Iooal oohomology its applioations
edited
and
by
Gennady Lyubeznik University of Minnesota Minneapolis, Minnesota
MARCEL
MARCELDEKKER, INC. DEKKER
NEWYORK" ]BASEL
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PREFACE
Local cohomology was introduced by A. Grothendieck in [2, 3]. Subsequent development to a great extent has been motivated by Grothendieck’s ideas. For example, ¯ The problem in [2, p. 79] has motivated important work on local coo homological dimension; see for example Hartshorne [4, 5], Ogus [11, 12], Hartshorne and Speiser [6], Faltings Ill, Huneke and Lyubeznik [7] and Lyubeznik[9]. ¯ The line of research initiated by the conjecture in [3, p. 173] has in the last decade culminated in the discovery of somestriking finiteness properties of local cohomologymodules. This development is surveyed in [10]. ¯ The local duality theorem [2, Section 6] has motivated generalizations and extensions explained from a unified point of view in Lipman[8]. In the last decade connections between local cohomologyand several quite diverse areas have been discovered, such as topology, D-modules, combinatorial analysis and others. Greenlees-Mayduality, a far-reaching generalisation of local duality, is one such exciting development. It was discovered by two topologists whonaturally cam~ to this commutative-algebraic result via their work in topology.
III
iv
Preface
Recently an intemational workshop on local cohomologyand its applications was held at the mathematical institute CIMAT in Guanajuato, Mexico. It was organized jointly by Xavier Gomez-Montof the CIMATand Gennady Lyubeznik. It was funded jointly by the USNational Science Foundation and the MexicanNational Council of Science and Technology. The workshopfeatured two minicourses consisting of three lectures each. One was by John Greenlees on some applications of local cohomology in topology and the other was by Joe Lipmanon foundations of local cohomology and duality. There were a number of other talks by invited participants on various aspects of local cohomology.Here is a list of the invited participants. From the US: Richard Belshoff, Southwest Missouri State University Anton Leykin, University of Minnesota Joseph Lipman, Purdue Universi~ty Gennady Lyubeznik, University of Minnesota Thomas Marley, University of Nebraska Ezra Miller, University of California at Berkeley Mircea Mustata, University of California at Berkeley Anuragh Singh, University of Utah Harrison Tsai, University of California at Berkeley Uli Walther, Mathematical Sciences Research Institute Janet Vassilev, Virginia CommonwealthUniversity Cameron Wickham, Southwest Missouri State University From other countries: Josep Alvarez Montaner, Barcelona, Spain Markus Brodmann, Ziirich, Switzerland Ricardo Garcia Lopez, Barcelona, Spain Xavier Gomez-Mont, Guanajuato, Mexico John Greenlees, Sheffield, UK Michael Hellus, Regensburg, Germany I-Chiau Huang, Taipei, Taiwan Leif Melkersson, Lund, Sweden Rodney Sharp, Sheffield, UK Kohji Yanagawa, Osaka, Japan Santiago Zarzue!a, Barcelona, Spain This volume contains expanded lecture notes of the two minicourses by John Greenlees and Joe Lipmanand papers by some of the other participants in the workshop. Namely, Markus Brodmann surveys some results on the cohomology of projective schemes; Harrison Tsai and Anton Leykin deal with algorithmic aspects of local cohomologyvia the theory of D-modules; Ezra Miller and Kohji Yanagawadiscuss some aspects of local cohomology over various kinds of graded rings; Anurag Singh deals with the question
v
Preface
of whether the set of the associated primes of a local cohomology module is finite; I-Chiau Huang deals with applications of local cohomology to combinatorial analysis and Gennady Lyubeznik surveys some of the developments in local cohomology in the last decade. Finally, it gives me pleasure to thank my co-organizer Dr. Xavier GomezMont and the CIMATstaff for having made the workshop an enjoyable experience. REFERENCES hoher Ordnung,J. f. d. reine und angew. [1] G. Faltings, ~ber locale Kohomologiegruppen Math., 313, (1980), 43-51. [2] A. Grothendieck, Local Cohomology,Lecture Notes in Mathematics, 41, Springer, Berlin, 1966. [3] A. Grothendieck, CohomologieLocale des Faisceaux Coherents et Theoremesde Lefschetz Locauxet Globaux (SGA2), Norht-Holland, Amsterdam,1968. [4] .R. Hartshorne, Cohomologicaldimensionof algebraic varieties, Annalsof Math., 88 (1968), 403-450. [5] R. Hartshorne, Amplesubvarieties of algebraic varieties, Springer lecture Notes156, (1970). [6] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p, Annals of Math., 105 (1977), 45-79. [7] C. Hunekeand G. Lyubeznik, On the vanishing of local cohomologymodules, Invent. Math., 102 (1990), 73-93. [8] J. Lipman,Lectures on local cohomologyand duality, this volume. [9] G. Lyubeznik,Etale cohomologicaldimensionand the topology of algebraic varieties, Annals Math., 137 (1993), 71-128. [10] G. Lyubeznik,A partial survey of local cohomology,this volume. [11] A. Ogus, Local cohomologicaldimensionof algebraic varieties, Annals of Math., 98 (1973), 327-365. [12] A. Ogus, On the formal neighbourhoodof a subvariety of projective space, Amer.J. Math., 97 (1976), 1085-1107.
Gennady Lyubeznik
CONTENTS
Preface Contributors
Ill
The Minicourses Local Cohomologyin Equivariant Topology J. P. C. Greenlees Lectures on Local Cohomologyand Duality Joseph Lipman
39
Other Papers CohomologicalInvariants of Coherent Sheavesover Projective Schemes: A Survey Markus Brodmann
5.
A Partial Survey of Local Cohomology Gennady Lyubeznik
121
p-Torsion Elements in Local CohomologyModulesII Anurag K. Singh
155
Algorithms for Associated Primes, WeylClosure, and Local Cohomology of D-Modules Harrison Tsai 7.
9.
91
169
Computing Local Cohomologyin Macaulay 2 Anton Leykin
195
Squarefree Modules and Local CohomologyModules at MonomialIdeals Kohji Yanagawa
207
Graded Greenlees-MayDuality and the (~ech Hull Ezra Miller
233
10. Residue Methodsin Combinatorial Analysis l-Chiau Huang vii
255
CONTRIBUTORS
Markus Brodmann University of Zurich, Zurich, Switzerland J. P. C. Greenlees Sheffield University, Sheffield, E. ngland I-Chiau Huang Institute of Mathematics,AcademiaSinica, Nankang,Taipei, Taiwan, R.O.C. Anton Leykin
University of Minnesota, Minneapolis, Minnesota
Joseph Lipman Purdue University, West Lafayette, Indiana GennadyLyubeznik University of Minnesota, Minneapolis, Minnesota Ezra Miller Massachusetts Institute
of Technology, Cambridge, Massachusetts
Anurag K. Singh University of Utah, Salt Lake City, Utah Harrison Tsai Cornell University, Ithaca, NewYork Kohji YanagawaOsaka University,
Osaka, Japan
ix
LOCAL
COHOMOLOGY
IN
EQUIVARIANT
TOPOLOGY
J. P. C. Greenlees Sheffield University, Sheffield, England
ABSTRACT. The article describes the role of local homologyand cohomologyin understanding the equivariant cohomologyand homologyof universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomenaare studied and illustrated in several rather different families of examples.Both topology and commutative algebra benefit from the connection, and many interesting questions remainopen.
2
Greenlees CONTENTS 1. Overview
Part I. Equivariant K-theory. 2. The complex representation ring 3. Free G-spaces. 4. The Atiyah-Segal completion theorem and its dual. 5. Duality for the representation ring. 6. A proof for p-groups. 7. Variations and extensions. Part II. Ordinary cohomology and graded connected k-algebras. 8. Cohomologyof some classes of groups. 8.A. Finite groups. 8.B. Compact Lie groups. 8.C. Arithmetic groups. 8.D. p-adic Lie groups. 8.E. Commonfeatures. 9. The local cohomology theorem for group cohomology. 10. An algebraic proof of the local cohomologytheorem for ordinary cohomology. 11. Structural implications of the local cohomologytheorem. Part III. Examples related to bordism. 12. The theorem for MU-modules. 13. The proof. 13.A. G-spectra. 13.B. Commutative algebra with G-spectra. 13.C. The map A is an equivalence. 13.D. The map B is an equivalence. 14. Homotopically Gorenstein rings. 15. The chromatic case. 16. Connective K theory. References
1. OVERVIEW The usefulness of local cohomologyin equivariant topology is not jllst a superficial phenomenon.It arises because similar structures occur in both contexts. The aim of these lectures is to explain one particular connection that I am especially familiar with, showing the commonstructures in the process. This connection is useful in both directions. I hope to expose :some
LocalCohomology in Equivariant Topology
3
interesting algebraic structures and recommendthem for further study, and to display some topological phenomena that may be amenable to methods of commutative algebra. The main ingredients on the topological side are a group G and a Gequivariant cohomology theory E~(.). On the commutative algebra side begin with an augmentedk-algebra S (i.e., the k-algebra S is equipped with a k-algebra map S --~ k called the augmentation), and the augmentation ideal J = ker(S ---~ k); in the topological examples S = E~ is the equivariant coefficient ring and k = E* is the non-equivariant coefficient ring for a suitable equivariant cohomology theory E~(.). Aside from the common structures, the reason for studying these phenomenais that the rings S have an interesting duality property generalizing the notion of Gorenstein rings. The paper is organized around families of examples of rings S arising from equivariant cohomologytheories. Westart in Part I with a rather simple instance (K theory and the complex representation ring), where quite complete results are available in very concrete terms. This should give somelife to the ideas. Wethen summarize the other examples we want to discuss, before considering each in more detail: these subsequent examples all correspond to co~nplete rings. First in Part II we consider the case of ordinary cohomology,where S is a complete graded ring over a field k. In this simpler situation it is possible to give general geometric results of somesubstance. In Part III we turn to chromatic examples, where S is finite over k; the geometry is more complicated here, so we restrict attention to rather crude structural pheno~nenaand to some very special cases. During the discussion, there will always be a group G in the background. Most results are interesting even for the group of order 2, and the reader may want to concentrate on this case to begin with. A few sections treat groups which are not finite, and readers will lose little by ignoring them. Part I. Equivariant K-theory. In Part I, we concentrate on a particular example which is not only very close to algebra, but can also be made completely explicit. Most of the phenomenawe are concerned with occur in this case in very concrete forms. 2. THE COMPLEX REPRESENTATION
RING
For further details of this section see [52, 53]. Let G be a compact Lie group and X a G-space (i.e., a topological space with continuous left G-action). Throughout the discussion the trivial group is permitted; this is referred to as the non-equivariant case, and notation for the group is often omitted. A complex G-equivariant vector bundle over X is a continuously varying family of co~nplex vector spaces, parametrized by X. Moreprecisely, it is a G-map~r : ~ -~ X so that for each x ~ X the fibre ~x := ~r-~(x) is a complexvector space, and for each g E G, the translation g : ~x ~ ~gx is linear. Wealso require that ~ is locally trivial: for each
4
Greenlees
point x E X there is an open neighbourhood U over which r is projection in the sense that there is a commutative diagram -
71.-1 (U)
U Welet VectG(X) denote the set of isomorphism classes of G-equivariant complex vector bundles on X. Direct sum of vector spaces extends to wector bundles, making VectG(X) into a monoid. The equivariant K-theory .of is obtained by the Grothendicek construction, adjoining inverses to form an abelian group K~(X) = AbGp(VectG(Z))/([~ [~] + [~?]). Tensor product of vector spaces extends to vector bundles, and this makes K~(X) into a commutative ring. Note that if X is a point, a vector bundle over X is just a vector space with linear G action, which is simply a complex representation: g~(pt)
= R(G)
where R(G) is the complex representation ring. Indeed, tensor product with a representation makes K~(X) into an R(G)-module, so from the algebraic point of view we are discussing the ring R(G) and modules over it. Example 2.1. (i) If G is cyclic of order n we may choose a faithful, one dimensional representation a and R(G) = Z[a]/(a n - 1). This is one dimensional, and using cyclotomic polynomials, we see that the irreducible components of spec(R(G)) correspond to the subgroups of (ii) If G is the dihedral group of order 8 (i.e., the symmetrygroup of square), character theory shows R(Ds)=Z[a,b,a]/(a2=b2=
1, aa=ba=a,
a2= l +a+b+ab).
For an arbitary compact Lie group G, R(G) is Noetherian. Segal has shown[52] that its dimension is 1 + rank(G) where rank(G) is the dimension of the maximal torus in G, and the irreducible components of spec(R(G)) correspond to the conjugacy classes of topologically cyclic subgroups of G (these are the closed subgroups generated by a single element, and are thus iso~norphic to the product of a torus and a finite cyclic group). Nowin fact the functor K~ extends to an equivariant cohomologytheory K~ : G-spaces ~ AbGp. We will not explain in detail what this means, but it is a contravariant functor with good exactness properties (analogous to those of functors which take exact sequences of modules to long exact sequences of cohomology groups), and which takes sums to pr(~ducts. The extension can be given by stating that K~ is 2-periodic in the sense that K~ = K~+~ and the odd part is given by K~I(x) = ker(K~(S ~ × X)
LocalCohomology in Equivariant Topology K~({1} × X)). The fact that this gives a suitably periodicity. For example
5 exact functor is Dott
Kh(pt ) = R(G)[~, where u is a unit of degree 2; the value K~(X) on a G-space X is a module over K~(pt), and multiplication by u gives the periodicity. Henceforth we adopt the convenient abbreviation K~ = K~(pt) for the coefficient ring. The other consequence of having a cohomologytheory is that there is an associated homology theory K,~(X). This is more complicated to define, so we will be content to say that it is related to cohomologyin such a way that a form of Poincar6 duality holds when X is a manifold. The most trivial example of this is that K~(pt) = KG_i(pt), so that, with the usual convention for relating upper and lower indexing, we have
= Ky= Where necessary we refer to lower indices as degrees and upper indices as codegrees; for example u E K~G = K~2 is of degree 2 and codegree -2. 3. FREEG-SPACES. For further details of this section see [53, 28]. The simplest sort of G-spaces are those with a free action (i.e., so that the identity element of the group is the only element fixing anything). In particular if X is a free G-space then equivariant vector bundles ~ --+ X over X correspond to non-equivariant bundles ~/~ X/G over the quotient: given ~ we take ~ = ~/G and given ~/we take ~ to be the pullback of ~ along the quotient map X ~ X/G. This passes to K theory to say that if X is free Kh(X ) = K~(X/G). For finite groups there is a similar statement K,G(X) = K,(X/G) for homology. Howeverthis is less elementary, and involves transfer arguments. Accordingly, if G is a general co~npact Lie group the statement must be modified by inserting a suitable kind of twisted shift. The other thing about free G-spaces is that there is a terminal free G space EG in the homotopy category. This means that for any free G-space X there is a G-map ~x : X ~ EG, unique up to homotopy. In fact EG is characterized by two properties: it is free and non-equivariantly contractible. For example if G is cyclic we mayview G as a subgroup of the unit complex numbers, and then EG = S(c~cC~) is the unit sphere in the direct sum of infinitely many copies of (:. This may be more familiar to shme through the quotient BG = (EG)/G, called the classifying space of G: for instance ~ BC~ . = ]lOP
6
Greenlees
Note in particular induces a diagram
that for any free G-space X the universal map vx K~(X)
K°(X/G)
~
K~(EG)
~ K°(BG).
Thus a knowledge of K~(EG)gives canonical characteristic classes in K~(X) K°(X/G). These can be useful invariants for distinguishing different Gspaces with the same quotient X/G. For example if G = C2 we may have X1 = G × ]I¢~P3 whilst X2 = S(2C) (i.e., the 3-sphere with the antipodal action). In both cases the quotient is projective space ]~p3, but the different characteristic classes distinguish them. A similar motivation for studying K,G(EG) can be given, but a more convincing one can be given because quite powerful torsion invariants of free G-manifolds belong to this group [56]. 4. THE ATIYAH-SEGAL COMPLETION THEOREM AND ITS DUAL. For further details of this section see [3, 25]. In this section we describe the calculation of K~(EG)and the associated homologyK.G(EG).Wepresent this in a quasi-historical way, as a series of theorems starting very concrete and proceeding to greater generality at the cost of some additional machinery. Reme~nberingthe periodicity of K theory, we may work with the degree 0 part of the coefficient ring S = R(G), and it suffices to describe the 0th and let K groups. The ring S = R(G) is augmented over k = R(1) by the map S = R(G) R(1) = Z ta king th e di mension of act ual rep resentations, and we need to consider the augmentation ideal J = ker(R(G) ---+ R(1) whose elements are said to be of virtual dimension 0. Theorem 4.1. (Atiyah (1961) [1]) The equivariant K theory of EG is K~(EG) = K°(BG)= R(G)~, where R(G)~ denotes the J-adic completion of R(G), K~,(EG) -~ KI(BG) Example 4.2. (i) When G = C2, we have seen R(C~) = Z[~]/(~ 2 - 1), and since 1 and o~ are one dimensional representations, we see J = (1 -- ~). Nowa short calculation allows us to identify the completion: (1 - ~)2 1 - 2c~ + c~2 = 2(1 - c~), so that (1 --"00 n+l = 2n(1 - o~), and
=
7
LocalCohomology in Equivariant Topology (ii) Moregenerally, if G is a p-group the J-adic and (p)-adic topologies J coincide [4, III.l.1] so that
(iii) For finite groups G, J-adic completion R(G) ~-~ R(G)~is injective if and only if G is a p-group: injectivity for a p-group follows from (ii), and is not hard to construct an element of the kernel otherwise. [] There is a more conceptual way of expressing the phenomena. Weconsider the projection map EG --~ pt and the induced map
* EG X) K*(EG×GX) Kb(X) --+ Ke( in K-theory. To avoid pathology we want to restrict attention to G-spaces X whichare locally nice: the simplest wayto do this whenG is finite is to consider simplicial complexes with an action of G induced by permutation of the vertices. Wecall these G-complexes: up to G-equivariant homotopy equivalence this gives the sameclass as if we require that any setwise fixed simplex is fixed pointwise, and it also gives the same class as G-CW-complexes.A G-complexis said to be finite if it has finitely manysimplexes. Theorem 4.a. (Atiyah-Segal (1969) [3]) For any finite G-complex X the natural map Kb(X ) -~ K*G(EG x X) = K*(EG XG is completion at the augmentation ideal J of R(G). Wepause for some remarks about the finiteness restriction and how to remove it. Firstly, Theorem4.1 is the special case X = pt, since we have seen that K~(pt) = R(G)[u,u-1]. Next, we comment that the codomain K~(EG × X) is a cohomology theory of X, so that for the statement to be plausible, it is necessary thatKe(X)j * ^is a cohomology theory on finite complexes X. However, for each i, the R(G)-module K~(X) is Noetherian provided X is a finite G-complex. This is clear for K~(G/H) = K°H(Pt) R(H), and the general case follows by induction on the number of cells, using the exactness properties of a cohomology theory. Accordingly, the * A is a Artin-Rees lemma implies that J-completion is exact, and so KG(X)j cohomologytheory on finite complexes X. Since J-completion is definitely not exact for arbitrary R(G)-modules, the statement must be modified it is to cover arbitrary complexes X. Atiyah and Segal used the device of pro-groups, but for our purposes we prefer an alternative solution. Theorem 4.4. (Greenlees-May [32]) For any G-space X there is a natural short exact sequence 0 --+ LJ~(KN(X)) ~ KN(EG × X) ---4
L~o(K~(X))
where L~ is the ith left derived functor of J-completion.
8
Greenlees
The Artin-Rees lemma implies that for a Noetherian module M, we have LJoM= M/) and L~M= 0 for i > 0, so that that if we take X to be a finite complex, we obtain the previous versions of the theorem. It is a general phenomenonin topology that statements in homologyhave better finiteness properties than in cohomology,so it is natural to seek statements about K homology. Finally, local cohomology is about to ,nake an appearance. Theorem 4.5. (Local Cohomology Theorem (Greenlees (1993) [25]).) For any finite group G and any G-complex X, there is a spectral sequence H~(K,G(X)) ~ K,~(EG x X) = K,(EG where J is the augmentation ideal of R(G). Weoutline a proof for p-groups in Section 6, and a rather different proof for all groups in Part III (Section 13). The next step is the major input from commutative algebra, and appears quite magical from the topological point of view. Grothendieck’s vanishing theorem states that local cohomologyvanishes above the dimension; .since R(G) is one dimensional, the spectral sequence collapses. Corollary 4.6. (Greenlees (1993) [25]) For any finite group G and any G-complex X, there is a short exact sequence ~ a X) --} H°j(KiG(X)) 0 ~ Hj(Ki+I(X)) K~(EG In particular if X = pt KoG(EG) = H~(R(G)) and K~G(EG) = H~(R(G)). Example 4.7. If G = C2 we may make the calculation completely explicit. Wework in R(C2) Z[a]/(a 2 - 1), an d J -- (1 - a ). Since (1 - a)( 1 1 - a2 = 0, inverting (1 - a) kills 1 + a. Similarly, since (1 - 2 = 2(1 a), inverting 1 - a inverts 2. Thus the stable Koszul complex R(C2) R(C2)[1/(1 a) ] be comes Z~Z --+ Z[1/2], ~. Z/2 showing H~(R(C2)) = Z and H~(R(C2)) = The calculation of K,G(EG)is quite explicit in general. Indeed, H~(R(G)) is easily calculated in terms of characters: J consists of representations whose characters vanish at the identity element e ~ G. Since characters separate conjugacy classes, H~(R(G)) consists of representations whose characters vanish except at e. This consists of the integer multiples of the regular representation. It is not hard to calculate H~(R(G))either. Indeed, since for.any p-group G the p-adic and J-adic topology on J coincide, H~(R(G)) = R(C)/Z °° .
Local Cohomoiogyin Equivariant Topology
9
Remark4.8. The theorem is also true for an arbitrary compact Lie group G. For this one must use the proof in Part III. The crucial ingredient is that the representing spectrum of K theory is a highly structured ring spectrum. Until recently this was only knownfor finite groups, but M.Joachim[44] tells me that he has proved the result for arbitrary compact Lie groups. Since the dimension of R(G) is 1 + rank(G), the spectral sequence will not normally collapse. Howeverit still collapses when K,a(X) has depth _> rank(G). Remark 4.9. An alternative way of motivating this whole connection is that by definition J acts as 0 on K~(G) = K*(pt). By induction J acts nilpotently on K~(X) and K,a(X) if X is finite and free. Since EG is a direct limit of finite free complexes, we expect K~(EG)to be the simplest possible "inverse limit" of J-power torsion complexes(hence it is calculated by the left derived functors of J-completion), and K,a(EG)to be the simplest possible "direct limit" of J-power torsion complexes(hence it is calculated by the right derived functors of J-power torsion). Here ’simplest possible’ has a rather complicated meaning, requiring homotopyinvariance, functoriality and exactness. 5.
DUALITY FOR THE REPRESENTATION RING.
In this section we introduce one of the central properties of equi~/ariant cohomologyrings; in this case we are considering the ring S = R(G) augmented over k = R(1) = Z. For the representation ring, this property appears rather commonplace,but at least it is visible to inspection. The idea is to combine the local cohomologytheorems with the universal coefficient theorem to obtain a duality statement. We are going to use the form appropriate to complete rings, and the completion theorem in the non-equivariant form: K°(BG) = R(G)~ and KI(BG) = Similarly, we use the non-equivariant form of the local cohomologytheorem; assuming G is finite, this takes the form of the identifications Ko(BG) = H~(R(G)) and KI(BG) = H~(R(G)). Wehave used the non-equivariant for~ns since there is a Universal Coefficient Theorem[2] for calculating K* (X) from K, (X). This is a short exact sequence 0 ~ Ext~(gi+~(X),Z)
~ Ki(X) Homz(K~(X), Z) -
-~
Weapply this to X = BG and substitute the algebraic expressions for its homology and cohomology to obtain a short exact sequence 0 ---~
Ext~(H3(R(O)),Z
) --~ R(G)~ ~ Homz(H~(R(O)),Z) This no longer mentions K theory: it is just a statement about the augmented Z-algebra R(G). It is reminiscent of local duality for Gorenstein
10
Greenlees
rings, but of course R(G) is not local or even Cohen-Macaulay. It is also notable that local cohomologyhas separated the Z-torsion free H~° from the torsion part Hi. The former gives the uncompleted part of R(G)J: (namely multiples of the regular representation), and the latter gives the completed part (namely J^ in the case of a p-group).
6. A PROOFFORp-GROUPS. In this section I will outline a proof of the Local CohomologyTheorem4.5 for p-groups. The reason for restricting to this case is that on the one hand it is a convincingly large family of examples, whilst on the other, it is possible to give a rather concrete version of the proof in which the correspondence between topological and commutative algebraic structures is highlighted. In Part III we will see one method for going beyond p-groups using :~ome quite sophisticated machinery. For the case of K-theory, and for finite groups, there is the more elementary approach used in [25]: the Burnside ring A(G) (i.e., the Grothendieck ring of finite G-sets) is a good approximation to the representation ring. Indeed the permutation representation homomorphism A(G) ~ R(G) has the property that V/J(A(G)) ¯ R(G) = J(R(G)) so that we may replace the representation ring by the Burnside ring in the proof, considering all R(G)-modules (such R(G) its elf) as modules ove A(G). For topological reasons, manipulations with the Burnside ring are much more elementary. For infinite compact Lie groups this method fails since A(G) is still 1-dimensional whilst- the dimension of R(G) is equal to the rank of G. For the present we return to a method applying to p-groups and complex orientable theories. Weintroduce notation by summarizing standard constructions in the commutative algebra of a ring S. If A and B are chain complexes of S-modules, and f : A --~ B is a chain map, one may define mapping cone C(f): it is the chain complex with kth term Ck(f) = Bk @Ak-~ and differential dk (b, a) = (d~ (b) + f (a), -d~_ ~ its hom ology lies in a long exact se quence with that that of A and B. Wealso need the suspension (or shift) operation on chain complexes: EnA is the chain complex with (EnA)a Ak-n, and differential (-1)ndA¯ Returning to the case of interest, the unstable Koszul complex is the cochain complex K(x) = (S -~ concentrated in codegrees 0 and 1. This is the desuspension of the mapping cone of x: K(x) -- E-’M(x) --+ S -~ S.
LocalCohomology in Equivariant Topology
11
To form the stable Koszul complex, we may now do the anMogousconstruction for powersof x, assemble them into a direct system and pass to liinits:
K(x) = $
$
=
3) K(x
= E-~M(x3) --~ S ~
E-1M(x
2)
2X
2) K(x
~ S
K(x~) = Z-IM(~) --~
S
--~
S
~
Now given an ideal J = (x~,x~,...,xT) in S we may form the stable Koszul complex K(J ~) = K(x~) ® K(x~°) ®... ® K(xr~). The notation is reasonable since the complexis independent of generators up to quasi-isomorphism (exercise). This allows us to define local cohomology by H}(M) := H*(K(J °~) ® M), and its dual local homologyby H~, (M) := H,(Hom(PK(J~), where PK(J~) is a complex of projectives quasi-isomorphic to K(J~). These definitions become interesting by Grothendieck’s theorem that, provided R is Noetherian, local cohoinology calculates right derived functors of J-power torsion: H~(M) --- R*Fj(M) where rj(M) = {~ ~ M I Jgx = 0 for N >> 0}, and the fact [33] that local homologycalculates left derived functors of completion H~,(M)=L,A~(M) where A~(M) = lim M/jkM. ~-k Note also that Grothendieck vanishing and the evident universal coefficient theorem E2 = Ext8(H~(S),M) ~ HJ, shows that the LiAJ(M) = for i > di m(S). Th is ex plains wh y on ly th zeroth and first derived functors entered into the Completion Theorem4.4. Nowin the topological context, the analogue of the stable Koszul co~nplex is the universal free space EG. Let us start with the case whenG is a cyclic
Greenlees
12
group, G C_ C×. Wenote that for each k the group G acts freely on the unit sphere S(kC) in the k-fold direct sum kC. Since S(kC.) is a (2k- 1)sphere (and hence (2k- 2)-connected), the union S(~cC) [. J~S(kC) is contractible. Thus EG = S(ocC.) in this case. Remark 6.1. For the best analogy with the above commutative algebra we need to work with G-spaces equipped with a G-fixed basepoint, so we need to describe the routine translation between the based and unbased context. For cohomology, it is in terms of the reduced cohomology: if Y is a Gspace with G-fixed basepoint Y0 we have maps {Y0} ~ Y ~ {Y0} so that if we define the reduced cohomology by ~(Y) = ker(K~(Y) --+ K~(y0)),
wehaveKS(Y) = ~5(Y)¯ KS(y0). If X is an unbased G-space we form the based space X+, where the subscript + denotes the addition of a disjoint basepoint fixed by G. Thus
~b(X+) = Kb(X ). The appropriate product for based spaces is the smash product defined by Y A Y’ = Y x Y’/(Y x {y6} U {~0} x Y’). This is not a categorical product, but it is commutative and associative up to homeomorphism.It is related to the unbased product by (X+) ~ (X~) = (X × X’)+. We need one more construction. In addition to the unit sphere S(V) of a finite dimensional inner product space V, we may form the one point compactification SV. A useful feature of the based context is that SV is the mapping cone of the map S(V)+ ~ pt+ taking S(V) to the non--base point. Consequently we may also say that S(V)+ is the desuspension of the mapping cone of the inclusion So --~ SV, as happened for the Koszul complex in algebra. Nowwe have the ingredients to construct a diagram exactly analogous to the one for the stable Koszul complex
EC )=
S(C)+
EC(3+)
S(2C)+
so ~
S ° 2c [-~
S
° S
EG )= :
:
:
EG+ = S(~C)+ ~ S ° ~c ~ .S To complete the analogy~ I should explain whyit is reasonable to consider So ~nd Skc ~] as analogues of the same object S, and whywe have written Ix
LocalCohomology in Equivariant Topology
13
for the inclusion map. For this we mayas well allow G to be arbitrary. The point is that equivariant Bott periodicity provides a specific isomorphism ~2~(Sv) ~- fi[~(S °) = R(G) for any complex representation Y. We then have a diagram
~) ~0~(s0)~ ~0~(s : [
0) R(G)
-~ 1 Bott
~0~(S
This defines the element x(V) E R(G) called the Euler class of V, and by transitivity of Bott periodicity isomorphisms, x(V~W) = x(V)x(W). Thus the map marked Ix k] above is x(kC.) = X(C)k. For an arbitrary group G and representation V, the construction of the Bott mapmeans that x(V) is the alternating sum of exterior powers of V. It follows that when we apply
~,~(.)to S(~V)+ --~
° ~S ° °y
we obtain ~5(S(c~Y)+) R( G) -~ + R( G)[1/x(V)] and it is in this sense that ~.G(S(c~V)+)is the analogue of the stable Koszul complex K (x(V)~). Now,just as most ideals are not principal, so ~nost groups are not cyclic. However, we can deal with this in the same way that we constructed a stable Koszul complex for an ideal with several generators. For example, if G = C1 × C2 × ..- × Cr is a product of cyclic groups, we have
E~ = S(~C1)× S(c~Ca)× ... × where C/ is the natural representation of the cyclic group Ci on C. It is not hard to use character theory to show that if G is a p-group (or more generally supersoluble) then it has complexrepresentations V~, V~,..., Vr for some r so that G acts freely on S(VI) × S(V~) × ... S(Vr), an d he nce
Ea= S(ooV~) x S(oo½) x ... x S(ooV~). (Equally it is not hard to show that most groups do not have such representations; the smallest example is A4, but see [50] for muchmore detailed information). Supposing that we do have such a model for EG, we may immediately outline a proof of the local cohomologytheorem.
14
Greenlees
Outline of proof of 4.5. The theorem was stated with an unbased space X, but the proof works with based spaces, such as X+. The translation is routine as described in Remark 6.1. In particular EG+AX+= (EG x Z)+ and ~,G(X+) == K,~(X). Weapply /~,G to the filtered space EG+A X+. Because the filtration precisely modelled that of the stable Koszul complex, we immediately find a spectral sequence E1*’* = K(£~) ®R(G) K.G(X) ~ K,~(EG+), where
£ = (x(yl),x(y2),..., is the ideal generated by the Euler classes. Again, the correspondence between topology and commutative algebra makes clear that dl is the Koszul differential, so that E~’* = H~(K,~(X)). Finally we need to explain why~ maybe replaced by J. First, the inclusion SO ---~ Sy is obviously null-homotopic non-equivariantly if V # 0, so 6 C_ J. In the cyclic case we had equality, and for a general p-group we have
vzg=J. This can be observed explicitly for the representation ring, since the primes are knownby [52], but in fact it follows from some finiteness assumptions
[29].
Since local cohomologyonly depends on the radical, H~ (M) as required.
it proves H~(/~//)
As far as any applications we have made are concerned, there is no particular advantage to using the augmentation ideal J rather than the Euler class ideal E. It is whencomparing different groups that J comes into its own. 7.
VARIATIONS
AND EXTENSIONS.
There are two directions to develop the above ideas. The first replaces free G-spaces (i.e., spaces with all isotropy groups trivial) by spaces with isotropy in some other family 9" of subgroups. On the ring theoretic side, the ideal J is replaced by the ideal J(9") of elements restricting to zero R(H) for all subgroups H in 9-. For equivariant K theory this goes smoothly. Howeverthere is no comparable expression in terms of the non-equivariant case, and hence no duality statement. Wewill not pursue this variation any further here. The second variation is to replace K theory by another cohomologytheory; on the ring theoretic side this replaces the augmentedring R(G) by anc,ther. If K theory is replaced by E-theory then R(G) is replaced by the coefficient ring S = E~ (or its degree zero part E~), R(1) is replaced k =E*(or i ts
LocalCohomology in Equivariant Topology
15
degree zero part E°) and J is replaced by the kernel J = ker(E~ ---4 E*) the map forgetting equivariance. Context 7.1. The correspondence between commutative algebra and topology is via
¯ S=E b
¯ k=E* ¯ J = ker(E~ ---+ E*). The remainder of this article is structured round a number of these. In this section we give an overview. In the following table the first row summarizesthe properties of K theory discussed above, and each subsequent row is an analogous example. To avoid discussion of special cases, the information refers only to finite groups G. The final two columns refer to the question of whether the homologyand cohomologyof EG can be calculated using local cohomologyand completions as was the case for K theory. Cohomology theory CoefficientsDimension Aug~nented over Homology Cohom. K-theory R(G) 1 R(1) = [25] [3] E~-theory E°n(BG) n Z~p)[[ul ..... un-~]] [25,21,23] T Stable cohomotopy A(G)nilp FALSE! [16] 1 A(1) = Ordinary T H*(BG) rankp(G) H*= Fp [27] ConnectiveK-theory ku*(BG) 2 ku*=Z[u] [25,21,23] T Remark7.2. (i) There are various ways of grouping these examples. First, K-theory and En-theory are periodic, so we may work with ungraded rings. In the cohomotopy example the Burnside ring A(G) is in degree 0, and all other elements are nilpotent (Nishida’s nilpotence theorem); accordingly we may work over the degree zero part A(G) in this case too. The other examples are graded. (ii) Moresignificant is the fact that apart from K-theory and cohomotopy, all the examples are complete for the augmentation ideal. This means that the completion theorem is a tautology for finite complexes(hence the entries T for ’true’ and ’tautology’) and concentrates attention on local cohomology and the duality statement. This is no disadvantage for present purposes, but in topology it is a major problem to identify natural rings of which these are completions. To see why it is valuable, one may imagine we only knew the completion of R(G). We maythen seek to refine it to R(G) itself, hoping thereby to invent representation theory. (iii) In the first three examples,the ring is essentially finite over its nonequivariant counterpart: in the other two we have finitely generated algebras over the non-equivariant coefficients.
16
Greenlees
(iv) All the examples for which the local cohomology theorem is known be true can be proved for p-groups using the method of Part I. They can be proved for finite groups by the method described in [25, Appendix], using [29] to establish finiteness properties. However,the technology necessary for implementing the strategy was not introduced until [21]; it was shown to apply to these cases in [23]. (v) Stable cohomotopyis quite exceptional. The fact that local cohomology of ~r~ does not calculate u,(BG) means that the type of proof for the completion theorem we will give in the complex orientable case (by formally deducing it from the local cohomology theorem) cannot apply. Carlsson’s proof of the completion theorem ("the Segal conjecture") is not a simple formality like the proofs we give here, and it seems that the substantial[ calculational input is necessary. Other examples of this type occur in algebraic K theory, but we shall not be concerned with them further. In Part II (Sections 8 to 11).we discuss ordinary cohomology: here the topological technology is at a minimum, but the commutative algebra is correspondingly increased. In Section 12 we discuss a family of examples arising from the topological theory of manifolds: this has the advantage of being a rather extensive family from the ring-theoretic point of view. Finally, we look at two special cases of this. In Section 15 we discuss the En-theory example because an interesting interaction of Cousin filtrations occurs. In Section 16 we turn to connective K theory; the interest here is that although it is very close to exampleswhere rather complete results are available, some rather intricate commutative algebra appears.
Part II.
Ordinary cohomology and graded connected k-algebras.
We spend Sections 8 to 11 considering ordinary cohomology. This has several special attractions, especially to algebraists. First, the cohomolo.gyof a point can be described in purely algebraic terms as an Ext algebra relevant to representation theory. Second, the ring S is a commutativegraded algebra over a field k and it is connected in the sense that it is concentrated in degrees _> 0 with So = k; this makes it especially easy to apply the methods of commutative algebra.
8.
COHOMOLOGY
OF SOME CLASSES
OF GROUPS.
For the duration of Part II, S is an N-cograded algebra over a field k, and SO = k. Properly speaking, S is graded commutative in the sense that ab = (--1)deg(a)deg(b)ba, and all the results we need can be proved in this context. Those not wishing to worry about graded commutativity can restrict attention to the case that char(k) = 2. For most of the discussion suffices to replace S by its even-graded commutative subring.
17
Local Cohomologyin Equivariant Topology
In this section the augmentation ideal J = ker(S ~ k) is the unique maximalideal of the graded local ring S. To emphasize this we write m for it. 8.A. Finite groups. The first
class of examples we discuss are the rings
S = H*(G;k):= Ext~a(k,k ) for a finite group G, where kG denotes the group ring, and k is given the trivial G-action. By the Evens-Venkovtheorem the ring S is a finitely generated k-algebra; general references for this subsection are [6, 12]. One of the principal reasons for studying this is representation theory. For example the rate of growth of a minimal resolution of k is exactly captured by the Hilbert series h(G;k) := E dimk(gi(G;k))ti; i>0
for Noetherian rings ~his power series is easily seen to be the expansion of a rational function about t = 1. The cohomology ring H*(G; k) is known explicitly for a large number of groups, and we tabulate some examples to bring the discussion down to earth. All the examples are 2-groups, and k is a field of characteristic 2. Weuse the convention that a subscript on a generator indicates cohomological degree. The Cohen-Macaulaydefect for a local ring S is defined by CM-defect(S) = dim(S)-depth(S). The groups elementary abelian, quaternion of order 8, dihedral of order 8, semidihedral of order 16, a certain group of order 32, and the extraspecial 2-groups. Group G
(c D8 SD16 21+n
H*(G;k)
ktx(lh,x(2h,...,
k[xl, Yl, z4]/( x2 + xy + y2, x3, y3)
k[x, See below k[z(1)t, z(2)~,..., x(n)t]/I ®
dim defect
r
o
1 2 2 3 r
0 0 1 2 0
Hilbert series
r1/(1- t)
(1 + t + t2)/(1 - t)(1 2) 21/(1 - t) 1/(1 - 02(1 + 2)
Remark8.1. (i) There is a copious supply of further examples on J.F.Carlson’s webpages[15] (the cohomologyof all but 5 of the 267 groups of order 64 are there, together with all 2-groups of smaller order). (ii) Quillen [47] has shownthat dim(H* (G; k)) = rankp(G), where the p-rank rankp(G) of G is the rank of the largest elementary abelian p-subroup (Cp)r in G. (iii) For p-groups of small order, the Cohen-Macaulaydefect tends to quite small. Indeed, for char(k) = p Duflot [17] has shownthat depth(H*(G; k)) >_ ranky(Z(G))
18
Greenlees
where Z(G) is the centre of G, and that in any case the depth is at least 1 (iv) The cohomologyring of the semidihedral group k[xl, Yl, z3, t4]/(xy + y2, y3, yz, x3z + x3y + y2t + z2). The cohomology ring of FTa2 has 8 generators and 18 relations, but the relevant information is well summarizedin [7]. (v) The case of the extraspecial groups of order 2 l+2m iS included because Quillen’s calculation [48] is so elegant (see also [6, 5.5] for a brief account). The ideal I is generated by a regular sequence. To be more precise, consider an extension 1 ---~ C2 ~ G ~ .E ---~ 1 where E is elementary abelian of rank n. If C2 is the commutator subgroup and the centre of G then G is called extraspecial, n = 2m is even and there are only two isomorphismtypes of extraspecial groups of this order, but the cohomologycalculation applies to any extension of the given form. As usual, r is the dimension of the cohomologyring, or equivalently the rank of the largest elementary abelian subgroup. To describe the ideal, we use the fact that any such extension is classified by the map q: E ~ C2 given by squaring the preimage in G of an element of E. This q is a quadratic form with associated bilineary form b. Nowif el, ¯ ¯ . , en is a basis of E as an F2-vector space and v = Eieixi then I is generated by the sequence q(v),q(v,F(v)),...
,q(v, Fn-~(v)),
where F is the Frobenius map, and the sequence turns out to be regular. The codegrees are 2, 4, 8,..., 2n-r+~, so it is easy to write downthe Hilbert series.
8.B. Compact Lie groups. If G is a compact Lie group, the ring S = H*(BG;k) is the cohomologyring of the classifying space of G that we met in Section 3, and it is again a finitely generated k-algebra by Venkov’s argument. One reason for studying this is exactly parallel to the motivation we gave ibr K theory: it tells us about characteristic classes. One often thinks of compact Lie groups such as the orthogonal and unitary groups, but of course finite groups are perfectly good examples, and using the bar construction for BG one sees that it can be constructed with cells corresponding to copies of kG in the bar resolution: thus H*(BG; k) = Ext~G(k,k), and our notation is consistent. Wemay give a good supply of examples here too.
