> 1 and X Eo be an element with dim 11f/xM d - 1. Then we have for 0: all n x~+lM:x1

xn+lM:o

+ O:x) n (x"+lM:Il)

+ 0:0) n (xll+lM:u) = 0:1l + x"(x",IM:u· x") 0:1l + x"(xM + O:x"):o) £; 0:1l + x"(xM:(m») = 0:(1 + x"(xM:Il).

£; (x"M

=

(xll+l.M:x) n (xll+lM:Il) = (xllM

266

Appendix

Therefore for each 1 ::;;

i::;; t -

1:

(Xl' •.. ,:t" .'" XI-I> X;·+l) M:a

= (Xl' .,"

:ti, .'"

XI-I)

M:a

+ X;"(Xt> ... ,:i;, .. " X/-I, XI) M:a),

This with the induction hypothesis results in the following (set M' := M/x;,+l M): [(X~l+l, , •• ,

~(~"'+1 = W} ,

-.

X?,+l)

M:X~"

.... x;']jx;.+IM

~"'_1+1)1011. ~"l •••• ' "'I-} ~"'-l)'·M' X'"I U 'M''''1

••• , WI_}

1-1

1-1

[( .J: (X}' , .. , :ii, ... , XI-I) M:a + x?'.J: (Xl' ... ,:ij, ... , XI_l> XI) M:a) .-1

'-1'

+ (Xl> ... , XI-I) M) :x~,]/ x;,+lM £;;;

[(Xl> .•• , XI-I) M:a· X;' [(Xl> ... , XI) M

+ i~\(XI' ..., :ii' ..., XI-I> XI) M:a)]1 x;,+lM

+ i~ (Xl' ..., :iil ... , X;) M :a)JI Xl',+1 M,

.since I

q.e.d.

(Xl' ... , XI) M £;;;.J: (XI> ... ,:i., ... , X,) M:a) '=1

for t

2,

Lemma 24. Let M be a Noetherian A~module of dimemrion d > 0 with 1... (H:n(M») for all i < d and let a c A be an m-primary ideal such that the canonical maps

).ir: Hi(a; M) ~ H~(M) 8uriect~Ve jor all ~. < r, where

<

00

H:n(M»)

are 0 r < d. For every part Xl, ••. , Xr of a 8Y8tem of parameters of M contained in a we have then a[(xI' ... , Xf) M:(m)J C (Xl' ... , Xf) M.

Proof: Note that by our assumption for every part Xb ... , Xf of a system of parameters of M, (Xb ••. , Xf) M is unmixed in M up to m-primary components. We use induction O:M(a). on r. Let r = O. Since 12t is the embedding O:Ma HO(a; M) £;;; H!(M) = O:M(m), hence a(O:M(m» = a(O:Ma) = 0, Therefore let 0 < r < d and assume that Xl' "" Xf is a part of a system of parameters .of M contained in a. By the induction hypothesis, a(Yl' ... , Y.) M:(m») £;;; (Yl' ... , Y.) M for every part YI, ... , Y. of a system of parameters of M contained in a with 0 8 < r. Choose an element yEa such that Xl' ... , X., Y is again a part of a system of parameters of M. Set q (Xl' ••• , X,) A, c( := (Xl> •.. , Xr , y) A. Consider the commutative diagram of Lemma 0.1.5: H'(a; M)

-4

H'(q' ; M)

1':; H~.(M)

On generalizations of Buchsbaum modules

267

By Lemma 22, p. is an isomorphism and since ).L- is surjective, )." is surjective. Now consider the right-hand part of the first commutative diagram of Corollary 0.1.7:

H'(q'; M) _ HO(y; H'(q; M») _ 0 ~l"

~1'

I¥q,(M) It shows us that

H~A(H~(M»)

_ O.

l' is surjective. By the remarks made in Chapter 0, § 1, 3., we get

H~AH~(M»):::: H:A(1~ ~j(x~, ... ,~) M) "-' ~ ~AM!(x~, ... , x;) M)

!!!!: [(x~, •••, x~) M :(yA)/(x~, ..., x:) M]. n

Take mE qM:(m), in:= m mod qM. Then the image of in in H:A(H~(M») can he represented by an element Xr(v) with v E HO(y; H'(q; M») '" HO(y; MjqM) qM:yjqM. Let v = in' with m' E qM:y. Then there is an integer n > 0 with x~