Local Cohomologyin Equivariant Topology
Group G T (u(1)) su(2) U(n) O(2n) Spin(n)
H*(BG;k)
19 dim defect
k[x(1)2,x(2)2,... ,x(r)2]
r
0
k[x4] k[cl, c2,..., cn]
1 n
0 0
k[pl,p2,... ,pn] k[~,~3...,~,,,]/s ~ ~[¢~.-.]
n ,-
0 o
Hilbert series
1/(1- t’~)"
1/(1 - 4) 1/(1 - t2)(1 - ... (1 - t 2n)
1/(1- t4)(1 - ... (1 - t 4’~)
Remark8.2. (i) By force of precedent, the Chern class ci is of cohomological degree 2i and the Pontrjagin class Pi is of cohomological degree 4i. Otherwise we have retained the convention that subscripts refer to codegrees. (ii) The field k can be of any characteristic except that char(k) ~ 2 for O(2n) and char(k) = 2 for Spin(n). (iii) The case O(2n) is included because its behaviour is slightly more plicated than its polynomial cohomologyring suggests (see Example9.5(iii)). (iv) The cohomologyof Spin(n) is also calculated in [48], and was Quillen’s motivation for considering the extraspecial 2-groups. The ideal I is generated by the regular sequence w2, Sq~ w2,... , Sq2 .... Sq2 .... ~ .. ¯ Sq2 Sql W2, where Sqi is the ith Steenrod square. Quillen deduces this calculation by comparison with the calculation for the extraspecial subgroup of Spin(n). One other feature will be important. The group G acts on itself smoothly by conjugation. Since it preserves the identity element e ~ G, a group element gives a self:map of the tangent space TeG, and we obtain the adjoint representation ad : G ~ GL(TeG), and hence the k-orientation representation G --~ GL(T~G) -~ ~* {+1,-1} --
~ k* .
If the imageis trivial we say that ad is orientable over k. Evidently if char(k) = 2, the adjoint representation is orientable for every group G. By continuity, each connected component maps to the same point, so that if G is connected, or if it has an odd number of componentsthen ad is orientable over any field. Howeverif char(k) ¢ 2 then ad(O(2n)) is not orientable over k (this is easy to check for n = 1, and follows in general). 8.C. Arithmetic groups. For an arbitrary sider the cohomologyring
discrete
group G we may con-
S = H*(G; k) = EXt~G(k , k). This can be arbitrarily unpleasant unless we place a restriction on G. We are interested in arithmetic groups, such as SLy(Z) and O,~(Z). However the appropriate level of generality is a little wider: virtual duality groups.
20
Greenlees
In this case the cohomologyring is again a finitely generated k-~Llgebra; a general reference for this subsection is [12]. Before reaching virtual duality groups we should discuss duality groups. The reader may like to construct an analogy with Gorenstein local rings. A Gorenstein ring can be characterized as one having finite injective dimension over itself, but perhaps the reason Gorenstein rings are so important is l;heir duality. Similarly, if we restrict attention to groups G with the finiteness condition that k admits a resolution by finitely generated projective., kG modules, we may characterize duality groups as torsion free groups G so that Hi(G;kG) := Ext~a(k, kG) = 0 unless i = n for some n. The number n is called the dimension of G and the module I= Hn(a;ka) is called the dualiz~ng module. Such groups then have a duality isomorp:hism Hi(G; M) TM Hn-i(G; M ® I). Wesay that G is a Dn-group; if I is one dimensional over k we say G is a Poincard duality group (PDn-group), and if in addition G acts trivially on I we say it is orientable. A group G is said to be a virtual duality group if it has a subgrou:p G’ of finite index which is a duality group. The virtual dimension of G i~,~ the dimension of G~ (and is well defined) and the dualizing module for G I = H’~(G’; kG’) ~- H’~(G; kG) where the isomorphis~n is Shapiro’s lemmaand this shows I is a G-module. Example8.3. (i) Anyfinite group is a virtual duality group since the trivial group is a Poincar~ duality group of dimension 0. (ii) Borel and Serre [14] show that any torsion free arithmetic group is duality group, and a general arithmetic group is a virtual duality group. (iii) Fundamental groups of a knot co;nplement are D3-groups. (iv) Mappingclass groups are virtual duality groups [42, 41]. (v) Automorphismgroups of free groups are virtual duality groups [11]. Thus the class of virtual duality groups is very extensive. Howeverthere are very few non-trivial examples where the cohomologyring is known explicitly. 8.D. p-adic Lie groups. It is possible to transpose the entire previous’, section into the category of profinite groups. If G is a profinite group we may consider modulesfor its (discrete) finite quotients. Howeverit is important to be able to discuss inverse limits of these (compact G-modules) and direct limits (discrete G-modules). It is possible to put both classes of modules into a reasonably well behaved category [55], and make most of the usual
LocalCohomology in Equivariant Topology
21
constructions of homological algebra in this category. In particular the colnplete group ring k[[G]] behaves like a free module and the cohomologyring may be defined by S := H*(G;k) = Ext~[[c]](k , k). Just as in the discrete case we maydefine duality groups and virtual duality groups. Virtual duality groups have a virtual dimension and a (colnpact) dualizing module I = Ha(G;k[[G]]). Example8.4. Lazard shows that any p-adic Lie group is a profinite virtual duality group [45]. As in the discrete case, the interesting examples that come up are automorphism groups of suitable structures. Familiar examples are the groups SLn(Z~p)), and another class of examples of considerable interest to topologists are the Moravastabilizer groups (automorphismgroups of certain formal groups). 8.E. Commonfeatures. All of these classes of groups include all finite groups, and the ring S is knownto share with the case of finite groups the properties that it is a finitely generated k-algebra and of dimensionrankp(G)
9.
THE LOCAL COHOMOLOGY THEOREM FOR GROUP COHOMOLOGY.
Wereturn to the local cohomology theorem, treated in Part I for the representation ring. Nowwe consider what it says for ordinary cohomology, which is to say for the augmentedk-algebras S discussed in Section 8. This case is much more interesting because S can be of higher dimension, and exhibits a wide variety of behaviours as the group varies. Weare writing m for the augmentation ideal: it is the maximal ideal of positive codegree elements. Wewrite DM= Soms(M, I(k))
Homk(M, k)
for the Matlis dual, whereI(k) is the injective envelope of k. Theorem9.1. [27] If S = H*(G; k) for a finite group G there is a spectral sequence E~’* = H,*~(S) =~ DS, with differentials dr : ESr’t ~ Ers+r’t-r+l. This is a spectral sequence of S-modules in the sense that dr : E~s’* ---+ Ers+r’* is a mapof S-modulesfor all r and s. Wewill outline the proof of the result in Section 10 below, but first we discuss somespecial cases. Example9.2. (i) If G is an elementary abelian 2-group of rank r and k of characteristic 2 then S = k[xx,...,Xr]. It is easy to calculate the local coho~nology from the stable Koszul complex, and we deduce H~,(S) : H,~,(S) E-rDS,
22
Gre¢,nlees
where the suspension operation E-r shifts degrees by -r as before. More generally, whenever S is Cohen-Macaulayof dimension r, we fi.nd H~(S) = H~(S) and hence the spectral sequence collapses to give H~(S) = E-rDS, and hence S is Gorenstein. This recovers a theorem of Benson and Carl~on [8]. (ii) The simplest non Cohen-Macaulayexample is the semidihedral group order 16, of dimension 2 and depth 1. The spectral sequence again collapses since the E2-term is concentrated in two adjacent columns. (iii) The simplest example with Cohen-Macaulaydefect 2 is the group I~’7a2. The cohomologyis calculated by Rusin [51], and the relevant features are highlighted by Benson and Carlson [8]. One may show that the spectral sequence does not collapse, and one mayidentify the differential. Turning next to compact Lie groups, recall the conjugation of G induces the adjoint representation G --+ k*, and let k(ad) denote k with this action. Theorem 9.3. [10] If S = H* (BG; k) for a compact Lie group G of dimension d, then there is a spectral sequence H~(S) ~ E-dH,(Ba; k(ad)), of S-~nodules where k(ad) is the coe~cient system in which each element o[ the group acts on k by +1 or -1 according to whether conjugation by g preserves or reverses the orientation in a neighbourhoodof the identity. Remark9.4. WhenG is orient~ble the spectral finite case with a shift
sequence is just like the
H~(S) ~ E-dH.(BG; Example 9.5. (i) All the finite group examples discussed before are also examples here. (ii) If G is the circle group (of dimension d = 1) then k[x2] and if G = SU(2) (of dimension 3) then S = k[x4]. Both groups are connecte,i, that k(ad) = k. These examples show how the dimension shift E-d comes into play. (iii) Wehave already commentedthat the adjoint representation of 0’(2n) is non-trivial if char(k) ~ 2. This accounts for the fact that the local cohomology of H*(BO(2n)) = k~,... ,Phi is a copy of its dual shifted by -2n(n + 1), whereas the dimension shift for an orientable group of the same dimension (namely n(2n- 1)) and rank (namely n) would -n- n(2n- 1). The difference of 2n is explained by the non-trivial adjoint action. Finally, we turn to virtual duality groups.
Local Cohomologyin Equivariant Topology
23
Theorem 9.6. [9] If S : H*(BG;k) for a (discrete or profinite) duality group G of dimension n there is a spectral sequence H~(S)
virtual
~ EnH,(G;
of S-modules where I is the dualizing module. Remark9.7. WhenG is an orientable Poincar~ duality group the spectral sequenceis just like the finite case with a shift
Note that the shift for a virtual duality group is in the opposite direction to that for a compact Lie group: virtual duality groups behave like negative dimensional compact Lie groups. Wewill discuss the implications of these theorems for the structure of the ring S in Section 11, and in Section 10 we sketch a proof.
10.
AN ALGEBRAIC
PROOF OF THE LOCAL COHOMOLOGY THEOREM FOR ORDINARY COHOMOLOGY.
In this section we outline the proof of the local cohomologytheorem in the case of finite groups. This is the proof given in [27]; it is possible to give topological proofs as in Part I or Part III and another proof is given in [20]. In fact we will prove a more general result for cohomologyof a kG-inodule M. This states that there is a spectral sequence H~(H*(G; M)) H.( G; M). The case M= k recovers the theorem. Similarly there are versions of the results for other classes of groups with modulesof coefficients. The simplest case is when G has periodic cohomology. This is knownto be a very restrictive assumption, but once this case is clear we should be able to describe the proof in general. The assumption means that there is a codegree n and an element x E S of codegree n so that multiplication by x gives an isomorphism Sm TM S m+nwhenever rn >_ 0. Thus S D k[x]. Now we may view x as an extension class x E Ext~a(k, k) and represent it by an exact sequence 0 "~’+ ]~ -----~[Cn-1-~-)’Cn-2----~’"----~
C1----~
Co]--~
k ~0.
If we begin a minimal resolution the Co, Ct,..., Cn-2 are projective by construction, and C~-1 is projective since Hn(G) ~ H°(G) = We thu s consider the complex
24
Greenlees
and form a projective resolution P of k by concatenating copies of C(x), we may do because Hn-1 (C(x)) = k = Ho(C(x)). We thus let
P:
,c._,
c, ~ ] . Cn-1
~ Cn_2
I ~...
~C1
~Co[
Nowwe may write the stable Koszul complex S ~ ~im ~/z~S = S[~/~]. ~k It mayhelp to visualize the term 1/x~S as a copy of S, displayed vertically, so that the bottom is in degree -kn~ and the maps in the direct system just include each column in the next. Wemodel this at the level of resolutions, at least after reversing arrows, to obtain P=L0~Ll=lim
L[-kn,~) ~k
= lim
E-knp. ~k
Again, one mayview L[-kn, ~) as the chain complex P, displayed vertically, with the bottom in degree -kn, and the maps in the inverse system just project each column onto the previous one. Thus L. = (L0 ~ L~) is double complex, and the proof proceeds by considering the double complex X = Homkv(L.,
M)
There are two spectral sequences for calculating the cohomologyof the double complex. The one in which we take Koszul cohomologyfirst collapses to show H. (X) = H. (H~oszut(Somka(L.,
M))) = H. (HomkG(H~°Szut(L.),
since H~°Szut(n.)
= H1(L.)
where P~ = L~/P is the part of L~ in negative codegrecs. Now Homk(P~, M) = P @k so that H,(X) H.(G; M) Taking the spectral sequence arising from the other filtration
we obtain
H~osz~t(H*(Homk~(no , M))) H.( X) = H.( G; M). This is the required spectral sequence. Indeed, by definition we have H*(HomkG(Lo, M)) .= H*(G;
LocalCohomology in Equivariant Topology
25
and for L1 we calculate H* (Hornka( L1, M)
= H*(Homka(line_~nL[-kn, = lim H*(Homka(L[-kn,
oo),M)) oc),M))
-+n
= lim 1/xnH*(G,M)) --~n
= H*(G,M))[1/x], where the second equality used the fact that the limit is achieved in each degree. Thus we obtain the stable Koszul complex, (H*(Hom~ca(L., M))) = (H*(G; M) --~ M)[1/x ]). The cohomologyof this complex is the local cohomologyof H* (G; M). WhenG does not have periodic cohomology, Noether normalization shows that S is finite over a polynomialsubring k[xl,..., Xr]. Wereplace the single complex C(x) by C = C(Xl) ® "" ® C(xr), which we can view as an dimensional box. This time the top modules in each complex C(xi) are not projective, but by the theory of support varieties [6, Chapter 5], because S is finite over the polynomial subring, their tensor product is projective. Nowthe proof is directly analogous. Wemimic the construction of the multigraded Koszul complex by stacking boxes. For example with r = 2 we have L2 1 =
P((-cx~, cx~) x (-cx~, P((-cx3, o~) × [0, o¢)) @P([0, o~) x (-o~, P([0, oo) x [0, e¢))
whereP([0, oe) x [0, o¢)), for instance, is the result of stacking boxes in first quadrant. 11. STRUCTURAL IMPLICATIONS OF THE LOCALCOHOMOLOGY THEOREM. This section summarizes the contents of [31], outlining the implications for a finitely generated k-algebra S of the existence of a spectral sequence H~(S) ~ r~°DS. Duality is exact in this context so it is equivalent to say there is a spectral sequence DH~n(S) ~ N-as. Wesay that S has a local cohomologytheorem with shift a, or that it is an LCTa ring. First we repeat the immediate observation that if S is Cohen-Macaulay of dimension r then the spectral sequence collapses to give an isomorphism DH~(S) so that S is Gorenstein.
26
Greenlees
Next, consider the case that S has Cohen-Macaulaydefect 1 (in the sense that its depth is one less than its dimension). Wewrite ~ = DH~(S)for the canonical module, and A = DH~-I(S) for the subcanonical module. Thus the spectral sequence collapses to the short exact sequence 0
---~
~a-r+lA
~ S
~ ~a-rg~
~ O.
Proposition 11.1. If S is an LCTa ring with Cohen-Macaulay defect then ¯ gt has depth r and H~(~) Er-aD~, and ¯ A has depth r - 1 and Hr~-I(A) = Er-I-aDA.
1
It is natural to say that a ring with Cohen-Macaulaydefect 1 is almost Cohen-Macaulayand to say that it is almost Gorenstein if it satisfies the conclusion of the proposition. Prooi!. Note first that the depth statements are equivalent to the vanishing of local cohomologyup to the top deg~’ee. To give the idea of proof, we will just explain the vanishing of lower local cohomologygroups. The isomorphisms for the top local cohomology groups follow by extending the analysis one more step. Grothendieck’s spectral sequence E~’q -- HPm(DH~-q(s)) (which applies to an arbitrary commutative ring) gives H~(A) for i _< r - 3. Applying the local cohomologyto the short exact sequence from the local cohomologytheorem gives H/m-l(~t) ---- EH/m(A)for i _~ r Combining these gives the statements about depth. [] Remark11.2. It is worth recording some consequences of these results for ~ ring. First, we recall that Grothendieckdefined a dual. loan arbitrary LCT calization functor which is useful for Artinian modules like local cohomology modules: this is given by L~M= D~(D(M)~), where D~ is Matlis duality for the local ring R~. The functor Lp is exact, and an easy consequence of local duality is the fact that if go is a prime with Sip of dimension d then L~H~(M) = H~-d(M~). Accordingly, we may apply L~ to see that if S is an LCTa ring then S~ is an LCTa-d ring. Hence we conclude that for any mini~nal prime go, the ring Se is Gorenstein (one says S is Gorenstein in codirnension 0), and that if go is of height 1, the ring Sp is almost Gorenstein (one says S is almost Gorenstein in codimension 1). On a more concrete level, if S is a finitely generated k-algebra in codegrees >_ 0, we mayobtain conditions on its Hilbert series [S](t) = E~ i. dim~(S~)t Corollary 11.3. (i) (Stanley [54]) If S is a Cohen-Macaulay LCTa ring then
=
27
LocalCohomoiogy in Equivariant Topology (ii)
If S is an almost Cohen-MacaulayLCTa ring then [S](1/t) - t-~(-t)r[S](t)
(- 1)r-1(1 + t) [h](t)
and [A](1/t) t~(-t)l-r[A](t). Remark11.4. In [31] the functional equation for A is misstated: two signs are negated, one in the statement and one in the proof. Proof: To give the idea, we present the proof of Part (i) in a way that suggests that of Part (ii), referring the reader to [31] for further details. first observe that the result is elementary for a polynomial ring. Nowby Noether normalization we may find a polynomial subring ~ _C S over which S is a finitely generated module. Wemay work entirely with ~-modules in the rest of the proof. By the Auslander-Buchsbaumformula, since S is Cohen-Macaulay,S = F0 ®k~ as ~-modules, where F0 is a finite dimensional graded vector space. Thus for Hilbert series, IS] = [~][F0]. Nowcalculate [S](1/t)
= [F~][~](1/t)
= (-1)r[F~][DH~(~)] = (-1)r[DH~(S)] = (-1)rtr-a[s],
where the last equality is the local cohomologytheorem. The proof in the almost Cohen-Macaulaycase uses exactly the salne ingredients, now working with a short exact sequence 0 -~ F1 ®k ~ -~ F0 ®k ~ + S ---> 0 of ~-modules. [] Example 11.5. We consider the semidihedral group SD16. It is of dimension 2 and has Hilbert series f(t) = 1/(1 - t)2(1 + t2). Wecalculate t4 t2 t2 2) 2) " f(1/t)-(-t)2f(t)= (1-t)2(l+t (1-t)2(l+t =-(l+t)(1-t)(l+t~) It is then easy to check that with 5(t) = t2/(1 - t)(1 ~)we have (~(1/t) = (-t)-l~(t) as required. It is interesting to note that f(t) also satisfies the single functional equation f(1/t) = t2(-t)2f(t) just as if it was the Hilbert series of a Cohen-MacaulayLCTring with shift a= -2. Part III.
Examples related
to bordism.
There is a rather general class of cohomologytheories for which one can give a uniform treatment very close to ideas from commutative algebra. These examples arise from the study of manifolds: a rather general construction forms a cohomologytheory from a suitable class of manifolds with additional structure. The additional structure is specified by a structure
28
Greenlees
group for the stable normal bundle; in our case the structure group is the unitary group U, so the cohomologytheory is called MU*(.) (the letter refers to the use of the Pontrjagin-Thom construction in the homtopytheoretic analysis of the theory). Curiously, these geometric antecedents are not relevant here. Even in topology, Quillen’s theorem [49] that there is ala intimate relation between bordism and the algebraic theory of formal groups means that much discussion of bordism is conducted in purely algebraic terms. For the purpose of this article, the important thing is that complex cobordism MU*(.) has a numberof universal properties, and it can therefore be thought of as the generic example of a cohomologytheory which behaves well for complexvector bundles. Wewill be more specific about some of its specializations below. 12.
THE THEOREMFOR MU-MODULES.
There is an equivariant version of bordism constructed from certain manifolds with group action. The cohomologytheory MU~(.) is called equivariant (homotopical) complex cobordism. Its value MU~(X) on a G-space X is a module over the coefficient ring S = MU~. Theorem 12.1. (Greenlees-May [38]) IrMa(.) is a module valued cohomology theory over MU~then for any finite group there are spectral sequences H~(M,G(X))
~ M,G(EG × X) = M,(EG
and HJ, (M~(X)) :=~ M~(EG × X) = M*(EG where J = ker(MU~---r MU*)is the augmentation ideal. Remark 12.2. (i) In fact MU~is not Noetherian, and J is not known to be of finite arithmetic rank. Accordingly there is work involved in showing the initial terms of the spectral sequences makesense: one shows that H~, (.) is independentof J~ for all sufficiently large finitely generated ideals J~ _CJ. (ii) Weshall only be applying this in the case that M~is a Noetherian ring in its own right, and one may replace J with JM ---- ker(M~ ---~ M*). view this as an example in the form of Context 7.1 by replacing S with SM ---- M~, and k with k M = M*. (iii) The sense in which M~(.) is required to be module valued will be plained in Section 13. Example 12.3. The easiest class of examples to describe are those which are J-complete in a suitable sense. Fortunately these are the ones we intend to discuss later. For this class we first define the non-equivariant theory. Indeed the coefficient ring of complexcobordism is a polynomial ring MU*= Z[x~, x~, x3,...] on infinitely manyvariables, where xi has degree 2i. The essential feature of a homologytheory is that it takes cofibre sequences to long exact sequences
LocalCohomology in Equivariant Topology
29
in the same way that homological functors of chain complexes do: we say that it is exact. For any flat module M, over MU,the definition M,(X)
= MU,(X) ®MU,
gives an exact functor of X and is therefore a homologytheory, and hence represented by a spectrum M. For topological reasons the flatness condition can be weakened to something much more easily verified (Landweber exact functor theorem). There are also many module valued theories for which M, is not flat, but other means must be used to construct them. Prom any such non-equivariant theory one may form the complete theory M~(X) = M*(EG xa One family of well knownexamples are the complete Johnson-Wilson theories M = En with E~=
Z~[[~I,...,
~tn-1]][~,
~-1],
where the ui are of degree 0 and u is of degree 2. This maybe constructed using the Landweber exact functor theorem. One may show that for any finite group G, E~(BG)is finitely generated as a moduleover E~*. Another example has ku* = Z[xl], and here ku*(BG) is only the completion of a finitely generated algebra over ku*. The coefficient ring does not satisfy the hypotheses of the Landweberexact functor theorem, so other methods must be used to construct it. Wewill return to an algebraic investigation of these two last examples in Sections 15 and 16, but first we spend Section 13 explaining the idea of the proof of the the local cohomologytheorem for Noetherian MU-algebras. This is so closely analogous to commutative algebra in the derived category that it inspires a numberof definitions of interest in algebra as well as topology, and we briefly introduce them in Section 14. 13.
THE PROOF.
The main thing is that there is a good category to work in [21]. Working there, the proof is essentially formal and just like working in an algebraic derived category. This is the category of G-spectra, briefly introduced in Subsection 13.A. In Subsection 13.B we transpose some commutative algebra into the category of G-spectra and in the final two subsections we complete the proof. 13.A. G-spectra. A G-spectrum may be thought of as a generalized based G-space. The purpose of the generalization is to ensure that any equivariant cohomologytheory E~(.) is represented in the category of G-spectra. This means that there is a G-spectrum E so that for any unbased G-space X, Eb(X ) = [X+, where the expression on the right denotes G-homotopyclasses of G-mapsof G-spectra. Sufficiently well behaved cohomologytheories (such as MU)are
30
Greenlees
represented by ring objects R in the category of G-spectra, and there is a category of module spectra over the ring spectrum R. These ring .,spectra R are analogous to differential graded algebras, and the homotopycategory of modules over R is directly analogous to the derived category of differential graded modules over the differential graded ring. This homotopy category is where we work. Thus for a module M over R, and an unbased G-space X, M~(X) [X+, M] ~ =- [R A X+, M] ~,a and M,G(X) = [S O X+ A M],~ = [R, X+ A R’a M1 where the decoration R, G refers to equivariant module maps.
homotopy classes of R-
Warning 13.1. Even ifR~(.) is represented by a ring R, it is not automatic that the representing G-spectrum Mof a module valued cohomology theory M~(.) is an R-module. Howeverthis is true in manycases, and specifically for the complete theories formed from En and ku discussed above. The proof of the local cohomologytheorem described in this section does not in fact apply to R = MU.It requires us to work with a cohomologytheory R~(-) represented by a ring G-spectrum R, with two properties. Firstly, the coefficient ring R~ must be Noetherian (and for all subgroups H the modules R~ must be finitely generated), and secondly, it must be complex oriented (this is equivalent to saying that R is an MU-algebra up to homotopy, but it has a more concrete meaning that will be described at the appropriate point). The coefficient ring of MUis not Noetherian, so the main obstacle is the construction of enough elements of the ideal J: this is interesting but not relevant to our applications. In fact the complete theories of both En and ku are both represented by ring spectra R to which the argument of this section does apply. 13.B. Commutative algebra with G-spectra. In the category of modules over R one can mimic most constructions in the derived category. For example we can construct the homotopy /-power torsion functor. If x e RG, = JR, RIG,’R we may form, F(z)R := fibre(R --~ R[1/x]) and then if/= (x~,...,Xn)
we take
FtR := F(z~)R ARF(z.,)R
...
/~R F(z~)R-
Up to equivalence, F~Rdoes not depend on the generators used, and it only dependson the radical of the ideal I: this is an easy exercise exactly as in the algebraic derived category of modules over a commutative ring (see Section 6). The case of a principal ideal is constructed as a fibre and so comeswith a filtration of length 1, and hence FIR has a filtration of length n. Because
31
LocalCohomology in Equivariant Topology
the construction is modelled on the stable Koszul complex it is clear that the homotopyspectral sequence of the filtration takes the form E2 HI(M~) = =t. [R,"
.,.I.~ViJ, ,,ln,G
¯
To prove the local cohomologytheorem it therefore suffices to establish the two equivalences R A EG+ ~ F jR A EG+ --~ F jR of equivariant R-modules. We may deduce the local cohomology theorem by applying (MAX+)An (’) to obtain the equivalence M A EG+ A X+ ~ F.~R An M A X+. The homotopy of the left hand side is M,~(EG× X) and that of the right hand side is calculated by a spectral sequence with E: term H~(M,~(X)). We may deduce the completion theorem by applying HomR(. A X+, M) to obtain the equivalence Homn(R A EG+ A X+, M) _~ Homn(FjR A X+, M). The homotopy of the left hand side is M~(EG× X), and the homotopy of the right hand side may be calculated by a spectral sequence with E2-term HJ.(M~(X)). It remains to prove that the maps A and B are equivalences of equivariant R-modules. 13.C. The map A is an equivalence. Since EG+is free, it is built from free G-cells Sn A G+, so it suffices to show that R A G+ e--- F jR A G+ is an equivalence, which is to say that R +-- F jR is a non-equivariant equivalence. Using res to denote restriction of the group of equivariance we have res~J : 0 by definition~ so that res~rgR
= r,,e~R
= r0R = R
as required. 13.D. The map B is an equivalence. We show the mapping cone of B is contractible. In fact we mayassume by induction on the group order that it is H-contractible for all proper subgroups H. This uses the fact that res~J has the same radical as J(H) = ker(R~/~ R*) as follows in the Noetherian setting from [29]. Fromthis we deduce resGHFjR = Fres~gR = Fj(H)R. Now the mapping cone of B is F jR A ~G where ~G is the mapping cone of EG+ ~ S °. Because (/~G) ~ = S°, there is unique map/~G ~4 S~y which is the identity in G-fixed points, where V is the reduced regular
32
Greenlees
representation. It suffices to show that P jR A S~V is contractible. if we define Q by the co fibre sequence ~,G
Indeed,
~ S ~V ~ Q,
we see that Q is built from cells G/H+for proper subgroups H. It follows from the inductive hypothesis F jR A Q -~ ,. Since FjR A S~V is obviously H-contractible for all proper subgroups H it suffices to show its G-homotopy is zero. Nowfor any R-module Mwe may define Euler classes x(V) exactly as was done for K-theory in Section 6. Indeed, it is immediate from the construction of MUthat MUA Sy ~_ MUA S Iyl, (where IVI denotes V with the trivial G-action) so that we yl = [sIVt, MU]* under the may take x(V) to be the pullback of 1 ¯ MUla a inclusion SO ~ Sy. From the definition we see that 7r,a(M AS~v) = lim 7ra, (M ASny) = r~, (M)[1/x(V)]. ~
n
On the other hand, since the inclusion SO --~ SV is non-equivariantly nullhomotopic, x(V) ¯ sothat inv erting it kil ls J-l ocal cohomology and ther efore
s = 0. This completes the proof.
[]
14. HOMOTOPICALLY GORENSTEINRINGS. For further details see [30, 20]. To smooth the transition between topology and commutative algebra we now write k*(X) -- E*(X) for the non-equivariant cohomology theory and S*(X) = E~(X) for the equivariant one. In view of the completion theorem we may also write ,~*(X) = E*(EG ×G for the complete theo ry. We restrict attention to finite groups G, and therefore to the case with shift a = 0 (in the notation of Section 11). Wehave obtained duality statements by combining the local cohomology theorem with the universal coefficient theorem. In the first instance these were spectral sequences H~(k*(BG))
~ k,(BG)
and *’*(k,(BG), k,) Extk, but we have already reformulated the first
k*(BG), as
F jF(EG+,k) ~ k A EG+, and the second can be written Homk(kA BG+,k) ~-- F(BG+, k),
LocalCohomology in Equivarlant Topology
33
which we would like to think of as the fixed points of an equivariant equivalence Ho~nk(k A EG+, k) "~ F(EG+, k). The analogy is with the algebraic derived category, so Horn corresponds to the total right derived functor of ordinary homomorphisms.Substituting in the local cohomology theorem expression for k A EG+~- F(EG+,k) A EG+, and writing ~ = F(EG+, k) we find Homk(Fd~,k) _~ Moregenerally we can consider this condition on any ring spectrum or differential graded algebra ~ equipped with ring maps k --~ ~ --~ k making it into an augmented k-algebra. We may say that such an augmented kalgebra ~ is homotopically Gorenstein if there is such an equivalence. If k is a field we may take homotopy and deduce that ~, is an LCTring. If k is not a field, the left hand side is the composite of two functors. Each of these functors could be calculated with a spectral sequence, but it is harder to extract information about their composite. Howeverwhen k is of small injective dimensionwe can still get striking duality properties for ~,. For complete rings it is quite often equivalent to consider the state~nent Fj~ ~-- Homk(~,k), and this is better behaved for non-complete rings, so we may say that an augmented k-algebra S is homotopically Gorenstein if FjS ~- Homk(S,k). It turns out [20] that under quite weak hypotheses this is equivalent to requiring Homs(k, S) "~ k as modules over Homs(k,k), which may be a more familiar form of the Gorenstein condition. This is related to the notion of Gorenstein differential graded commutative algebras considered by Avramov and Foxby [5], and to the ideas of F~lix, Halperin and Thomas[24] but they only require the isomorphism as S-modules. This circle of ideas is investigated further by Dwyer, Greenlees and Iyengar [19, 20]. 15. THE CHROMATIC CASE. For further details see [40]. Wehave explained that it is of interest to calculate the local cohomology modules HI(S) where S = R~ = E~(BG), because they give the E2-term of a spectral sequence for calculating the more subtle invariant (En),(BG). Playing this spectral sequence off against the universal coefficient theorem implies that the ring R~ has very special duality properties.
34
Greenlees
Nowhere we have k = R* = Z/~[[~l,...,~n_l]][~,~t-l], sider the Landweber sequence of prime ideals
o c_(p) c_(p,
and we may con-
...
It turns out that a natural ring of endomorphisms acts on R* and that the Landweber sequence is the unique maximal sequence of invariant prime ideals. For any module M we may form the Cousin complex C(M) = [M[1/p] ~ M/(p~)[1/u~]
~ ~, uF)[ 1/u~] ~ .. ¯ ..
~ Ml(P~,U~,...,u~_l)].
If (p, ul,..., un-1) is M-regular we say Mis good; in this c~e M~ C(M). It is then natural to use this filtration to ~pproach the calculation of local cohomology. Theorem15.1. [40] ff E~(BG) is good, then local cohomology is trivial pure chromatic strata in the sense that H~(E~(BG))
= H*(r~C(G(Bg)).
Remark 15.2. (i) The module E~(BG) is known to be good in m~ny cases, including all abelian groups G, and all symmetric groups [43]. (ii) The complex rxC(E;(BG)) is highly non-trivial. Indeed, F~C(E~(BG)) is usually non-zero up to degree n: for exampleif G is abelian then in degree i it is IGi~Ci(E~). On the other h~nd if G is abelian of rank r, then J has arithmetic rank ~ r, nnd hence the local, cohomologyis trivial above degree r. If n > r the exactness of the complex above degree r must involve interesting differentials. (iii) The case of periodic K-theory gives something we have already seen. states ¯
A
= H*
= H*
This is very effective, and recovers the local cohomologycalculations of Section 4. 16.
CONNECTIVE
K THEORY.
For further information see [13, Chapter 4]. In this section we present a very concrete example where the commutative algebra is very intricate and very striking. As in Example12.3, ku*(X) and ku, (X) denote the connective K cohomology and homology of a space X. The important feature of this example is the fact that ku. = Z[v] is graded but not periodic. For such examples the extended Rees ring construction is relevant. Given any commutative ring R and ideal J we may form the graded ring R[u, u-i], where u is of degree 2; the extended Rees ring is the graded subring Rees(R, J) C_ R[u, u-I] generated by R, u and 1/u ¯ J. Thus Rees(R, J) is R in each even degree _> 0 and jn in degree -2n <_ 0.
Local Cohomology in EquivariantTopology
35
The local cohomology theorem states
there is a spectral
H}(ku*(BG))
sequence
~ ku.(BG),
for any finite group G, but we want to discuss a case where the entire behaviour is understood. We take G -- V an elementary abelian 2-group of rank r. In fact there is a short exact sequence 0 ---~
T --~ ku*(BV) --~ Q ~
Here Q is the extended Rees ring for the J-completion/~(V) sentation ring R(V)
=
Z[~I,...
,Olr]/(o~
~
.....
of the repre-
2 = 1).
O~ r
This completion has the effect of 2-adic completion on J, ~nd leaving alone the Z summand generated by the regular representation. Thus Q = ~(V)[v, yl, y2,...,
Yr]
where Yi = (1 - ai)/u, and we may take J to be generated by elements mapping to Yl,... ,Yr. One may calculate H~(Q) without too much difficulty: H~(Q) Z[u] an d H~(Q) is zer o bel ow deg ree -2r and the orde r of i ts Z-torsion increases with degree. On the other hand T is ~ bit more complicated. It is a P-submodule of F~[x~,...,Xr] where P = F~[yl,...,y~] and x i 2= Yi. It turns out that it is generated by the elements i~S
j,k~S
~s S ranges over subsets of {1, 2,..., r} with at least 2 elements. Now we can decompose T = T~ ~ T3 $ ... $ Tr, where Ti is generated by the qs with IS[ = i. These P-modules T~ show some remarkable behaviour. Proposition 16.1. (i) Ti is of dimension r and depth (ii) H~(T~) = 0 unless j = i or (iii) H~a(Ti) ~ . is 1-dimensional ff i < (iv) Hm(~) ~ v: Tr-i+2(-r +4)ff < v (v) H~(Tr) = P(-r + 4). Thus in codimension r-2, the module T is Cohen-Macaulay and Hr(T) v = T(-r + 4): it is very nearly Gorenstein. However it turns out the structure of ku* (BV) is bound even more tightly. Lemma16.2. The modules H/m(Ti) v are the subquotients in the 2-adic filtration of H~(Q): i-1 Hj(Q)) 1 v Hj(Q)/2 H~(T/)V(i - 2) =(2 ~-2 ~ for i = 2, 3, . . . r -1.
[]
36
Greenlees
It turns out this is just what is required for ku, (BV) to be well-behaved. There is a short exact sequence 0 ~ E-4T v ~ ~u,(BV)
~ E-I(2r-IH~(Q))
and furthermore this corresponds under the universal coefficient theore:m to the exact sequence above for ku* (BV) v, in the sense that Ext~u* (E-2T ku,) = and ku,) = Q/(Z[u]). Ext~u. (2r-1Hj(Q), 1 This shows that even whenwe are not working over a field, of the homotopyGorenstein condition are very striking.
the implications
REFERENCES [1] M. Atiyah "Characters and cohomology of finite groups." IHES Pub.Math. 9 (1961) 23-64 [2] M. Atiyah "Vector bundles and the Kiinneth formula." Topology 1 (1962) 245-248 [3] M. Atiyah and G.B.Segal "Equivariant K theory and completion." J. Diff. Geom. 3 (1969) 1-18 [4] M. Atiyah and D.O.Tall "Group representations, A-rings and the J-homomorphism" Topology 8 (1969) 253-297 [5] L.L. Avramov and H.-B. Foxby "Locally Gorenstein homomorphisms." American J. Math. 114 (1992) 1007-1047 [6] D.J.Benson "Representations and cohomology II" Cambridge University Press (1991) [7] D.J.Benson and J.F.Carlson "Functional equations for Poincar~ series in group cohomology." Bull. London Math. Soc. 26 (1994) 438-448 [8] D.J.Benson and J.F.Carlson "Projective resolutions and Poincard duality complexes." Trans. American Math. Soc. 342 (1994) 447-488 [9] D.J.Benson and J.P.C.Greenlees "Commutative algebra for cohomology rings of classifying spaces of virtual duality groups." J.Algebra 192 (1997) 678-700. [10] D.J.Benson and J.P.C.Greenlees "Commutative algebra for cohomology rings of classifying spaces of compact Lie groups." J. Pure and Applied Algebra 122 (1997) 41-53. [11] M.Bestvina and M.Feighn "The topology at infinity of Out(F~)." Inventiones Math. 140 (2000) 651-692 [12] KS.Brown "Cohomology of groups." Springer-Verlag (1982) [131 R.R.Bruner and J.P.C.Greenlees "The connective K-theory of finite groups." (Submitted for publication) 126pp (With an appendix by A. Douady and L.H~rault) "Corners [14] A.Borel and J.-P.Serre and arithmetic groups." Comm.Math. Helv. 48 (1973) 436-491 [15] J.F.Carlson "The rood 2 cohomology of 2-groups" Website, http://www.math.uga.edu/~jfc/groups/cohomology.html [16] G.Carlsson "Equivariant stable homotopy and Segal’s Burnside ring conjecture." Ann. of Math. 120 (1984) 189-224 [17] J.Duflot "Depth and equivariant cohomology." Comm.Math. Helv. 56 (1981) 627-637 [18] J.Duflot "The associated primes of H~(X)" JPAA 30 (1983) 131-141 [19] W.G.Dwyerand J.P.C.Greenlees "The equivalence of categories of complete and torsion modules." (Submitted for publication) 20pp [20] W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar "Duality in algebra and topology." (In preparation) 21 pp
Local Cohomologyin Equivariant Topology
37
[21] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and Algebras in Stable Homotopy Theory, Volume 47 of Amer. Math. Soc. Surveys and Monographs. American Mathematical Society, 1996. [22] A.D.Elmendorf, J.P.C.Greenlees, I.Kriz and J.P.May "Cominutative algebra in stable homotopy theory and a completion theorem." Mathematical Research Letters 1 (1994) 225-239. [23] A. D. Elmendorf, and J. P. May "Algebras over equivariant sphere spectra" J. Pure and Applied Algebra 116 (1997) 139-149 [24] Y. F~lix, S. Halperin and J.-C. Thomas"Gorenstein spaces" Advances in Mathematics, 71, (1988) 92-112 [25] J.P.C.Greenlees "K-homology of universal spaces and local cohomology of the representation ring" Topology 32 (1993) 295-308. [26] J.P.C.Greenlees "Tate cohomology in commutative algebra." J. Pure and Applied Algebra 94 (1994) 59-83 "Commutative algebra in group cohomology." J.Pure and Applied [27] J.P.C.Greenlees Algebra 98 (1995) 151-162 [28] J.P.C.Greenlees "An introduction to equivariant K-theory." CBMSRegional Conference Series 91 American Math. Soc. (1996) 143-152. [29] J.P.C.Greenlees "Augmentation ideals of equivariant cohomology rings." Topology 37 (1998) 1313-1323 [30] J.P.C.Greenlees "Tate cohomology in axiomatic stable homotopy theory" Proc. 1998 Barcelona Conference (to appear) 25pp [31] J.P.C.Greenlees and G.Lyubeznik "Rings with a local cohomology theorem and applications to coholnology rings of groups." J. Pure and Applied Algebra 149 (2000) 267-285. [32] J.P.C.Greenlees and J.P. May "Completions of G-spectra at ideals of the Burnside ring" Proc. AdamsMemorial Conference II, Cambridge University Press (1992) 145178. [33] J.P.C.Greenlees and J.P. May "Derived functors of I-adic completion and local homology" Journal of Algebra 149 (1992) 438-453. and J.P.May "Completions in algebra and topology" Handbook of [34] J.P.C.Greenlees Topology (ed. I.M.James) North Holland (1995) 255-276. and J.P.May "Equivariant stable homotopy theory." Handbook of [35] J.P.C.Greenlees Topology (ed. I.M.James) North Holland (1.995) 277-323. [36] J.P.C.Greenlees and J.P.May "Brave new equivariant algebra." CBMSRegional Conference Series 91 American Math. Soc. (1996) 299-314. and J.P.May "Localization and completion in complex bordism." [37] J.P.C.Greenlees CBMSRegional Conference Series 91 American Math. Soc. (1996) 315-326. [38] J.P.C.Greenlees and J.P.May "Localization and completion theorems for MU-module spectra." Annals of Maths. 146 (1997) 509-544 [39] J.P.C.Greenlees and H. Sadofsky "The Tate spectrum of v,~-periodic complex oriented theories." Math. Zeits. 222 (1996) 391-405. [40] J.P.C.Greenlees and N.P.Strickland "Varieties and local cohomology for chromatic group cohomology rings." Topology 38 (1999) 1093-1139. [41] J.L. Harer "The virtual cohomological dimension of the mapping class group of an orientable surface." Inventiones Math. 84 (1986) 157-176 [42] W. Harvey "Boundary structure for the modular group." Ann. Math. Stud. 97 (1978) 245-251 [43] M.J.Hopkins, N.J.Kuhn and D.C.Ravenel "Generalized group characters and complex oriented cohomology theories." J. American Math. Soc. 13 (2000) 553-594 [44] M.Joachim (private communication) [45] M. Lazard "Groupes analytiques p-adiques." IHES Pub. Math. 26 (1965)
38
Greenlees
[46] L.G.Lewis, J.P.May and M. Steinberger (with contributions by J.E.McClure) "Equivariant stable homotopytheory." Lecture notes in maths. 1213 Springer-Verlag (:[986) [47] D.G.Quillen "The spectrum of an equivariant cohomology ring I,II" Ann. of Math. 94 (1971) 549-572, 573-602 [48] D.G.Quillen "The mod 2 cohomology rings of extra-special 2-groups and the spinor groups." Math. Ann. 194 (1971) 197-212 [49] D.G.Quillen "On the fomal group laws of unoriented and complex cobordism theory." Bull. American Math. Soc 75 (1969) 1293-1298 [50] U.Ray "Free linear actions of finite groups on products of spheres." J. Alg. 147 (1992) 456-490 [51] D.Rusin "The cohomology of the groups of order 32." Math. Comp. 53 (1989) 359-385 ring of a compact Lie group." IHES Pub.Math. 34 [52] G.B.Segal "The representation (196S) 113-12S [53] G.B.Segal "Equivariant K theory." IHES Pub.Math. 34 (1968) 129-151 [54] R.P. Stanley "Hilbert functions of graded algebras. " Adv. in Maths. 28 (1978) 157-83 [55] P.A.Symonds and T.Weigel "The cohomology of p-adic analytic groups." Progress in Math. 184, Birkh~iuser (2000) 349-410 [56] G.Wilson "K-theory invariants for unitary G-bordism." QJM24 (1973) 499-526 DEPARTMENT OF PURE MATHEMATICS,HICKS BUILDING, SHEFFIELD $3 7RH. UK. E-mail address: j. greenleesOsheffield, ac. uk
LECTURES
ON LOCAL
COHOMOLOGY AND DUALITY
JosephLipman PurdueUniversity, WestLafayette, Indiana
ABSTRACT. In these expository notes derived categories and functors are gently introduced, and used along with Koszul complexes to develop the basics of local cohomology. Local duality and its far-reaching generalization, Greenlees-May duality, are treated. A canonical version of local duality, via differentials and residues, is outlined. Finally, the fundamental Residue Theorem, described here e.g., for smooth proper maps of formal schemes, marries canonical local duality to a canonical version of Grothendieck duality for formal schemes.
CONTENTS
Introduction 1. Local cohomology, derived categories and functors 2. Derived Hom-Tensor adjunction; Duality 3. Koszul complexes and local cohomology 4. Greenlees-May duality; applications 5. Residues and Duality References
39
Lipman
40 INTRODUCTION
This is an expandedversion of a series of lectures given during the Local Cohomology workshop at CIMAT, Guanajuoto, Mexico, Nov. 29-Dec. 3, 1999, and at the University of Mannheim,Germany, during May, 2000. ][ am grateful to these institutions for their support. Rings are assumed once and for all to be commutative and noetherian (though noetherianness plays no role until midwaythrough §2.3.) Wedeal for the most part with modules over such rings; but almost everything can be done, under suitable noetherian hypotheses, over commutative graded rings (see [BS, Chapters 12 and 13]), and globalizes to sheaves over schemes or even formal schemes (see [DFS]). In keeping with the instructional intent of the lectures, prerequisite~’~ are relatively minimal: for the most part, familiarity with the language of categories and functors, with homologyof complexes, and with some basics of commutativealgebra should suffice, theoretically. (Little beyondthe flatness property of completion is needed in the first four sections, and in §5 someuse is madeof power-series rings and exterior powersof xnodules of differentials. The final section 5.6, however, involves formal schemes.) Otherwise, I have tried to makethe exposition self-contained, in the sense of comprehensibility of the main concepts and results. In proofs, significant underlying idea.’~ are often indicated without technical details, but with ample references to where such details can be found. These lectures are meant to complement foundational expositions which have full proofs and numerous applications to commutative algebra, like Grothendieck’s classic [Gr2], the book of Brodmannand Sharp [BS], or the notes of Shhenzel [Sch]. For one thing, the basic approach is different here. Onegoal is to present a quick, accessible introduction--inspiring, not daunting--to the use of derived categories. This we do in §1, building on the definition of local cohomology. Derived categories are a supple tool for working with homology, arising very naturally when one thinks about homology in terms of underlying defining complexes. They also foster conceptual simplicity. For example, the abstract Local Duality Theorem(2.3.1) is a framework for several disparate statements which appear in the literature under the name "Local Duality." The abstract theorem itself is almost trivial, following immediately :from derived Horn-Tensor adjunction and compatibility of the derived local cohomology functor with derived tensor products. The nontrivial fun comes in deducing concrete consequences--see e.g., §2.4 and §5.3. Section 3 shows howthe basic properties of local cohomology,other than those shared by all right-derived functors, fall out easily from the fact that local cohomologywith respect to an ideal I is, as a derived functor, isomorphic to tensoring with the direct limit of Koszul complexes on powers of a syste~n of generators of I. A more abstract, more general approach is indicated in an appendix.
Local Cohomologyand Duality
41
Section 4 deals with a striking generalization of local duality--and al,nost any other duality involving inverse limits--discovered in special cases in the 1970s, and then in full generality in the context of modules over rings by Greenlees and Mayin the early 1990s [GM1]. The derived-category formulation, that left-derived completion (= local homology)is canonically rightadjoint to right-derived power-torsion (= local cohomology),is conceptually very simple, see Theorem4.1. One application, "Affine Duality," is discussed in §4.3; others can be found in §§5.4 and 5.5. While we stay with modules, the result extends to formal schemes [DGM],where it plays an important role in the duality theory of coherent sheaves (§5.6). Further, it is through derived functors that the close relation of local duality with global Grothendieck duality on formal schemes is, from the point of view of these lectures, most transparently formulated. The latter part of Section 5 aspires to ~nake this claim understandable, and perhaps even plausible. As before, however, the main challenge is to negotiate the passage between abstract functorial formulations and concrete canonical constructions. The principal result, one of the fundamental facts of duality theory, is the Residue Theorem. The final goal is to explain this theorem, at least for smooth maps (§5.6). A local version, which is a canonical form of Local DuMityvia differentials and residues, is given in Theorem5.3.3. Then the connection with a canonical version of global duality is drawn via flat base change (§5.4)--for which, incidentally, Greenlees-Mayduality is essential-and the fundamentalclass (§5.5), developedfirst for power-series rings, then, finally, for smooth maps of formal schemes. 1.
LOCAL COHOMOLOGY, DERIVED CATEGORIES
AND FUNCTORS
1.1. Local cohomology of a module. Let R be a commutative ring and 2t4 (R) the category of R-modules. For any R-ideal I, let F! be the I-powertorsion subfunctor of the identity functor on Ad(R): for any R-module FIM = {me M [ for some s > 0, ISm = 0}. If J is an ideal containing I then Fj C FI, with equality if jn C I for some n > 0. Choose for each Man injective resolution, i.e., a complex of injective 1R-modules E~ : ...
-~ O ~ O-+ E°M-~ E~ --~ E2M--~ ...
1A complex C° = (C°, d°) of R-modules (R-complex) is understood to be a sequence of R-homomorphisms
...
(i e z)
such that did i-~ = 0 for all i. The differential d° is often omitted in the notation. The ° is ker(di)/im(di-1). i-th homology H~C The translation (or suspension) C[1] ° °of C i ¯ C[1]i --4 C[ll i+’ is i i+~ is the complex such that C[1] := C and whose differential dc[ q. -die +~: C~+~2. __+Ci+
Lipman
42
together with an R-homomorphismM-~ E~4 such that the sequence
is exact. (For definiteness one can take the canonical resolution of [Brb, p. 52, §3.4].) Then define the local cohomologymodules H~M:= Hi(FIEf)
(i
e
Each H} can be made in a natural way into a functor from A//(R) to J~A(R), sometimesreferred to as a higher derived functor of FI. Of course H} == 0 if i < 0; and since FI is left-exact there is an isomorphismof functors HI I. :~ F To each "short" exact sequence of R-modules (a) : 0 ~. M’ -~ M -+ M" --~ there are naturally associated connecting R-homomorphisms 5~(a):
H~M" ~ i+1 HI M (i
e
varying functorially (in the obvious sense) with the sequence (a), and that the resulting "long" cohomologysequence ¯ .---~ H~M’--~ H~M-+ H~M"--~ H~+1M’--~ H~+tM--~ ¯ ¯ is exact. i,)i>0, in which ° is lef t-exact, tog ether wit h A sequence of functors ( H connecting maps 5~ taking short exact sequences functorially to long exact sequences, as above, is called a cohomological functor. Amongcohomological functors, local cohomology is characterized up to canonical isomorphism as being a universal cohomological extension of F~--there is a functorial isomorphism H~ ~ F~, and for any cohomological functor (n~,5,/), every functorial map¢0 : H~~ H,° has a unique extension to a family of functorial maps (¢i: H~ -~ H/,) such that for any (a) as above, ’) H~(M")
H/,(M ’’)
~ H~+I(M
’) ~ H/,+~(M
commutesfor all i _> 0. Like considerations apply to any left-exact functor on ~/~(R), cf. [Grl, pp. 139ff]. For example, for a fixed R-moduleN the functors Ext/~(N,
M):= HiHomR(N, E~) (i >_
with their standard connecting homomorphismsform a universal cohomological extension of HomR(N,-). ~, E~4) one gets the canonical identificaFrom F~E~= li_~ms>0 HomR(R/I tion of cohomological functors (1.1.1) U~i = lira ExtiR(R/IS, i).
43
Local Cohomology and Duality 1.2.
Generalization
to complexes.
Recall
that
a map of R-complexes
¢: (C°, °) - ~ is a family of R-homomorphisms (¢i: Ci ~ Ci,)iez such that di,¢ i = ¢i+ldi for all i. Such a map induces R-homomorphisms HiC° -+ Hic:. We say that ¢ is a quasi-isomorphism if every one of these induced homology maps is an isomorphism. A homotopy between R-complex maps ¢1: C° °--~ C: and ¢2: C is a family of R-homomorphisms (h~: i ~C,~-1) su ch th at
-
= d ,-lh + h +ld (i
If such a homotopy exists we say that ¢1 and ¢2 are homotopic. Being homotopic is an equivalence relation, preserved by addition and composition of maps; and it follows that the R-complexes are the objects of an additive category K(R) whose inorphisms are the homotopy-equivalence classes. Homotopic maps induce identical maps on homology. So it is clear what a quasi-isomorphism in K(R) is. Moreover, i can b e t hought o f a s a fu nctor from K(R) to J~(R), taking quasi-isomorphisms to isomorphisms. 2 An R-complex C" is q-injective if any quasi-isomorphism ¢: C" --~ has a left homotopy-inverse, i.e., there exists an R-map ¢, : C: -~ C" such that ¢,¢ is homotopic to the identity map of C°. Numerous equivalent conditions can be found in [Spn, p. 129, Prop. 1.5] and in [Lp3, §2.3]. One such is (#): for any K(R)-diagram C,* ~-¢ X" ~ C" with ¢ a quasi-isomorphism, there exists a unique K(R)-map ¢, : C: --~ C" such that ¢,¢ = i i s a n i nFor example, any bounded-below injective complex C" (i.e., i jective R-module for all i, and C = 0 for i <~ 0) is q-injective [Hal, p.41, Lemma4.5]. And if C" vanishes in all degrees except one, say Cj ¢ 0, then C° is q-injective iff this cJ is an injective R-module[Spn, p. 128, Prop. 1.2]. A q-injective resolution of an R-complex C" is a q-injective complex E" equipped with a quasi-isomorphism C° -~ E’. Such exists for any C’, with E" the total complex of an injective Cartan-Eilenberg resolution of C° lEG3,
p.32,
An injective resolution of a single R-module M can be regarded as a qinjective resolution of the complex M* such that M° = M and Mi = 0 for all i ¢ 0. ~K-injective in the terminologyof [Spn]. ("q" connotes "quasi-isomorphism.") ~It has been shownonly recently that a q-injective resolution exists for any complex in an arbitrary Grothendieckcategory, i.e., an abelian category with exact direct limits and havin~ a generator [AJS, p.243, Thm.5.4]. Injective Cartan-Eilenberg resolutions alwaysexist in (~rothendieckcategories; and their totalizations--which generally require countable direct products--give q-injective resolutions when such products of epimorphisms are epimorphisms (a condition which fails, e.g., in categories of sheaves on most topological spaces), see [Wb2, p. 1661].
44
Lipman
After choosing for each R-complex C" a specific q-injective resolution C" -~ EL, we can define the local cohomology modules of C" by: (1.2.1)
H/~C" := Hi(F~ES)
It results
from (#) that
(i
for any K(R)-diagram
C2 with ¢I and ¢2 q-injectiveresolutions, thereis a unique¢, : E~, ~ E~ such that ¢,¢, = ¢2¢. ~om this follows that the H} can be viewed as functorsfrom K(R) to ~(R), independent(up to canonicalisomorphism) of the choicesof E~, and takingquasi-isomorphisms to isomorphisms. It willbe explained in ~1.4,in the contextof derivedcategories, how short exact sequenceof complexesin ~ (R)--a sequenceC~ ~ C" ~, with 0 ~ long exactcohomo]ogysequence ¯ .- ~ H~C~ ~
~+~" H~+~C" ...
Similarconsiderations leadto the definition of Extfunctors of complexes: (1.2.2)
Ext~(D’,C’):= H HomR(D
, E~)
wherefortwoR-complexes (X’,d}), (Y’,d~), thecomplex Hom~(X’:,Y’) is givenin degreen by Hom~ (X’, Y’):= ", Y’) specifiedby with differential tin: Hom~(X’,Y’) ~ Hom~+~(X d~f:= There is a functorial (1.2.3)
identification,
H~C" = lim
compatible with connecting maps,
Ext~(R/I
s,
where R/I ~ is thought of as a complex vanishing
C*) outside
degree 0.
1.3. The derived category. An efficacious strategy in studying the behavior of and relations among various homology groups is to regard them as shadows of an underlying play among complexes, and to focus on this more fundamental reMity. ~om such a point of view arises the notion of the derived category D(R) of ~(R). Whenour basic interest is in homology, we needn’t distinguish between homotopic maps of complexes, so we start with the homotopy category K(R). Here we would like to regard the source and target of as isomorphic objects because they have isomorphic homology. So we for~naIly adjoin to K(R) an inverse for each such ¢. This localization :procedure
45
LocalCohomology andDuality
produces the category D(R), described roughly as follows. (Details can found, e.g., in [Wbl, Chap. 10].) The objects of D(R) are simply the R-complexes. 4 A D(R)-morphism C -~ C~ is an equivalence class ¢/¢ of K(R)-diagrams C ~ X -~ ~ with ¢ a quasi-isomorphisIn, the equivalence relation being the least such that ¢/¢ = ¢~bl/~b~bI for all such ¢, ¢ a,nd quas,i-iso~norphisms ¢1:X1 ~ X. The composition of the classes of C~ ~ X~ -~ C" and C ~ X ~ C’ is given by
(¢’/¢’)(¢/¢)=
where(¢2 : X2--~ I, ¢2 : X2--~XI) is a ny pairwith ~b2 aquasi- isomorphism and ¢’¢2 = ¢¢2- (Such pairs exist.) X2
There is a canonical functor Q: K(R) -~ D(R) taking any complex to self, and taking the K(R)-map¢: C -~ C’ to the D(R)-~nap ¢/1c (where 1c is the identity mapof C). This Q takes quasi-isomorphisms to isomorphisms: if ¢ is a quasi-isomorphism then the inverse of ¢/lc is 1c/¢. The pair (D(R), Q) is characterized up to isomorphism by the following property: (1.3.1) For any category L, composition with Q is an isomorphism of the category of functors from D(R) to L (morphisms being functorial maps) onto the category of those functors from K(R) to L taking quasi-isomorphisms to isomorphisms. (If F: K(R) --~ L takes quasi-isomorphisms to isomorphisms then corresponding functor FD: D(R) --~ L satisfies FD(¢/¢) F(¢)o F( ¢)-1.) D(R) has a unique additive-category structure such that Q is an additive functor. For instance, to add two maps ¢1/¢1, ¢2/¢2 with the same source and target, rewrite them with a commondenominator--which is always possible, because of [Hal, pp. 35-36, proof of (FR2)]--and then just add the numerators. The characterization (1.3.1) of (D(R), Q) remains valid restricted to additive functors into additive categories. The homologyfunctors Hi are then additive functors from D(R) One shows easily that--in accordance with the initial motivation--a D(R)map ~ is an isomorphism if and only if the homology maps Hi(o~) (i ~ are all isomorphisms. 4Asa rule wewill no longeruse ° in denotingcomplexes. Butthe degree-ndifferential of a complex Cwill still be denotedby d’~ : C’~ TM. --4 C
46
Lipman
Example. WhenR is a field, any R-complex (C°, d°) splits (non-canonically) into a direct sum of the complexes im(di-1) ~, ker(d i) (concentrated in degrees i - 1 and i), whence (exercise) C canonically D(R)-isomorphic to thecomplex
Consequently, the functor C ~ @~ez ¯ H~Cfrom D(R) to graded R-vector spaces is an equivalence of categories¯ Example. A commontechnique for comparing the homology of two boundedbelow complexes C and Cr is to map them of them into a first-quadrant double complexas (respectively) the vertical and horizontal zero-cycles. Thus if Y is the totalized double complex, then we have K(R)-maps ~: C -~ Y, ~’: r - + Y. I f the appropriate spectral sequence of the double complex degenerates then ~’ is a quasi-isomorphism, and so one has the D(R)-map (lc,/~’)o(~/lc):C -~ from which one gets homology maps HiC -~ HiC’. In some sense, the role of the spectral sequence is taken over here by the conceptually simpler D(R)-map. The real advantage of the latter becomes more apparent when one has to work with a sequence of comparisons involving a variety of homological constructions--as will happen later in these lectures. It follows at once from definitions that for any R-complexes D, E, H°Hom~(D, E) = HomK(R)(D, Furthermore, (#) in §1.2 implies that for q-injective E the natural map HomK(R)(D, E) -~ HomD(R)(D, E) is bijective. Hence, C -+ E: =: Ec the previously used q-injective resolution and [i] denoting /-times-iterated translation (see footnote in §1.1), Ext~(D,
C) = HiHom~(D,E)
(1.3.2)
= H°Hom~(D,E[i]) = HomD(R)(D,E[i])
The following illustrative and any m ~ Z, M[-m] is vanishes elsewhere.
HomD(R)(D, C[ i]).
Proposition will be useful. For any R-module M the R-complex which is M in degree m and
Proposition 1.3.3. If C is an R-complex such that Hic = 0 for all i > m then for any R-module M, the homology functor Hm induces an isomorphism HomD(R)(C,M[-m])
-~
HomR(HmC,
Hm(M[-m]))
= HomR(HmC,
If, moreover, HiC = 0 for all i < m, then the D(R)-map corresponding this way to the identity map of Hmc is an isomorphism C --~ (n’~c)[-m]. Proof. ¯ ..
Let C<m C C be the "truncated" _+ Cm-2
d rn-~
m-~ crn_ ~ 1 d --~
complex
ker(C m ~ d.~
Gin+l) ~
0 --~ 0 -~ ...
47
LocalCohomology andDuality
The inclusion C<m ’-’) C is a quasi-isomorphism, so we can replace C by C<m, i.e., we may assume that Cn = 0 for n > m. Then for any injective re~olution 0 -~ M~ I ° -~ 11 ~ ... we have natural isomorphisms HomD(R)(C,M[--m])
-~ HomD(n)(C,I°[-m]) ~-- HomK(R)(C,I°[-m])
--~ HomR(HmC,
(Bijectivity of the second mapfollows, as above, from (#) in §1.2. Showing bijectivity of the third map--induced by Hm--is a simple exercise.) The first assertion follows. The second results from the above characterization of D(R)-isomorphisms via their induced homology maps. (More explicitly, the D(R)-mapin question is represented by the natural diagram of quasiisomorphisms C ~-~ C<_m -~ (HmC<_m)[-m].) Corollary 1.3.4. The functor taking any R-module M to the R-complex which is M in degree zero and 0 elsewhere, and doing the obvious thing to R-module maps, is an equivalence of the category .M(R) with the full subcategory of D(R) whose objects are the complexes with homology vanishing in all nonzero degrees. A quasi-inverse for this equivalence is given by the functor 0. H For a final example, we note that as the above-defined local cohomology functors H}: K(R) --~ M(R) (i ¯ Z) take quasi-isomorphisms to isomorphisms, they may be regarded as functors from D(R) to M(R). In of (1.3.2), (1.2.3) yields an interpretation of these functors in terms of D maps, viz. a functorial isomorphism H~C ~ lim HomD(R)(R/I s, C[i])
(C ¯ D(R)).
1.4. Triangles. As we have seen, exact sequences of complexes play an important role in the discussion of deri~,ed functors. But D(R) is not abelian category, so it does not support a notion of exactness. Instead, D(R) carries a supplementary structure given by certain diagrams of the form E -~ F --~ G -~ E[1], called triangles, and occasionally represented in the typographically inconvenient form G
E
~F
Specifically, the triangles are those diagrams which are isomorphic (in the obvious sense) to diagrams of the form (1.4.1)
X -% Y ~+ Ca -~ X[1]
where a is an ordinary ~nap of R-complexesand Ca is the mappingcone of a:
48
Lipman
as a graded group, Ca := Y @X[1], and the differential C~ --+ C~I+1 is the sum of the differentials of d~. and d~.[~] plus the mapan+~: Xn+l ~ yn+l, as depicted: C~+1 = yn+~ ~ X~+~
~+~ C~ = Y~ ~ X For any exact sequence 0~
(T)
Z ~ Y £ Z ~0
of R-complexes, the composite map of graded groups Ca ~ Y ~ Z turns out to be a quasi-isomorphism of complexes, and so becomes an isomorp:hism in D(R). Thus we get a triangle X ~ Y ~ Z ~ X[1]; and up to isomorphism, these are all the triangles in D(R). (See e.g., [Lp3, Example(1.4.4)].) The operation E ~ E[1] extends naturally to a functor on R-complexes, which preserves homotopy and quasi-isomorphisms, and hence gives rise to a functor T: D(R) ~ D(R), called translation, an automorphism of the category D(R). Applying the/-fold translations Ti (i a Z) to a triangle A: E ~ F ~ G ~ E[1] and then taking homology, one gets a long homology sequence (1.4.2)
...
~ HiE ~ HiF ~ HiG ~ HiE[l] = Hi+~E ~ ...
This sequenceis exact, as one need only verify for triangles of the form (1.4.1). If A is the triangle coming from the exact sequence (z), then this homology sequence is, after multiplication of the connecting maps HiG ~ Hi+~E by -1, precisely the usual long exact sequence associated to (~). This is why one can replace short exact sequences of R-complexes by triangles in D(R). And it strongly suggests that whenconsidering functors between derived categories one should concentrate on those which respect triangles, as specified in the following definition. Let A~, A2 be abelian categories. In the same way that one constructs the triangulated category D(R) from M(R), one gets triangulated derived categories D(AJ, D(A2).5 Denote the respective translation functors by
T~, T~. Definition 1.4.3. A A-functor ~: D(A~) ~ D(A2) is an additive functor which "preserves translation and triangles," in the following sense: ~modu]o someset-~heoreticconditionswhichweignorehere. (See [Wbl,p. 379, 10.3.3].)
49
LocalCohomology andDuality ¯ comes equipped with a functorial
isomorphism
O: OT1 ~ T2O such that for any triangle E --~ F --~ G __T_+E[1] =TIE in D(A1), the corresponding diagram (I)E
-~ OF -~ (bG ~ ((I)E)[1]
T2cbE
is a triangle in D(J[2). These A-functors are the objects of a category whose maps, called A-]unctorial, are those functorial maps which commute(in the obvious sense) with the supplementary structure. In what follows, those functors between derived categories which appear can always be equipped in some natural way with a 0 making them into Afunctors; and any noteworthy maps between such functors are A-functorial. For our expository purposes, however, it will not be necessary to fuss over explicit descriptions, and 0 will usually be omitted fi’om the notation. In summary: if (I): D(~41) --~ D(A2) is a A-functor, then to any exact sequence of complexes in J(~ (7"1)
O-~X
-~V
~-~
Z -~0
there is naturally associated a long exact homologysequence in ~42 ¯ .. -+ Hi(~X) -~ W(OY)~ Hi(OZ) -~ Hi+l(OX) that is, the homologysequence of the triangle in D(A2gotten by applying to the triangle given by (T1). Wewill also need the notion of triangles in the homotopycategory K(R). These are diagrams isomorphic in K(R) to diagrams of the form (1.4.1). to isomorphism, K(R)-triangles come from short exact sequences of complexes which split in each degree as R-modulesequences: for such sequences, the quasi-isomorphism following (’r) (above) is a K(R)-isomorphism,see [Lp3, Example (1.4.3)]. (One might also think here about the common of such a sequence of co~nplexes to resolve an exact sequence of modules-see e.g., [Wbl, p. 37)] for the "dual" case of projective resolutions). The canonical functor Q: K(R) -~ D(R) is a A-functor: it commutes translation and takes K(R)-triangles to D(R)-triangles. Anyadditive functot from A//(R) into an additive category extends in an obvious sense to A-functor between the corresponding homotopy categories. 1.5. Right-derived functors. RHomand Ext. Here is how in dealing with higher derived functors we lift our focus from homologyto complexes. The q-injective resolutions qc: C -~ Ec being as in §1.2, set (1.5.1) Then by Definition 1.2.1,
Rr~C := r~Ec. H~C= HiRFtC.
50
Lipman
The point is that RFI can be made into a A-functor from D(R) to D(R). For, the characterization (#) (§1.2) of q-injectivity implies that ~myquasiisomorphism between q-injective complexes is an isomorphism, and then ~that any K(R)-diagram C X-~ C’ wit h ¢ a quas i-isomorphism embe ds uniquely into a commutative K(R)-diagram, with ~--and hence F1~---an isomorphism:
c
x
c,
Ec~Ex~Ec, Furthermore~ the equivalence class(see ~1.3)of the K(R)-diagr~m
r, Ec r,E r, Ec, dependsonly on that ofC ~ X ~ C’. Thus we can associateto the D(R)m~p ¢/¢: C ~ C’ the m~p F/~/F/~: RFIC ~ RFIC’. This ~sociation respectsidentitiesand composition, makingRFI into a functor.And with Q: K(R)~ D(R) as before~a ~-structure on I isgiven bythefunctorial isomorphism O(C): aS(C[~]) ~ (aSC)[~] obtainedby applying QFI to the unique isomorphism¢: Ec[~] ~ Ec[I] suchthat¢oqc[~ (Fordetails, cf.[Lp3,Prop.(2.2.3)]). ~ = (qc)[l]. Thereis a functorialmap ~: QFI ~ RF~Q such that for each C, ~(C) the obviousmap F~C ~ F;Ec.The pair (RFI,~) is right-derived nctor fu of FI, characterized up to canonicalisomorphism by the propertythat~ is ~n initialobjectin the categoryof ~ll functori~lmaps QFI ~ F where F rangesover the categoryof functorsfrom K(R) to D(R) whichtake quasiisomorphisms to isomorphisms. In otherwords~for each suchF composition with~ mapsthe set [RFIQ~F] of functorial mapsfrom~FIQto F bijectively ontotheset [QFI~F]. Moreover~ (1.3.1)givesa uniquefactorization F =: for some F: D(R) ~ D(R)~ and a bijection[~FI~ F] ~ [RFIQ~ SimHarly~ one has via q-~njective resolutions a right-derived ~-functor RF of any ~-functor F on K(R).The characteristic initial-object propertyholds with"~-functor" in placeof "functor." SuchF arisemostoftenas extensions of additivefunctorsfrom~(R)to someabe]i~ncategory(seeend of ~1.4). For example~ for any R-complex D one has the functor RHom~(D~-) with RHom~(D, C) = Hom~(m, 6 Ec) andthen~as in Definition 1.2.2~ (1.5.2)
Exth(D, C) = H~RHom~(D,
6which, with somecaution regarding signs, can also be made into a contravariant Afunctorin the first variable, see e.g., [Lp3,(1.5.3)].
51
LocalCohomology andDuality
To illustrate further, let us lift the homologyrelation (1.2.3) to a relation among complexes in D(R). A first guess might be that RFIC s, C); but that doesn’t makesense, because the lim__~ of ~__~ms>oRHom(R/I sequence of complexes in D(R) doesn’t always exist. It is however possible to replace lira--thought of naively as the cokernel of an endomorphism of an infinite direct sum--by the summit of a triangle based on such an endomorphism, thereby expressing RF! as a ’~homotopycolimit." For this purpose, let hs : D(R) -+ D(R) be the functor described hsC:= RHom~(R/I s, C) (s >_ 1, C D(R)). There are natural functorial mapsPs : hs --~ hs+l and qs : hs -~ RFt, satisfying qs+lPs = qs. The family (1, -Pm): hm -~ hm @ hm+l C ~gs>_lhs (m >_ 1) defines a D(R)-map p: (~s>lhs ~ ~s>lhs. (Details, including the interpretation of infinite direct sumsin D (R), are left to the reader.) Proposition 1.5.3. Under these circumstances, there is a triangle
Proof. Replacing C by an isomorphic complex, we mayassume C q-injective, ¯ 8 so that hsC = HomR(R/I ,C) and RFIC = FIC. Since (~qs)op = O, it follows, with Cp the mapping cone of p, that there exists a map of Rcomplexes (t:
Cp = ( (~s>_l
hsC)
0 ( (~s>_l
hsC)[1]-~ F~C
restricting to ~ qs on the first direct summandand vanishing on the second; and it suffices to show that ~ is a quasi-isomorphism. But from the (easilychecked) injectivity of Hip and exactness of the homologysequence of the triangle (1.4.1) with (~ replaced by p, one finds that the homologyCp is H/~Cp = limHih~C = limHiHom~(R/I~, C) = Hi limHom~(R/Is, C) = i F~C, whence the conclusion.
[]
52
Lipman 2. DERIVEDHOM-TENSOR ADJUNCTION;DUALITY
2.1. Left-derived functors. Tensor and Tor. Dual to the notion of right-derived functor is that of left-derived functor: Let 7: K(R) --+ K(R) be a A-functor. le ft-derived fu nctor of ~/ is a pair consisting of a A-functor L~/: D(R) -~ D(R) and a functorial ~: L’yQ-+ Q~/ which is a final object in the category of all A-functorial maps F -+ Q’)’ where F ranges over the category of A-functors from K(R) to D(R) which take quasi-isomorphisms to isomorphisms. In other words, for each such F composition with ~ maps the set IF, L’yQ] of functorial maps from F to L3’Q bijectively onto the set IF, QT]. Moreover, (1.3.1) gives unique factorization F --- FQ for some F: D(R) -+ D(R), and a bijection [F, LT] --~ IF, Example. Recall that the tensor product C ®RD of two R-complexes is such that (C ®RD)n = @~+j=nCi ®RDJ, the differential (~n : (C ®R D)n ~ (C ®R D)n+l being determined by
6’*(x®y)=d~cz®y+(-1)~z®dJDY
(xeVi,
yeDJ).
Fixing D, we get a functor ~fD: .... ®RD: K(R) -+ K(R), which together with t~ = identity is a A-functor. To make -~ := C ®R--. (C fixed) a functor, one uses the unique 0~ (5 identity) such that the R-isomorphism C®RD~ D®RCtaking x®y to (-1)iJy®x is A-functorial [Lp3, (1.5.4)]. One gets a left-derived functor ... ~nD of ~Das follows (see [Spn, p. 147, Prop. 6.5], or [Lp3, §2.5]). An R-complexF is q-flat if for every exact R-complexE (i.e., HiE = 0 for all i), F ®RE is exact too. It is equivalent to say that the functor F®~t... preserves quasi-isomorphism, because by the exactness of the homology sequence of a triangle, a map of complexes is a quasi-isomorphism if and only if its cone is exact, and tensoring with F "commutes"with forming 7cones. Anybounded-aboveflat complexis q-flat (see, e.g., [Lp3, (2.5.4)]). Every R-complexC admits a q-fiat resolution, i.e., there is a q:flat ,complex F equipped with a quasi-isomorphism F ~ C. This can be constructed as a lim of boundedflat resolutions of truncations of C (loc. cir., (2.5.5)). Aft--~r choosing for each C a q-flat resolution Fc -~ C, one shows there exists a left-derived functor, as asserted above, with C~=RD = Fc ®RD (loc.’cit.,
(2.5.7)).
Taking homologyproduces the (hyper)tor functors Tori(C, D) H-i(c @=R D)
If F~) -+ D is a q-flat resolution, there are natural D(R)-isomorphisms C ®R ZExercise:
FD ~
Fc
®R FD
An R-complex E is q-injective
iff
-’~
Fc
®R D,
Hoxn~(-, E) preserves
quasi-isomorphism.
LocalCohomology andDuality
53
so any of these complexes could be used to define C ~R D. Using Fc ®RFD one can, as before, makeC __@RD into a A-functor of both variables C and D. As such, it has a final-object characterization as above, but with respect to two-variable functors. 2.2. Hom-Tensor adjunction. There is a basic duality between RHom~ and =@,neatly encapsulating a connection between the respective homologies Ext and Tot (from which all other functorial relations between Ext and Tot seem to follow As we’ll soon see, this duality underlies a simple general formulation of Local Duality. Let ~ : R -~ S be a homomorphismof commutative rings. Let E and F be S-complexes and let G be an R-complex. There is a canonical S-isomorphism of complexes: (2.2.1)
Hom~(E®sF, G) -~ Hom’s(E,
Hom’n(F,G)), which in degree n takes a family (fij: Ei ®s FJ -+ i+j+n) t o t he f amily (fi: Ei --~ Hom~n(F,G)) such that for a E Ei, fi(a) is the family of maps (gj : Fj -+ i+j+n) with gj(b) = fi j(a ® b)(b ~ F This relation can be upgraded to the derived-category level, as follows. Let ~. : D(S) --4 D(R) denote the obvious "restriction of scalars" functor. For a fixed S-complex E, the functor Hom~(E,G) from R-complexes G S-complexes has a right-derived functor from D(R) to D(S) (gotten q-injective resolution of G), denoted RHom~(~,E,G). If we replace G in (2.2.1) by a q-injective resolution, and F by a q-flat one, then the S-complex Hom~(F,G) is easily seen to become q-injective; and consequently (2.2.1) gives a D(S)-isomorphism (2.2.2) o~(E, F, G): RHom~((p. (E __@s F), G) -~ RHom}(E,RHom~(~,F, of which a thorough treatment (establishing canonicity, A-functoriality, etc.) requires some additional, rather tedious, considerations. (See [Lp3, ~2.6].) Here "canonicity" signifies that a is characterized by the property that it makes the following otherwise natural D(S)-diagram (in which H" stands for Horn’) commutefor all E, F and G: H~(E ® F, G) --~ RH~(v.(E
® F),
G) ~ RH~(~o.(E
H’s(E,H’R(F,
G)) --~ RH’s(E,H*R(F, G)) ~ RH’s(E, RH’R(~.F, Application of homologyH° to (2.2.2) yields a functorial isomorphism
(2.2.3)
HomD(R)(~,(E@=sF), G) HOm D(s)(E, RHo m~(~.F,G)),
see (1.5.2) and (1.3.2). Thus the functors ~.(... RSom~(~.F,-): D(R) --~ D(S) are adjoint.
~sF): D(S) --~ D(R) and
54
Lipman
2.3. Consequence: Trivial Duality. The following proposition is a ’very general (and in somesense trivial) form of duality Proposition 2.3.1. With ~o: R -~ S, ~,: D(S) --+ D(R) as above, let E E D(S), let G ~ D(R), and let F: .A/[(S) --~ .All(S) be a funetor, right-derived functor RF: D(S) --> D(S) (see §1.5). Then there, exit~ts a natural functorial map (2.3.1a)
E RrS
-~
whence, via the isomorphism (2.2.2)
RFE,
with F = RFS, a functorial
map
C) -~ RHom’s(E, RHom~(~,RFS, G)) , whence, upon application of the homology functor H°, a functorial map (2.3.15)
aHom[(~,arE,
(2.3.1c) nomD(R)( .arE,
G) -~ HOmD(S)(E,
RHom[( .RrS, G)).
This being so, and E being fixed, (2.3.1a) is an isomorphism ~:~ (2.3.1b)
is an isomorphism for all G
*==~(2.3.1c) is an isomorphism for all G. Proof. For fixed E’, the functor RHom}(FE’, RF-): K(S) --~ D(S) quasi-isomorphisms to isomorphisms. So the initial-object characterization of right-derived functors (§1.5) gives a unique functorial mapUE’ makingthe following otherwise natural D(S)-diagram commutefor all S-complexes Horn;
(E’,
E)
nom;(rE’, rE)
> Rnom~ (E’,
E)
Rnom;(rE’, rE)
Taking E’ to be a q-injective resolution of S, one has the map pE,(E) : E = RHom~(S, E) -~ RHom~(RFS, which gives, via (2.2.3) (with R = S and ~ = identity), (2.3.1a). It is clear then that for any E, G: [(2.3.1a)
is an isomorphism] ~ [(2.3.15) ~ [(2.3.1c)
the n~tural
is an isomorphism] is an isomorphism].
Conversely, if (2.3.1c) is an isomorphismfor all G then using (2.2.3) sees that (2.3.1a) induces for all G an isomorphism HomD(R)((p,
, G) ~ HomD(R)(~o,(E whence ~,(2.3.1a) is an isomorphism. Thus (2.3.1a) induces homology morphismsafter, hence before, restriction of scalars, and this means that (2.3.1a) itself is an isomorphism(§1.3). s [] s(2.3.1a) is an isomorphism iff I~F commuteswith direct sums, see Prop. 3.5.5 below.
55
LocalCohomology andDuality
The map (2.3.1a) is the obvious one when F is the identity functor and it behaves well with respect to functorial maps F -~ Ft, in particular the inclusion Fj ~-~ 1 with J an S-ideal. For noetherian S it follows that (2.3.1a) is identical with the isomorphism¢(S, E) in Corollary 3.3.1 below (with I C R replaced by J C S), whence (2.3.1b) and (2.3.1c) are isomorphislns. Thus: Theorem 2.3.2 ("Trivial" Local Duality). For ~: R ~ S a map of commutative rings with S noetherian, J an S-ideal, and ~. : D(S) -4 D(R) the restriction-of-scalars functor, there is a functorial D(S)-isomorphism RHom[(~,RFjE,
G) -~ RHom~(E, RHom[(~,RFjS,
G))
(E E D(S), G E D(R)); and hence with ~): D(R) ~ D(S) the functor
~(-) := RI-Iom[(~,RFjS,-) ~ RHom~(Rr:S,RHom[(~,S,-)) there is a natural adjunction isomorphism HOmD(R)(~,RFjE,
G) ~ HOmD(3)(E,
#
Now with (S,J) and ~: R -+ S as above, let ¢: S --~ T be another ring-homomorphism, with T noetherian, and let ~b, : D(T) --~ D(S) be corresponding derived restriction-of-scalars functor. Let K be a T-ideal containing ¢(J). Then ¢,D~((T) C Dj(S), and therefore by Corollary below, the natural map is an isomorphism RFg¢,RFK -~ ¢,RFIi , giving rise to a functorial isomorphism ~,RF~¢,RF K -~ ~,¢,arK = (¢~),Rr~ whencea functorial isomorphismbetween the right adjoints (see Thm. 2.3.2): (¢~)~ -~¢;~. # #
(2.3.3)
2.4. Nontrivial dualities. From now on, the standing assumption that all rings are noetherian as well as commutativeis essential. "Nontrivial" versions of Theorem2.3.2 convey more information about Suppose, for example, that S is module-finite over R, and let G ~ Dc(R), by which is meant that each homology module of G ~ D(R) is finitely generated. (Here "c" connotes "coherent" .) Suppose further that Ext~ (S, G) is a finitely-generated R-modulefor all i ~ Z, i.e., RHom~(~,S,G) ~ Dc(R). (This holds, e.g., if HiG= 0 for all i << 0, cf. [Hal, p.92, Prop. 3.3(a)].) Then Greenlees-May duality (Corollary 4.1.1 below, with (R,I) replaced by (S, J)--so that ~’ denotes J-adic completion of S--and F replaced RHom~(~,S,G)) gives the first of the natural isomorphis~ns ItHom~(~o,S, (2.4.1)
G) ®s ~ RH om}(RFjS, RHom~(~,S, G) (4.1.1) (~.2.~)
56
Lipman
Moreparticularly,
for S = R and 9~ = id (the identity map) we get id~G = G ®R ~ (G E Dc(R)).
Specialize further to where R is local, ~ = id, J = m, the maximal iideal of R, and G E Dc(R) is normalized du alizing co mplex, 9 sotha t in D(R), RFmG --- 2: with Z an R-injective hull of the residue field Rim [Hal, p. 276, Prop. 6.1]. Then there are natural isomorphisms RHom~(RF,~E, G)
" " - RHOmR(RFoE RF~G) =~ RHomR(RF,~E Z) , , (3,2.2)
Substitution into Theorem2.3.2 gives then a natural isomorphism (2.4.2)
RHom~(RFmE, ) -- ~ RHom~(E,G®R~)
(E ~ D (R )).
For E ~ De(R)this is just classical local duality [Hal, p. 278 ], moduloMatlis dualization. 10 Applying homology H-i we get the duality isomorphism (2.4.3)
HomR(H/mE,2:) -~ Ext~i(E,
G ®R/~).
If R is Cohen-Macaulay,i.e., there is an m-primary ideal generated by an R-regular sequence of length d := dim(R), then by Cor. 3.1.4, HInR = 0 for i > d; and in view of [Gr2, p. 31, Prop. 2.4], (1.1.1) gives H~R= 0 for i < (Or, see [BS, p. 110, Cor. 6.2.9].) Since/~ is R-flat, (2.4.3) nowyields 0 = Ext~i(R,
G ®R/~) = H-i(G ®R/~) = (H-iG) ®R/~ (i
Hence the homologyof G vanishes outside degree -d, so by Proposition 1.3.3 there is a derived-category isomorphism G -~ w[d] where w := H-dG (a canonical module of R). In conclusion, (2.4.3) takes the form HomR(H/mE,Z) -~ Ext~-i(E,&). Another situation in which ~o] can be described concretely is whenS is a power-series ring over R, see §5.1 below. For more along these lines, see [AJL, pp. 7-9, (c)] and [DFS, §2.1].