..... x:(m

m') E (x~+l, ... , X;+l) M,

, qM:a c qM:y by Lemma 23. Therehence m - m' E I: (Xl' "., :t" .'" X,) M :a) fore mE qM:y. 1=1 t qM:a, i.e. Now take an (MjqM)-basis Yl> "., Yt of a. Then mEn (qM:YI) ;=1 a{qM:(m» qM, q:.e.d. Now we are able to prove our Theorem 20. Proof of Theorem 20: (i):::;. (iii) is trivial and (iv):::;. (i) is a' consequence of Theorem and Definition 17 since for any system of parameters Xl>"" Xd of M contained in a, xi', _._, X:;d is an (Xl' •. " Xd) A-weak M-sequence fot all nl> ... , nd 1 if it is Jm a-weak M-sequence. To prove the implication (iii) :::;. (ii) we use by applying Theorem

and Definition ~7 word for word the proof of the implication (iii) :::;. (ii) of Theorem 1.2.15 replacing only m by a. Finally we prove (ii):::;. (iv). Thesurjectivityof ).1-implies that l(Hk(M») l(HI(a; M») < 00 for all i < d. Let Xl> ••• , Xd be a system of parameters of M contained in a and let 1 d. Then

a(xb ... , Xi-I) M:Xi) S;;;; a(Xl' •••, Xi-I) M:(m») ~ (Xl' ••. , XH) M by Lemma 24, i.e.

Xl> ••• ,

Xd is an a-weak M-sequence and the proof .is finished, q.e.d.

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II

Notations

a,....., b 156 a(lIt) 36, 99 AnnN 21 am V 184 AssAM, Ass.Y" 21, 34 Ass Rq (M) 201 Blq(R)

124 On,m 185, 186 codimL 113 conep(L) 114 cx 108 c·sphere 124 ~ 161 C.(L; K) 143

176

(Def. 0.2.2) elM) 36 emb R 53 eo(q, M) 23 24

38

lz:

G*

129 g(M) 98 Gq(M) 201

H~(M) 26 hO(/T) (= dim lJO(X, /T), /T a sheaf on X) 189 ho(M), h 1 (M), hd(M) 36 .!p(mR, 21f) 39

37

112

K[H], K[S] 229 Ki(x1 , ••• , x r ; .,w) 27 K(x l , ••• , x r ; M) 26 Koo(x 1 , ••• , x r ; lIf) 28 KM 51 K[L] 115, 122

liP)

38

35

i(C) = 1(K[x1 , ••• , xa]/l(C») 157, 161 lIM) 14,62 inj dimR M 54 Int(P) 152 Iv 37

Ik.!.' (f

IFI 108

!Ji(R, M)

36

HM(n)

H~(X, /T)

e(!!~("Y"») 99, 101

Extk(,)

hM

38

Hom(X', Y") 45 II q(X; A) 109 1l,,(Lt; A) 108 llQ(A; A) 110 hx(O) 104 mIX, /T) 36

deg hj}[ 36 depthA M, depth M 23, 35 dim A M, dim M 21 dM'[I) 132 ·Di; 143

e(xl' ..• , x"llJt)

H~,,(M)

27 27

H(m, kELt]) 148 HomR(M, N) 37 HomR(M, N) 37

199

c-disc

E(C)

lli(x 1 , ... , x r ; M) Hi(xp ... , X r ; M) II!(X,) 37

lR(M)

113 111

52

11f il 34 If(il) 34 Mil.1I 34

M/q.M 205 M'[t] 132 11f" 56

!f 205 N:Ma 25 N:M(a) 25

P

111 152 P/l(X) 17,104 pdRM 54 Polar m 107 P xQ 152

p*

Pq.M(n), PQ.M(n)

R d •m

231

R~:~ 231

R(f) 34 R,(M, ) 34 RG 119 r(lJf) 36, 101 RNQM 125 rIP) 111 R q (lIf) 201 Rrm(X')

49

S

229 229 Sing L 124 Spec A 22

f;

MX M*

59

94 M(C) 157

Star.!.' (f 113 SUPPA .Y", Supp M

M d •m

231 231 34 33 33

Supp Rq(M) 201 Supp(x) 148

M~:~ MU ) [.Y"].. M(p)

23

Proj R 36 P s 153

P-lM 34 Tot K" 215

21, 35

Notation&

15,22

U(o), U(Y)

LIn .1(P)

EX

123

111

A~ 28

46

Z(P,m)

153

cx(P, S)

153

(3(P, S)