3.
KOSZUL
COMPLEXES
AND LOCAL
COHOMOLOGY
Throughout, R is a commutative noetherian ring and t = (t~,..., tin) is a sequence in R, generating the ideal I := tR. The symbol ® without a subscript denotes ®R, and similarly for @. 9which exists if R is a homomorphic image of a Gorenstein local ~°which is explained e.g., in [BS, Chapter 10]. For more details,
ring [Hal, p. 299]. see [AJL, p. 8].
LocalCohomology andDuality
57
3.1. RFI -- stable Koszul homology. Before proceeding with our exploration of local cohomology,we must equip ourselves with Koszul complexes. They provide, via ~ech cohomology,a link between the algebraic theory and the topological theory on Spec(R)--a link which will remain implicit here. (See [Gr2, Expos~II].) For t E R, let E(t) be the complex... --~ 0 -+ R 3+ Rt --~ 0 -+ ... which in degrees 0 and 1 is the natural map from R =: E°(t) to its localization Rt =: ~ (t) by powers of t, and which vanishes elsewhere. For ~ny R-complex C, define the "stable" Koszul complexes ~(t):=~(t~)@’"~(tm),
~(t,C):=~(t)~C.
Since the complex~(t) is flat and bounded, the functor of complexes~(t, takes quasi-isomorphisms to quasi-isomorphis~ns (apply [Hal, p. 93, Lemma 4.1, b2] to the mapping cone of a quasi-isomorphism), and so may~and will--be regarded as a functor from D(R) to D(R). Given a q-injective resolution C ~ Ec (~1.2) we have for E = E~ (j ~ F~E= ker(~°(t,E)
= E ~ ~=~Et~ = ~(t,E)),
whence a D(R)-map 5(C): RF~C(1.5.DFIEc ~ ~(t,
Ec) ~ ~(t,C),
easily seen to be functorial in C, making the following diagra~n commute: RF~C
~ ~(t,C)=~(t)@C
(3.1.1)
natural 1 ~ C > R~C where r(C) is obtained by tensoring the projection ~(t) ~ E°(t) (which is a map of complexes) with the identity map of 1~ The key to the store of properties of local cohomologyin this section is: Proposition
3.1.2.
The D(R)-map 5(C) is a functorial
isomorphism
RF~C~ ~(t,C). Proof. (Indication.) Wecan choose Ec to be injective as well as q-injective (see footnote in ~1.2), and replace C by Ec; thus we need only showthat if C is injective then the inclusion map ~RC ~ ~(t, C) is a quasi-isomorphism. Elementary "staircase" diagram-chasing (or gument) allows us to replace C by each ~ ( i ~ Z), re ducing th e pr oblem to where C is a single injective R-module. In this case the classical proof can be found in [Gr2, pp. 23-26] or [Wbl, p. 118, Cor. 4.6.7] (with arrows in the two lines preceding Cot. 4.6.7 reversed). There is another approach when C is a bounded-below injective complex (applying in particular whenC is a single injective ~nodule). Every injective l~Butsee ~3.5for a Koszul-free,moregeneral, approach.
58
Lipman
R-module is a direct sum of injective hulls of R-modules of the form R/P with P C R a prime ideal, and in such a hull every element is annihilated by a power of P [Mtl]. It follows that for every t E R, the localization map C -+ Ct is surjective, 12 so that the inclusion FtRC ~-~ ]C((t), is a quasi-isomorphism; and that the complex FtRC is injective, whence FtRC is bounded-below and injective, therefore q-injective (§1.2). Moreover,/C(t, C) is bounded-below and injective, hence q-injective, since for any flat R-module F and injective R-module E, the functor HomR(M, F ® E) ~- F ® HomR(M, E) of finitely-generated R-modules M is exact, i.e., F ® E is injective. One shows now, by induction on m >_ 2, that with t’ := (t2,..., tm)~, the top row of
r lRr ,Rc --+
rtRc RrIC
R~tl~Rrt,~ C
RFtlRK:(t’, C)
E(t, C)
is a D(R)-isomorphism. For R-ideals I and I’ there is, according to the initial-object characterization of right-derived functors (§1.5), a unique functorial map X making the following otherwise natural D(R)-diagram commute
rI+x, =rxr, I, Corollary
3.1.3.
The preceding
natural
X: RFI+t,
functorial
map is an isomorpi~ism
-~ RFIRFI,.
Proof. LetI=tR(t:=(t~,., ¯ , tm))andI’=t’R(t’:=(t~l ~ ¯ , , . n)),sothat tI I + I’ (t V t’)R (t V t’:= (t~,... tm, t~, , n))" It is a routine exercise to deduce from Proposition 3.1.2 an identification of X(C) with the natural isomorphism/C(t V t’, C) -~ /C(t,/C(t’, C)). We see next that the functor RFI is "bounded"--a property ~3 able importance in matters involving unbounded complexes. Corollary 3.1.4. (resp. i
of consider-
Let C be an R-complex such that HiC = 0 for all i > it Then HiRFfC=0foralli>i~+m (resp. i
~2Thereare easier waysto prove this. ~3Thisboundednessproperty, called "way-outin both directions" in [Half, often .enters via the "way-out"lemmas [loc. cir., p. 69, (iii) and p. 74, (iii)]. See, for instance, the proof of Corollary 3.2.1 below.
LocalCohomology andDuality
59
Proof. If HiC = 0 for all i > it, then replacing Ci by 0 for all i > il and Cil by the kernel of Cil -~ Ci1+1 produces a quasi-isomorphic subcomplex C1 C C vanishing in all degrees above il. There are then isomorphisms RFIC ~ RFIC1 .---~ ]C t, C1), (3.1.2) ( and Hi/C(t, C1) (indeed,/C(t, C1) itself) vanishes in all degrees above il A dual argument applies to the case where HiC = 0 for all i < i0. (More generally, without Prop. 3.1.2 there is in this case a surjective quasiisomorphism C ~ Co with Co vanishing in all degrees below i0, and a quasiisomorphism C0 -* E0 into an injective E0 vanishing likewise [Hal, p. 43]; and so HiRP~C~ HiFtEo vanishes for all i < i0.) [] 3.2. The derived torsion category. Wewill say that an R-module Mis I-power torsion if F~M= M, or equivalently, for any prime R-ideal P ~ I the localization Mp= 0. (Geometrically, this means the corresponding sheaf on Spec(R) is supported inside the subscheme Spec(R/I).) For any Rmodule M, I’~M is/-power torsion. Let DI(R) C D(R) be the full subcategory with objects those complexes whose homologymodules are all/-power torsion, i.e., the localization Cp is exact for any prime R-ideal P ~ I. For any R-complex C, (1.5.1) implies that RF~CG DI(R). The subcategory DI(R) is stable under translation, and for any D(R)triangle with two vertices in DI(R) the third must be in D~(R) too, follows from exactness of the homologysequence (1.4.2). Corollary 3.2.1. The complex C is in DI(R) if and only if the natural map ~(C): RFIG--* C is a D(R)-isomorphism. Proof. (~=) Clear, since RF~rCE DI(R). (0) The boundednessof RFt (3.1.4) allows us to apply [Hal, p. 74, (iii)] to reduce to the case where C is a single/-power-torsion module. But then ]C(ti) ® C = for i = 1,..., m, whence (by induction on m ) / C(t, C) = and so by Proposition 3.1.2 and the commutativity of (3.1.1), ~(C) is isomorphism. [] Weshow next that RF~ is right-adjoint Proposition
3.2.2.
to the inclusion DI(R) ~ D(R).
The map ~(G): RFIG ~ G induces an isomorphism
RHom°(F, PriG) -% RHom°(F,G) (F e D~(R), G e.D(R)), whence, upon application of homology H°, an adjunction isomorphism o(F, G): HomD~(R)(F, RFIG) = HomD(R)(F, RFIG) ~ HomD(R)(F, G). Proof. Since D(R)-isomorphism means homology-isomorphism (§1.3), since (see (1.3.2)) HiRHom°(F’,G ’) = HomD(R)(F’,G’[i])
(F’,G’
D(R)),
60
Lipman
we need only show that p(F, G) is an isomorphism for all F E DI(R) and G E D(R). Referring then HomD(R)(F, G) HomD(R)(RFIF,
HomD(R)(F ) , RFIG where ~ is the natural map and where p is induced by the isomorphism ~(F): RFIF ~ F (Corollary 3.2.1), let us show that p-~, is inw~rse to That pp-~,(~) = ~ for any e HOmD(R) (F , G)amounts to the (obvious) commutativity of the diagram
F
~
RFIG) ~
G
That p-~’e(~) = ~ for ~ e HomD(R)(F , RF~G) amounts to commutativity of RF~ F ~ RF~RFI
G
and so (since ~ is functorial) it suffices to show that RFI~(G) = ~(R~G). Wemay assume that G is injective and q-injective, and then the second paragraph in the proof of Prop. 3.1.2 shows that FIG is injective and that is a D(R)-isomorphism. It follows that RFIe(G) and ~(RFIG) are canonically isomorphic to the identity map F/F~G ~ F~G, so that they are indeed equal. ~ 3.3.
Local cohomology and tensor
product.
Corollary 3.3.1. There is a unique bifunctorial
isomorphism
whose composition with the natural map RF~(C~ C’) ~ C ~ C’ is the nat~. ural map RFIC ~ C’ ~ C ~ C Proof. Replacing C and C~ by q-flat resolutions, we mayassume that C and C~ are themselves q-flat. Existence and bifunctoriality of the isomorphism¢ are given then, via Prop. 3.1.2 and commutativity of (3.1.1), by the natural isomorphism It follows in particular that RF~C~ C~ e D~(R),~ and so uniqueness of ¢ results from Proposition 3.2.2. ~ 14Thisis easily shownwithoutusing/(:.
61
LocalCohomology andDuality Here is a homological consequence. (Proof left to the reader.) Corollary 3.3.2. For any R-complex C and fiat natural isomorphisms
R-module M there are
H~(C) ® M -~ H~(C ® M) Here is an interpretation of some basic properties of the functor RF! in terms of the complexRFIR~-/~(t). (Proof left to the reader.) Corollary 3.3.3. Via the isomorphism ¢(R, -) of the functor RF~R@=(-) with RF~(-) the natural map RFIC’ ~ C’ corresponds to the map
1:
RF R@ C’= C’,
and the above map¢(C, 15 C’) corresponds to the associativity
isomorphism
(RF,R ~= C) ~ C’ -% RFIR ~ (C ~= c’). 3.4. Change of rings. Let ~o: R -+ S be a homomorphismof noetherian rings. The functor "restriction of scalars" from S-complexesto R-complexes preserves quasi-isomorphisms, so it extends to a functor ~o, : D(S) --} D(R). As in §2.1, we find that the functor M ~-+ M ®RS from R-modules to S-modules has a left-derived functor ~*: D(R) --+ D(S) such that ter choosing for each R-complex C a q-flat resolution Fc --~ C we have ~o*C = Fc ®R S. If S is R-flat, then the natural map is an isomorphism ~p*C -~ C ®R S. There are natural functorial isomorphisms (3.4.1)
B@=R~,D -% ~o,(~o*B~=sD)
(BeD(R),
(3.4.2)
~o*(B@RC) -% ~o*B~=s~*C
(B,
D e D(S)),
CeD(R)).
Proofs are left to the reader. (In view of [Lp3, (2.6.5)] one mayassume that all the complexes involved are q-fiat, in which case ~ becomes ®, and then the isomorphisms are the obvious ones.) For example, there are natural isomorphisms(self-explanatory notation):
ClCR(t)= ~ ]OR(t) So putting B = ]C(t) in the isomorphisms(3.4.1) and (3.4.2) we obtain, Propositions 3.1.2 and 3.2.2, and commutativity of (3.1.1), the following two corollaries. Corollary 3.4.3.
There is a unique D(R)-isomorphism ~o, RF/sD -% RFI~o,D (D e D(S))
whose composition with the natural map RFI~o,D-~ ~o,D is the natural map ~o, RFIsD -~ ~o,D. Thus there are natural R-isomorphisms ~o,H~sD -% H~o,D (i
e Z).
l~derivedfromassociativityfor tensorproductof R-complexes as in, e.g., [Lp3,(2.6.5)].
62 Corollary
Lipman 3.4.4.
There is a unique D(S)-isomorphism
v*RF,C
Rrxs *C
(C e D(R))
whose composition with the natural map RF, rs~o*C --~ ~*C is the natural map ~*RFIC ~ ~*C. Consequently, if S is R-flat then there are na~ural S-isomorphisms H}C @ S ~ H~s(C
@ S)
(i
Z)
If M is an/-power-torsion R-module, for example, M = H~C (see ~3.2), and ~ is the I-adic completion of R, then the canonical map ~: M ~ M@~ is bijective: indeed, since this map commutes with lira we may assume that Mis finitely generated, in which case for large n the natural map n) M @ ~ ~ M @ (~/In~)
: M @ (R/I
as well as its composition with 7 is bijective, S = ~ in the preceding Corollary we get:
so that 7 is too. Thus putting
Corollary 3.4.5. For C ~ D(R) the local cohomology modules H~C (i ~ Z) depend only on the topological ring ~ and C @ ~, in that for any defining ideal J (i.e., ~ = ~ ) there are natural isomorphisms H}C ~ H}(C ~ ~)
= H~(C@
Remark. For (~,J) as in 3.4.5, the functor Fj = H~ on ~-modules depends only on the topological rin~ ~ : FjM consists of those m ~ M which are annihilated by some open R-ideal. Exercise. (a) Let F be a q-injective resolution of the S-complex D. Showthat applying HiFI to a q-injective R-resolution ~.F ~ G produces the homology maps in Corollary 3.4.3. (b) Suppose that S is R-flat. Let C ~ E be a q-injective resolution of the complex C and y: E @ S ~ F a q-injective S-resolution. Showthat the homology maps in Corollary 3.4.4 factor naturally ~ H~C~S ~ H~F~E~S ~ H~(F~E@S)
H~F~s(E~S) n’r,s’~
HiF~sF ~ H~s(C@S ).
3.5. Appendix: Generalization. In this appendix, we sketch a more general version (not needed elsewhere) of local cohomology,and its connection with the theory of "localization of categories." In establishing the corresponding generalizations of the properties of local cohomologydeveloped above, we make use of the structure of injective modules over a noetherian ring together with some results of Neeman about derived categories of noetherian rings, rather than of Koszul complexes. At the end, these local cohomologyfunctors are characterized as being all those idempotent A-functors from D(R) to itself which respect direct sums. Let R be a noetherian topological ring. The topology ~ on R is linear if there is a neighborhood basis ~ of 0 consisting of ideals. An ideal is open iff it contains a member of ~.
63
Local Cohomology and Duality
\Ve assume further that the square of any open ideal is open. Then 11 is determinedby the set O of its open prime ideals: an ideal is open iff it contains a power product of finitely many members of O. Thus endowing R with such a topology is equivalent to giving a set O of prime ideals such that for any prime ideals p C p’, p E O =~ p’ E O. The case we have been studying, where A~ consists of the powers of a single ideal I, is essentially that in which O has finitely manyminimal members (namely the minimal prime ideals of I, whose product can replace I). Let F’ = -~ be the left-exact subfunctor of the identity functor on J~(R) such that for any R-module M, /~’M = { x ~ MI for some open ideal I, Ix = 0 }. The functor/~’ commutes with direct sums. If p is a prime R-ideal and Ip is the injective hull of R/p, then F~Ip = Ip if p is open (because every element of Ip is annihilated by a power of p), and I"Ip = 0 otherwise. Thus /" determines the set of open primes, and hence determines the topology 11. Moreover, /~’ preserves injectivity of modules, since every injective S-moduleis a direct sum of Ip’s, and any such direct sum is injective. Conversely, every left-exact subfunctor F of the identity which commuteswith direct sums and preserves injectivity is of the form ~. Indeed, since Ip is an indecornposable injective, the injective module l~(Ip) must be Ip or 0. If p C then by left-exactness, F(Ip) F(Ip,); an d hence th e se t of p su ch th at /’ ([p) is the set of open primes for a topology 11. One checks then that F = F~ by applying both functors to representations of modules as kernels of maps between injectives. Lemma3.5.1. isomorphism
If F is an injective complex, then the natural D(R)-map is ~(C): FtF -~ RF~F.
Proof. The mapping cone C of a q-injective exact, and as RF~F= F~EF, it suffices to consider for any ideal I = (tl,... ,tn)R the form a neighborhood basis of 0, so that with
resolution F -~ EF is injective and show that F’C is exact. To this end, topology tl I for which the powers of I previous notation,/7,’ a, = F~’. Then
which reduces the problem to where ~2 = g,; and Ft ~ = ~t~... ~.~ gives a further reduction to where I = tR (t ~R). Finally, exactness of the complex C and of its localization C~ in the exact sequence 0 -~ ~C -~ C -~ C~ --+ 0 (see proof of Proposition 3.1.2) imply that ~C is exact. [] Since any direct sumof q-injective resolutions is an injective resolution, and since _r" commuteswith direct sums, one has: Corollary 3.5.2. morphism
For any small family (E~) in D(R), the natural map is an iso-
From Lemma3.5.1, and the fact that/’~ preserves injectivity of complexes, one readily deduces the ("colocalizing") idempotence of R/": Proposition 3.5.3. (i) For an R-complex E, with q-injective resolution E -~ Itv, the maps e(R/~’E) and RF’~(E) from RF’R/"E to RF’E are both inverse to the isomorphism RI’~E --~ R/~’R/~E given by the identity map of F~IE = F’I~IE, and so are equal isomorphisms.
Lipman
64 (ii)
For E, F E D(R) the map ~(F) : RF’F --~ F induces an isomorphism HomD(R)(RF’E, RF’F) ~ HomD(/~)(RF’E,
F),
with inverse HomD(R)(RF’E,
F) ~ HomD(n)(RF’RF’E, ~ HomD(n)(RF’E,
RF’F).
The properties given in Corollary 3.5.2 and Proposition 3.5.3 (i) characterize functors of the form RF~ among A-functors from D(R) to itself. This will shownat the end of this appendix (Proposition 3.5.7). Next we generalize §3.2. Let A/In(R) = F~AJ(R) be the full abelian subcategory of M(R) whose objects are the it-torsion R-modules--those R-modules M such that F’M= M, i.e., the localization Mp= 0 for every non-open pri~ne R-ideal p. The subcategory M/In(R) C A4(R) is plump, i.e., if M1-> M2-+ M--> M3-~ M4 is an exact sequence of R-modules such that Mi E A4u(R) for i = 1, 2, 3, 4, then also M~ A~tu(R). (To see this One reduces to the case where M1= M4 = 0, uses that the product of two open ideals is open.) One can think of F’ as a functor from A4(R) to A4u(R), right-adjoint to the inclusion functor fl4u(R) ¢-~ A4(R). Upgrading to the derived level, let Du(R) C D(R) be the full subcategory objects those complexes C whose homology modules are all in A4a(R), i.e., the localization Cp is exact for every non-open prime R-ideal p. The exact homology sequence (1.4.2) of a triangle, together with plumpness of A4u(R), entails Du(R) is tr iangulated su bcategory ofD(R), tha t is, if t wo vert ices of a D(R triangle lie in Du(R)then so does the third. In fact Du(R)is localizing su bcategory of D(R) (= full triangulated subcategory closed under arbitrary D(R)-direct snms). If C -+ Ec is a q-injective resolution then RF~C = F~Ec ~ Da(R), and RF’D(R) C Du(R). Thus (i) in the following Proposition implies that is the essential image of the functor RF’ (i.e., the full subcategory whoseobjects are the complexes isomorphic to one of the form RF’C); and (ii) says that RF’ can be thought of as being right-adjoint to the inclusion functor D~(R)~-~ D(R). Proposition 3.5.4. (i) An R-complex C is in Du(R) if and only if the natural map t(C): RF’C -~ C is an isomorphism. (ii) For all E ~ D~t(R) and F ~ D(R) the natural map e(F): RF’F .-+ F induces an isomorphism HOmD(R)(E, RF’F) --~ HOmD(R)(E, Proof. (i) "If" is clear since, as noted above, RF’C~ Du(R). As for "only if," by Corollary 3.5.2 those E E Du(R) for which ~(E) is isomorphism are the objects of a localizing subcategory L C Du(R). Now[Nml, p. 528, Thm. 3.3] says that any localizing subcategory L’ C D(R) is comp][ete!y determined by the set of prime R-ideals p such that the fraction field ap of Rip is in L~. As ~p ~ Du(R) ~:~ ap is ll-torsion ~:~ p is open, it follows that L = Du(R) only e(~p) is an isomorphismfor any such p, which in fact it is because a;p admits quasi-isomorphism into a bounded-belowcomplex of it-torsion R-injective modules, as follows easily from the fact that if an it-torsion module Mis contained in an injective R-moduleJ then Mis contained in the it-torsion injective module F~J. (ii) In view of (i), the assertion results from Proposition 3.5.3 (ii).
65
Local Cohomology and Duality
To generalize the results of §3.3--details left to the reader--one can use the next Proposition (cf. Brownrepresentabflity [Nm2,p. 223, Thin. 4.1].) Proposition 3.5.5. Let F: K(R) ~ K(R) be a A-functor, with right-derived functot RF: D(R) --~ D(R). Then the following conditions are equivalent. (i) RF commuteswith direct sums, i.e., for any small family (E~) in D(R), the natural map is an isomorphism
Rr( oE). (ii)
For any E E D(R) the natural map (2.3.1a) is an isomorphism E @ RFR --~ RFE.
(iii)
RFhas a right adjoint.
Proof. One verifies that the map (2.3.1a) respects triangles and direct sums. Hence if (i) holds then the E for which(ii) holds are the objects of a localizing subcategory E C D(R). Since R E E (easy check), therefore by [Nm2, p. 222, Lemma3.2], E = D(R). Thus (i) =~ (ii). Derived adjoint associativity ((2.2.3), with ~ the identity map of R) gives bifunctorial isomorphism, for E, F ~ D(R), HOmD(n) (E __@ urn, F) -~ Homo(R)(E, RHom’(RrR, F)). Hence (ii) ~ (iii); and the implication (iii) ~ (i) is straightforward. Weconclude this appendix with a remarkably simple characterization of derived local cohomology(Proposition 3.5.7), of which a more general form--for noetherian separated schemes--can be found in [Sou, §4.3]. Definition 3.5.6. An R-colocalizing pair is a pair (/~, e) with F a A-functor from D(R) to D(R) respecting direct sums and ~: F -~ 1 a A-functorial isomorphism (Def. 1.4.3) which is "symmetrically idempotent," i.e., the two maps /~ and t(/~) are equal isomorphisms from/~/~ to/" =/’1 = 1/’. For example, if e a : RF~~ 1 is the natural map, then (RF~, ca) is a colocalizing pair (see Corollary 3.5.2 and Proposition 3.5.3 (i)). This is essentially the only example: Proposition 3.5.7. Every R-colocalizing pair (1",~) is canonically isomorphic one of the fo~n (RF~, e~) for exactly one topology ~ = ill.. More precisely, ~ factors (uniquely, by Proposition 3.5.4(ii)) as eair where it: /’ ~ RF~ is a A-functorial isomorphism. Remarks. The set of topologies on R is ordered by inclusion, so maybe regarded as a category in which Hom(il, ~U) has one memberif il C ~ and is empty otherwise. The colocalizing pairs form a category too, a morphism(/~, ~) -+ (F’, e’) being functorial map ¢: /" -~ /~’ such that e’¢ = ~. Proposition 3.5.7 can be amplified slightly to state that the functor taking H to (RF~[, ca) is an equivalenceof categories. It follows from Propositions 3.5.7 and 3.5.5 that by associating to a colocalizing pair (/~, ~) the pair (/’(R), ~(R)) one gets another equivalence of categories, colocalizing pairs and pairs (A, ~) with A e D(R) and ~: A -+ R a D(R)-map that 1 ~ ~ and ~ __@1 are equal isomorphisms from A ~ A to A. The quasi-inverse association takes (A, ~) to the functor F(-) := - @A together with the functorial map~:=l~.
Lipman
66
Proof of Proposition 3.5.7. There is at most one/dr, since a prime R-ideal p is /d-open iff with Ip the R-injective hull of the fraction field ap of R/p, R]~Ip ~- O. Let us first construct/dr. Since/’ is a A-functor commuting with direct sums and e is A-functorial, therefore the complexes E for which FE = 0 are the objects of a localizing subcategory Lo C D(R) and the complexes F for which e(F) is isomorphism are the objects of a localizing subcategory L~ C D(R). If ~gp ~ 0 then ~(av) ¢ 0, since F~(ap) : ~ ~g p is a n i somorphism; and so the natural commutative diagram, with bottom row the identity map of ~p,
shows that Pap ~ an ~ 0. Idempotence of e gives that Fan ~ L1, whence, as in the proof of [Nml, p. 528, (1)] (with X = Pap), ap e L~. But ape L~ (resp. implies the same for Ip ([Nml, p. 526, Lemma2.9]). So we have Ifp C p’ are prime ideMs and ~Ip ~ 0 (so that Ip e L~), the naturM surjection Rip ~ Rip’ extends to a non-zero map y : I, ~ In,, and the commutative diagram
shows that Pip, ~ O. Thus those p satisfying the equivalent conditions in (*) are the open prime ideals for a topology/d =/dr on R. Now,keeping in mind that every injective R-moduleis a direct sum of Ip’s, one sees that for any injective complex E, the Ip’s appearing as direct summands(in any degree) of the injective complex F~Ecorrespond to open p’s--so that by [Nml, p. 527, Lemma2.10], F~E E L1; and that the Ip’s appearing as direct summands of E/F~Ecorrespond to non-openp’s, i.e., p’s such that ap E L0--so that by loc. cir. again, E/F~E ~ L0. From this follows that the maps ~(F~E): FF~E-~ F~]~ and Feu(E) : FF~E ~ FE are both isomorphisms. Thus e(E) factors in D(R) FE -~ F,~E ~ E , wit h i(E ) fun ctorial to the ~(~) ~ ~,(E) extent that if v : E -~ F is a homomorphism of injective q-injective complexes then the following D(R)-diagram commutes:
rE FF
~(E) ~
i(F)
F~F
u TM
E RF~F
~
F
(For the right square and for the outer border, commutativity is clear; and then Proposition 3.5.4 (ii) gives it for the left square.) Onefinds then that the q-injective resolutions qc: C --> Ec of §1.2 give rise to the desired A-functorial isomorphism it(C):
rCI~qc~ PEG i(Ec~ F~Ec
= R~C (C
D( R)).
LocalCohomology andDuality
67
4. GREENLEES-MAY DUALITY;APPLICATIONS This section revolves about a far-reaching generalization of local duality, first formulated in the 1970s by Strebel [Str, pp. 94-95, 5.9] and Matlis [Mt2, p. 89, Thin. 20] for ideals generated by regular sequences, then proved for arbitrary ideals in noetherian rings--and somewhatmore generally than that--by Greenlees and May in 1992 [GM1]. While we approach this topic from the point of view of commutative algebra and its geometric globalizations, it should be noted that Greenlees and Maycame to it motivated primarily by topological applications, see [GM2]. The main result globalizes (nontrivially) to formal schemes [DGM],where it is important for the duality theory for complexeswith coherent homology. Brief mention of such applications is madein Sections 5.4 and 5.6 below. Here we confine ourselves to the case of a noetherian commutative ring R and an ideal I C R, to which as before we associate F[, the/-power-torsion subfunctor of the identity functor on R-modules M, such that FIM = lim HomR(R/Is, M). Dually, the I-completion functor is such that ¯ ®R(R/IS)) AIM ~- lira (M These functors extend to A-functors from K(R) to itself. With 1 the identity functor, there are natural A-functorial mapsF~ -~ 1 -~ At. The basic result is that AI has a left-derived functor (§2.1) which is naturally right-adjoint to the local cohomologyfunctor RP~. In brief: left-derived completion is canonically right-adjoint to right-derived power-torsion. We know from Prop. 2.3.1 (with S = R and ~ the identity map) that RFt has the right adjoint RHom~(RFIR,-), which Greenlees and May call the "local homology"functor. So local homology= left-derived completion. Throughout §4, Horn" (resp. ®) with no subscript means Hom~(resp. ®R). Theorem 4.1. With Q: K(R) -~ D(R) as usual, there exists a unique Afunctorial map ~(F): RHom’(RF~R, QF) --+ QAIF (F K(R)) such that (i) the pair (RHom’(RFIR,-), is a l ef t-derived fun ctor of hi, and (ii) for any R-complex F the D(R)-composition F = Hom*(R,E)
p(g) ) RHom*(RFtR, F) ~ AtE via RF/-~1 is the canonical completion mapF -+ AIF. Moreover, ~(F) is an isomorphism whenever F is a q-flat complex.
For a complete proof--which plays no role elsewhere in these lectures-see [AJL]. (The generalization to formal schemes is in [DGM].)The mildly curious reader can find a few brief indications at the end of this subsection.
68
Lipman
Duality statements in which inverse limits play some role are often consequences of the following Corollary of Thm. 4.1. Twosuch consequences, Local Duality and Affine Duality, are discussed in succeeding subsections. (For more, see [AJL, §5].) We write D for D(R) and let Dc C D be the full subcategory whose objects are those R-complexes all of whose homology modules are finitely generated. (Here "c" signifies "coherent.") The I-adic compl~etion /~ of being R-fiat, we can identify the derived tensor product F __@R (§2.1) with the ordinary tensor product F ® &. Corollary 4.1.1. (i) There exists a unique functorial 0(F):
F ® ~-~
RHom’(RFIR,
map
~- RHom’(RFIR, R
)
(3.2.2)
FIF ) ( F~D)
whose composition with the natural map ~(F): RHom°(R, F) = F -~ is the map p(F) induced by the natural map RFIR -+ (ii) If F E Dc then O(F) is an isomorphism. Proof. (i) Extension of scalars gives a fimctorial/~-map k(F): F®~--~ such that k(F)~(F) is the completion map AF: F -~ AIF. Since /~ is fiat, the functor __ ® ~ takes quasi-isomorphisms to quasi-isomorpisms, so by Theorem4.1(i) and the definition of left-derived functors there exists unique functorial map O(F) : F ® [~ RHom’(RFIR, F) suc h tha t in k(F) = ~(F)O(F). Then ((F)O(F)a(F) = k(F)g(F)
=
,~F4.~(ii)~(F)p(F),
and therefore--by the definition of left-derived functors--O(F)a(F) = p(F). For uniqueness, note that a(F) induces an isomorphism RF[R @= F --~ RFrR~ (F®~). (apply the isomorphism¢(R, -) of Cor. 3.3.1, and then use Cor. 3.4.5 or just co~nbine the remarks preceding it with Prop. 3.1.2), whence the top row of the following commutative diagram ~nust be an isomorphism: HomD(F ®/~,
RHom’(RF~R,F))
HomD((F ®/~) ~ RF~R,F)
HomD(F, RH om’(RF~R,F))
~i---~+
HomD(F ~ RF~R,F)
(ii) To show that O(F) is an isomorphism whenever F E De, use the fact (nontrivial, cf. [AJL, Lemma(4.3)]) that the functor RHom’(RFtR,-) is bounded to get a reduction to the case where F is a single finite-rank free R-module[Hal, p. 68, Prop. 7.1]. In this case ~(F) (( F)O(F) is an isomorphism, whence, by the last statement in Theorem4.1, so is O(F).
69
Local Cohomology and Duality
Here is an outline of the proof of Theorem4.1. For details, see [AJL, §4]. Uniqueness of ~. Set A~F := RHom’(RFIR,F). If ~’: A~Q-~ QA~is such that (At, ~’) is a left-derived functor of At then by definition (§2.1) there is a functorial map v~: At -~ At inducing v~Q: AtQ -~ AIQ such that (’v~Q = ~; and if ~’ also satisfies (ii), so that ~’p = ~p ~’~Qp, th en p = ~)Qp. But p( F): F ~ AtF in duces a bijection from HomD(AIF,AtF) to HomD(F, AIF). (This, and other relations involving RF~ and At, all following formally from adjointness and from "idempotence" of RFt, are given in [DFS, §6.3].) Thus 0Q = identity and ~’ = ~. As for the existence of ~, one first establishes that At has a left-derived functor LAI such that for any R-complexC, with q-flat resolution Fc -~ C as in §2.1, LA,(C)A,(Fc). This is given by [Hal, p. 53, Thm.5.1], for if F is q-flat and exact then so is At(F), the lim of the surjective system of exact complexes F ® (R/I~), see lEG3, p. 66, +_.__ (13.2.3)]. (If Fs -~ R/I s is a q-flat resolution then F ® Fs is quasi-isomorphic to F ® R/I ~ and exact.) Nowwe may assume that F is q-flat. With R -> G an injective resolution (so that in D, F ® G ~ F) and s > 0, the natural map (F ® R/I s) ® Hom’(R/Is, G) ~- f ® (R/I~ ® nom’(R/I s, G)) --+ F ® corresponds under Horn-® adjunction to a functorial map F ® R/I s --~ Hom’(Hom’(R/Is, G), F ® G). So there is a natural composition, call it : LAIF --~ AIF = lira s) (F ® R/I ~ lim Hom"(Hom" (R/Is,
G), F ®G)
~ Hom’(lim Hom’(R/IS,G),
F ®
~ Horn°(FtG, F ® G) --~ RHom’(FtG, F ® G) ~ RHom’(RFIG, The essential problem is to show that (I)(F) is an isomorphism. The next step is to apply "way-out" reasoning (a kind of induction, [Hal, p. 69, (iii)]) to reduce the problem to where F is a single fiat R-module. A nontrivial prerequisite is boundedness (cf. 3.1.4) of the functors LAt and RHom’(RFtG, Then F -) F ® G is an injective resolution (so that ~ is an isomorphism). With t = (tl,... ,tm) such that I = tR, one uses that K:(t) = li__~ms>0 of the ordinary Koszul complexes K(t s) = K(t~,. , m) (defined by replacing R --> Rt in §3.1 with R t/~ R, the maps K(t u) --~ K(tv) (v >_ u) being derived from the maps complexes K(tu) ~ K(tv) which are identity in degree 0 and multiplication by t in degree 1) to turn the basic problem into showing for all i that the’natural map is an isomorphism H~RHom"(RF~G, F)~.e i ! imHom°(K(t~), EGG) - -~ l imHiHom°(K(t~), F®G). (This is used to show that a certain map ~I,(t, F): RHom’(RF~G, F) --~ LA~F pending a priori on t is an isomorphism. One must also show that ¢ = ~I’(t, F)-I.) Treating such questions about the interchange of homologyand inverse limits requires somenontrivial "Mittag-Leffier conditions," see lEG3, p. 66, (13.2.3)].
70
Lipman
4.2. Application: local duality, again. In §2.4, Greenlees-Ma~y duality was used to relate a form of classical local duality (2.4.2) to "~h,~ivial"’ local duality (2.3.2). More directly (and more generally), for E E D(R) F E De(R), and with /~ the I-adic completion of R, apply the functor RHom°(E,-) to the isomorphism in Corollary 4.1.1, and then use the isomorphisms(2.2.2) (with R = S, ~ = identity) and (3.3.1) to get a nal~ural isomorphism RHom’(E, F ®/~) --~
RHom’(RFIE, -~ RHom’(RFIE, RFI F). (3.2.2)
4.3. Application: affine duality. a natural D(R)-map
For any R-complexes F and G there is
a(F, G): F --~ RHom’(RHom’(F,G), corresponding via (2.2.3) to the natural composition F @= aHom’(F, G) _L~ aHom’(F, G) @ F _2+ °(F, G), and where ~/ corresponds via (2.2.3) to the identity map of RHom ~- is the map(clearly an isomorphism) determined by the following property: replacing F by a q-fiat resolution and G by a q-injective resolution, one can ’, G), change ~ to ® and drop the R’s, and then for x ~ Fi and ¢ ~ HomJ(F T(X ® ¢) = (--1)iJ(¢ ® X). (Proving the existence of such a ~---by e.g., of the general technique for constructing functorial maps in derived categories given in [Lp3,Prop. (2.6.4)]--is left as an exercise.) With ¢ = (¢n : Fn --~ Gn+J)neZ, we have then
G)(x)](¢) Let D be a bounded injective R-complex such that for any F ~ Dc(R), a(F, D) is an isomorphism. For example, D could be dualizing co mplex ([Hal, pp. 257-258]), which exists ~6 R is a homomorphic image of a f in itedimensional Gorenstein ring [Hal, p. 299]. Define the I-dualizing functor by T),(F) := RHom’(F, ariD) (F E The following result "double-dual=completion" is called Affine Duality. ([Ha2, p. 152, Thm. 4.2]; see also [DFS, p.28, Prop. 2.5.8] for a formalscheme-theoretic version). Theorem 4.3.1. Let ~ be the I-adic functorial isomorphism
completion of R.
Then there is a
F ®/~ --~ T)~:D~F (F ~ De(R)) whose composition with the natural map F ~ F ®Rf{ is a(F, RF~D). 16and only if--[Kwk,
Cor. 1.4].
LocalCohomology andDuality
71
Example. WhenR is local with maximal ideal I and D is a normalized dualizing complexof R then RFIDis an R-injective hull of the residue field R/I (see §2.4), and Theorem4.3.1 is a well-known componentof Matlis Duality [BS, p. 194, Thm.10.2.19(ii)]. Proof of Theorem 4.3.1. One checks (see below) that a(F, RFID) is following natural composition: F -~ F ®R ~ --~ RHom’(RFrR, F) (4.1.1) °~ RHom RF~ R, RHom"(RHom"(F, D), via a °--~ RHom RFIR ~= RHom’(F,D), D)) (2.2.2) °-~ RHom RF, RHom ° (F, D), D)) (3.3.1) °~ RHom RFIRHom" ( F, D ), RF~D ) (3.2.2) via u RHom°(RHom’(F, RFID), RFID)) = 7)fl)zF where ~, is the isomorphismgiven by: Lemma4.3.2.
There is a unique map ~,:
RHom’(F, RFID) ~ RFIRHom’(F, D)
whose composition with the natural map RFrRHom’(F, D) -~ RHom’(F, is the map induced by the natural map RF~D-~ D; and this ~ is an isomorphism. Proof. By Prop. 3.1.2, RF~Dis D(R)-isomorphic to a complex K:(t) which is bounded and injective; and hence (4.3.3)
RHom’(F, RFID)"~-Hom’(F,I~(t)®D)eDx(R),
as one sees by "way-out" reduction to the simple case where F is a finiterank free R-module[Hal, pp. 73-74, Prop. 7.3]. Then Prop. 3.2.2 ensures the existence of u. For ~ to be an isomorphism it suffices that for an arbitrary A E Di(R), the image of ~, under application of the functor HOmD(R) (A, be an iso morphism. By (2.2.3) and Prop. 3.2.2, this amounts to the natural map HOmD(R) (A @__F, RFID) --~ HomD(R)(A ~ F, being an isomorphism,so, by Prop. 3.2.2, it suffices that A @=F E D~(R),i.e., (Cot. 3.2.1) that the natural mapRFI(A =@F) -+ A =@F be an isomorphism, which it is, by Cor. 3.3.1, since RF~A~ A (Cor. 3.2.1, again). The patient reader mayapprehendmore of the functorial flavor of our overall approachby perusing the following details of the check mentionedat the outset of the proof of Theorem4.3.1.
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Lil~,man
Consider the following natural diagram, in which 7) is the dualizing functor RHom°(-, D) and the functorial map 7)I ~ I) is induced by the canonical map RF~ -> 1, as are the horizontal arrows preceding the right column, which along with the top row is as in the sequence of maps near the beginning of the proof of Theorem 4.3.1. F
~
RHom’(R,
RHom°(R,
(h)
F)
7)~F)
RHom°(R __@ 7)F,
RHom"(:DF, D)
~ RHom"
--~
(RFIR,
RHom° (RFIR,
7)7)F)
D) --~ RHom°(RFIR DF, D)
--~
RHom" (RFIT)F,
0
The unlabeled squares obviously commute. To verify commutativity of subdiagram (A) one checks (exercise) that the isomorphism (2.2.2) for E = S = ~o = identity is naturally isomorphic to the identity map of RHom~(F,G). Commutativity of (B) follows from Corollary 3.3.3. Commutativity of (C) follows Lemma4.3.2, as one sees by drawing the arrow induced by u from the upper right to the lower left corner. Thus the whole diagram commutes. Since DIF E DI(R) (see (4.3.3), Proposition 3.2.2 gives that the mapt) diagram is an isomorphism. It remains only to show that the left column followed by 0-1 is a(F, RF~D), and this is straightforward.
5.
RESIDUES AND DUALITY
This section begins with a concrete interpretation of the duality functor ~ff of Theorem 2.3.2, for ~o the inclusion of a noetherian commutative ring R into a power-series ring S := R[[t]] := R[[t~,... ,tm]] and J the ideal tS = (t~,... ,tm)S. The resulting concrete versions of Local Duality lead to an introductory discussion of the residue map, its expression through the fundamental class of a map of formal schemes, and hence to canonical versions of, and relations between, local and global duality--at least for smooth residually separable maps. Henceforth we omit "~," from the notation for derived functors when the context makes the meaning clear. For example, for G ~ D(R) we ’write RHom~(RFjS, G) in place of RHom~(~, RFj S, G), and G @Rwt [m] in :place of G ~R ~,wt[m].
LocalCohomology andDuality
73
5.1. The duality functor for power series rings. and J = tS are as above. Wefirst give some concrete representations of the duality functor ~o~: D(R) -~ D(S) (see Theorem2.3.2). Using the definition of the stable Koszul S-complex]C(t) (§3.1), one finds that ~t~ := Hm/c(t)= c°ker[K:m-1(t) = @~=1S~l[2...ii...~ "~ S~li(:2’"~m : ](:m(t)] m
is a free R-module with basis {t~ -ul... t~n nm [ nl > 0,...,nm > 0}, and an S-submodule of S~I~2...tm/S. Since the sequence t is regular, /~(t) exact except in degree m lEG3, p. 83, (1.1.4)]. Hence by Propositions 3.1.2 and 1.3.3 there are natural D(S)-isomorphisms
arjs
(5.1.1)
-m -m
and so there is a functorial (5.1.2) ~G= RHom~(RFjS,
D(S)-isomorphism
G) --~
¯ RHOmR(t[-m],G) b’
(G e D(R)).
Since ~t is R-free the functor Hom~(pt[-m],-) preserves exactness, and so takes quasi-isomorphisms to quasi-isomorphisms (as quasi-isomorphisms in K(R) are just those maps whose cones are exact), so that it may be regarded as a functor from D(R) to D(S). Replacing G in (5.1.2) quasi-isomorphic q-injective complex, we see then that the canonical mapis a functorial D(S)-isomorphism (5.1.3)
Hom~(-t[-m],
Thus we have a functorial (5.1.4)
G) -~ RHom~(~t[-m],
D(S)-isoraorphism
~G -~ Hom~(~t[-m],G)
(G ¯ D(R)).
Here is another interpretation of ~G, for G ¯ De(R) (i.e., modules of G are all finitely-generated). Set
the homology
wt = ~ := SomR(-tS, R),
(5.1.5)
a "relative canonical module." This wt is a free rank-one S-~nodulegenerated by the R-homomorphism7t: Pt -~ R such that nm=l, otherwise. {10 ifnl ..... That’s because the map (~n~>0 rn~...n~t~ 1-~""" ,n~-l~ ~m )’~’t . takes t~~ . . . n’~ t~ (5.1.6)
~t(t~-nl"""
t~nnm)
=
to rn~...n m.
For any R-complex G there is a unique map of S-co~nplexes
Xm(G): G ®R wt[m] --> Hom~(ut[-m], whose degree-n componentX~nsatisfies
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Lipman
Since wt is S-flat, the functor ... ®Rwt takes quasi-isomorphisms to quasiisomorphisms, so may be viewed as a functor from D(R) to D(S), and Xm(G) is a functorial D(S)-map. "Way-out" reduction to the trivial case where G is a finite-rank free R-module ([Hal, p. 68, 7.1(dualized)], with A~ C A := Ad(R) the category of finitely-generated R-modules), shows that for G E Dc(R), Xm(G) is a D(S)-isomorphism. In conclusion, for G E De(R) we can represent 99]G concretely via the functorial D(S)-isomorphisms (5.1.7)
99~G ~ Hom~(vt[-rn],G) (5.1.4)
---1 ) G ®R wt[rn]. Xm(G)
5.2. Funetors represented via relative canonical modules. We continue with a nontrivial instantiation of Trivial Local Duality (2.3.2). Set, as above, wt := HomR(~t,R), so that there is an "evaluation" map ev: wt ®s/]t ~-} R. Moreover, t~t being R-free, if F is a finitely-generated natural map is an isomorphism (see also above)
R-modulethen the
(5.2.1)
X0(F): F ®R wt = F ®RHomR(b’t, R) ~ HomR(b’t, The local cohomology functor H~n on the category J~(S) of S-modules can be realized through the functorial S-isomorphism (5.2.2)
~t(E):
H~E -~ E ®s "t (E 6 A4(S)),
defined to be the composition H’~E = HmI:tFjE -~ Hm(E ®s RFjS) -~+ Um(E ~=s pt[--m]) (3.3.1) =
:
E ®s ~’t.
Via (5.2.1) and (5.2.2), the natural isomorphism HomR(E®R~t, F) --~ Horns(E, HOmR(~t, (see (2.2.1)) gets transformed into the following down-to-earth duality, whose substance comes then from Proposition 3.1.2 and the structure of Hm~(t). (Details are left to the reader.) Insofar as this duality involves a choice power-series variables t it lacks canonicity, a deficiency to be remedied in Theorem 5.3.3. Proposition 5.2.3. For any finitely-generated torial isomorphism
R-module F there is a func-
HomR(H’~E,F) -~ Homs(E,F ®Rwt) which for E--F ®R wt takes the composite map ~/t(F)
H~n(F ®R wt) ~ >
(E e M(S))
F ®Rwt®S "t ~ F l®ev
et (F®R wt)
to the identity map of F®Rwt. In other words, the functor HomR(H~E, of S-modules E is represented by the pair (F ®Rwt, */t(F)).
LocalCohornology andDuality
75
Complement.Bymeansof 3.4.5 and 3.4.3, Proposition 5.2.3 extends as follows (exercise). Let T be an R-algebra, u := (Ul,..., Um)a sequence in T, I := uT, ~ the I-adic completion of T, and fi = (t~,...,tim) the image of u in ~. ~ is S(= R[[t]])-algebra via the continuous R-homomorphism taking ti to ~i for all i. As above, set J := tS, so that for any ~b-moduleE considered as a T-moduleand S-module, respectively, H~IE= H~E. Let ev’: Homs(~, F ®/t wt) ~ F ®Rwt be the S-horaomorphism"evaluation at 1." Thenfor any finitely-generated R-moduleF, the functor HomR(H~’1E, F) of T-modulesE is represented by the pair (Homs(~,F ®Rwt), tit(F)oH~(ev’)). The next Proposition provides a canonical identification of the duality isomorphism of Proposition 5.2.3 with the one coming out of Theorem2.3.2, namely HomR(H~E,F) ~-> HomD(s)(E, ~F[-m]). Proposition 5.2.4. For any S-module E and any R-module F the following sequence of natural isomorphisms composes to the map given by (2.2.1): HomR(E ®~ ~t,
F) ~ HomR(H~nE, (5.2.2)
(1.3.3) (2.3.2)
HomD(R)(RFjE,F[-m]) (see Cor. 3.1.4) HomD( s) ( E, ~ F[- m]
(5a.4) HomD(s)(E,Hom~(ut I-m], F[- m]) HOmD(s)(E,Hom~(,t, Horns (S, HOmR(ut,F)). (1.3.3) Proof. The proof, left to the reader as an exercise in patience, is a matter of reformulating the assertion as the commutativity of a certain diagram, which can be verified by decomposing the maps involved into their elementary constituents, as given by their definitions, thereby expanding the diagram in question into a patchwork of simple diagrams all of whose commutativities are obvious. [] 5.3. Differentials, residues, canonical local duality. Let ~s/n be an Smodule equipped with an R-derivation d: S -+ ~S/R such that (dtl,..., dtm) is a free S-basis of ~ts/R. Then for any u = (Ul,U2,... ,urn) such that S = R[[u]], it holds that (dul,..., dum) is a free basis of ~/R" This follows e.g., from the fact that the pair (~ts//~,d) has a universal property which characterizes it up to canonical isomorphism: for any finitely-generated Smodule M and R-derivation D: S ~ M there is a unique S-linear map 5: ~S/R~--~ Msuch that D = 5d. Let ~m (m > 0) be the m-th exterior power of ~ts/t~ , a free rank-one S-modulewith basis dtl A dr2 .. ¯ A dtm. Let Ct : ~rn --~ wt be the isomorphismwhichtakes dtl A dr2 ¯ ¯ ¯ A dtm to the generator 7t of wt (see (5.1.6)).
Lipman
76 Let rest
be the composition
(5.3.1)
H~r~
m ~
H~swt
~
(5.2.3)
R.
For any u as above, resu is similarly defined. Moreover, if 0 is a bicontinuous R-automorphism of S (t-adically topologized) and u = 0t, then H~ = Hums (see remark following Corollary 3.4.5). Proposition 5.3.2. The R-linear map rest: H~m --> R depends only on the R-algebra S = R[[t]] and its t-adic topology: if a bicontinuous Rautomorphism of S takes t to u (so that S = R[[u]], the t-adic and u-adic m topologies on S coincide, and H~ = Hus ) then rest = resu. The proof of this key fact will be discussed below. In summary, there is given a complete topological R-algebra S having an ideal J such that: (i) The topology on S is the J-adic topology, and (ii) J generated by an S-regular sequence t = (tl,..., tin), and (iii) the natural map is an isomorphism R -~ S/J. It follows that the continuous R-algebra homomorphism from the powerseries ring R[[T1,..., Tm]] to S taking Ti to ti (1 < i < m) is an isolnorphism. Then the S-module ~m and the local cohomology functor H~n depend only on the R-algebra S and its topology, as does the R-linear residue map ress/R :: rest:
H~n~m -+ R.
This being so, and by the definition (5.3.1) of rest, Cor. 5.2.3 gives the following canonical version of local duality for power-series algebras: Theorem 5.3.3. In the preceding situation, the functor S-modules E is represented by the pair (~m, reSs/R).
HomR(H.~E, R) of
Remark. Again, J = tS. Recall that the stable Koszul S-complex ]C(t) is the direct limit of ordinary Koszul complexes K(t~’,..., t~nm) (cf. paragraph immediately preceding §4.2). So we can specify any element of
= n~,,~ (3.1.2)li___~m
n’~K(t~ ,...,t,~
,~’~)
by a symbol (non-unique) of the form It~
tnm := t~n~,...,nmTrn~,...,nmLt ~’’’, m J
for suitable u fi ~mand positive integers nl,..., am, with ~ and ~ the natural maps ,...,t,~ , ~m) _. H,~K(t~,,...,t,~ ,[~m), (5.3.4) ~,~,.....n,. : HmK ( t~ ¯ .., t,~ , Then, recalling that Cry ~ wt = HomR(vt,R) and that t~- ...... ress/~
t~,...,t~ ~
t~n"~ ~ vt, we get
LocalCohomology andDuality In particular, since (5.3.5)
ress/R
Ctdtl"
77 ""
dtm="Yt wehave
nm=l, [dtl...dtml ~1 if nl ..... [t~, t~/ = , ma[0 otherwise.
Whenm = 1, H~~ is the cokernel of the c~nonical map ~ ~ ~ (localization w.r.t, the powers of t:= t~), and [~] = r(dt/t ~) with ~: ~ ~ g~~ the natural map. Then(5.3.5) yields the formula resn[[~ll/~ ~((~0 r~t~)dt/ whichhas an obviousrelation to the classical formulafor residues of one-variable meromorphicfunctions. Exercise. (i) Using Prop. 3.1.2, or otherwise, establish for R-modulesF and modulesG a bifunctorial isomorphism ~(F,G):
F@~H~(G) ~ H~(F~G)
such that, with notation as in Proposition 5.2.3, 5t(F @RWt) o~(F, wt) 1 ~net( wt): F (ii) Showthat for any finitely-~enerated R-moduleF, the functor Hom~(H~E, of S-modulesE is represented by the pair (F @n~, (1 @ress/~) o~(F, ~-~)). Next, let ~: R ~ S be any fiat (hence injective) local homo~norphism complete noetherian local rings with respective maximal ideals m and such that S/mS is a Cohen-Macaulaylocal ring with residue field S/~ finite over Rim. Then any sequence t := (t~,..., tm) in S whose image in S/mS is a system of parameters is S-regular, and P := SItS is a finitely-generated projective R-module. (See [EG4, p. 18, Prop. (15.1.16)]) and [ZS, p. Cor. 2].) After ~(R) is identified with R, it follows that the R-homomorphism from the formal power-series ring R[[T~,...,Tm]] to S taking T~ to ti is an isomorphism onto R[[t]] C S, and that S is R[[t]]-module-isomorphic to P @aR[[t]] (see [Lp2, ~3]). To such a ~ there is ~ssociated a finitely-generated S-module~ together with ~n R-derivation d: S ~ ~ which has the universal property that for any finitely-generated S-module M, composition with d maps Homs(~, M) bijectively onto the S-moduleof R-derivations frown S into M(see [SS, ~1]). There is also a $race map ~m ~m T: A~ :: ~ ~ ~R[[tl][R , see [Knz, ~16], [Hfi, ~4]. The definition of this mapis somewhat subtle. However, in the special case when ~ = mS + tS ~nd in addition S/~ is finite separable field extension of Rim (so that S is formally smooth over R lEG4, p. 102, (19.6.4) and p. 104, (19.7.1)]), ~nd P is a finite fl~t unramified (= ~tale) R-algebra, it follows e.g., from lEG4, p. 148, (20.7.6)] (5.3.6)
~ ~ S @R[[t]l ~R[[t]I/R ~ P
a free S-modulewith basis (dt~,..., dtm). (In other words every R-derivation of R[[t]] into a finitely-generated S-moduleextends uniquely to S.)
78
Lipman
So
~R[[t]]/a’ by the usual trace map tr: = @R
Now define H~m
~m
Proposition
Rest:
5.3.2’.
T becomes the map induced
m^m
H~fl~ -+ R to be the composite
natural ~t +
and correspondingly P --~ R.
R[[t] ~ts~
~m~m ]~ (3~3)
_
~m
~m
~
map
~m m H~R[[t]]~R[[t]]/R
~ (5.3.1)
R.
This map Rest does not depend on the choice of t.
Thus we have a residue
map Res~
m^m
: H~ -+ R.
There are several approaches to the proofs of Propositions 5.3.2 and 5.3.2’. For 5.3.2, the most elementary one, bru’~e-force calculation, is rather tedious (cf. e.g., [Lpl, pp. 64-67]), and not particularly illuminating. It is more satisfying first to find an a priori intrinsic definition of the residue map, and then to show that it agrees with the above one. For example, such a definition via Hochschild homology is the foundation of [Lp2]. (See [ibid., §4.7], or [Hii, §7], for the connection between residues and traces.) Another, richly-textured, intrinsic approach is undertaken in [HiiK]. In fact Hiibl and Kunz prove Theorem 5.3.3 in a more general situation., for maps R -~ S factoring as R -~ R[[tl,...,tm]] I_~ S with f a finite generic complete intersection. In such a situation, it is easy to generalize Corollary 5.2.3, with the representing object wt replaced by HOmR[[t]](~’, cot); but the trick is to find a canonical representing object, not depending on t. For this Hiibl and Kunz use the module of "regular differential forms," constructed via the theory of traces of differential forms. For example, if ~ : R -+ S as above makes S formally smooth and residually separable over R then the trace ,nap tr: P -+ R gives rise, via (5.3.6), ^m an R[[t]]-isomorphism ~ ~ HOmR[[t]](S, ~la[[t]]/R). In the non-separable case the same isomorphism obtains by means of the general trace map for differential forms. There results a canonical local duality theorem for formally smooth local algebras: Theorem 5.3.3’. /f ~: (R,m) -~ (S, if)l) is a formally smooth local homomorphism of complete noetherian local rings making S/9~ finite over and m := dimS/m~S, then the functor HomR(H~E,R) of S-modules is represented by (~’~, Rest). We will now outline yet another approach to residues, which is perhaps the "least elementary," but has the advantage of connecting immediately with the global theory of duality on formal schemes [DFS], through the fundamental class of certain flat maps of formal schemes. There result canonical realizations of, and relations between, local and global duality, summarized by the Residue Theorem. The introductory discussion here will be confined to smooth maps.
LocalCohomology andDuality
79
5.4. Flat base change. Our definition of the fundamental class makes use of a basic property of duality, having to do with its behavior under flat base change, (Proposition 5.4.2). Henceforth ring homomorphismswill be continuous maps between noetherian topological rings, mostly adic. That is, we work in the category of pairs (R, I) with R a commutative noetherian ring and I an R-ideal such that R is complete and separated with respect to the I-adic topology, morphisms ~o: (R, I) --~ (S, J) being ring homomorphisms ~o: R -+ S such ~o(I) C x/7. (Pairs (R, It) and (R, I2) are considered identical if I1 define the same topology, i.e.,vOT~ =x/~-~.) For such for the functor ~o~ of Theorem2.3.2, because it depends only on the J-adic topology, whichis a part of (the target of) Consider then a coproduct square in this category, i.e., a commutative diagram of morphisms (R, I) __5__+(S,
(g, ~) ~ (~, such that the resulting mapinto V from the complete tensor product S ~R U (the completion of V0 := S ®RU with respect to Mo := LVo + JVo) is an isomorphism, and where M := LV + JV. (For simplicity we proceed as if V0were noetherian. Usually this is not so, and a more complicated approach is needed, cf. [DFS, p. 76, Definition 7.3; p. 86, Theorem8.1].) Let ~: V0 -~ V, ~0: U -+ V0, and ~0: S -+ V0 be the natural maps, so that ~ = ~0 and ~, = ~0" Suppose #,. hence ~0 and z~, to be flat. Then the functor ... ®RU from R-modules to U-modulesis exact, so takes quasiisomorphisms to quasi-isomorphisms, and consequently extends to a functor /~*: D(R) --~ D(U) (cf. §3.4). Similarly we have ~: D(S) --~ D(V0) ~* = ~*~: D(S) -+ D(V). For any A-functor F: K(R) -~ K(S), K(R)-quasi-isomorphism C -~ Ec with Ec q-injective, there is an isomorphism ~Rr(C)-~ .~r(Ec); hence .~Rr: D(R) --+ D(V0) is a right-derived functor of F(-) ®RU: K(R) -~ K(V0)(see The base-change map/5: ~*~o~ -> ~##*, that is, the functorial map fl(G):
u*RHom~(RFjS, G) -+ RHom~(I=tFMV , #*G) (G (D(R)),
is defined as follows. First, as noted above, t,~l:tHom~(RFaS,-) is a right-derived functor Hom~(RFaS,-) ®RU; so by the characteristic universal property of rightderived functors (§1.5), there exists a unique functorial map/~’(G) making
80
Lipman
the following otherwise natural D(V0)-diagram commute: Hom~(RFjS, G) ®R U ------+
u~RHom~(RF~S,G)
Hom~(RFjS ®R U, G ®R U)
~ RHom~](~RF~S,
#*G);
and the natural composition
RrMoV0 RrjvoV0(3.4.4) . RFjS combines with ri’(G) to give a functorial map rio(G):
,~*G ,~ RHom~(RrjS, G)~ R Homb(RFMo Vo, ,*G) = ~*G
Second, for any F ~ D(V0), we have a natural isomorphism n.RFMn*F
~ n.n*RFuoF. (3.4.4)
Also, the natural map is an isomorphism RFMoF~ ~,~*RFMoF: to verify this, since the functors ~, and ~* are both exact and isomorphism means "homology isomorphism" (~1.3), we can replace RFMoFby its homology, and then the assertion follows because the homology is M0-powertorsion (see ~3.2). The resulting composition ~,RFM~*F~ RFMoF~ F is dual to a map(see 2.3.2) (5.4.1)
~(F): n*F ~ n*F.
Finally, fl(G) is defined to be the composite map
=
~* (Zo(a))
4~’a)
(2.~.~)
Let D+(R) (resp. D-(R)) be the full subcategory of D(R) with those complexes G whose homology HiG vanishes for i << 0 (resp. i ~ 0). The full subcategories D~ (R) and D~(R) of Dc (R) are defined similarly. before, Dc(R) C D(R) is the full subcategory whose objects are complexes having finitely-generated homologymodules.) Theorem 5.4.2 (Flat Base-Change). In the preceding situation, if S/J is (via ~) a finite R-module and G Dr(R) th en ~( G) is an iso morphism. Proof. (Outline.) The finiteness of S/J over R means that ~ = ~2~1 where ~1: R ~ R[[t]]:= R[[tl,... ,tm]] is the natural mapof R into a power-series ring, whichis completefor the I ~ := (I, t)R[[t]]-topology, and ~2 : R[[t]] .~ is such that J = I~S, so that ~2 makes S into a finite R[[t]]-module having the I~-adic topology ([ZS, p. 259, Cor. 2]). A readily-established transitivity property of the base-change map fl then reduces the problem to the two cases ~ = ~ and T = ~2.
81
Local Cohomologyand Duality
When~ = ~1 then ~ is the natural mapU -~ U[[t]], and so (5.1.7) reduces the problem to identifying/~(G) with the natural isomorphism (notation in (5.1.5)) t,
an exercise in unraveling definitions. When~ = ~2, i.e., S is a finite R-moduleand J = IS, then Vo = S®RUis a finite U-modulewith L-adic topology, and so is complete, i.e., V -- V0. Now Greenlees-Mayduality enters crucially to yield, via (2.4.1), an identification of/~(G) with the natural map ~3~(G) : RHom~(S, G)®RU-~ RHom~(V, G®RU) = RHom~(S®R U, G®RU) whoseexistence is shownsimilarly to that of fl’ (see above). That fl~(G) is an isomorphism for any G ¯ D+ (R) becomes clear upon replacement of by a (finite-rank) R-projective resolution P, in view of the simple fact that the functor Hom~(P,-) takes quasi-isomorphisms to quasi-isomorphisms, fact whoseapplication to an injective resolution of G shows that RHom~(S, G)- RHo~n~(P, G)
"~= HOmR(P ," G),
and similarly (since U is R-flat) RHom~(S ®R U, G ®R U) ~ RHom~(P ®R U, G ®R -~ Hom~(P®RU, G ®RU).
5.5. Residues via the fundamental class. Specialize square (n,I) ~ (S,J)
(S,J)
now to a coproduct
~ (V,M)
with ~ fiat, and S/J finite over R--so that Theorem5.4.2 is applicable. Let 5: V -- S ~n S -~ S be the continuous extension of the map S ®RS -~ S taking sl ® s~ to SlS~. Let n : S ®RS --+ V be the completion ~nap, so that u(s) = ~(s 1)and~(s) = n(1® s) ( s ¯ S ), and le t L be t heV-ideal(closed, since V is assumed noetherian) 17 generated by all the elements ,(s) - ~(s). For any f ¯ V, f - ~Sf ¯ L (check this first with V replaced by its dense subring n(S®RS), then pass to the limit); and it follows that L is the kernel of 5. One shows then that the S-module ~ := L/L ~ together with the Rderivation d: S -~ ~ such that d(s) = ~,(s) - ~(s) (mod ~) f or a ll s ¯ S is universal (cf. §5.3) for R-de~vat~onsof S ~nto fin~tely-~ene~ted S-modules. ~In fact the noetherianness of V ~ollows from that of R and S plus the R-finiteness
~/J laD, ~. 414,(10.6.4)].
of
82
Lipman
Let m be the least non-negative integer all i > m (see Corollary 3.1.4). Then
such that
HiS = HiRFjS = 0 for
H-i~R = H-iRHom’n(RFjS, R) = 0 (i > Set ~m = ~ := A~t~. The fundamental linear map (5.5.2)
class
of ~o is a canonical
S-
f~o: ~m _.~ w~o :: H-m~#R,
defined with the assistance of flat base-change, as follows. Since 5~ = 5~, = ls, we have, clearly, ~,5, = 1D(S); and with 5" as §3.4, there is a natural isomorphism 5*u* --- 1D(S). There results a natural D (V)-composition (2.3.2)
(3.4.3)
-+ ~#S -~ ~*R ~-~ u*~#.R, (5.4.2)
to which application
whence a natural (5.5.2)
of 5* gives a natural
D(S)-map 5"5,S -~ 5* ~ * ~ R ~- ~# R,
map
TorYm(S, S) = H-ms*5,S H-m~’R = w~ .
Now with L = ker(5)
as above, there
is a natural
isomorphism
~ = L/L 2 ~- ToqY(s, S). Moreover, $i>0ToqY(s, S) has a canonical alternating ture (for which the product arises from the natural
graded-algebra maps
struc-
H~(S @__vS)®v H~(S@vS)--+ H~+~((S~= vS) @__v(S @=vS)) H~+J(S where p is induced by two copies of the composition of the natural maps S @=vS~ S ®y S -~ S). The universal property of exterior algebras gives then a canonical map (5.5.3) The fundamental class [~:
~m-~ WorVm(S,S). W~ ° is the composition of (5.5.2) and (5.5.3).
~m __~
We can now define the R-linear formal residue map p~: H~n~TM --~ R to be the canonical composition (where the unlabeled map comes from a dual form of Proposition 1.3.3): p~: n~n~m = H°RFj~m[m] ~ H°RF~,w~[m] --~ H°RF~o#R(2.3.--~D I-[°R via [
:: R.
The local Residue Theorem states that under the conditions considered in §5.3, the formal residue map is the same as the residue maps defined ti~ere. As the formal residue depends only on ~, Theorems 5.3.3 and 5.3.3 ~ result.
Local Cohomologyand Duality
83
A complete proof of the local Residue Theorem will appear elsewhere. For the case whenS = R[[t]] is a power-series R-algebra all the necessary definitions have been spelled out, so no further new ideas are needed, just painstaking work. For example, for connecting the "abstract" formal residue p~ with the "concrete" residue rest, one needs commutativity of the diagram HORFjwt[m]natural ~ H°RF~Uom~(~t[-m],R)
~
H°RF~#R
1(2.3.1) wt ~S ~t
>
evaluation
which can be seen by detailed consideration of Proposition 5.2.4 with E = wt and F = R. A full treatment involves more about the relation between fundamental classes and tr~ces of differential forms. Consider, for example, a pair of continuous maps with ~ the canonical map, and T a finite R[[t]]-module (via ¢). ~om(2.4.1) we find that the integer m used to define [¢~ is the same as that used for [~ (namely, the number of variables in t). There is then, by the abovementioned dual form of Proposition 1.3.3, a natural map w¢~ := H-m(¢~)~R ~ (¢~)~R(~3) and as pnrt of the proof of the local Residue Theoremone needs: Theorem 5.5.4.
The fundamental class [~ is the composite isomorphism ~m
~ ~ wt ~ H-m~*R =: w~. Ct (~.~.~) So there is a unique S-linear map ~ making the following D(S)-diagram commute:
T 1
(2’3’2)1(2"4’1)
and this ~ coincides with the trace mapfor differential
forms.
5.6. Global duality; the Residue Theorem. This culminating section introduces the connections between residues and global duality theory on noetherian formal schemes. A key advantage of working in the category of formal schemes~rather than its subcategory of ordinary schemes~is that local and global duality then becometwo aspects of a single theory. Wefirst set up some notation and briefly review necessary background material. (The prerequisite basics on formal schemesare in [GD, Chap. I, ~10].)
84
Lipman
Let 27 = (1271, (-gx) be a noetherian formal scheme, with ideal of definition 3. ([27] is a topological space and 03: is a sheaf of topological rings.) Let A(37) be the abelian category of C0x-modules, and D(27) the derived category of A(2:). Let Dqc(%) C D(%) (resp. De(%) C D(27)) subcategory with objects those A(27)-complexes whose homology sheaves are quasi-coherent (resp. coherent), i.e., locally cokernels of maps of free (resp. free, finite-rank) (93c-module s. In D... (37) the homologicallyboun.dedbelow complexes--those £ whose homology sheaves Hi£ vanish for i <<: 0are the objects of a full subcategory denoted by D.. +. (X). The torsion subfunctor _P:~ of the identity functor on A(27) is given F~£ = lim 7-lom~((gx/3S,£)
(£ ¯ ‘4(27)).
F:~ depends only on 27, not 3. It has a derived functor RF:~: D(27) ~ D(27), which satisfies RF~Dqc(27)C Dqc(27) [DFS, p. 49, Prop. 5.2.1(b)]. To any noetherian adic ring R--i.e., R is a complete noetherian topological ring with topology defined by the powers of some ideal/--is associated an affine formal scheme Spf(R), whose underlying space is the same as that of the ordinary scheme Spec(R/I). Any noetherian formal scheme 37 has a finite open covering by affine formal schemes, with structure sheaves obtained by restricting (gx. Continuous maps 9~: R ~ S of noetherian adic rings correspond bijectively to for~nal-scheme maps ~: Spf(S) --> Spf(R); and any map f: 3: --~ ~ of noetherian formal schemes is locally of this form. The direct image functor f.: ‘4(3:) --~ ‘4(~) has a right-derived functor P~f.: D(37) --~ D(~); and the inverse image functor f*: ,4(~) has a left-derived functor L f*: D(~) For any such f : 3: --~ t3, there are ideals of definition 3 C (.9~ and ~J C COx such that ~(93c C ~J [GD, p. 416,(10.6.10)]; and correspondingly there is a of ordinary schemesf0: (IX], O3c/3) -~ (l~l, 0~/3) [GD, p. 410, (10.5.6)]. say f is separated (resp. pseudo-proper) if f0 is separated (resp. proper), condition independent of the choice of (~, 3). For example, a map~ as above is pseudo-proper iff S/J is, via 9~, a finite R-modulefor some (hence any) S-ideal J defining the topology of S. Wesay f is proper if f is pseudo-proper and for some(hence any) ~1, ~Oxis an ideal of definition of 37. Wesay f is fiat if it is locally ~ for some ~o as above making S a flat R-module. For flat f the functor f*: A(t~) -+ .4(27) exact (se e [DFS, p. 72, Lemma7.1.1]), so may be thought of as a functor from D(~) naturally isomorphic to L f*. One has then the following globalizations of Local Duality (Theorem2.3.2) and Flat Base-Change (Theorem 5.4.2). (Despite the obvious formal similarities, however, fully elucidating the connection between the global and local versions requires more than a little work.) Wenote in passing tha.t for proper maps Greenlees-Mayduality plays a basic role in the proofs.
85
LocalCohomology andDuality
Theorem5.6.1 ([DFS, p. 64, Cot. 6.1.4; p. 89, Thin. 8.4]). If f : T~ ~ ~ is a separated map of noetherian formal schemes then the functor
Ry,RF:
RF:~-t(Dqc(X)) --~ D(t3)
has a right adjoint f#. Moreover, if f is proper (hence separated) then this f# induces a right adjoint for R f,: D:(X) Theorem5.6.2 ([DFS, p. 89, Cot. 8.3.3]). diagram of noetherian formal schemes
Let there be given a commutative
with the i~d~eed map V ~ ~ x~ ~ an isomorphism, f (hence 9) pseudoproper (hence separated), and ~ (hence fla t (see [D~ S, p. 71,Prop. 7.1] ). Then ~here e~is~s ~ f~nc~orial base-change isomorphism
The fundamental class If of any flat pseudo-proper map f can now be defined, as follows: with respect to the diagram
where ~ is the diagonal map and ~, ~ the canonical projections (so ~hat ~ = 1~ and ~ = 1~), there is a sequence of natural D(~)-maps
to which application of the left-derived functor L6* produces If:
L6*6.Oz
~ L6*~f"Oy u.
~ f"O
Let £ be the kernel of the canonical map Ox~u~ ~ 6.Oz, and let be the coherent O~-module 6"£, i.e., after identification of with ~I : (~/£2)11~1"This ~I is closely related to the universal finite differential modules ~ of ~5.5, thus: if f looks locally like ~ with ~ : R ~ S as above, then P(Spf(S), fiI) = As in ~5.5, Ii determines for each integer m a map
86
Lipman
When~ is the inclusion of R into a power-series ring R[[t]] (t := (tl,.. ¯, tm)) and f = ~, one shows (with some effort) that modulo the standard correspondence between modules and sheaves, f~ agrees with the fundamental class defined in §5.5. More generally, global fundamental classes "restrict" to local ones, as we shall now illustrate--without proof--for formally smooth pseudo-proper maps of relative dimension m. For a pseudo-proper map f to have these properties means that for any closed point y E f(2~) and any closed point x ~ f-ly, the corresponding map of completed local rings (~,y =: R ~ S := ~x,z is formally smooth and if is the maximal ideal of R then the local ring S/mS has dimension m. For simplicity, one may assume in what follows that, furthermore, S is residually separable over R. For any such x, y, there is a natural commutative diagram
Sp(S)
x
(5.6.3)
1 Spf(R)
~ ~y
The maps ~ and ~y are flat, and both ~ and g := ~y~ = f~x are pseudoproper.As the topological space]Spf(R)] consists of the singlepointm, the categoryA(Spf(R)) can be identified with the categoryof R-modules, and particular ~pf(R) R.It is similar for Spf( S), One veri fies that ~a =: ~#, and that ~I = ~so that ~I is locally free of rank m (see (5.3.6)). It is a consequence of Greenlees-May duality that for any ~ ~ Dc(~), the map in (5.4.1) is an isomorphism
and similarly
for ny. Thus (and cf. (2.3.3))
(5.6.4)
~#R=~-#nvO*~=~-#av
So, a~ being exact, ~ m
=
~x
there are natural
#O~=~g#O~=n~afrO~=n~
isomorphisms *f#O~.
we have for each closed x ~ ~ the map ~ H ~ O~ = H-m~#(R);
The assertion relating globalto loc~lfundamental classesis: Lemma 5.6.5.The precedingcompositemap is f~ (see (5.5.2)). ~om this we see, first, that [~: ~ ~ H-mf#o~ is an isomorphism. Indeed,Lemma 5.6.5localizesthe problemto showingthat [~ is an isomorphism(since then the kernel and cokernelof [~ would each h~ve at everyclosedpointa stalkwhosecompletion vanishes,and hencethey would both vanish). When ~ : ~t is the inclusionof R into a power-series ringR[[t]](t :: (Q,...,tm)), the local~ssertion is givenby the first of Theorem5.5.4.In the generalcase write ~ = ¢~t with ¢: R[[t]] ~ S 6tale(seeremarkspreceding (5.3.6)), and use the followingdiagram,whose
87
Local Cohomology and Duality top row comes from the trace bottom row comes from (2.4.1) Theorem 5.5.4, commutes:
A m ) ---~-~ ÷ HomR[it]](S, f/~t
f~t^ m
H-m~#~R
(remarks preceding Theorem 5.3.3’), whose applied to ¢, and which, as a corollary
~ HomR[[t]](S,H-m~R)
Second,note thatthereis a naturalisomorphism (H-mf#og)[m]
(5.6.6)
resulting via Proposition 1.3.3fromthe v~nishing of H~f#Oyfor a]] j ¢ -m: since~ is exactforallx, theisomorphisms (5.6.4) reduceverification of this vanishingto the corresponding vanishingfor ~#R, whichholdsby (5.1.7) whenS is a power-series R-algebra, andthenfollowsvia(2.4.1) in thegeneral case when S is an ~t~le extension of ~ power-series algebra (see remarks preceding (5.3.6)). Using (5.1.7) one shows the same true with any coherent ~y-module ~ in place of Oy. So we have the D(X)-isomorphisms
Hence,by Whm.5.6.1, ~ represents the functor HOmD(,)(Rf.RF~[m], of quasi-coherent O~-modules $; and when f is proper, ~ represents the functor HOmD(y)(Rf,~[m], Oy) of coherent O~-modules For such ~, there is an n such that RJfi~ := HJRfi~ = 0 for ~ll j > n [DFS, p. 39, Prop. 3.4.3(b)]. Then with 6 the O~-module HnR£~, which is coherent [DFS, p.40, Prop. 3.5.2)], there are isomorphisms HomD(~ )(~
f’g[-n])
~ HOmD(9)(Rf,~,~[
5.6.1
-n
~ Homo,
1.3.3
(~,~) .
But as noted ~bove, HJf#g[-n] = HJ-Uf#6 = 0 if j -n < -m, i.e., if j < n - m; and hence if n > m then Homo,(G,6) = 0, i.e., ~ = 0. conclude that RJfi~ = 0 for all j > m, ~nd therefore, by Proposition 1.3.3, (5.6.7)
HomD(y) (Rf.~[m],
O~) ~ Homo, (Rmf.~,
In summ~ry: Theorem 5.6.8. Let f : ~ ~ ~ be a formally smooth pseudo-proper map of noetherian formal schemes, of relative dimension m. Then ~ represents the functor HomD(~)(R£~[m], O~) of quasi-cohereni O~-modules this f is proper, then ~ represents coherent Ox-modules ~.
the functor
Homo,(Rmfi~,
Oy)
88
Lipman
To completethe discussion, we review howthe mapRf, RI’~g~n[m]-~ Oy (resp., whenf is proper, the mapRmf,~n ~ (9~) implicit in the proof of Theorem5.6.8 is uniquely determined by residues. Weneed only look at the first of these maps,since in the proper case, they correspondunder the composite isomorphism t^m HOmD(~)(Rf, l:tF2~f~ / [m], (D~) --~ HOmD(x)(~n[m], f*O~) (5.6.1)
--~ HOmD(~)(Rf,(~a[m], (5.6.~) ~
HO~o~
m ~m
(~.6.~) Tha~first mapcorresponds by duality to the Mndamentalclass and so is determined by [~: ~l ~ H-~f*O~, which is in turn uniquely determined by its completions ~[~ at all closed points x; and Lemma 5.6.5
implies that a~[~ is dual to the formal residue mapp~: H~fl~~ R of ~5.5. *** The foregoing provides for formally smooth pseudo-proper maps a canonical version of abstractly defined (by Theorem 5.6.1, but only up to isomorphism~) global duality, a version which pastes together all the canonical local dualities~vi~ residues~associated to closed points of ~. When ~ is a perfect field and ~ is an ordinary variety, not necessarily smooth, this is essentially the principal result in [Lpl], Theorem (0.6) on p.24. (See loc. cit.,~ll for the smooth case, and for a deduction via traces of differential forms of the main theorem.) A more general relative version, Theorem (10.2), involving a for~nal completion, starts there on p. 87. Another generalization, to certain maps of noetherian schemes, is given by Hfibl and Sastry in [HfiS, p. 752, (iii) and p. 785(iii)]. These results should all turn out to be special cases of one Residue ’Theorem for arbitrary pseudo-proper maps of noetherian formal schemes, for which the constructions sketched in this section provide a foundation. (~Vork in progress at the time of this writing.) REFERENCES
[AJL] L. Alonso Tarrlo, A. Jeremlas L6pez, J. Lipman, Local homologyand cohomology on schemes, Ann. Scient. 1~,c. Norm.Sup. 30 (1997), 1-39.
[DFS] __, Duality and fiat base change on formal schemes, Contemporary Math., Vol. 244, Amer.Math. Soc., Providence, R.I. (1999), 3-90.
[DGM]__, Greenlees-May duality on formal schemes. ContemporaryMath., Vol. 244, Amer.Math. Soc., Providence, R.I. (1999), 93-112.
[AJSl L. AlonsoTarrfo, A. JeremiasL6pez, M.J. Souto Salorio, Localization in categories IBs/
of complexesand unboundedresolutions, CanadianMath. J. 52 (2000), 225-247. N. Bourbaki, Alg~bre, Chapitre 10: Alg~bre homologique,Masson,Paris, 1980. M.P. Brodmann,R. Sharp, Local Cohomology,CambridgeUniv. Press, 1998.
89
Local Cohomologyand Duality
A. Grothendieck, J. Dieudonn~, ~lements de G6omdtrie Algdbrique III, Publications Math. IHES 11, Paris, 1961. __, ~lements de G~omdtrie Alg6brique IV, Publications Math. IHES 20, Paris, 1964. __, ~lements de Gdomdtrie Algdbrique I, Springer-Verlag, New York, 1971. J. P. C. Greenlees, J.P. May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), 438-453. __, Completions in algebra and topology, in Handbook of Algebraic Topology, Edited by I.M. James, Elsevier, Amsterdam, 1995, pp. 255-276. A. Grothendieck, Sur quelques points d’alg~bre homologique, Tdhoku Math. J. 9 (1957), 119-221. ~, Cohomologie locale des faisceaux coh~rents et Th~oremes de Lefschetz locaux et globaux (SGA2, 1962), North-Holland, Amsterda~n, 1968. R. Hartshorne, Residues and Duality, Lecture Notes in Math., no. 20, SpringerVerlag, NewYork, 1989. __, Affine duality and cofiniteness, Inventiones Math. 9 (1970),145-164. R. Hiibl, Traces of Differential Forms and Hochschild Homology, Lecture Notes in Math., no. 1368, Springer-Verlag, NewYork, 1966. R. H/ibl, E. Kunz, Integration of differential forms on schemes, J. Reine u. Angew. Math. 410 (1990), 53-83. R. Hiibl, P. Sastry, Regular differential forms and relative duality, American J. Math. 115 (1993), 749-787. E. Kunz, K6hler Differentials, Vieweg, Braunschweig~ 1986. T. Kawasaki, On arithmetic Macaulayfication of noetherian rings, preprint. J. Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Ast~risque, vol. 117, Soc. Math. France, Paris, 1984. ~, Residues and traces of differentials forms via Hochschild homology, Contemporary Mathematics, vol. 61, AMS,Providence, 1987. __, Notes on Derived Categories, preprint, wuw.math.purdue, edu/~lipman/ E. Matlis, Injective modules over noetherian rings, Pacific J. Math. 8 (1958), 511-528. __, The higher properties of R-sequences, J. Algebra 50 (1978), 77-112. A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519-532. __, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205-236. G. Scheja, U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Annalen 197 (1970), 137-170. P. Schenzel, Local cohomology in algebra and geometry, in Six Lectures on Commutative Algebra, Edited by J. Elias et. al., Birkhaiiser, Basel, 1998, pp. 241-292. M. J. Souto Salorio, Localizacidn de Bousfield en categor(as derivadas de categorias de Grothendieck, Alxebra no. 63, Depto. de/~lgebra, Univ. de Santiago de Compostela, Spain, 1998. [Spn] N. Spaltenstein, Resolution of unbounded complexes, Compositio Math. 65 (1988), 121-154. R. Strebel, On homological duality Pure and Applied Algebra 8 (1976), 75-96 [Wbl]C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994. [Wb2]__, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 1655-1662. [zs] O. Zariski, P. Samuel, Commutative Algebra Vol. II, Van Nostrand, 1960. DEPT. OF MATHEMATICS,PURDUEUNIVERSITY, W. LAFAYETTEIN 47907, E-mail address: lipmantmath.purdue, edu URL: ~ .math .purdue.edu/~lipman/
USA
COHOMOLOGICAL INVARIANTS SHEAVES OVER PROJECTIVE A SURVEY
OF COHERENT SCHEMES:
MarkusBrodmann University of Zurich, Zurich, Switzerland
ABSTRACT. We give a survey on certain results related to the cohomology of projective schemes with coe~cients in coherent sheaves. In particular we present results on cohomological patterns, cohomological Hilbert functions and cohomological Hilbert polynomials. Bounding results for Castelnuovo-Mumford regularities, Severi coregularities and cohomological postulation numbers are discussed. Moreover, a number of open questions is presented.
CONTENTS
1. 2. 3. 4. 5. 6.