153

Fa, l'a(M)

.1 108 1.11 108

111

EPolar I

45

X"@Y"

Y'[n]

E"X

112 112

26

TTM' T8 M'

IlR(M) 53 Jl(M) 162 +a, -a a(ml' m 2) a(Ml' M 2 ) a(Rl' R 2 ) ak(R 1 , R 2 )

E 112 E[ 106

T~(M')

126 99 101

39,99 39

tpk

26

X(P s )

X(X)

X(.1)

'Pk

132 132 132

154 109 109

27

283 -

Index

Absolutely superficial element 70 system of parameters 70 Acyclic 110 Affine semigroup ring 229 Almost Cohen-Macaulay poset 152 Annihilator of a module 21 Arithmetic genus 17, 104 Associated graph of a simplicial complex 128 graded module 201 Arithmetically Buchsbaum 15, 17 Arithmetically Buchsbaum curves 17, 156 Associated prime 33,201 Associated prime ideal 21,201 Augmented chain complex 108 Auslander, M. 24 Bezout, E. 9 Bezout's Theorem 10 Bouquet 152 Buchsbaum, D. A. 12, 13, 14, 18, 23, 62 Buchsbaum complex 106, 143, 144, 149, 154 Buchsbaum ideal 182 Buchsbaum module 14, 15, 63, 64, 66, 79, 84, 132, 135, 136, 138, 142, 145, 207, 211, 215, 220, 222, 224 for Segre products 101 Buchsbaum poset 111, 152, 153, 154 Buchsbaum ring 14,15,63,135,138,142,143, 157 Canonical module 51, 138 Catenarian 253 Chain complex 108 Om. It-configuration 186 Codimension of a finite simplicial complex Cohen-Macaulay ideal 182 Cohen-Macaulay module 23, 35 Cohen-Macaulay poset 152 Cohen-Macaulay ring 23 Coherently oriented 126 Combinatorial disc 124 manifold 124 sphere 124

113

Complex 45 bounded 46 bounded above 46 bounded below 46 Configurations of lines 181 Convex hull 108, 113 Co-primary module 21 Curve (in P~) 55 Cylinder 122, 126 Degree of a homogeneous element 33 of a variety 10 Depth of a module 23 Desingularization 199,200 Diameter 129, 130, 131 Disc 124 d-sequence 69 Dual of a module 56 Dual graph 129, 130, 131 Dualitv Theorem 58 Dualizing complex 47, 132, 133, 143 Embedded prime ideal 22 Equidimensional 59, 104 Equidimensional module 23 Euler characteristic 109 Evaluation map (of complexes)

47

Face 108 Face rings 106 Facet 113, 129 Filter-regular (f-regular) sequence 252 Formula of Deligne 30 F-pure 117 Free resolution of a (graded) module 54 I-vector 148, 149, 151, 153 Geometric genus 17, 104 Geometric realization 108, 113, 116, 122 Glueing (over a prime ideal) 238 Gorenstein k-algebra 58 Gorenstein ring 23 Goto, S. 18 Goto's Lemma 139

285

Index Graded Buchsbaum module 95, 96 k-algebra 35 Koszul'cohomology 39 Koszul compl~x 39 module 33 ring 33 Grabner, W. 9 Grothendieck, A. 25 g-vector 150, 151, 154 h-Buchsbaum module 95, 96, 97 Hilbert coefficients 36 Hilbert function 35, 101 Hilbert polynomial 36 Hilbert-Samuel function of a module 23 Hilbert-Samuel polynomial of a module 23 Hironaka, H. 18 Homogeneous element 33 ideal 33 Homology of the Koszul complex (= Koszul homology) 27 Homology manifold 122 h-vector 106, 149, 150, 151, 153, 154 Hyperplane sections 89

Local cohomology functor 25 Local duality (compare local duality theorem) 50 Local duality theorem 49 Local ring 22 Localization 34 Locally Cohen-Macaulay 59, 104 Locally nilpotent element 21 Macaulay, F. S. 9, 12, 15, 16 H-basis of an ideal (with respect to a module H) 63 Maclaurin, C. 9 Manifold (combinatorial) 124 Minimal injective complex 46 graded free resolution 54 set of generators of a module 53 Mobius band 123, 126, 127 "'fabius function 112, 152 Monomial space curve 166 M-sequence 22, 35 Multiplicity of an ideal (with respect to a module) 23 of a local ring 23 Multiplicity symbol 24 Mumford, D. 18