Introduction Cohomological patterns Cohomological Hilbert functions Purely diagonal bounds b-sheaves and Hilbert coefficients A few specific bounds
91
92
Brodmann 1. INTRODUCTION
Let R = ~n>_ORnbe a homogeneous noetherian ring. Let X := Proj(R) be the projective scheme induced by R. Let i ¯ No. For a sheaf of (.0x-modules ~, the i-th Serre cohomologygroup Hi(X, ~) of X with coefficients in $" carries a natural structure of R0-module. If ~ is coherent, it follows from Serre’s finiteness theorem, that the R0-moduleHi(X, ~) is finitely generated (cf. [42, III, Theorem5.2], [72, §66, Th~or~me 1]). Now,fix a coherent sheaf of (2x-modules ~. For n ¯ Z consider the coherent sheaf of (2x-modules J:(n) := .~ ®ox (2x(n) = .T (2X( ®n, e.g. t he n-th twist of J: (with respect to the very ample sheaf (22;(1)). A considerable number of results in algebraic geometry can be expressed in the form of vanishing statements for some cohomologygroups Hi (X, 9V(n)). Correspondingly, there are several numerical cohomological invariants of the pair (X, ~’) related to the vanishing and non-vanishing of the groups
Hi(X, Let us inention first the cohomological dimension of (X, ~), which is given by (1.1) cd(Z,j ~) = cd(:F) :-- sup{/e No I 3n E Z: H~(X,J~(n)) 0}. (Throughout this paper we use the convention that sup 0 = -oc and inf0 oc.) It is important to notice that cd(5v) < oc. In fact the invariant z) cd(J depends only on the topological behaviour orbv along the fibres of the natural morphism ~r : X ~ X0 := Spec(Ro), (s. (2.3)), in accordance with Vanishing Theorem of Serre-Grothendieck (cf. [72, §66, Th~or~me1], [42, III, Theorem2.7]). Next, for each i E N, we maydefine the i-th cohomological right vanis.hing order of (X, r) by (1.2)#x,~:i = #i:r :=sup{n z lgi(x,f(n
- 1))# 0}
By the Vanishing Theorem of Castelnuovo-Serre (cf. [72, §66, Th~or~me (b)]) we have #~: < oc for all i ¯ For k ¯ No, the (Castelnuovo-Mumford) regularity of (X,~’) above level is defined by
(1.3)
=
:= sup(, + ill >
As #~ < cx~ for all i > 0 and as cd(f) < co, we have regk(5 r) < oc. Moreover, reg(~:) := rego(J:) is the (Castelnuovo-Mumford) regularity originally introduced in [65]. In 1893 G. Castelnuovo proved a result which, in our language, says that rego(,~) <_ d- 1, where ,~ C_ O~is the sheaf of vanishing ideals of a smooth projective curve of degree d in I~, (s. [25]). This is apparently the first bounding result for cohomological right vanishing orders and regularities. For this reason, Mumforddid speak of Castelnuovo regularity and for the
Cohomological Invariantsof Coherent Sheaves
93
same reason we associate the nameof Castelnuovo to all vanishing or bounding results on the groups Hi(X, ~(n)) in the range n _> -i. The behaviour of the groups Hi(X, Jr(n)) for n << 0 is more subtle than in the "Castelnuovo" range: in general, these groups do not vanish if n << 0. The problem of vanishing in this range essentially can be attacked only if i does not exceed the cohomological finiteness dimension (1.4)
f(X, ~) = f(~:) ¯ = inf{i E No I Hi(X, iT(n)) for infi nitely manyn _< 0}
of the pair (X,.T). Under certain assumptions on R0, the finiteness dimension f($’) depends only on the behaviour of 7" along the fibers of the natural morphism~r: X --+ Xo = Spec(Ro), (s. (2.4) (iii)), in accordance with Vanishing Theoremof Severi-Enriques-Zariski-Serre (cf. [72, §74, Th~or~me 2], [42, III, Theorem7.6 (b)]). Now,obviously for each i < f(~’), we maydefine the left-vanishing order of (X, .T) (1.5)
Uix,.r = ~’} := inf{n e Z IHi(X,~(n 1)) =~
and thus have u~- > -cx~ for all i < f(gv). For k ~ {0, 1,..., f(~) - 1} we define the (Severi-) coregularity of (X, ~) and below level k by (1.6)
coregk(X, ~) = coregk(~) := inf{u~= + i [ i <_ k} .
As ,~= > -~ for all i < f(~’) we have coregk(~) > --~ for all k < If f(~) > 0 we call the number coreg(~) := coregy(:r)-I (~’) the (Severi-) coregularity of (X, In 1942 F. Severi proved a result which, in the language of sheaf cohomology, says that f(wz) > 1 for the canonical sheafwx of a smooth projective surface X _C F~, (s. [73]). In 1949, F. Enriques gave an attempt to generalize this to the case where X C_ ]P~ is a smooth projective variety of dimension > 1, (s. [29]). In 1952, Zariski proved that for an arbitrary invertible sheaf £: over a normal projective variety X of dimension > 1, we have f(£:) > 1, (s. [82]). In 1955, J.P. Serre brought this result to its final form by showing that for an arbitrary coherent sheaf 9r over a projective variety X, f(9 v) equals the minimumof the depths of the stalks of 7" at closed points x ~ X, (cf. [72, §74, Th6or6me2]). Apparently, Severi was the first to prove a non-trivial vanishing result for some of the groups Hi(X, :~(n)) in the range n << 0. Therefore, we speak of Severi coregularity. Moreover, for the same reason, we associate the name of Severi to all vanishing or bounding results on the groups Hi(X, :~(n)) in the range n _< -i, i < f(gv).
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Brodmann
In order to study and to formulate vanishing results, introduce the so called cohomological pattern (1.7)
P(X,H) = P(H):= {(i,n)
it is convenient to
e No Z] Hi(X,H(n)) ~
0}
of the pair (X, H). Weshall discuss this concept in Section Besides the mere vanishing and non-vanishing of the cohomology groups Hi(x, .~(n)) one mayalso ask for the "size" of these R0-Inodules. A natural approach to this is to assume that the base ring R0 is artinian: then the R0modules Hi(X,Y{n)) are of finite length, so that this length may be used to measure their size. So, assume now, that R0 is artinian. For any i E No, any n E Z and any coherent sheaf of Ox-modules H we consider the length (1.8)
hix,a~(n)
= h~=(n) := lengthRo(Hi(Z,H(n))
of the R0-module Hi(x,H(n)). For fixed i ~ No, the function
(1.9)
¯
i = h~: Z ---~ No; (n ~--~ h}(n)) hx,:~
is called the i-th cohomologicalHilbert function of (X, ~). Now, fix k ~ N0. Then, there are two easy vanishing constraints numbersh}(n) (cf. (2.4) (i), (ii)). The first of these constraints
for the
If hk(-0 = 0 for then h~=(n) = O Eor all i > k and a11 n >_ -i. The second constraint says: 0.11)
thenh~:(n)=0forall
i h k and all n < -i.
Observe that the vanishing assumption" in the constraint (1.11) implies particular that k < f(H). In sections 3 and 4 we shall present a couple of bounding results, which extend the above constraints. Weactually have to distinguish three types of such results. The first type gives bounds on the invariant regk(~) and on the numbers h}(n) in the range i > k, n >_ -i in terms of the cohomology diagonal
,n:7:[--,J}i=k+ i , .,,~(:~)
10f H abo~e level k. Werefer to these bounds as (diagonal) bounds of Cas~elnuovotype - in accordance with our earlier convention. The second type of result applies if k < f(H) and bounds the invariant coregk(H) and the numbers h~(n) in the range i _< k, n _< -i in terms of the cohomology diagonal (h~=(-i))~_ o of ~ at and below level k. Here, we speak of (diagonal) bounds of Severi type. The third type of result is referred to as bounds of extended Severi l~,ype. Their aim is to extend the bounds of Severi type beyond the situation in
Cohomological Invariantsof Coherent Sheaves
95
which k < f(~). This first of all needs some conceptual modification of the ideas underlying the previous type of bound. The crucial point is to keep in mind that for each i E No there is a (unique) polynomial
the i-th cohomological Hilbert polyr~omial of (X,.T), (s. [21, (20.4.12)]). Moreover, deg(p)) ~ Now,we may define the i-th cohomological de~ciency function of (X,
and the i-th cohomologicMpos~ula$ion number of (X, ~): ’
,~ inf{n e Z ~ A~(n + 1) ~ 0}.
Then, clearly ~ > -~ for all i ~ H0. As p~~ ~ 0 for all i < f(~), we have A~= h~ for all these i. Therefore, the concept of cohomologic~l postulation number naturally extends the concept of cohomological left-vanishing order (cf. (1.5)) to the range i ~ f(~). But now, it is clear, what bounds extended Severi type should achieve: They should bound the invariant s and the numbers A~(i ) nin the range n ~ -i for ~rbitrary values of WeshMl discuss a result of this type, which gives a bound on the numbers ~ for i ~ k in terms of the cohomology diagonal (h~(-i))~= o of ~ at and below level k and the cohomological Hilbert polynomial p~ in section 3. In section 4 we consider the case in which the base ring R0 is a field. In this particular situation we are able to bound the cohomologicMpostulation numbers ~ in terms of the £~11 co5omology diagonal [n~[-~)h=o of ~. As a consequenceof this we obtain that there are only finitely manychoices for each of the cohomologic~l Hilbert functions h~ if the cohomologydiagonal (~ ~ ~d(~)is fixed. ’~Y~--~=0 All the bounds mentionned so far, are a priory bounds, v~lid for arbitrary pairs (X, ~) (with appropriately chosen base ring R0). Moreover, they (part of) the cohomology diagonal as a bounding system. In section 5 also shall consider bounds for the regularity which depend on the so called Hilbert coe~cients and hold for b:shea,es in the sense of Kleiman[38, Exp. XIII]. In section 6 we sh~ll briefly discuss a few speci~c bounds, e.g. bounds concerning special pairs (X,~). Our interest is focussed on the classical cases, in which X is a projective space over an algebraically closed field and ~ is an algebraic vector bundle or a sheaf of ideals defining a projective variety. Wealso consider the case in which X is a projective variety and ~ = Ox is its structure sheaf. Most of the results we present, are originally formulated and proved in terms of local cohomologyrather than in terms of sheaf cohomology.So, we briefly recall the link between these two concepts. To do so, let the base ring R0 be
96
Brodrnann
arbitrary noetherian. Then, the coherent sheaf.T is induced by some finitely generated graded R-moduleM(s. [42, II, Proposition 5.11]). Now, let R+ := ~n>OT~nC R be the irrelevant ideal and let DR+denote the R+-transform functor, e.g. the linear left exact functor on the category of ’~, .). R-modules given by DR+(.) := li_~mHomR((R+) For i ¯ N0, let T~iDR+denote the i-th right derived functor of DR+.Then, the R-modulesT~iDR+(M) carry a natural grading (s. [21, (12.4.5)]). Moreover, there are isomorphisms of R0-modules (1.15) Hi(X,,~(n)) ~- T~iDR+(M)n, (Vi ¯ N0, Vn ¯ Z) in which "n denotes the formation of n-th graded parts (s. [21, (20.4.4)]). iNow, keep in mind that the local cohomology modules HR+(M) carry a natural grading (s. [21, (12.3.3)] or consult one of [23] or [26]) and that natural exact sequence 0--~
H~+(M) ~ M ---+
DR+(M)---~ H~R+(M)---~
and the natural isomorphisms i (M), (Vi N) T~iDR+ (M) "~ HR+ respect these gradings (s. [21, (12.4.2), (12.4.5) (iii)]). So, altogether, each n ¯ Z we obtain a short exact sequence of R0-modules (1.16)
0 ~ H°R+ (M)n ~ Mn "--+ H°(X,.Tr(n))
~ H1R+ (M)n
and isomorphisms of R0-modules (1.17)
gi(x,~(n))
"~ g~++l(M)n, (Vie N).
The relations given by (1.16) and (1.17) are a version of the so called SerreGrothendieckcorresponde~ice (cf. [21, Chap. 20], [37]). 2. COHOMOLOGICAL PATTERNS Let X = Proj(R = @n>_oRn)and 9v be as in the introduction. In this section we shall discuss a few properties of the cohomological pattern P(~) introduced in (1.7). Let us start with the following definition. 2.1. Definition. (i) Let w ¯ No. A set P C_ No x Z is called a combinatorial pattern of width w, if it satisfies the following five conditions: (~) ~n,~¯~:(0, , (w, n) ~ (i,n) ¯P~i_<w; (~r2) ~P~Bj_
i: (k, n + i - k - 1) (r5) i>O==--=~(i,n) C_P, Vn>O. The width of a combinatorial pattern P is denoted by w(P).
Cohomological Invariantsof Coherent Sheaves
97
(ii) Let i E N0. A combinatorial pattern P is said to be tame at level i, if either (i,n)
ePforall n<<0, or (i,n)
C P for all n <<
Observe that any combinatorial pattern P is tame at level 0 and at level w(P). A combinatorial pattern P is said to be tame if it is tame at any level i ENo. (iii) A combinatorial pattern P is said to be minimal if there is no combinatorial pattern Q c P. (iv) Fix w G No and let ~ be the set of all monotonically decreasing functions # : Z ~ {0,...,w} for which 0, w E #(Z). For any/t e M~, let us introduce the "skew graph" of #, e.g. the set P[#] := {(#(n),n - #(n)) I n e Z} c_ N0 It is easy to verify that P[#] is a minimalcombinatorial pattern of width w. In the following remark we summarize a few properties of combinatorial patterns. For a more detailed and complete presentation of the listed facts werefer to [13, Sec. 2]. 2.2. Remark, (i) Let C N0x Zbea combi natorial patte rn and l et ( i, n ) E P. Then, there is a integer w _> i and a function # E 1V~wsuch that (i,n) Pitt] C_P. (ii) In view of what we just remarked, it is easy to see, that the minimal combinatorial patterns of width w are precisely the patterns of the formP[#] with tt E 1~¢~. (iii) On use of the previous observation one nowproves immediately: A set P _C N0 × Z is a tame combinatorial pattern if and only if it is the union of finitely many minimal combinatorial patterns. ¯ After these combinatorial preliminaries, let us turn back to cohomological patterns. 2.3. Proposition. In the notation of (1.1) and (1.7) we have: p(~z) is a combinatorial pattern of width cd(~) -- sup{dim(Supp(~) ~-l(x0) ) I x0E X0}.
Proof.. s. [13,(3.5)].
[]
2.4. Remarks. (i) Observe that according to Proposition (2.3) the pattern P(~) satisfies in particular the two conditions ~r4 and ~r3, which correspond respectively to the following two vanishing constraints (in which k E N0 and r ~ Z are fixed): (*)1 Hi(X,Y(r-i))=O, (*)2
Vi > k~ Hi(X,:lZ(n-i))
=0, Vi
Hi(X,~r(r - i)) = O, Vi <_ k ~ Hi(X,J:(m - i)) Vi <_ k, Vm<_ r.
98
Brod~nann
In the special case, whereR0 is a field, the constraint (*)1 is found[ in Mumfords work [65]. For a proof of both constraints in the present for!m see [13, Sec. 3]. Obviously, if R0 is artinian and r = 0, the constraints (*)1 and (*)2 respectively correspond to the constraints (1.10) and (1.11). (ii) The constraints (*)1, (*)2 tell us respectively: Hi(x,~r(n)) vanishes "along a diagonal above (resp. "at and below") level k" then it vanishes everywhere"to the right" (resp. "to the left") of this diagonal. In particular, these constraints give alternative descriptions of the concepts of regularity above level k and of coregularity at and below level k (s. (1.3), (1.5)), namely: regk(Y) in f{r E z[ g~(x,Y(r - i )) = O,> k}; coregk(~ :) = sup{c E Z lH~(X,~(c- i)) ~ O, Vi <_k}, (k < f(~)). (iii) Clearly, the cohomologicalfiniteness dimensionf(~’) r i s t he lowest level i at which there are infinitely many (i, n) ~ P(~’) with n < 0. invariant f(.T’) is related to the local behaviour of r along t he f ibres o f t he morphism ~r : X ~ X0 by the inequality f(~=) <_ inf{depthox,~(~) di m(~r-l(~r(x)) f~ {x}x e X} - -: 5(b~). Moreover, if R0 is a homomorphic image of a regular ring, equality holds (cf. [21, (20.4.20)]). Now, let us say a few words about the tameness of the pattern P(~). First of all, let us mentionthe following result (cf. [13, (4.3)]). 2.5. Theorem. If the base ring Ro is semilocal and of dimension <_ 1, the cohomological pattern P(J:) of the coherent sheaf ~: is tame. 2.6. Comment and Problem. We do not know, whether the conclusion of (2.5) holds without any restriction on the (noetherian) base ring R0. let us pose the following tameness problem: Is P(~:) always tame? 2.7. Remark. Obviously, one might try to answer the tameness problem at certain particular levels i. Indeed, for certain values of i the requested tameness is easily verified. So, by the axioms~3 and ~r4 and in view of (2.3) it is clear, that P(br) is tame at level 0 and at all levels i >_cd(~:). Moreover, P(bv) obviously is tame at any level i < f(~-). In view of the previous remark, i = f(b ~) is the first level at which the tameness question is non-trivial. At this paxticulax level P(~) is indeed tame. Actually, we have the following result, which is an easy consequence of [13, (5.6)] and the Serre-Grothendieck Correspondence(1.16), (1.171}.
Cohomological Invariantsof Coherent Sheaves
99
2.8. Theorem. The sets AssRo (HI(~:)(X,~’(n))) are asymptotically for n ~ -c~, e.g. there exists an integer no such that ASSRo(HY(~:)(X,~F(n))) = AssRo(Hf(~=)(X,J:(no))) In particular P(~) is tame at level f(~).
[]
2.9. Remark and Problems. (i) For arbitrary values of i E N0, the sets AssRo(Hi(X,.T(n))) need not be asymptotically stable for n --~ -cx3. Indeed, using a construction of Singh [75] one can easily write downa positively graded homogeneousnoetherian domain R --- @n>_oRn such that with X = Proj(R) we have cd(Ox) = 2 and n<’O (ii) On the other hand, we do not know an example for which the sets AsSRo(Hi(X,.T(n))) are not "asymptotically increasing" for n ---~ -c~. So, let us ask the following question: Is there some no E Z such that for all n <_ no AssRo(Hi(X, .~(n 1))) _~ ASSRo(Hi(X, .~( n)))? Clearly, an affrmative answer to this question implies that P(~’) is tame (at level i). (iii) The base ring R0 in the example mentionned in part (i) is not local, and in fact we do not know an example with local base ring R0 for which the sets AssRo(Hi(X,.~(n))) are not asymptotically stable for n --~ So, let us pose the following problem: Assume that Ro is local. Is there some no ~ Z such that ASSRo(Hi (X, Y(n))) -- ASSRo(Hi(X, .T:(n0))) for all n _< (iv) Let us also mention here the following fact, which is an easy consequence of [13, (5.5)] and the Serre-Grothendieck Correspondence(1.16), (1.17): If ASSRo(Hi(X,.T(n))) is asymptotically stable for n --~ and if.T is induced by the tlnitely generated graded R-module M, then AssR(HiR++I(M)) is a tlnite set. In the special case, where (R0, m0) is local and m := m0 + R+ denotes the 1 (M)) and graded maximalideal of the ring R, the two sets ASSR(HiR++ AssR,~ (n(R+)n,~(Mm))are in natural bijection. So, the "stability problem" mentionnedin part (iii) is closely related to the tlniteness problem for the sets of associated primes ASsR(Hi~(M)) of the local cohomology modules HI(M) of a finitely generated module Mover a noetherian (local) ring with respect to someideal a C_ R, (cf. [47]). This problem recently has found an affirmative answer for someparticular values of i (cf. [14], [19] and also [12]) and is knownto have a negative answer in general (s. [75]). In special case, where R = Mis a local ring, there are somepartial results (cf. [43]), but even in the case of a local Cohen-Macaulayring R, the problem
I00
Brodmann
is open. Only in the case where R = Mis a regular local ring, there are complete (and further reaching) results (s. [49], [55], [56]). Onemight whether the ideas developed in [74] could lead to some progress in the study of the "graded case" of the above finiteness problem. ¯ It is natural to ask, whether the pattern axioms (2.1) (~rl - ~5) (together with the requirement of tameness) are the only general constraints to which cohomologic’al patterns are subject. More precisely: Is any tame combinatorial pattern the pattern of a pair (X,~)? This question has indeed affirmative answer. 2.10. Theorem. Let Ro be a semilocal noetherian ring of dimension <_ 1 and let w E No. Then, the tame combinatorial patterns of width <_ ~ are precisely the combinatorial patterns of coherent sheaves of (9~o-modules. Proof: (s. [13, (4.8)]).
[]
2.11. Remark. (i) The crucial step in the proof of (2.10) is to show for any field K and any minimal combinatorial pattern P of width w, there is a coherent sheaf ~" of O~-modules with P(~) = P. In view of the vanishing Theorem of Severi-Enriques-Zariski-Serre, such a ~ must be an algebraic vector bundle. Indeed, we can choose the requested sheaf ~ as an (indecomposable)algebraic vector bundle of rank _< w! (s. [13, (4.5)]). let us ask the following problem: Whatis the least possible rank of an algebraic vector bundle ~ over ~¢ for which P(~) = In [13, (4.5)] a candidate for v is constructed as t he direct i mage r7,£ o f a li ne
bundle £ = over the w-told Segre product = x...
of a projective line I1~ under a generic projection ~r : Y ~ I?~.. (Here ~ri : Y ~ Ii~7 denotes the projection to the i-th factor.) The twisting orders ri are determined appropriately on use of the Kiinneth formulas (s. [77], [31]). To answer the above question one likewise has to use a different method of construction. (ii) It is easy to write down examples of tame combinatorial patterns width w which are realizable only by decomposable vector bundles (s. [13, (4.9)]) over F~. This makes arise the following question: Is there a purely combinatorial characterization of those patterns, which may be realized by an indeco~nposable algebraic vector bundle over ~( ? Let K be a field. By Theorem (2.10) (and by the Vanishing Theorem Severi-Enriques-Zariski-Serre) the combinatorial patterns of algebraic vector bundles over lI~ are precisely the combinatorial patterns P of width ~ such that (i, n) ~ P for all i ~ w and all n K~0. Inspired by this observation one may be tempted to ask whether there is a purely combinatorial description of the patterns P((-gx) of a smooth projective variety X C_ I~.~ ower an
Cohomologica| lnvariants
101
of Coherent Sheaves
algebraically closed field K (of characteristic 0). In order to say a few words about this question, we introduce some notions: 2.12. Definition and Remark. (i) A combinatorial pattern P is said be positive, if (0, 0) E P and if (i, n) g P for all w(P)and all n < 0. A combinatorialpattern P is said to be s~raight, if it is positive and if (i, n) ~ whenever 0 < i < w(P) and n > 0. (ii) If P is a combinatorial pattern of width w, we set a(P) := sup{n e Z I (w,n) e If P is positive, the axiom (2.1) Qr3) implies that a(P) >_ -w(P)- 1, wheras axiom(~r4) gives (i,0)¢P
for
all
i>w+a(P)+l.
Now, we are ready to show: 2.13. Proposition. Let P be a straight combinatorial pattern of width w > 0 and let K be an algebraically closed field. Then, there is a smoothprojective variety X over K such that P(Ox) = Proo[: (Induction on w) First, let w = 1, so that ~(P) _> -2. If c~(P) = choose X = ~¢. Otherwise, let X C_ £~ be a smooth curve of degree
+ 3.
Now, let w > 1, ~ := ~(P) and m = max{/ e N0 I (i,0) e P}. ~ _> m - w - 1 (s. (2.12) (ii)). Assume first, that c~ = m-w- 1. Then m < w and c~ < 0. Ifm = 0, choose X = l?~. So, let m > 0. Then, there is a straight pattern Q with w(Q) = and Q N (N0 x {0}) = P N (No x {0}). So , by induction, the re is a smooth projective non-degenerate variety Y C_ ~( with P(Oy) = Q. Let Z:=]?K w-m . Then, the SegreproductX :=YxZ C ~_ +l)(~-m+~)-~ IS a ~smooth projective variety. As P((gz) ({0} × N0)tO ({ w -m} × Z_<m-w-1), the Kiinneth formulas give for all i ~ N0 and all n E Z Hi(X, Ox(n))) ~ Hi(Y, Oy(n)) ®~ H°(Z, @ Hi-m-w(Y, Or(n)) ®K Hw-m(z, Oz(n)) (with the convention that Hj :_ 0 for j < 0). Using again the shape of P(Oy) and of P(Oz) it follows that P(Ox) = Next, assume that~n-w-l
102
Brodmann
P(Oz) = ({0} × No) U ({1} × Z<0), the Kfinneth formulas give for all i E and all n E Z
Hi(X,Ox(n))-~ Hi(Y,Oy(n))®~~0(~, ~ H~-~(~,~(~)) ~ ~(~, Oz(~)). Using once more ~he shape of P(Oy) and of P(Oz) it follows P(Ox) := P. So, let a ~ 0. Then, there is a straight pattern Mwith a(M) ~ {0~-1}, w(M) : w-land such tha~ M~(N0 x {0}) : P~({0,...,w-1} x {0}). induction, there is some smooth projective non-degenerate variety T C F~ such that P(OT) : M. Now, let W ~ F~ the third Veronesean of the projective plane, so that P(Ow) = ({0} x N0) ~ ({2} x Z<0). Moreover, w~0r+9 be the Segre product of T and W. Then, U is a U:=TxWc~ K smoo~hprojective variety &nd~he Kfinne~h formul&s give Hi(U, Or(n))Hi(T, OT(n)) ~K H°(W, Ow(n))
~ m-~(T, O~(~)) .~H~(mOw(~)), forM1 i, n 6 Z. It follows, thatP(O~)is a straight pattern withw(P(Ou)) ~ + ~, a(P(O~))= -~ ~nd such that
P(O~) ~ (~o~ {0}) : ~ ~ ({0,... ,~- 1}~ {0}). be ~ generic hypersurface of degree 2~ + i. Then,by Now, let H Cmlor+9 ~ ~10r+9 is a smooth projective variety (cf.[50]). Now, Bertini,V := U~H C ~K theexactsequences
H{(U, Ov(~- 2~- ~))~ H{(U,O~(~)) ~ H{(
Ov(n))=O
if
2a+l}
Hi(V, Or(O)) ~ Hi(U, Ou(O)) for 0 < i < w ; Hw(V, Ov(2a)) 0; Hw(V, Ov( n)) : 0 for all n > 2a. Now, let X C-- ~50r2+105r+54 be the second Veronose transform of V. As "~g X - V, X is smooth. It now follows easily from the above statements on the groups Hi(V, Ov(n)) that P(Ox) : P. 2.14. Remark and Problems. (i) In view of the Kodaira Vanishing Theorem (cf. [52]) the pattern P(Ox)of the structure sheaf of a smoothproje, ctive complex variety X is always positive. So one may be lead to ask: Is each positive pattern the cohomological pattern of the
103
Cohomologicallnvariants of Coherent Sheaves
structure sheaf Ox of a smooth projective variety X? (ii) The methodof construction used in the proof of (2.14) realizes straight patterns by smooth projective varieties X C IP~( of rather large degree. So, we ask:
What can be said about the minimal possible degree of a smooth projective variety X C_ arK whose structure sheaf Ox realizes a given straight pattern? (iii) Let us also mentiou the relation of the above problems to the following Non-Rigidity Theorem of Evans-Griffiths (s. [30] and [63]): Given r - c 1 graded modules L1,... ,Lr-c-~ of finite length over the polynomial ring K[x0,..., xr] over the algebraically closed field K, (2 _< c _< r - 2), there is a normal projective variety X C_ F~. of codimension c and an integer t such that there are graded isomorphisms ~n~zHi(X, Ox(n)) ~- Li(t) for l
3.
COHOMOLOGICAL HILBERT
FUNCTIONS
Let X=Proj(R = ~n>oRn)---~Xo~r = Spec(Ro)and Y be as in the introduction, but assume from nowon, that R0 is artinian. Wenow shall present a few results on the cohomological Hilbert functions h~r introduced in (1.9). Let us start with a few preliminary remarks. 3.1. Remark. (i) As R0 is artinian, we have sup{dim(Supp(Y) n ~r -l(x0)) I x0 E X0} = dim(Supp(Y)) dim(Y), so that (cf. (2.3)) cd(Y) di m(Y’). (ii) Clearly, nowthe invariant 6(Y) = inf{depthox,~(Y~) di m0r-l(~r(x)) f~ {x}) [ x of (2.4) (iii) takes the value inf{depthox.,(G) l x ~ X,{x} = {x}}. As R0 is artinian, its localizations are complete and hence homomorphic images of regular local rings, by Cohens Structure Theorem(s. [60]). (2.4) (iii) it follows noweasily f(Y) = 6(Y) = inf{depthox,, Now,let us formulate a first nuovo type.
(Yz) [ x {x} = {x}}. ¯
bounding result - a diagonal bound of Castel-
3.2. Theorem. Let d := dim(Y). Then a) reg~c(Y)
i (2 dEi:k+l ( d- a-l~h~=(_i)) \i-k-l!
2d-k-1
~
(0 --< k < d).
104 b) hJ_r(n)
Brodmann -- <
½(2~di= j¢d-j~hi "~-~]2 ~i-j] .7
~ ;~2d-J (O< --j -j) ~ --
"
Proo~See [16, Sec. 4]. Our next result gives a diagonal bound of Severi type. 3.3. Theorem. Let k < ~(~).
b) h~(n) ~ ~(2 ~i=o (~)h~(-i))
Then
, (O ~ j < k,
Proo~See [16, Sec. 5]. 3.4. Definition. (i) Let C denote the class of all pairs (X,~) in which X = Proj($nEoRn) is a projective scheme over an artinian base ring R0 and ~ is a coherent sheaf of Ox-modules. Let ~ G C be a subclass. Now, in the spirit of [16, (4.8)] and [21, (16.4.1)] we define numerical invariant on ~ to be an ~signement p : ~ ~ Z U {~}. We say that the numerical invariant is ~nite, if ~(~) G (it) Let C,O be as in part (i) and let ~,... ,~s,P be numerical invariants on ~. Wesay that ~1,-. ¯, ~s form an upper (resp. lower) bounding system for p on ~, if the invariants p~,... ,~s are finite and if there is a function B : Zs ~ Z such that v(V) ~ "(,~(V),...,gs(g)) (resp. p(V) (iii) In the situation of part (it) we call B an upper (resp. lower) bounding function for p in terms Of~l,..., ~s. Instead of saying that ~,..., ~s form an upper (resp. lower) bounding system for p on ~, we also say that p bounded above (resp. below) by (or - in terms of) ~l,... ,~s on ~. ~e that p is polynomially boundedabove (resp. below) on ~ by ~,..., ~:, if is bounded by a polynomial bounding function B. (iv) Let ~,C and ~l,.-.,~s,P : ~ ~ Z be as above. Wesay that ~l,..-,~s form a minima/upper (resp. lower) bounding system ~or p on ~ if they form such a system, but if none of the s systems ~1,...,~i-1,~i+1,... ,~ does
(i = 3.5. Lemma.Let K be a field, let d E N, let i ~ {0,... ,d} and let :D be the class of all pairs (]~dg,E) .for which ~ is an indecomposablealgebraic vector bundle over ]?dK. Then ¯ i+1(-z-1),...,h.d(-d) ¯ a) IfO _< k < i, the invariants h.°(0) ,...,h. i-1 (-z+l),h. do not form an upper bounding system for the invariant regk(.) on the class b) I.f i <_ k < d, the invariants h°.(O),hi.-l(-i 1), h~+l(-i- 1) ,... ,h ~ (-d do not form a lower bounding system for the invariant coregk(*) on the class
Cohomological lnvariants of Coherent Sheaves
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Proof: Let t E N. It is easy to see, that there is a minimal combinatorial pattern Pi,t of width d with the properties that (i, -i - t), (i, -i + t) ~ P~,t and (j, -j) # Pi,t for all j ¢ i. By(2.10) and in view of (2.11) (i) there indecomposablealgebraic vector bundle $~,t over Fd~; such that P(Si,t) : Pi,t. In particular we have hJ~,t(-j) = for al l j # i. Moreover,if 0 < k < i we conclude from (i,-i + t) ~ Pi,t that regk($i,t) > If i _< k < d, we use (i, -i - t) ~ Pi,t to show that coregk($i,t) < -t. This proves our claim. E] 3.6. Notation. For r ~ N0, let C(r) denote the class of all pairs (X, ~’) ~ with dim(~) = r and let (r)c denote the class of all pairs (X, ~-) E
> r. 3.7. Corollary. Let k ~ No. a) For each d >_ k, the invariant regk(.) is bounded above polynomially terms of the invariants h~.+l(-k-1), h.~+2(-k- 2),..., hd.(-d) on the class C(d). Moreover, these latter invariants form a minimal bounding system for regk(.) on (d). b) coregk(.) is boundedbelow polynomially in terms of the invariants h°.(O), h.~(-1),..., h~.(-k) on the class C. Moreover,these latter invariants form minimal bounding system for coregk(.) on Proof" Immediateby (3.2), (3.3) and (3.5).
[]
3.8. Remarks and Problems. (i) The polynomial bounds of Theorems (3.2) and (3.3) are not the sharpest possible bounds. In [16] we give sharper bounds in terms of certain piecewise polynomial and recursively defined bounding functions B(i) N~+I-~ --(d+l)
:
, (O
----~_>-i
and BIi;+I)
: [k~ +1 x0 Z_<_i--~l~
, (o <_i < k) C(~) (k+~) : No/+ such that ~(~1 < --B!~! -,(h~(-i), --[a+~)
¯ ¯¯
,hd:r(-d);n), Vn
v(x, regi-l(~)
i ( ~ < C(i) rh ~’~-i ],’’" ~--(d+l)(
, h~=(-d)) +
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and B(i) ~h o ~a~ h~(-i);n), <_ (~+~)~ ~=~-~,...,
hi:r(n)
Vn <
v(x, ~) ~(~) coregi(~)
<_ C~)+l)(h°~(O),...hi~(-i))
(ii) In spite of the sharper bounds mentioned in part (i), one may whether the polynomial bounds of Theorem (3.2) and (3.3) reflect the of growth for the invariants regk(~) (resp. coregk(~)) in terms of the corresponding diagonal values h~(-i). More precisely, let d, k ~ N0 with k < d. Then, we ask Is there a sequence (Xn, Yn)n~ ~ (d) and a constant c > 0 su ch th at
i=k+l
Is there
a sequence (Y~, ~)ns~ ~(~) C and a constant
b > 0 such that
k
_
h~. (-~))
for all
~ ~
i=1
(iii) In the special case where k = 0, diagonal bounds of Castelnuovo type similar to those in Theorem (3.2) are given in [21, Chap. 16]. There, the main tool is an algebraic version of the Lemmaof Mumford-Le Potier (cf. [21, (16.1.4)]). This lemma applies only to the function n ~ h~(n) in the range n _> 0 and thus is not of great help to bound rega (~’) for k > 0 and to bound corega(~) at all. So, we use another idea to derive the bounding results mentioned under (3.2), (3.3) and (3.4): a generalized version principle of//near systems of hyperplane sections. This principle has been used to derive a-priori bounds of Castelnuovo type and of Severi type in the case where R0 = K is an algebraically closed field in [5], [7], [9], [18]. In [4] and [16] versions of this principle adopted to the case where R0 is local (and artinian) are used. Now, let us say a few words on a-priori bounds o~ extended Se~eri type, e.g. bounds on the cohomological deficience functions n ~ A~(n) (cf. (1.13)) and the cohomological postulation numbers ~ (cf. (1.14)) for i ~ f(~), without any restriction on i. 3.9.
Theorem. Let i ~ N0. Then, in the notation
a) u} ~ - (2(1+ b)
~:o
IA~(n)l ~2(1 + ~j:o
(~)(h~(-j)
(})(h~(-j) + IP~(-J)I))
Proo£" See [17, Sec. 3].
of (1.12-14):
, (n _<
CohomologicalInvariants of Coherent Sheaves
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3.10. Remarks and Problems. (i) Here again, the polynomial bounds of Theorem (3.9) are not the sharpest possible bounds. In [17] we give sharper bounds in terms of certain piecewise polynomial and recursively defined bounding functions. (ii) Similar as in (3.8) (ii) one mayask Does the polynomial bound (3.9) a) "reflect the rate of growth" of the invariant ~? (iii) It is easy to see that the methodof linear systems of general hyperplane sections (cf. (3.8) (iii)) cannot be used to derive bounds of extended type. So, in order to prove (3.9), one has to apply a different method: the method of sequences of admissible linear forms. This method was introduced by Matteotti [61] and turns out to be successfull even in greater generality than here. (iv) It turns out, that the 2(i + 1) numerical invariants h.°(0),...,hi(-/), p~.(0),... ,p~.(-i) do not form a minimal lower bounding system for the variant ~oi on the class C (s. (3.4) (i)). In fact, one has (s. [17, (4.12)]): Let i E N0 and let k ~ {0,..., i - 1}. then, ~he 2i invariants hJ.(-j), (j =O,...,i), andpi.(-j), (j =O,...,i-1; j#k), form a lower bounding system for the invariant Uoi on the class C. This observation gives rise to the following question: For which sets I~ C_ {0,..., i) do the invariants j . ( -j), (j -- 0,..., i), p~. (-/), (l E 1~) form a minimallower bounding system for the invarian~ ~oi on the class C? ¯ 4.
PURELY DIAGONAL BOUNDS
Keep the previous notations and hypotheses. Weshall discuss in this section ,--i , .~ ~ cd(.T’):-dim(.T’) as a bounding the role of the full cohomologydiagonal (n~:(-z))i= o system in the special case where the base ring R0 is a field. For these considerations the case where R = K[x] = K[x0,...,Xr] is a polynomial ring over a field K is crucial. Webegin with someprerequisites. 4.1. Definition and Remark. (i) Let Mbe a finitely generated and graded module over the positively graded homogeneous ring R = Sn>ORn. Let R+ := ~>oRn C R be the irrelevant ideal of R and let k ~ No. For a graded R-module T = @nezTn let end(T) := sup{n ~ Z Tn# 0} denote the end ofT. Wedefine the regularity of Mat and above level k by (cf. [21, (15.2.9)]) sup(end(H/R+(M)) q- i [ H~t i _> bearing in mind regk(M) that the:~-local cohomologymodules + (M) carry a natural grading. Then red(M) :-- reg°(M) is the Castelnuovo-Mumford regularity of the module M, as it was introduced by Ooishi [68].
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(ii) Keep the notations and hypotheses of part (i). Let Proj (R) and let ~ := AT/be the sheaf of Ox-modules induced by M. Then, on use of the Serre-Grothendieck Correspondence (1.16), (1.17) we rega(~) = rega+~(M), (Vk >_ In particular we have (cf. [21, (20.2.4)])
= e 2(M) reg(i)
= max{reg(:F),end(U°R+ (M)), end(H~+ (i))
(iii) Now,let K[x] = K[x0,..., xr] be a polynomial ring over the field K.. Let Mbe a finitely generated and graded K[x]-module with a minimal graded free resolution of the form by (p) 0 ~ @i=lK[X_](ai ) @i=l b0 K[x](ai(0) ) ~ M ~ ~"" --~ ~i=1 Then, the well knownsyzygetic characterization of regularity (cf. [21, (15.3.7)], [27]) gives reg(M) max{-a~j) - j [ 0 _<j _ ~ p, 1 ORn be as in the introduction. Let M = ~nEZMnbe a finitely generated and graded R-module. Wedefine the generating degree d(M) of Mas the smallest integer d such that Mis generated over R by homogeneouselements of degree _< d. Thus d(M) := inf{d e Z[M = @n<_dMnR}. Now, we may formulate the following bounding result. 4.3. Theorem. Let r ~ N. Then, there is a polynomial Pr ~ Q[s, t] such that for each s ~ N, each field K, each polynomial ring K[_x] = K[Xo,.... , xr] and for each graded submodule M C_ $~ K[x] reg(M) <_ Pr(s, d(M)). Proo~qSee [15, Sec. 2].
[]
4.4. Remark and Problems. (i) One way of obtaining a bounding polynomial Pr ~ Q[s, t] as in (4.3) is to use the classical Hentzelt-Hermann bound for the generating degree of the kernel of a polynomial matrix, (s. [44], [45] but also [59], [71] and [76]). This bound says that the kernel Ker(F) C g[x] ~t of a matrix sx F t= [f~j [ 1 <_ i <_ s, 1 <_ j <_ t] ~ K of size s × t with entries fij ~ K[x] is generated by elements of degree < (2sdeg(F)) 2~, where deg(F) := max{deg(fij) I 1 < i < s, 1 _~ j :~ Starting form this bound one obtains P~(s, t) ---- (2st)2~(~+’)~ Q[s, t] as a possible choice for Pr.
Cohomological Invariantsof Coherent Sheaves
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(ii) A considerable smaller bounding polynomial Pr than in part (i) is tained if one proceeds as in the thesis of A.F. Lashgari [53]: First, the regularity criterion of Bayer-Stillman [3] is extended from the case of graded ideals of K[x] to the case of graded submodulesof K[_x]~s for arbitrary s E N. Then adopting the arguments which furnish the Bayer-Mumford regularity bound for graded ideals in K[x] (cf. [2]) one obtains Pr(s,t) = Set (2t) r! e Q[s,t] aS a possible candidate for Pr, wherethe exponents er are defined recursively by eo :-- 0 ; em= mem-I q- 1 for m ~ N. (iii) There is some evidence, that the bounding polynomial of part (ii) for apart from giving sharp bounds either: Namely, for a graded ideal a C_ K[_x] we may apply theorem (4.3) with the bounding polynomial Pr(s,t) mentionnedin part (ii) and obtain reg(a) <_ (2d(a)) r!. This is the regularity bound of Bayer-Mumfordfound in [2, (3.8)] and mentionned already in part (ii). It is still an open problem, whether in our special case one has the sharper estimate reg(a) < (2d(a)) 2~-1. For base fields K of characteristic 0, this inequality is in fact true (cf. [2, (3.7)], [32], [33]). On the other hand, examples due to E. Mayrand A. Meyer[62] show, that this latter bound is "fairly close" to be sharp. Moreprecisely, for each r ~ l~, there is a graded ideal ar C_ C[x0,... ,Xr] such that d(ar) = 4 and reg(ar) >_22[(~-1)/1°1 + 1 (cf. [2, (3.11)]). (iv) In view of the above observations it seems natural to ask, whether Theorem (4.3) one might expect a mere "doubly exponential" bound reg(M). More precisely: Are there constants a, b > 0 such that for each r ~ N, each s ~ N, each polynomial ring K[x] = K[x0,... , Xr] and each graded submodule M C_ K[x_] ~s we have reg(M) <_ (asd(M))b~? Now, let X = Proj(@n>_oRn= R) and ~- be as in the introduction. Let ¯Tv := Homox (~, Ox) be the dual of ~’. Our next result says that, over projective spaces, the "cohomology diagonal of a coherent sheaf bounds the regularity of the dual sheaf’. 4.5. Theorem. Let r ~ No. Then, there is a polynomial Qr ~ Q[t0,..., t~] such that for each field K and each coherent sheaf of O~,k-modules ~ we have reg(Y:v) <_ Q~(h~-(0), h~-(-1),..., h~(-r)). Proof." See [15, Sec. 3].
[]
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Brodmann
4.6. Exampleand Problems. (i) For a locally free sheaf9r of (.9~.-modules (e.g. an algebraic vector bundle over ~.) Serre-duality gives hr:r-i(-n- r- 1) for all i E {0,...,r} and all n e Z. Fix j E {0,...,r - 1} and let t ~ No. Consider the minimal combinatorial pattern Pj,t:=({O} xN) U({j} x {k l -j - t _< k _< -j}) W({r} xZ <-t-r) of width r. Then, there is an indecomposablealgebraic vector bundle br),t of rank _< r! such that P(~j,t) = Pj,t (cf. (2.11) (i)). Consequently,P(~’jJ,
({0} x z_>z)u - j} × {k I J - - < k < j - In particular, we have reg(.~j,t) = 0 and h)y.t(-i) = 0 for all i 7~ j and reg(.~,t) = t. This shows at the same time that the regularity of a coherent sheaf of Oe~¢-modules .7" need not bound the regularity of 9vv and that the full cohomology diagonal is a minimal upper bounding system for the regularity on the class of all these sheaves. (ii) In view of the previous observation one is lead to ask: Which invariants of lP bound the regularity of ,~v? As a consequence of Theorem(4.5) one now gets that the "full cohomology diagonal bounds the cohomological postulation numbers". To formulate this result, let us introduce the following notation. 4.7. Notation. Let ~ denote the class of all pairs (X,~’) in which Proj(@~>oR~) is a projective schemeover a field R0 = K and .~ is a coherent sheaf of (_gx-modules. For r ~ No let ~(r) denote the class of all pairs (X, r) ~C with di m(~)(= cd(:~)) = Obser ve that in th e n otat ion of (3.4) (i) and (3.6) wehave ~ _CC and if(r) Now, we are ready to formulate the announced bounding result. 4.8. Theorem. Let r ~ No. Then, there is a polynomial Tr ~ Q[t0,..., such that for each pair (X,.T) ~ d(r) and each i ~ {0,..., r} we
tr]
t/~ >_ Tr(h~(O),h~(-1),...,h)(-r)). [] Proof." See [15, Sec. 4] or [53]. As an interesting consequence of the previous result we get that "the fuI1 cohomology diagonal numerically bounds cohomology". 4.9. Corollary. Let i, r ~ No and let h°,..., hr ~ N0. Then, there are only tinitely many cohomologica]Hilbert functions h~ with (X, ~) ~ d(r) and hJ~(-j) = hJ for j = 0,...,r. Proof." See [15, Sec. 5] or [53].
[]
4.10. P~emark and Problems. (i) For r = 2, the conclusions of (4.8) (4.9) even hold on the class (2), e.g. o ver a rbitrary a rtinian b ase r ings R0 (cf. [17, Sec. 4], (3.10) iv)). (ii) The observation madein part (i) gives rise to the question:
111
Cohomological Invariantsof Coherent Sheaves
Do the results (4.8) and(4.9) hold for arbitrary v on the class C(r)?
5. _b-SHEAVESANDHILBERTCOEFFICIENTS A fundamental issue of algebraic geometry is the relation between the so called Hilbert coefficients and the regularity of coherent sheaves - notably for sheaves of ideals or for invertible sheaves. Results of this type are of basic significance for the theory of Hilbert schemes and Picard schemes (cf. [34], [40], [57], [58], [38], [51], [65]). 5.1. Reminder and Remark. (i) Let X = Proj(R = (~n>_ORn) and $r ~ 0 be as in the previous sections and assumein addition that the base ring R0 is artinian. For i E {0,... ,cd(~v) = dim(.T’)} let ei(.T’) E Z be the i-th Hilbert coettlcient of $’, so that the Hilbert-Serre polynomialof .T" is given by dim(Y) t +dim(.~)-i) X~-(t) = E (-1)iei(’~)( dim(S’) i=o Keep in mind that e0(~) E N and that r) dim(b X~:(n) : ~ (-1)ih~-(n) i=0
e Q[t].
for all
Sometimesit is useful to introduce the invariants ai(~) -~ (--1)dim(~’)-iedim(.~)_i(~’), (i = 0,..., so that dim(~) (t + i)ai(~)
x (t)
i
’
(ii) Let K be an algebraically closed field and let r ~ N. A fundamentalresult of Mumford[65] says that the regularity reg(‘7) of a coherent sheaf ‘7 C_ (.9~( of ideals is boundedpolynomially in terms of the Hilbert coefficients eo(‘7),..., er(‘7) of For each m ~ N let ~T"m := O~(-m) @O~(m),so that eo(~m) = 2, el (.T’I) = 0 and reg(~m) = m. This example shows that, even on the class of fully decomposable algebraic vector bundles over ll~(, regularity is not bounded by the Hilbert coefficients. . In spite of the failure just observed, there is an extension of Mumfordsbounding result mentionnedin (5.1) (ii) to a larger class of sheaves. This extension is due to S. Kleiman(s. [38, Exp. XIII]) and based on the notion of b-sheaf. Werecall this notion for the readers convenience.
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Brod~nann
5.2. Definition and Remark. (i) Let X Proj(R = @n >oRu) and ‘T be as in the introduction. A sequence of global sections fl,..., fT E H°(X, (9~ (1)) is said to be ‘T-regular, if Hi NAssx(‘T[H~N...NHi_~)= 0 for i = 1,...
,r,
where Hi C_ X is the subschemedefined by fi. It is equivalent to say, that the natural homomorphismsof sheaves
./~ : .~" tH~n...nH,_, (n) ~ ‘T [H,n...nH~_~(n + 1) are injective for all i E {1,...,
r} and one (resp. all)
If R0 contains an infinite field, there are .T-regular sequences fl,..-, H°(X, Ox(1)) of arbitrary length
fr
(ii) Now, let X Proj(R : @n >_Ol~n) and ‘T be as above andassu me, that R0 is in addition artinian. Let r E No and let b = (b0,..., br) E l~0+1¯ Then, ‘T is said to be a b-sheaf, if dim(~’) _< r and h°:r
IHln...nHi
(-71)
~ bi for i : 0,...,
r,
where again Hy C_ X is the subscheme defined by fj. Then, Kleimans Main Theorem on b-sheaves says 5.3. Theorem. (cf. [38, Exp. XIII, Th~or~me 1.11]) Let r ~ No. Then, there is a polynomial U~ ~ Q[t0,..., tT] st~ch that for each algebraically closed field K, each projective scheme X over K and each coherent sheaf of Ox-modules which is a b = (b0,..., br)-sheaf of dimension r we have reg(.T) <_Ur(b0- a0(.T’),..., br -- ar(~)) 5.4. Remark and Problems. (i) Let X be a purely r-dimensional reduced projective scheme of degree d over an algebraically closed field K. Then, on use of Bertini, it follows easily that Oxis a (0,..., 0, d)-sheaf. As the property of being a _b-sheaf is inherited by coherent subsheaves(cf. [38, Exp. XIII, Prop. 1.6])it follows from (5.3) that reg(~7) <_ U~(-a0(~7),...,-ar-l(~), ar(~)) for each coherent sheaf of ideals ~7 _COx.This clearly recovers the bounding result of Mumfordmentionnend in (5.1) (ii). (ii) Let K be an algebraically closed field, let ml,...,m, ~ Z and let ‘T t @i=~O77¢(mi) be a coherent subsheaf. It follows easily from (5.3) that reg(‘T) is polynomially boundedin terms of the Hilbert coefficients ei(‘T) of ~: and the integers rnj. In [21, Chap. 17] we proved an algebraic version of this, in which K maybe replaced by an arbitrary artinian ring. (iii) Now,let X Proj(R = (gn>oR,~) and ‘T as in (5. 1) (i) . Ass ume tha f~,...,fr ~ H°(X, Ox(1)) form an ‘T-regular sequence. For j ~ {0,...,r}
Cohomological Invariants of Coherent Sheaves
113
let Hj C_ X denote the subschemeof X defined by fj. Then, it follows easily by induction on k, that k
hi~Hln...nHk
(n)<_ E (~)h~ j(n-
j);
j=0
k k(j) h j~(-3), ¯ (k = 0,... it f oll ows So, if dim(~’) _< r and with bk := ~-~j=o that 7"(1) is b = (b0,..., bT )-sheaf. Thus, if R0 contains an infinite field, for each coherent sheaf 2: of Oxmodules of dimension r with given cohomology diagonal the twisted sheaf .T(1) is a b-sheaf, where__b = (b0,..., bT) is defined as above. It is easy to that the converse of this does not hold: If K is an algebraically closed field, one maychoose a sequence (‘Tn)n~N of coherent sheaves of ideals ‘Tu C_ O~,~ such that limn-~ reg(‘Tn) = o~. Then ¯ r clearly the set of cohomologydiagonals i e N} is unbounded by (3.2), whereas the sheaves (~Tn(-1))(1) = ~Tn are all (0,..., sheaves (cf. part (i)). (iv) For r ~ No let ~(r) be the class of all pairs projective schemeover an algebraically closed field and ~- is a coherent sheaf of (gx-modules with dim(~’) = The observations of part (iii) makeus ask the following question: Let b = (b0,..., br) e ~0 +1 , (c0,... , Or) e ~0+l and let D be the class of a11 pairs (X, .T) ~(r) inwhich .T is a b-sheaf and for which [ei(gr)[ <_ci for i = 0,... ,r. Is the set of cohomology i " r diagonals {(h~=(_~)(-~))~:0 [(X,.T) fi nit e? Obviously, the best answer to this would be given by polynomial upper bounds for the numbers h~(_D(-i i 1) - in temns of the numbers ) --- h~(-i b~ and the Hilbert coefficients ej (~). (v) In view of [38, Exp. XIII, Th~or~me(6.4), (6.7)] one also could ask following modified version of the previous question: Let b, c_ be as in part (vi), let m ~ No and let ~)~mresp. Z)~ the classes of all pairs (X, 5") ~ ff(~) for which .T(m) is generated by globMsections and :~ is a b-sheaf resp. [ei(~)l <_ ci for i =0,..., r. Are the sets {(h~-(_~) ~ " ~ [ (X,.T) (-z))~: 0 and {(h~:(_~)(-i))[: o I (X,$’) e :D"m}finite? 6. A FEW SPECIFIC
BOUNDS
Now, we shall say a few words about the cohomological invariants of some specific classes of pairs (X, ~’). Westart with pairs (l?~:, .7) with r where K is an algebraically closed field and ~7 C_ C9~ is the sheaf of
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Brodmann
vanishing ideals of a projective variety Y C_ ~, e.g. a closed reduced irreducible subscheme. Wealways shall assume in this situation., that Y is non-degeneratedly embedded, e.g. that it is contained in no hyperplane
_: 6.1. Reminder, Remark and Problem. (i) Keep the above hypotheses and notations. The function ha : Z -~ No is called the Hartshorne-Rao function of the variety Y C_ l~. As Y is reduced and K is algebraically closed, we have hl~(n) = for al l n _<0. Write @n~zH°(F~(, (.9 ~(n)) = K[x0,..., xr] = K[x], let I := ~n~zH°(IP~:, J(n)) C_ K[x] be the homogeneous vanishing ideal of Y and let A := K[Y] := K[x]/I be the homogeneous coordinate ring of Y. As J = f, the Serre-Grothendieck correspondence (1.16), (1.17) yields ha(n ) = dimK(H~[~+(I)n) for all n e Z. So, graded short exact sequence 0 ~ I ~-~ K[x] ---~ A ---~ 0 shows that ha(n ) = dimK(H~+(A)n) for all n e Z. (ii) Keep the previous notations and hypotheses, and let us briefly recall the geometric meaningof the numbersha(n) . First of all, let us recall that ha(1 ) is the projection excess of Y C_ 17~, e.g. the largest number e ~ No for which there is a non-degenerate projective variety Y’ _C IP~. +e and a projection 7r: I~(+e \ £~(-~ --~ l?~( from a linear subspace ~7 -1 ~_ ]~-t-e (with the convention that I~ ~ = ~) such that ]~7 -1 if)Yt : ~ and ~r induces an isomorphism ~r [: Y’-~Y. For .arbitrary n ~ N let y(n) C_ ~(~+~)-~ the n-th Veronesean of Y and let ~ ,- 1 (~+’~ ~t(n) ]~K be the linear subspace spanned by y(a). Then ha(n ) is ~g _C nothing else than the projection excess of y(n) t(n __Cl~~: So here, the Hartshorne-Rao function has a description in purely geometric terms. (iii) The Hartshorne-Rao function of space curves Y C_ ~X has found ~nuch interest recently and here, very satisfactory results have been proved (cf. [64] for example). In [20] the structure of the Hartshorne-Rao module ~(A) ~ ~ (~rK, 3"(n)) of a curve Y C_ ~ of degree r+2 is completely described. @nezH The fact, that in this situation only 4 different Hartshorne-Rao functions occur if r >_ 4 allows a fairly good understanding of these curves. The :ideas of [20] have been used in [10] to study surfaces Y C_ I~( of degree r + 1. But here, we have not been able to describe all occuring Hartshorne-Rao functions. So, let us ask the following question: Which are the possible Hartshorne-Rao functions of surfaces Y C_ ~rK of degree r + 1? A related problem (open even in the surface case) is: What is a sharp upper bound for the Hartshorne-Rao function of a projective variety Y C_ ~rK of degree ~_ r - dim(Y) + 3 ?
Cohomological Invariantsof Coherent Sheaves
115
6.2. Remarkand Problems. (i) Let K be an algebraically closed field and let Y C_ lP~ be a closed subscheme, ff ~_ O?~its sheaf of vanishing ideals. The study of the regularity reg(Y) := reg(ff) is an extremely active field of algebraic geometry. One of the most appealing problems in this field is a conjecture of Eisenbud-Goto [27] saying that reg(Y) <_ deg(Y) - codim (Y) whenever Y is reduced and irreducible. For smooth curves in I?~, the above estimate corresponds to Castelnuovo’s classical result (cf. [25]). For curves Y C_ F}(, the above inequality has been established by Gruson-LazarsfeldPeskine [39]. For smooth surfaces and threefolds Y C_ I?~ in characteristic 0, the stated inequality has been shownby Lazarsfeld [54] and Pinkham[69] respectively by Ran [70]. (ii) Besides the proves of the above inequality a tremendeous amount upper boundsfor the regularity of specific classes of projective schemeshas been established (s. [46], [67] , [78] for example). For investigations of the same kind in some modified context see also [80] or [81]. The regularity of projective schemes is of fundamental interest for computational algebraic geometry (s. [2] for example) and thus related to Grbbner base techniques (cf. [24], [26], [33], [46], [57], [81]). (iii) For arbitrary l-Buchsbaum surfaces in characteristic 0 the EisenbudGoto inequality of part (i) has been established in [22]. The essential problem was to establish this inequality for l-Buchsbaumsurfaces Y ~_ IP~ of degree r + 1. In [10], the Eisenbud-Gotoinequality has been established in arbitrary characteristic for a larger class of surfaces Y C_ P~ of degree r + 1, but not for all of them. So we ask the following question - closely related to the problemin (6.1) (iii): Do all surfaces (resp. all projective varieties) Y C_K of degree r + 1 (resp. of degree <_ r - dim(Y)+ 3) satisfy the Eisenbud-Goto inequality? ¯ 6.3. Remark and Problems. (i) Elencwaig and Forster [28] have shown that all cohomological Hilbert functions h~ of an algebraic vector bundle £ over a complexprojective space I?[2 are boundedin terms of the first two Chernnumberscl (£), c2 (~) and the span a(£) of the generic splitting type ~. If ~ is in addition semistable, the Grauert-Miilich theorem gives bounds on the functions h} in terms of c~(~),c2(£) and rank(£), thus generalizing a result of [41]. In [5] and [18] such bound of Elencwaig-Forster type are studied over algebraically closed base fields K of arbitrary characteristic by means of the method of linear systems of hyperplane sections. (ii) In (2.11) we have asked already a few questions related to the cohomology of algebraic vector bundles over a projective space. Remember,that for a coherent sheaf ~" of O~-modules ~" with a given cohomology diagonal only finitely many different cohomological Hilbert functions h~- may occur
116
Brod~nann
(cf. (4.9)). This makes us ask, under which conditions on the cohomology diagonal all the occuring functions h~- with i < r are "left vanishing": Is there a non-trivial criterion on the full cohomology i diagonal ~~h ~:(-i) ~rJi=o of a coherent torsion free sheaf of -modules ~ which induces that .~ is a vector bundle? 6.4. Remarkand Problems. (i) One also can just study the cohomological Hilbert functions hi where X C P~c is a projective variety (e.g. a c][osed Ox ’ -reduced and irreducible subschemeof ?~.) over the algebraically closed field K. Bearing in mind the short exact sequence 0 -4 ,.7 -~ O?~¢ ---40x -~ 0 (in which ,.7 _C O~¢is the sheaf of vanishing ideals of X) this essentially comes up to study the cohomological Hilbert functions h~ for i > 1, e.g. different from the Hartshorne-Rao function. Using the method of linear systems of hyperplane sections, some bounding results on these functions are deducedin [5], [6] and [8]: (ii) Amongthe fundamental vanishing theorems for (local) cohomologygroups on maydistinguish results of Hartshorne-Lichtenbaum type (cf. [9,1, Chap. 8] and [48] notably) which concern cohomologydefined with respect to open subsets of projective schemes and results of Kodaira type (cf. [38], [82], [66]) which are closely related to the vanishing of the numbers h~x (n) the range n < 0, i < dim(X), where X is a projective variety over an algebraically closed field K. Wefocus only on the second type of result, tlere, the situation considered in the geometric context actually is more general: On consideres an ample invertible sheaf £ of Ox-modules and studies the vanishing of the groups Hi(X,£®~) for i (dim(X) and n ( 0. Choosing £-. = Ox (1) one obtains vanishing statements for the numbershiOx (n) in the requested range. Let us recall the main result of [~2] which says that the above cohomology groups Hi(X,£®’~) vanish if X is smooth and K :-- C. Let us also recall a vanishing result of Mumford[66], which says tha~ the groups H~(X, £®~) vanish for all n < 0 if X is normal of di~nension > 1 and if Char(K)= 0. In [66] it is also shownthat this result fails if Char(K) (iii) In positive characteristic one has at least someboundingresults for the dimensions h~(X, £~) := dim,v H~(X,£®’~). So, in [1] it is shownthat ¯with e(X)
:= ~ dimn(H~x.,(Ox,p) ) ~x,{~}=(p}
we have e(X) <_ h~(X, £®~) _< min{e(X), h~ (X, Ox) + n(h°(X, £) - 1)} for all n < 0, if X is of dimension ~ 1 and has only finitely many :nonnormal points. In [1], we actually consider a more general situation, "with the aim to prove a relative version of the mentionned bounding result: The canonical morphism X --4 Spec(K) is replaced by a projective morphism X -4 Xo = Spec(Ro) with geometrically normal fibres (cf. [36]), where
Cohomological Invariants of Coherent Sheaves
117
is a domainessentially of finite type over a perfect field. Wehave not been able to prove similar bounds for higher cohomologygroups. So let us ask the following question: Are there upper bounds of the above type for the numbers hi(X, £.®n) in the range n < 0, i < dim(X) ifX is smooth? (iv) One consequence of the bound mentionned in part (ii) is h1 (X, (fix (n)) = for a ll n on-degenerate surfa ces Z C_ 17~ with secti onal genus < r and having only finitely many non-normal points. In [11] we have applied this fact in order to give a characteristic free approach to the cohomological behaviour of non-degenerate surfaces X C_ ~r of degree <_ 2r - 2 and having only finitely many non-normal points. This leads to a generalization of a corresponding result of Chern and Griffiths for smooth complex projective surfaces. It seems natural to ask whether these results may be generalized to amply polarized surfaces: Let X be a projective surface with only finitely manynon-normal points and let £. be an ampleinvertible sheaf with deg(£:) 2h°(X,/:) - 4. Is it true that hl(X,£. ®n) = e(X) for all n < ~h°(X’£)-I induced What is the nature of the morphism X ~ --K by £,
ifdeg(£)
: 2h°(X,
£)
-
4?
REFERENCES
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ALBERTINI,C., BRODMANN, M., A bound on certain local cohomology modules and application to ample divisors, to appear in Nagoya Math. Journal. BAYER,D., MUMFORD, D., What can be computed in algebraic geometry? in: "Computational Algebraic Geometry and Commutative Algebra", Proc. Cortona 1991 (D. Eisenbud, L. Robbiano. Eds.) 1-48, Cambridge University Press (1993). BAYER,D., STILLMAN, M., A Criterion for dedecting m-regularity, Invent. Math. 87 (1987) 1-11. BRODMANN M., A lifting result for local coh0mology of graded modules, Math. Proc. Cambridge Phil. Soc. 92 (1982) 221-229. BRODMANN M., Bounds on the cohomological Hilbert functions of a projective variety, Journal of Algebra 109 (1987) 352-380. BRODMANN M., A bound on the first cohomology of a projective surface, Arch. Math. 50 (1988) 68-72. BRODMANN M., A priori bounds of Castelnuovo type for cohomological Hilbert functions, Comment.Math. Helvetici 65 (1990) 478-518. BRODMANN M., Sectional genus and cohomology of projective varieties, Math. Zeitschrift 208 (1991) 101-126. BRODMANN M., A priori bounds of Severi type for cohomological Hilbert functions, Journal of Algebra 155 (1993) 298-324. BRODMANN, M., Cohomology of certain projective surfaces with low sectional genus and degree, in: "Co~nmutative Algebra, Algebraic Geometry and Computational Methods" (D. Eisenbud Ed.) 173-200, Springer (1999). BRODMANN, M., Cohomology of surfaces X C P" with degree _< 2r - 2, in: "~’~om w mutative Algebra and Algebraic Geometry" (F. van Oystaeyen Ed.) 15-33; M. Dekker lecture notes 206, M. Dekker (1999).
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A PARTIAL
SURVEY
OF LOCAL
COHOMOLOGY
Gennady LyubeznikUniversity of Minnesota,Minneapolis,Minnesota
The word ’partial’ has two meanings: not complete and not impartial. Bo~hare intended here. This paper is a survey of some of the developments in local cohomologyin the last decade in which we took some part. Other relatively recent expositions of some aspects of local cohomologyare found in the book by Brodmannand Sharp [14] and the papers by Schenzel [84] and Hunekc[46]. The discovery of numerous connections between local cohomology and other fields - from D-modulesto topology and group cohomology- has been a striking feature in the last decade. It works in both directions: while local cohomology sheds some new light on topology and group cohomology, the theory of D-modules helps understand the structure of local cohomology modules. Local cohomologywas introduced by A. Grothendieck in [33, 35]. Among the manyapplications given in [35] was a proof of Samuel’s conjecture to the effect that a local complete intersection ring of dimension at least 4 is parafactorial. A muchsimplified version of this proof is given in our joint paper with F. Call [17]. Weassume the reader is familiar with basic concepts of commutativealgebra, homological algebra, algebraic geometry and topology. Most proofs are omitted. A number of open problems and questions ~re stated throughout the p~per. Wehope this survey will be interesting both for the novice and for the expert.
121
122
Lyubeznik
This survey covers four broad topics, namely, some foundational, material (Sections 1 - 3), finiteness properties of local cohomologymodules(Sections 3 - 7), F-modules(Sections 8 - 13) and finally, cohomological dimension topology of algebraic varieties (Sections 14- 17). Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
Definition and some elementary properties of local cohomology. Some uges of local cohomology. D-modules. Finiteness properies of local cohomologymodules: statements of results. Finiteness properties of local cohomologymodules: a discussion of proofs. Associated primes of local cohomology modules. Numerical invariants Ap,~ of local rings¯ F-modules. F-finite modules. F-finite modules have finite length in the category of F-modules. Applications of F-modules: cofinite A{f}-modules. Applications of A{f}-modules: the stable part and the F-depth. Application of F-modules: D-modules. Upper bounds on cohomological dimension. l~tale cohomology and cohomological dimension¯ Topology of algebraic varieties of small codimension in n-space. The Picard group of algebraic varieties of small codimension in projective space. 1.
DEFINITION
AND SOME ELEMENTARYPROPERTIES OF LOCAL COHOMOLOGY
If X is a topological space, a subset V C X is locally closed iff V is the intersection of an open subset and a closed subset of X. Let F be an abelian sheaf on X. If V is a locally closed subset of X and U ~ V is an open subset of X in which V is closed, one sets Fv(X, F) to be the subgroup of the sections s E F(U, F) of F over U supported on V, i.e. s E Fv(X, F) iff the restriction of s to the complementU \ V is zero. It is not hard to check that this definition is independent of U and the functor Fv(X,--) left exact. One sets the i-th local cohomology group H~(X,F) of F with support in V to be the i-th right ’ derived functor of Fv(X, -) evaluated F. Since Fv(X,-) is left exact, H~(X,-) ~- Fv(X,-). Proposition
1.1.
V C X is locally
closed
iff
there
exists
an abelian
sheaf
Zv, x on X such that the restriction of Zv, x to V is the constant sheaf of integers 7~y and the restriction of Zv, x to the complement X \ V is zero.
It is not hard to check that Fv(X, F) Hom(Zy, x,F), H~(X,F) ~- Exti(Zy, x, F). Proposition of X, there
1.2. a) If V = V2 \ V1, where VI C V2 are two closed is a long exact sequence 0--~ H~I(X,F ) --+ H~2(X,F ) --~ H~(X,F)--~ --~ H~ (X, F) --~ ~ ( X, F ) - -~ H ~(X, F ) -
subsets
123
Surveyof LocalCohomology b) If V is a closed subset of X, there is a long exact sequence 0 --~ H~(X,F) ~ H°(X,F) -+ H°(X \ V,F) -~ H~(X,F) --+ HI(X,F) --} HI(x \ V, F) Proof. The first long exact sequence results Horn(-, F) to the short exact sequence
from applying the functor
0 -+ Zva,x -+ Zv~,x -+ Zv, x -~ 0 while the second is the special case of the first considering that H~x(X,-) ~- Hi(X,-)
and H~x\v(X,-)
with V1 = V and V2 = X
~- Hi(X \ V,
because ~- F(X \ V, -). Uom(Zx,x,-)-~ F(X,-)and HOE(Zx\v,x,-) If X = SpecRis an affinescheme,whereR is a colnmutative Noetherian ring,V = V(I),whereI is an idealof R, and F = ~ is the quasicoherent sheafon X associated to the R-module M, we denoteFv(X,/~/) andH~I(X,I~I) by,respectively, Ft(M)andH~(M).SinceX is affineand ~ is quasi-coherent, Hi(X,I~I) = for i> 0,so by 1.2b weget Corollary 1.3. There is an exact sequence 0 --~ H~(M) -+ M -~ H°(X \ V, IQ) ~ H~(M) and isomorphisms Hi~(M)~- Hi-I(X \ V,l~/I) for i > O. Corollary 1.4. Let fl,.., fn generate I. Then H~(M)is the i-th cohomology module of the complex 0 --~ M~ @iMf~-~." ~-~ @io<...
e~ : Mi~o...~...y~,+, ~ Mf~o..%+ ~
is the natural localization map. Proof. This is a consequence of the previous corollary considering that the complexin question coincides in degrees greater than or equal to 1 with the ~ech complex for the computation of Hi(x \ V, l~/I) by means of the covering of X \ V by the open affines SpecRA. []
124
Lyubeznik
Proposition 1.5. Let R be a commutative Noetherian ring and let I C R be an ideal. a) H}(-) is the i-th right derived functor in the category of R-modules the functor FI(-). b) Let I1 D I2 D ... be a descending chain of ideals of R cofinal with I, i.e. such that for each n there is m with I n ~ Ira and In D Ira. Then Hi~(M) "" F~___~Exti(R/I~ Proof. a) If 0 -~ M-+ 27° --~ Z1 --~ ... is an injective resolution of Min the category of R-modules, the associated sequence of quasi-coherent sheaves 0 -+ ~ -~ ~70 _~ ~1 _~ ... is an exact sequence of flasque sheaves. Flasque sheaves are Fv(X,-)-acyclic for any topological space X and. any locally closed subset V C X, so this resolution maybe used to compute the local cohomology groups of M. [] b) follows from a) and the fact that FI(-) = li.i_~Hom(R/In,-). 2.
SOME USES
OF LOCAL
COHOMOLOGY
Let k be an algebraically closed field and let V C A~ be an algebraic set in the affine n-space, i.e. the set of solutions of a system of equaLions fl = 0,..., fv = O, where fi E kiT1,..., Tn] are polynoInials in n vari~bles T1,...,Tn with coefficients in k. What is the minimumnumber of equations defining V? That is, what is the minimumnumber s such that there exists a systemof s equations gl = 0,... , gs = O, wheregi ~ k[T~ , . . . , Tn], whose solution set is V? Hilbert’s Nullstellensatz implies that this minimumnumber of equations s equals the minimum number of polynomials g~,...gs ~ k[T~,...,Tn] that generate the ideal (fl,...,fv) up to radical, i.e. vf] = v/-~, where I = (f~,...,fv) and g = (g~,...,gs). The arithmetical rank of an ideal I of R, denoted araI, is the smallest s such that there exist s elements g~,...,gs that generate I up to radical. To prove that araI _< s all one has to do is find s elements that generate I up to radical. But how does one find a lower bound on araI? A trivial lower bound is araI _> height/ because adding one more element to the set of generators of an ideal increases the height of the ideal by at most one. Local cohomology provides a much better lower bound on araI as Proposition 2.2 shows. Definition 2.1. If X is a scheme and Y C X is a locally closed subscheme, the cohomological dimension of X with support in Y, denoted cd(X,Y) (resp. the cohomological dimension of X, denoted cd(X)),is the biggest such that there exists a quasicoherent sheaf .~ on X with H~,(X, r) ¢0 (resp. gi(x,J 7) ~ 0). If R is a ring and I C R an ideal, the cohomological dimension of R with support in I, denoted cd(R, I), is the biggest i such that there, exists R-module M with H~(M) ~ O. Clearly, cd(R, I) = cd(SpecR, V(I)). Proposition 2.2. araI >_ cd(R, I) : 1 + cd(SpecR, V(I)).
Survey of Local Cohomology
125
Proof. By definition, Hit(M) depends only on the closed subset V = V(I) C SpecR, i.e. on x/7, not on I itself, if fl,..., fs generate I up to radical, the length of the complexfrom 1.4 is s, so its cohomologyvanishes in degrees greater than s. This proves the inequality. The equality follows from the isomorphisms of 1.3. [] Example 2.3. Let V C A~ be the set of solutions of the system A1 = 0, A2 = 0, A3 = O, where A1, A2, A3 E C[Tll,T12,T2t,T22,T31,T32] are the 2 × 2-minors of the 2 × 3-matrix (Tij). The ideal I generated by A1, A2, A3 in C[Tii,T~2,T2i,T22,T31,T32] is prime of height 2. Hence 3 >_ araI > 2. But H~(R) ~ 0 (cf. [48, 3.13]) which shows that cd(R,I) = 3 and therefore araI = 3, i.e. V cannot be defined by fewer than 3 equations. While an inequality cd(R, I) >_ provides a lo wer bound onara I, an inequality cd(R, I) < s is also useful as the next proposition shows. Proposition 2.4. Let V C l?~ be an algebraic subset of the complex projective n-space and let I C R = C[T0,...,Tn] be the homogeneousdefining n V, . C) --- 0 for i <_n - cd(R, I). ideal of V. ThenHs~ing(~c, i n . V, C) is the singular cohomologyof the pair (F~:, V) with Here HsZing(l~c, i complexcoefficients. In fact Ogus[77], 4.4 gives a precise characterization n . of cd(R,I) in terms ofHs~ing(~C, i V, C) and the singular topology of V. characteristic p > 0 analogue is given by Hartshorne and Speiser [40], 4.2. Proof of Z.4. It follows from 1.3 that H~(A~ \ C(V), F) for i > cd(R, I), where A~ and C(V) are, respectively, the affine cones over ]P~ and V, and F is any quasicoherent sheaf on A~ \ C(V). Since the natural morphismA~\ C(Y) -~ ~ \ is aff ine, it fol lows that H~(II~ \ V , F) = 0 i > cd(R, I) and any quasicoherent sheaf on ~ \ V. There is the spectral sequence of algebraic deRham cohomology E~’q = Hq(HP(]~: \ ~5~/c)) ~ "’d. \ v; C), where ~/c is the standard deRham complex of F~. Since Hi(F~: \ V, f~/c) = 0 for i >_ cd(R, I) and the length of the complex ~/c is n, we see that E~’q = 0 if either q > n or p >_ cd(R, I). Hence i n H~a(I? c\V;C)
=0 for
i>n+cd(R,Z).
Since algebraic deRhamcohomologyis isomorphic to singular cohomology, i Y~ing (~ \ V; C) = 0 for i >
n + cd(R, I).
Finally, Lefschetz duality i n H~ing(F c \ V; C) "" 2n-i . C) = H~ing (]~C, n V,
completes the proof. For general information on algebraic deRhamcohomology the reader is refered to [39]. []
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The above shows that the knowledgeof cd(R, I) (indeed, even inequalities cd(R, I) >_ and cd(R, I) <_ s) yields int eresting inf ormation. Buthow,does one compute cd(R, I)? More generally, how does one decide whether or not H~(M)0?To thi s dayther e is n o algorithm to d eci de this (say’ if R i s a finitely generated algebra over a field, explicitly defined by generators and relations, I is defined by an explicit set of generators and Mis defined by an explicit presentation). However,if R is a ring of polynomials in a finite number of variables over a field, an algorithm to decide whether or not H~(R) = has re cently be en fo und. Actually, th ree di fferent al gorithms, two in characteristic 0 by U. Walther [92] and Oaku and Takayama[76], and the third one in characteristic p > 0 by us [65] (see Section 9 below). The characteristic zero algorithms have been implemented and they show that H~(R) ~ O, where R and I are as in the above example. For more examples of actual computations using these algorithms see Leykin [58]. The two characteristic zero algorithms do not extend to characteristic p and the idea of our characteristic p algorithm is completely different from them. Our algorithm has not been implemented because its complexity grows very rapidly with p; for an actual implementation a more practical algorithm is needed. It is not hard to show that cd(R,I) is the largest integer i such that H~(R) ~ O. Hence the above-mentioned algorithms lead to algorithms for the computation of cd(R, I), where R is a ring of polynomials in a finite number of variables over a field. The characteristic zero algorithms are based on the theory of D-moduleswhich we discuss in the next section. 3. D-MODULES Let k C R be a subring. A k-linear differential operator of R of order _ n is a k-linear map6 : R -~ R such that for every r E R, the commutator [6, ~] is a k-linear differential operator of order _< n - 1, where ~ : R -~ R is the multiplication by r (a differential operator of order zero is just P, for some r E R). It is not hard to see that the k-linear differential operators of order _< n form an R-module.If ~ and 61 are k-linear differential operators of order _< n and _< n’ respectively, then the composition ~ o ~ is a k-linear differential operator of order _< n + n~. Thus the k-linear differential operators form an associative (but in general not commutative!) ring in which the multiplication is defined via the composition. This ring is called the ring of k-linear differential operators of R and is denoted D(R, k). For further foundational material on rings of differential operators see [34], Section 16.8. Let R : k[T~,..., Tnl, or R = k[[T~,..., Tn]] be either the ring of polynomials, or the ring of formal power series in n variables T~,..., Tn over k and ~ oq~ let Di,t = 1Hb~, where ~ isO the partial differentiation of order t with re~ 1 0
spect to Ti. Note that the formal expression ~. b~ defines an honest k-linear differential operator even if some integers are not invertible in k. One can
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show that D(R, k) is the k-algebra generated by the Di,ts. By a D(R, k)module, or simply a D-module, we always mean a left D(R, k)-module. If k is a field of characteristic 0, Di,t is an integer multiple of d~ where di = o-~, hence D(R, k) is the ring extension of R generated by the n elements dl,...,dn. It is not hard to show that in this case D(R,k) is left and right Noetherian, hence every finitely generated D(R, k)-module is Noetherian. But if k is a field of characteristic p > 0, then D(R, k) is not Noetherian! For the rest of this section k is a field of characteristic 0 and R k[T~,... ,Tn] is the ring of polynomials in n variables over k. It is easy to see that diTi = 1 + Tidi while didj = djdi and diTj = Tjdi for i ¢ j. Using these identities one shows that every element 5 E D(R,k) can be ~, where uniquelywritten as a finite sum ~cit,...,in,jl ..... inTO1...AnTin,4J~l ¯ .. d~n cit,...,in,j~,...,j ~ ~ k. The Bernstein filtration on D(R, k) is the filtration 0 : F-1 C k : F0 C F1 C ... where F~, for t _> 0, is the k-submoduleof D(R, k) spanned by the ~with monomials T~I... q,in.4J~.n ~¯ "’din ii +""+in +jl +""+jn _< t. If5~Ft and 5’ ~ F~,, then 55’ E Ft+t, and 55~ - 5’5 ~ F~+t,_l. This implies that the graded object grrD --- @t>_0Ft/Fe-1 is a graded commutative ring, namely the ring of polynomials in the 2n variables ~1,..., ~, ~1,-.., ~, where and ~i denote the images of Ti and di in If Mis a D(R, k)-module generated by a finite set of elements m~,..., ms, let ftt = EiFtmi. Then 0 : ~-~--1 C ~’~1 C ... is a filtration of Mby finite-dimensional k-vector spaces and the graded object grnMis a finitely generated grrD-module. A fundamental result of Bernstein says that the support of the module graM is independent of the choice of the generating set rn~,..., ms and the dimension of this support is greater than or equal to n. The dimension of Mis by definition the dimension of grf~M as a grrD-module. A D(R, k)-module of dimension n is called holonomic. Example 3.1. R with its natural D(R,k)-module structure is holonomic. Indeed, R is generated by m = 1 ~ R and di ¯ 1 = O, hence ~i" ] = O, i.e. the k[~,..., Tn, dl,..., ~]-modulegraR is annihilated by the ideal ((~l,..., 0~n). Hence di~ngr~R <_ n. Holonomic modules have a number of remarkable properties: (i) The category of holonomic modules is a full subcategory of the category of D(R, k)-modules closed under formation of submodules, factor modules and extensions. (ii) Every holonomic module has finite length in the category of D(R, k)modules. (iii) If Mis a holonomic D(R, k)-module, then MI is holonomic for every
eR. For proofs of the above properties modules we refer to BjSrk’s book [7].
and further properties of holonomic
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Corollary 3.2. If M is a holonomic D(R, k)-module, then so is .H~(M). Proof. All modules in the complex 1.4 are holonomic by (iii), (i) so are all the cohomologymodules.
hence
Since R with its natural D(R, k)-action is a holonomicD(R, k)-module, all local cohomology modules H~(R) are holonomic. In particular, all H~(R) are finitely generated D(R~ k)-modules. This is quite remarkable because H}(R), as R-modules, are huge. They are finitely generated as R-modules and only if they are zero. This precludes an algorithmic approach to these modules via the standard GrSbner bases technique in R. However,D(R, k) is a finitely generated k-algebra and HiI(R) are finitely generated D(R,k)-modules. This makes these modules accessible to algorithmic methods of D-module theory. Using a version of GrSbner bases technique for non-commutativefinitely generated algebras one can construct an algorithm for explicitly writing downa presentation of HI(R) in terms of generators and relations and deciding whether Hi~(R) = 0. This is the basis of Walther’s algorithm [92] (and of that of Oakuand Takayama[76]). Wenote the remarkable fact that local cohomologyis defined without any reference to differential operators, or non-commutativealgebra, yet the only currently knownalgorithms (in characteristic 0) to decide whether H~(R) vanishes use D-modules! No algorithm within the framework of commutative algebra is knownat this time. 4. FINITENESS PROPERTIESOF LOCALCOHOMOLOGY MODULES: STATEMENTS OF RESULTS The structure of local cohomologymodulesis still full of mystery. In most cases one cannot even tell if a given local cohomologymoduleH~(M) is zero. Whenit is non-zero, it is rarely finitely generated, even if Mis. Yet over the last decade considerable progress has been made in our understanding of H~(M)in the special case where R is regular and M~ R. If R is local and dimR/I = O, the structure of H~(M), for a finitely generated M, is well-understood [33, 35]. Although not necessarily finitely generated, it is Artinian. In particular, HomR(K,Hi~(M)), where K is the residue field of R, is finitely generated. But if dimR/I > 0, this is not necessarily the case as is shownby the following example. Example 4.1. (Hartshorne [38], Section 3). Let K be a field, let A K[[X, y, u, vii be a formal power series ring in four variables x, y, u and v over K, let R = A/(xu + yv) and let I = (u,v). Then H~(R) is supported only at the maximal ideal of R but is not Artinian, so HomR(K,.H~(R))is not finitely generated. On the other hand the finiteness of HomR(K,H~(R)), where R is reg:ular, has been knownin some important cases; see Ogus [77], 2.7 and Hartshorne and Speiser [40], 2.4 where it is also proven that if R contains a field of characteristic p > 0 and H~(R) is supported only at the maximal ideal,
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then H~(R) is injective. This line of inquiry has culminated in Theorem4.2 below. Let T = T1 o T2 o ... o Ts be any composition of functors Tj such that ij
each Tj is either Tj(-) = (-) (where = \
closed subsets of SpecR) or the kernel, the image, or the cokernel of any of the three natural transformations appearing in 1.2a): Hi~,~,l(-)
-~ H~,~.:(-),
~,~ -~ H~.~(-)
or H~(-) -~ ..y~.~
(-).