Ideal of (ype (r, d) 83, 237 Idealization 244 Ideally intersection of d hypersurfaces 161 Idealtheoretic intersection multiplicity 11, 12 Index of regJllarity 36, 101 Intersection multiplicities 10, 12 Interval poset 152 Irregularity 17

Nakayama's Lemma 37 Ne"\\-ton, 1. 9 Noetherian graded R-module 33 Normal flatness 199 Normalization of a submonoid of N Normalized dualizing complex 48

Koszul cohomology (compare Koszul homology) 27, 134 Koszul complex 26, 118, 134 Koszul homology 27 Krull dimension of a module 21 of a graded module 35 Kiinneth relation 43

Order complex 112 Order-dual 152 Order-filter 112 'Order-ideal 112 Orientable 111, 126 monomial ideal 130 simplicial complex 130 Oriented simplex 126 q-simplex 108

Lasker, M. 9 Leibniz, G. W. 9 Length of a module 52 Liaison 56, 157, 159 Liaison addition 177, 178 Liaison addition problem 174,176 Liaison realization 180 Lifting the Buchsbaum property 87,88,90 Link 113, 114 Linkage class 156 Linked 156, 157, 159 Local cohomology 25, 26, 38

Parameter ideal 14,35 of a module 22 Part of a system of parameters 22 Perfect 135 Poincare Duality Theorem 111 Polar 107, 123, 128 Polarization 107, 123, 128, 130 Poset 111, 152 Power formula 179 Primary decomposition 22 Primary submodule 21

229

,

286

Index

Severi, F. 9 Sheafification 36 , Shifting of degrees 33 Simplicial complex 108 cone 114 q-face 108 Simultaneously Cohen·Macaulay 206 q-simpl,ex 108 Singh, B. 63 Quasi-Buchsbaum ring 74 Quasi-isomorphism of complexes 46, 132, 133, Singular chain complex 109 134,135 Quasi-manifold 125, 126, 130 q.simplex 109 Singularities q-weak .JI-sequence 257 of cones 125 of stars 125 Rank selected subposet 153 Rational twisted cubic 16 Sip.gularity of a simplicial complex 124 SoMan's example 121 Reduced Euler characteristic 109, 151, 154 Spectrum of a ring A (Spec A) 22 homology group 108, 115 Sphere 124 Mayer-Vietoria sequence 109 Splitting lemma 107 simplicial chain complex 143 Standard ideal 264 singular homology group 109 Standard n.simplex 112 Rees module 201 Standard system of parameters 261 Regular Star 113, 114 local ring 23 Submodule of a graded module 33 non-quasi-manifold 125 Support of a module 21 sequence (see .."I-sequence) 69 Suspension 111 Reisner complex 123, 128, 130, 131 Suzuki's proof 139 Reisner's Cohen-Maca ulay criterion 115, 123 System of parameters 35 Reisner's example 120 of a module 22 Reisner's locally Cohen-Macaulay criterion 116, 120, 132, 142 Topological n-manifold 110 Relative Torus 122 homology 110 Total ideal 37 regular sequence 70 Triangles 129, 130 Residue field of a local ring 22 Triangulation 108, 122, 123 Ring of invariants 119 Twisted global sections 36 Projections of Veronesian varieties (ideals) 231 Projective spectrum 36 Pseudomanifold 111 Pure homomorphism 117

Saturated ideal 37 Schwartau's Cohen-Macaulay criterion 128 Segre product 39, 99 Seidenberg, A. 18 Self· linked 193, 197 curve 192 Serre cohomology 38 Set·theoretic complete intersection 184

Universal-coefficient theorem for cohomology 110 Upper Bound Conjecture 19, 106 van der Waerden, B. L. 9, 12 Weak A-sequence (M-sequence) 69,71 Wei!, A. 9

14, 62, 63,

Stiickrad . Vogel

Buchsbaum Rings and Applications

This book describes interactions between algebraic geometry, commutative and homological algebra, algebraic topology and combinatorics. The main object of study are local Buchsbaum rings. The roots for this theory are precisely located in the desire to have a deeper understanding of the connection between "length" and "multiplicity" . Insight into these questions has an impressive record of solving old and new problems with the theory of Buchsbaum rings. An extensive bibliography is also included. This book provides both the versed specialist and the advanced student with an excellent working text.

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GHP: 41 TEG3061+1 ISBN 3-540-16844-3 ISBN 0-387-16844-3