Theorem 4.2. Let R be a commutative Noetherian regular local ring containing a field. Then i) inj.dimRT(R ) <_ dimSuppnT(R). In particular, fiT(R) is supported the maximalideal of R, then T(R) is injective. ii) The set of the associated primes of T(R) is finite. iii) The Bass numbersof T(R) are finite. Here inj.dimn and dimSupp/~denote the injective dimension and the dimension of the support of the corresponding module in the category of R~nodules and the i-th Bass number of an R-module Mat a prime ideal P is #~(M, P) = dim~(R/~,)Extin(R/P, where k(R/P) is t he fiel d of f rac tions of RIP [3]. In particular, (iii) implies that if R is local with residue field and maximal ideal m, then lengthK(HOmn(g, Hi~(R))) = #0(m, HiI(R)) is finite. The best result currently knownfor regular local rings of mixed characteristic is the following. Theorem4.3. Let R be a commutative regular local mixed characteristic unramified ring. Then T(R) has finite Bass numbers and the set of its associated primes is finite. As for property i), in the mixed characteristic unramified case only the weaker inequality inj.dimRT(R) <_ dimSuppT(R)+ 1 is known; see Zhou [99] and our paper [68]. For the ramified case of a regular local ring of mixed characteristic all three properties i) - iii) are open. For some related finiteness properties of local cohomologymodules including cofiniteness and the local-global principle see Belshoff and Wickham [4], Belshoff, Slattery and Wickham[5, 6], Brodmann[10], Brodmannand Lashgari [12], Brodmann,aotthaus and Sharp [13], Chiriacescu [16], Delfino [20], Delfino and Marley [21], Hellus [41], Huneke and Koh [47], Kawasaki [53, 54], Khashyarmaneshand Salarian [51, 52], Melkersson [72], Raghavan [81] and Yoshida [97]. For related finiteness properties of local cohomology with support in monomial ideals including some connections with combinatorics and toric varieties see Alvarez Montaner [1], Alvarez Montaner, Garcia Lopez and Zarzuela [2], Eisenbud, Mustard and Stillman [23], Helm and Miller [42], Miller [73], Mustata [75], Reiner, Welker and Yanagawa[82], Terai [91] and Yanagawa[94, 95].
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Theorem 4.2 was first proven by Huneke and Sharp [49] in the special case that T(R) = HiI(R) and R contains a field of characteristic p > 0. We gave a proof of Theorem4.2 in characteristic 0 by a completely different method [64]. While Huneke and Sharp used the Frobenius functor, we used D-modules in what may have been the first application of D-modules to a problem from commutative algebra (we have since learned that local cohomology had been treated in the context of D-modules for what appears to be the first time in Mebkhout’spapers [70, 71]). To find a characteristic-free proof of Theorem4.2 has been a major goal ever since. This goal has almost been reached by now, via D-modules. By completion the proof of Theorem4.2 reduces to the case where R is complete with respect to its maximalideal, i.e. (by one of Cohen’s structure theorems) R = k[[T1,...,Tu]] is a. ring of formal power series in a finite numberof variables over a field k. Theorem5.1. Let R : kilT1,... ,Tuff be the ring of formal power series in a finite numberof variables over a field k and let M be any D(R, k)-module. Then inj.dimnM _< dimSuppRM. Corollary 5.2. Let R = k[[T~,. . . , Tn]] be the ring of formal power series in a finite numberof variables over a field k and let G be a covariant additive functor from the category of abelian sheaves on SpecR~to the category of abelian groups. If M is an R-module, we denote G(M) by G(M). inj.dimRG(R ) <_ dimSuppRG(R). Proof. The action of 5 ~ D(R, k) on R induces an action on /~ which induces an action on G(/~), so G(R) acquires a natural structure of D(R, k)module. Nowwe are done by Theorem 5.1. [] Since T is a functor on the category of abelian sheaves, the corollary implies 4.2i) at once (see also Zhou[99]). This proof of 4.2i) is completely characteristic-free because in [66] we gave a completely characteristic-free proof of Theorem 5.1. It goes by induction on dimSuppRMwith the help of the following lemma. Lemma5.3. Let k be a field, let R = k[T~,... ,Tn] be the ring of polynomials in n variables TI,. . . , Tn over k and let m C R be a maximalideal. a) Hn~(R) is an irreducible D(R,k)-module (i.e. it does not contain non-trivial D( R, k )-submodules). b) Let K = R/m, let ks C K be the separable closure of k in K, let d = [K : ks] be the degree of the purely inseparable field extension K/ks and let D(R, k)m be the left ideal olD(R, k) generated by m. Then the D(R, module D(R, k)/D(R, k)m is isomorphic to the direct sum of d copies of D(R, k)-module H~(R). c) Every D(R, k)-module M supported at m is isomorphic as a D(R, module to a direct sum of copies of H~n(R ).
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In [67] wegave proofs of 4.2ii) and 4.2iii) that are characteristic-free modulo the following result whosestatement is characteristic-free but whichdoes not have a characteristc-free proof at this time. Theorem5.4. If R = k[[T1,...,Tn]] is the ring of formal power series in a finite number of variables over a field k, then Rf, for every f E R, has finite length in the category of D(R, k)-modules. Openproblems. 1. Find a characteristic-free proof of Theorem5.4. 2. Is the length of Rf in the category of D(R, k)-modules independent of the choice of the coefficient field k C R? This theorem and 1.4 imply that H~(R) has finite length in the category of D(R, k)-modules. This makesa proof of 4.2ii) especially simple: Proof of 4.2ii). It is enough to prove that every D( R, k )-module Mof finite length has a finite number of associated primes. Weuse induction on the length of M. If the length is zero, then M= 0, so the set of the associated primes is empty, hence finite. If P C R is an associated prime of M, then F~,(M) is a non-zero D(R, k)-submodule of M, so M/Fp(M) is a D(R, k)-module of smaller length than M. If P is maximal among the associated primes of M, then P is the only associated prime of Fp(M), and M/Fp(M) has a finite number of associated primes by induction on the length. But an associated prime of Mis either P or an associated prime of M/Fp(M). Theorem5.4 is proven by two completely different methods in characteristic 0 by Bj6rk [7], III.4.1 and in characteristic p > 0 by us [65], 5.9. A proof of the mixed characteristic Theorem4.3 is given in our paper [68]. It involves a fascinating interplay of the properties of D-modulesin characteristic 0 and in characteristic p > 0. Wedon’t knowof any proof of 4.3 that does not use D-modules. Corollary 5.2 fails in mixed characteristic. Zhou[98] pointed out that if R is a regular ring of mixedcharacteristic and residual characteristic p > 0, and G is the functor that takes an abelian sheaf F on SpecRto F(SpecR, F/pF), then G(R) ~- R/pR, so the inequality inj.di,nRG(R ) <_ dimSuppRG(R ) fails. 6.
ASSOCIATED
PRIMES
OF LOCAL COHOMOLOGY MODULES
If R is not regular, Example 4.1 shows that H~(R) need not have finite Bass numbers. If R is local of dimension d, Cohen-Macaulaybut not Gorenstein, local duality shows that H](R) is supported at the maximal ideal, but is not injective. Thus properties 4.2i and 4.2iii fail if R is not regular. As for property 4.2ii, its failure in the non-regular case has been shownonly very recently by the following amazing example of Anurag Singh [85]. Example 6.1. Let R : Z[U,V,W,X,Y,Z]/(UX + VY q- WZ) and I (X,Y,Z)R. Then H~(R) has p-torsion elements for every prime integer p ~ Z. Hencethe set of the associated primes of H3I(R) is infinite.
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On the other hand, a nice and simple argument shows that if R is a Noetherian ring, Mis a finitely generated R-moduleand i is the smelliest integer such that HiI(M) is not finitely generated, then the set ,af the associated primes of Hi~(M) is finite; see Brodmannand Lashgar~ [12] and Khashyarmaneshand Salarian [52]. For some more positive results on the finiteness of the associated primes of H~(M),where Mis finitely generated, see Brodm~.nn [10], erodmann, Rotthaus and Sharp [13], Hellus [41] and Marley[69]. For a closely related problem in the graded case see [9], 2.9. If R is regular, the finiteness of the associated primes of H~(R) is known not only if R is local and contains a field, but also if R contains a field of characteristic p > 0 [49] or if R is a finitely generated algebra over a field of characteristic 0 [64]. Openquestions. 1. If R is regular, is the set of the associated primes of H~(R) finite? 2. If R is either local, or contains a field, and Mis a finitely generated R-module,is the set of the associated primes of HiI(M) finite? It would be especially nice to prove at least that if R is a regular ring containing a field of characteristic 0, then the set of the associated primes of Hi~(R)is finite. 7.
NUMERICAL INVARIANTS )~p,i
OF LOCAL RINGS
The finiteness of the Bass numbers of local cohomologymodules has led to the discovery of new numerical invariants of local rings introduced in the following theorem-definition in our paper [64]. Theorem7.1. ([64] 4.1) Let A be a local ring which admits a surjective ring homomorphism~r : R --~ A, where R is a regular ring of dimension n containing a field. Set I : Kerr and let m be the maximal ideal of R. Then #p(m, H~-~(R)) is finite and depends only on A,i and p, but neither nor on ~r. Wedenote this iuvariant by Ap,i(A). A complete local ring containing a field is always a surjective image of a regular local ring containing a field. So, if A is any local ring containing a field, one can set £p,i(A) -- )~p,i(~), where ~ is the completion of A with respect to the maximalideal. It is not hard to show that this coincides with the definition given in 7.1 in the case that A is a surjective imageof a regular local ring containing a field. Set d = dimA. Here are some elementary properties of Ap,i(A) [64]: i) Ap,i(A) = if i > d. ii) A~,i(A) = 0 ifp > iii) Ad,d(A)~: Openquestion. Let FA be the graph obtained as folows. The vertices of FA are the minimal primes of -~, the completion of A. Twovertices _P’ and Q are joined by an edge if and only if the height of the ideal P + Q equals
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1. Is it true that Ad,d(A ) equals the number of the connected components of This is open even if ~i is a domainin which case the question is whether Ad,d(A) = 1. Someof the evidence in support of a positive answer to this question is as follows: a) It follows from Walther [93] and Kawasaki[55] that the answer is positive in the case that dimA= 2. b) It follows from Yanagawa[94], Corollary 3.16, that if A is the quotient of a regular ring by a monomialideal, then A~t,d = 1 if and only if FA is connected. c) Kawasaki [56] has shownthat if A is $2, then )~d,d : 1. It should be mentioned that for an $2 ring A the graph FAis knownto be connected; in fact Hochster and Huneke[43] have shown that I~A is connected if and only if the S2-ification of A is local. If V is a scheme of finite type over the complex numbers C and A is the local ring of V at a closed point q E V, the numbers Ap,i(A) are related the singular topology of V in a neighborhood of q. For example, if q is an isolated singular point of V, then Garcia Lopez and Sabbah [28] have shown that a) :ko,i(A) = dime H~(V,C), for 0 < i < dimA, b) Ap,g = dimcH~+dimA(v,c), for 2 ~_ p ~_ dimA, and c) all other Ap,i vanish, where H~(V, C) is the i-th singular local cohomologygroup of V with support in q and with coefficients in C and dime denotes the dimension as a complex vector space. Of course, it would be interesting to express the numbers in terms of the topology of V near q without assuming that q is an isolated singularity. Openquestion. Let V be a projective variety over a field k and let A be the homogeneouscoordinate ring of V for some embeding of V in a projective space over k. Is it true that &p,i depends only on V, i and p and not on the embedding? A positive answer to this question would produce a new set of numerical invariants of projective varieties. 8. F-MODULES In Sections 8 - 13 we summarize the theory of F-modules deveioped in our paper [65]. This theory has so~ne striking applications to local cohomology and/)-modules in characteristic p > 0. Standard assumption. Throughout Sections 8 - 13 R is a commutative Noetherian regular ring containing a field of characteristic p > 0. The Frobenius functor F : R-mod --~ R-mod
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is defined as follows. If Mis an R-module, F(M) = R’ ®R where R’ is the R-bimodule whose underlying additive group is R, the left R-module structure is the usual one and the right R-module structure is defined by r’r = rPr~ for all r E R, r ~ E R~. Weregard F(M) as an modulevia the left R-modulestructure on R’. If ¢ : M--~ N is an R-module homomorphism, F(¢): F(M) ida+ F(N). Since R is regular, a theorem of Kunz [57] implies that F is an exact functor. It has been known that local cohomology modules of R with support in any ideal I C R have the property that they are isomorphic to their own images under the ~obenius functor and this fact has been used, for example, by Hartshorne-Speiser [40], Huneke-Sharp[49], Peskine.Szpiro [79] and Sharp [83] to study local cohomology, but a systematic theory of modules having this property has not been constructed. The theory of F-modules is precisely such a theory. An FR-module (or simply an F-module, if this causes no confusion) an R-module ~ equipped with an R-module isomorphism 0 : ~ ~ F(~) which we call the structure morphism of ~. A homomorphismof F-modules is an R-module homomorphism f : ~ ~ ~’ such that the following diagram commutes (where 0 and 0’ are the structure morphisms of ~ and
The set of all F-module homomorphisms ~ ~ ~ forms a subgroup of the additive group of HOmR(~,~) but in general not an R-submodule. It not hard to see that the category of F-modulesis abelian. The kernel (resp. cokernel) of a morphism of F-modules f : ~ ~ ~ is the kernel (resp. cokernel) of f in the category of R-modules with the F-module structure induced by 0 (resp. 0~). In general, the same R-module ~ can support more than one F-module structure via different structure isomorphisms 0. Amongthe many different F-module structures on R there is the canonical one with structure isomorphism 0 : R r~ R~@RR= F(R). We will always regard R as an F-module with this canonical structure. If ¢ : R~ ~ Rn’ is an R-module homomorphism given by an n ~ n~ matrix (rij) with rij ~ R, then F(¢) Rn~ Rn’ (where we identify F(R~) with R~ via 0) is given by the matrix (r~j). If ~ is an F-module and S C R is a multiplicative set, there exists a unique F-module structure on ~8 such that the natural localization map g : ~ ~ ~8 is a homomorphism of FR-modules. This implies that if ~
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is an F-module, the complex from 1.4 is a complex in the category of Fmodules, hence the local cohomologygroups H~(Ad) inherit a structure F-module. In particular, the canonical F-module structure on R induces canonical structures of F-module on the local cohomologymodules H~(R). A generating morphism of an F-module A4 is an R-module homomomorphism /3 : M--~ F(M), where Mis some R-module, such that f14 is the limit of the inductive system in the top row of the commutative diagram M ----~
F(M)
~ F(M)
F2 (/) F_
F2 (M)-~ ..
~ F3 (/) F_
~ ..
and 0 : A4 --+ F(J~), the structure isomorphism of A/I, is induced by the vertical arrows in this diagram. Note that since the tensor product commutes with direct limits, the limit of the inductive system of the bottom row is indeed F(M), so this definition of 0 makes sense. Theorem 8.1. Let ./M be an F-module. Then inj.dimRfl4 _< dimnSupp.hd. In particular, if dimRSupp.hd= 0, then .Ad is an injective R-module. 9. F-FINITE MODULES An FR-moduleA4 is FR-finite, (or simply F-finite if this does not cause confusion) if A4 has a generating morphism/3 : M ~ F(M) with M a finitely generated R-module. If, in addition,/3 is injective, we call Ma root of A4 and we call/3 a root morphismof A4. The image of Min .4/[ will also be called a root of For example, R with its canonical F-module structure is F-finite with generating morphism/3 = 0 : R r~.~l R’ @RR -~- F(R). F-finite modules play a fundamental role in the theory of F-modules. Theorem9.1. The F-finite modules form a full abelian subcategory of the category of F-modules which is closed under formation of submodules, quotient modules and extensions. Theorem 9.2. If A4 is an F-finite module and f E R, then A4 y, with its induced F-modulestructure, is F-finite. The two preceding theorems and 1.4 imply the following corollary. Corollary 9.3. If .M is an F-finite particular, H~(R)are F-finite.
module, then HiI(.h4) is F-finite.
The following two propositions provide an algorithm for deciding whether Hi~(R) = O. For an example of this algorithm in action see Katzman[50]. Proposition 9.4. Let fl : M ~ F(M), where M is a finitely generated module, be a generating morphismof an F-finite module M. Let fli : M --~
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Fi(M) be the composition M~-+ F(M)F(_+~)... ~i~(f~) Fi(M). Then (a) The ascending chain ker~l C ker~2 C ... of submodules of M eventually stabilizes. Let C C M be the commonvalue of kerfli for sufficiently big i. (b) If i is the first integer such that ker/~i = ker~i+l, then ker/~i = C, i.e. ker/~i = kert~t for all t > i. (c) imfli = M/Cis a root of A4. Hence, every F-finite module has a root. (d) A/[= 0 if Ad has a zero root and .Ad 76 0 if ./M has a non-zero root. The above proposition leads to an algorithm for deciding whether the F-finite module generated by/~ : M -~ F(M) vanishes. Namely, for each integer r one should computethe kernel of fir and compareit with the kernel of/~r-1, until one finds r such that kerflr = ker/~r-~. One then should check whether ker/~r and Mcoincide. The F-finite module in question is zero if and only if they do coincide. If R is a polynomial ring in several variables over a field, these operations are implementable on a computer by means of standard software like Macaulay. In particular, we get an algorithm for deciding whether HiI(R) = provided we have an explicit generating morphism/~ : M -~ F(M) of Hi~(R) with a finitely generated M. Such a morphism is given by the following proposition. Proposition 9.5. Let I be generated by fl,...,fs the Koszul cocomplexof R on f~,..., fts, i.e.
E R. Let K’(ft;R)
do d1 d2 ds - 1 0 "--> R --> (31<_i<_sRi--~ ~l<_il
where each Ril,...,ij the differentials
s -->
is just a copy of R indexed by the tuple (i~,...,ij)
dj : (~l<_il<...
are given by ~_ V?v=j+l( ~v ct~,~
(dJ(m))i~,...,i~+~ "--’v:l ,-’~ av"°i~ ..... ~......i~+i" Let /~: Hi(K°(_f; R)) F(Hi(K°(f_; R))) = Hi(F(K°(f_; R))) be the map induced by the chain map /3" : K’(f_; R) --> F(K’(f_; R)) K’(f~_"; R) that sends each Ril,
morphism of Hit(R).
,ij
.... to Ril,.. ¯ .. ,ijf~jvia ).(f~-i Thenp-1~5 is
be
a generating
and
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Proof. The complexfrom 1.4 is the direct li~nit, over e, of the Koszul cocomplexes K° (fP~; R). Namely, K° (fP~; R) is isomorphic to the subcomplex of the complexfrom 1.4 whose (il,..., ij)th componentis the free R-module [] of rank one generated by (f~ ... f~) E Rfq ...fij. Another generating morphism of Hi~(R) is I~ : ExtiR(R/I,R) --~ Ext~(R/I[P],R) induced by the natural surjection R/I [p] -~ R/I where I [pt] is the ideal generated by the pt-th powers of all the elements of I (considering that H~(R) li__~ExtiR(R/I[P~},R) by 1.5 while Ft(Ext~(R/I,R)) ~ ExtR(R/I ],R)). Theorem 9.6. a) The set of the associated primes of an F-finite module is finite. b) All the Bass numbers of an F-finite moduleare finite. 10. F-FINITE MODULES HAVEFINITE LENGTHIN THE CATEGORY OF F-MODULES The main result of the theory of F-modules which underlies the applications of this theory discussed in Sections 11 - 13 below, is the following. Theorem10.1. If R is a finitely generated algebra over a Noetherian regular local ring of characteristic p > O, then every FR-finite modulehas finite length in the category of FR-modules. If MC A4 is a root of an F-finite module A/t, we call Ma minimal root of A4, if no other root of A4 is contained in M. As a consequence of 10.1 one gets Theorem10.2. If R is complete local, then every F-finite a unique minimal root.
module f14 has
A minimal root morphism is an injective R-module homomorphismfl : M-~ F(M) where M is a finitely generated R-module such that ~(M) F(N) for any proper submodule N of M. The preceding theorem immediately implies the following corollary. Corollary 10.3. If R is complete local, the set of the isomorphism classes ofF-finite modules is in a one-to-one correspondencewith the set of the isomorphism classes of minimal root morphisms. (Of course, we consider two minimal root morphisms ~ : M ~ F(M) and ~’ : M’ ~ F(M’) isomorphic if and only if there exists an R-module isomorphism h : M ~ M~ such that the diagram
M’ is commutative.)
~
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Throughout this section A is a commutative Noetherian loc~l ring of characteristic p > 0 and A{f} is the associative ring extension of A with one generator f and relations fa = a’f for all a E A. A cofinite A{f}-module is a left A{f}-modulewhich is cofinite (i.e. Artinian) as an A-module. this section we describe how the theory of F-modules leads to a structure theory for cofinite A{f}-modules. Typical examples of cofinite A{f}-modules are the local cohomologymodules Him(A), where m is the maximal ideal of A. The A{f}-modulestructure on Him(A)has been used one way or another, explicitly or implicitly, whenever the modules Him(A) were discussed specifically in the characteristic p > 0 setting; see, for example, Hochster-Roberts [44, 45], HartshorneSpeiser [40], Fedder-Watanabe [25] and Smith [86, 87, 88, 89]. However, the structure of Him(A) as A{f}-modules remained a mystery. The results discussed in this section to a great extent dispel this mystery. If M is an A{f}-m0dule, we denote by Mf the A-submodule of M generated by all the elements of the form fi(x), where x E M, and we set M*= y~. Clearly, all the Mfi, and hence also M*, are A(f}-submodules of NiM M. If Mis cofinite, the descending chain M ~ Mf ~ Mf2 ~ ... eventually stabilizes, so M*= Mfi for all sufficiently large i. If My~ = 0 for some i, we call M a nilpotent A{f}-module. Wecall an element x ~ M nilpotent if fi(x) = 0 for some i. For each i the kernel of fi is an A{f}-submodule of M. Following Hartshorne-Speiser [40] p. 46, we set Mn = U~kerfi to be the union of the nilpotent elements of M. Clearly, Mn is an A{f}-submodule of M. We set the reduced quotient module of M to be M~ : M/Mn. Clearly, (M*)r (Mr)*, so we den ote this module M,~. Our results from [65] on the structure of cofinite A{f}-modulesare based on the following theorem which enables one to apply the theory of F-tinite modules to the study of cofinite A{f}-modules.
Theorem11.1. Let R be complete local and let R ~ A be a surjective ring homomorphism.There exists a contravariant additive functor 7-~R,A from the category of cofinite A{f}-modules to the category of FR-finite modules such that (i) 7-lR,d is exact. (ii) 7tR,A(M) = 0 if and only if i is nilpotent. (iii) The minimal root morphism of TiR,A(M) is tiM: : D(Mr*) -~ FR(D(M;)) induced by the A{f}-module structure on M~. Here D(Mg) is the Matlis dual of M~. (iv) If M and M’ are cofinite A{f}-modules, then 74R,A(M)is isomo.rphic to 7tR,A(M~) in the category of FR-modulesif and only if M~is isomorphic to (i’); in the category of A{f}-modules.
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As an immediate application of this relationship between the FR-finite modules and the A{f}-moduleswe give a simple proof of the following result of Hartshorne and Speiser [40] 1.11. Proposition 11.2. If every element x of a cofinite A{f}-module M is nilpotent (i.e. fi(x) = 0 for some i), then the whole module M is nilpotent (i.e. one i works for all x). Proof. If/i, is the completion of A with respect to its maximalideal, then every cofinite A-module is cofinite as an A-module and every cofinite Amodulehas a natural structure of cofinite ft.-module. This implies that the categories of cofinite A{f}-modulesand cofinite ~{f}-modules are equivalent, so we can replace A by .~, i.e. we can assume that A is complete. In this case there exists a surjection R -~ A, where R is a complete regular local ring. Since Mr = 0, ll.liii implies that the minimal root of ~R,A(M) is 0, so v]-gR,A(M) : 0, so we are done by ll.lii. [] Cofinite A{f}-modules need not have finite length in the category of A{f}-modules. For example, if Mis a cofinite A-module of infinite length and Mf = 0, then Mis a cofinite A{f}-moduleof infinite length. Our main results on the structure of cofinite A{f}-modules (contained in Theorems 11.3, 11.4 and 11.5 below) show that up to nilpotency they do have finite length and satisfy the Jordan-HSlder property. Theorem11.3. There is a unique way to associate to every cofinite A{f}module M a non-negative integer qg(M), which we call the quasilength M, such that (a) q~(M) = 0 if and only if M is nilpotent. (b) q~(M) = 1 if and only if M~is a non-zero simple A(f)-module. (c) If 0 --~ ~ -+ M-+M"--~0 is a short exact sequence in the category of A(f}-modules, then qg(M) = qg(M’) + q~(M"). Let 0 = Mo C ... C Ms = M be a filtration of a cofinite A(f}-module Min the category of A(f}-modules. Wecall this filtration quasimaximal if (Mi/Mi-1)~ is non-zero and simple for all i. Theorem 11.4. Let M be a cofinite A{f)-module. There exist quasimaximal filtrations on M. The length s of any quasimaximal filtration 0 = MoC ¯ .. C Ms = M of M equals q~(M). The set o.f the isomorphism classes the simple A(f)-modules (Mi/Mi-1); depends only on M, but not on particular quasimaximalfiltration. The quasilength of a simple A{f}-moduleMis always _< 1 and it equals 0 if and only if Mis a simple A-module(i.e. Mis the residue field of A) and f : M-+ M is the zero map. Hence the quasilength of a simple A{f}module Mequals 1 if and only if f : M--~ Mis non-zero. This fact is used in the statement of the following theorem.
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Theorem11.5. Every cofinite A{f }-module M has a (not necessarily unique) filtration O= Lo C No C L1c N1c ... C Ls C Ns = M in the category of A{f }-modules such that Nj/Lj is nilpotent for each j while each Lj/Nj_I is simple with f : Lj/Nj_I --~ Lj/Nj_I non-zero. The integer s depends only on M and is equal to q~(M) and the set of the isomorphism classes of the simple subquotients Lj/Nj_~ also depends only on M. Weconclude this section with the following fundamental example. Example 11.6. Let R --~ A be a surjection
with R complete, local,
regu-
lar and n-dimensional. The Frobenius homomorphismA a__~__~pA induces a structure of cofinite A{f }-module on each local cohomologymodule Him(A), where m is the maximal ideal of A. Let I be the kernel of the surjection R --~ A. Then 7-IR,A(Him(A)) = H~-i(R), with canonical FR-module ture. Hence the quasilength of Him(A)equals the length of H~-i(R) in the egory of FR-modules and the simple subquotients of Him(A) are the Matlis duals of the minimal roots of the simple components of H~-i(R). In particular, H?-i(R) = 0 iff Him(A) is a nilpotent A{f}-module. 12. APPLICATIONSOF A{f}-MODULES:THE STABLEPART ANDTHE F-DEPTH AssumeA contains a coefficient field k. Let Mbe a cofinite A{f}-module, let imfj be the set of elements of Mof the form fJ(x), where x E M, and let k-imff be the k-vector subspace of Mspanned by imf j. Weset the stable part of Mto be the k-vector space Ms = ~qjk-imf j (cf. Hartshorne and Speiser [40], p.47). This definition dependson the choice of a coefficient field k; nevertheless we denote the stable part simply by Ms (see 12.2 below). In the case that k is perfect, Hartshorne and Speiser proved that Msis finitedimensional over k and f : Ms-~ Msis bijective [40], 1.12. If R is locale A4 is an FR-finite module and A/[’, A4" C A4 are two Fasubmodules such that dimRSuppR(fl4/A/[’) = dimRSuppR(.A4/fl4") = then their intersection also has this property and, since f14 is Artinian in the category of Fn-modules, there exists a s~nallest FR-submodule Af of J~/[ with this property. Wecall £ = .A4/Af the maximal zero-dimensional quotient of A/I. Since ~: is an FR-module,8.1 implies that it is injective. Since it is FR-finite, its socle has finite dimension(equal to #0(m, ~:), which is finite by 9.6). Hence/2 as R-moduleis isomorphic to Er, where r is the dimensionof the socle of ~2 and E is the injective hull of the residue field of R. Weset the corank of A// to be crk.A4 = r. The theory of F-finite modules sheds some new light on the properties of the stable part of cofinite A{f}-modulesvia the following proposition. Proposition 12.1. Assume A is complete with respect to its maximal ideal m and let k C A be a coefficient field (which always exists by one of Cohen’s
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structure theorems). Let R -+ A be a surjection with R complete local and regular. If M is a cofinite A{f}-module, then dimkMs = crk~R,A(M). The preceding Proposition may be viewed as a reinterpretation of the Hartshorne and Speiser result on the finite-dimensionality of Ms in terms of F-finite modules; in this interpretation their result says that the corank of an F-finite module is finite. From this point of view Theorem10.1 on the finiteness of the length of F-finite modules is a generalization of the Hartshorne and Speiser result on the finite-dimensionality of Ms. The preceding proposition immediately implies the following corollary. Corollary 12.2. dimkMs, for every cofinite A{f}-module M, is finite depends only on Mand not on the particular coefficient field k.
and
The question of whether dimkMsis independent of the choice of the coefficient field k was left open in [40]. Hartshorne and Speiser introduced the. following definition which westate in a so~newhatsimplified form taking into account 12.2. Definition 12.3. (~40], p. 60) Let Y be a Noetherian scheme of finite dimension whose local rings are all of characteristic p. Let y E Y be a (not necessarily closed) point. Let d(y) be the dimension of the closure of point y in Y as a Noetherian topological space. Let (.gy, y be the local ring of y and let k C O~.,y be a coefficient field. The F-depth of Y is the largest integer r (or -boo) such that for all points y ~ Y we have (Him(Oy,y))s for all i < r - d(y). Hartshorne and Speiser madethis definition by analogy with Ogus’ characteristic 0 notion of DeRhamdepth, namely Definition 12.4. (Ogus [77], 2.12) Let Y be a Noetherian scheme of equicharacteristic O. Then the DeRhamdepth of Y is the largest integer r such that for each (not necessarily closed) point ye Y, we have that Hb(Y) = 0 i < r-d(y), where HI(Y) is the local algebraic DeRhamcohomology ofOy, y in the sense of [39]. In equicharacteristic 0 one can interpret the DeRhamdepth in terms of closed points only [77], p. 340. Namely,let Si(Y) = {y e Y: gJy(Y) ~ 0for so me j < i - 2d(y)}. Then each Si(Y) is constructible of dimension <_ i/2 and the DeRhamdepth of Y is the largest integer r such that dimSi(Y)_< i - r for all i [77], 2.15. The following proposition shows, in particular, that in characteristic p > 0 there does not exist a similar criterion for expressing the F-depth of Y in terms of the dimensions of some sets of closed points y ~ Y such that (HJm~¢(((~y,y)K))s for somej, be cause all s uch sets are z ero- dimensionM. Proposition 12.5. Let Y be a scheme of finite type over a regular local ring of characteristic p > O. Then there are only finitely ~nany points y of Y such that (Him((~y,y))s
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Lyubeznik 13. APPLICATIONS OF F-MODULES:/)-MODULES
Let D = D(R,Z/pZ). It is not hard to show that every differential operator d E D of order _< pn _ 1 is RP~-linear, where Rp~ C R is the subring consisting of all the pU-th powersof all the elements of R; see Yekutiely [96], 1.4.8a. In other words, D is a subring of the ring U = U(R) = Un_>0HomRp~ (R, R). In particular, this implies that if K is not just Z/pZ, but any perfect subfield of R, then every ~ E D is K-linear, i.e. D = D(R, K). Let A4 be an F-module with structure isomorphism 0 : A4 -+ F(A//). give .A// a structure of U-moduleas follows. Set Ot = Ft-l(O) o Ft-2(O) o... o 0 : M--4 (t) Let R be the additive group of R regarded as an R-bimodule with the usual left R-action and with the right R-action defined by r’r = rPtr ’ for all r ~ R and r’ E R(t). Then Ft(.A4) = (t)®R.A4 a nd e very e lement u ~ Ut = Homnpt(R, R) acts on Ft(A4) = (t) ®R . All via u ®R idM. We let -1 u act on A/[ via 0~ o (u ®RidM) Or. This action is well-defined, i.e. it depends only on u, but not on the particular t. The action thus defined is compatible with the addition and multiplication operations of U and makes A/[ a U-module. An F-module homomorphism.A// --~ .A//’ is automatically a U-module homomorphism,so we have a covariant additive and faithfull (but not necessarily full) functor F-mod ~ U-mod. Since D(R, K) C ar e su brings of U, composing thi s fun ctor wit h the restriction of scalars functors U-mod~ D(R, K)-mod and U-mod -4 D-rood gives functors ~R,K : F-mod --> D(R, K)-mod ~ = ~(R) : F-mod ~ D-mod. Since D = D(R, Z/pZ), the functor ~ is a special case of the functor ~R,K. It is clear from this construction that the underlying R-modulesof A//and ~R,K(J~) are the same. If a D(R, K)-module .A// is in the image of ~R,K, we will say that the D(R, K)-module structure on 3//is induced by an. Fnmodule structure. Theorem 13.1. Assume that R is a finitely generated R’-module and every FR-finite module has finite length in the category of Ft,-modules. If A/i is an FR-finite module which is simple in the category of FR-modules, then in the category of D-modules, ~(A/I) is a direct sum of Af~,Af2,... ,A/s, where each Af~ is simple in the category of D-modules. Theorem 13.2. Assume that R is a finitely generated R’-module and every FR-finite module has finite length in the category of FR-modules.If.A4 is an FR-finite module, then ~(A/I) has finite length in the category of D-modules. In particular,
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(a) Rf with its canonical D-modulestructure has finite length in the category of D-modules for every f ¯ R. (b) T(R) with its canonical D-modulestructure has finite length in category of D-modules, where T is any functor defined in Section 4. Corollary 13.3. Let R be a finitely generated algebra over a commutative Noetherian regular local ring A of characteristic p > 0 such that A is a finitely generated AP-module.If .A4 is an FR-finite module, then ~(A/[) has finite length in the category of D-modules.In particular, (a) Rf with its canonical D-module.structure has finite length in the category of D-modules for every f ¯ R. (b) T(R) with its canonical D-modulestructure has finite length in category of D-modules, where T is any functor defined in Section 4. If K is any field of characteristic p > 0, let K* = I~1/p~ be the perfect closure of K. If A = K[[X1,...,Xu]] is the ring of formal power series in X1,...,Xn over K, let A* = K*[[X1,...,Xn]]. We regard A* as an Aalgebra via the natural inclusion K[[X1,... ,Xn]] C g*[[Zl,... ,Xn]]. If R is an A-algebra, let R* = A* ®AR. If R is a finitely generated A-algebra, then R* is a finitely generated A*-algebra, in particular it is Noetherian. Theorem 13.4. With notation as above, let R be a finitely generated Aalgebra such that R* is regular. If.All is an FR-finite module, then ~R,K(J~) has finite length in the category of D(R, K)-modules. In particular, (a) RI with its canonical D(R, K)-module structure has finite length the category of D(R, K)-modules for every f ¯ (b) T(R) with its canonical D(R, K)-module structure has finite length the category of D(R, K)-modules for any functor T defined in Section The following special case of the preceding theorem deserves to be stated separately. Corollary 13.5. Let R be a finitely generated K-algebra, where K is a field of characteristic p > O. Let K* = K1/"~ be the perfect closure of K and assume that R* = K* ®I( R is regular. If .A4 is an FR-finite module, then ~R,K(A4)has finite length in the category o.f D(R, K)-modules. In particular, (a) Rf with its canonical D(R,K)-module structure has finite length the category of D(R, K)-modules for every f ¯ (b) T(R) with its canonical D(R, K)-module structure has finite length the category of D(R, K)-modules for any functor T defined in S.ection Wedo not know whether the preceding theorem and its corollary remain valid without the additional assumption that R* is regular (i.e. whether our standard assumption that R is regular is enough). B~gvad[8] has developed an interesting theory of ’filtration holonomic’ moduleswhich yields 13.5 for an algebraically closed K (see [8] 3.2, 3.7). From now on we drop our ’standard assumption’ of Sections 8 - 13 that R is regular and contains a field of characteristic p > 0.
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Lyubeznik 14.
UPPERBOUNDS ON COHOMOLOGICAL DIMENSION
The main reference for this section is our joint paper with Huneke[48]. Werefer to 2.1 for the definition of cohomological dimension. Let R be a commutative Noetherian ring of dimension d. A result of Grothendieck [33], 1.12 says that cd(R,I) <_ for ev ery id eal I C R (a nd, more generally, cd(X, y) _< d for any d-dimensional scheme X and any locally closed subscheme Y). It is not hard to see that cd(R, I) equals the maximumof cd(Rp, Ip) as P runs through all the maximal ideals of R. If R is local, it is not hard to see that cd(R, I) = cd(~t, I~), where/~ is the completion of R with respect to the maximal ideal. Furthermore, cd(R, I) equals the maximumof cd(R/P, (I ÷ P)/P) as P runs through all the minimal primes of R. This reduces the computation of cd(R, I) to the case where R is a complete local domain. The Hartshorne-Lichtenbaumtheorem [36], 3.1 says that if R is a complete local domain, cd(R, I) _< d - 1 if and only if I is not primary to the maximal ideal of R. Perhaps the nicest proof of this result is due to Brodmannand Huneke [11]. For an analogous result for algebraic varieties see our paper
[62]. Open problems. Let R be a d-dimensionM complete local domain. 1. Find necessary and sufficient condition for cd(R, I) <_ d2. Let I be prime. Is it true that cd(R, I) <_ d-2 iff (P+I) is not primary to the maximal ideal for every prime P of R of height 1? Theorem14.1. Let R be a complete regular local tin9 of dimension d with a separably closed residue field that it contains. Thencd(R, I) <__d- 2 if and only if dim R/I >_ 2 and the puctured spectrum of R/I is connected. This was proven by Ogus [77], CoL 2.11 in equicharacteristic 0 and by Peskine and Szpiro [79], III, 5.5 in equicharacteristic p > 0. A characteristicfree proof was given in our joint paper with Huneke[48], 2.9 on the basis of a sharpening of the argument of Faltings [24]. The same idea leads to a proof of the following. Theorem14.2. ([48], 5.1) Let R be an excellent regular local tin 9 of dimension d containing a field and let I be an ideal of R such that every minimal prime of I has height at most b. i) (cf. [24], gor. 2) cd(R, I) <_ [(d - 1)/ b]. ii) If, moreover, [ is formally geometrically irreducible, then cd(R, I) < d - 1 - [(d - 2)/b]. iii) If I is formally geometrically irreducible and (R/I)p is normal for prime ideals P ~ I such that dim(R/P) >_ 3, then cd(R, I) <_ d - [d/(b 1)] - [( d - 1)/(b +
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Here ’formally geometrically irreducible’ meansthat the extension of I in (_R)sh, the completion of the strict Henselization of the completion of R, has a unique minimal prime. Similar results are also available for non-regular rings R. The corresponding upper bounds on cd(R, I) are in terms of the dimension of R, the embedding dimensions of the localizations of R at various primes and the height of the minimalprimes of I. For precise statements see [48]. The following examples showthat 14.2i) and 14.2ii) are sharp for all d, ^
Examples 14.3. Let R be as in Theorem 14.2. I. Let Po,... , P[(d-1)/b] be primesof Tt of height b with Po +"" + P[(d-1)/b] primary to the maximal ideal. Then cd(R,I) = d- [(d- 1)/b], where I Po¢q " " N P[(d-1)/b][48], 5.2, 5.3. 2. Let I C R be a prime ideal of height b such that for some prime P of dimension 1 the extension of I in Rp, the completion of the localization of R at P, is the intersection of 1 + [(d - 2)/5] primes of height b that add up to an ideal which is primary to the maximal ideal of Rp. Then cd(R, I) = d- 1 - [(d - 2)/5]. Such an ideal is constructed in [48], 5.6. Since araI _> cd(R, I), Theorem14.2i) suggests the following. Open question. Let R = k[[X1,...,Xd]] or R = k[X1,...,Xd] be either the ring of formal power series, or the ring of polynomials, in d variables over a field k and let I C R be an ideal whose minimal primes have height at most b. Can I be generated up to radical by d - [(d - 1)/b] elements? Somesupporting evidence for a positive answer to this question is provided by our paper [59]. Our generalization [61], 5.9, 5.11 of the celebrated result of Cowsik and Nori [18] gives a positive answer to this question if b = d- 1 and the characteristic of k isp > 0. This is the only case in which the answer is known(other than the trivial case b --- 1). This circle of problems has been surveyed in our paper [60]. Ifk is a field, V C ~ is a Zariski closed subset, R = k[Xo,..., Xn](x0,...,x~) is the local ring at the vertex of the affine cone over l?~ and I C R is a defining ideal of V, then cd(R, I) = 1 +cd(F~\ [36], p.4 12. Thi s andTheorem 14.2 imply the following. Theorem14.4. Let k be a separably closed field and let V C I?~ be a closed subset consisting of irreducible components of codimensions at most b. Then i) cd(]P~ \ < n- [n/b] if n o a dditional assu mptions on Yaremade, ii) cd(I?~ \ V) _< n - 1 - [(n - 1)/b] if Y is irreducible, iii) cd(I?~ \ V) <_[( n + 1)/( b + 1][n/(b+ 1)] if V is n ormal. Projective analogues of Examples 14.3 show that 14.4i) and 14.4ii) are exact for all n, b. Andfinally 2.4 and 14.2 imply the following Corollary 14.5. Let V C ]P~ be an algebraic set consisitng of irreducible components of codimensions < b. ThenHs~ing( i C, V; C) = provided --
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i) (cf. [80]) i <_ [n/b] - 1, if no additional assumptions on V are made, ii) i <_[(n - 1)/b], if Y is irreducible, iii) i <_[(n + 1)/(b + 1)] [n/(b + 1)] - 1,if Y is normal. Projective analogues of Examples 14.3 show that 14.5i) is exact for all n, b and 14.5ii) is exact for all n, b with b ~’(n - 1). Openquestions. 1. Is 14.2iii) sharp? 2. Is 14.4iii) sharp? 3. Is 14.hiii) sharp? The vanishing of H~ing(~, V; C) tells us that H:ing(]~, V; Z) is torsion. But one naturally expects that under the assumptions of 14.5 H:ing(]~, V; Z) vanishes in the same range as doesHs~ing(l?o i n V; C). This was the motivation behind our paper [63] where the vanishing of H~ing(F~, V; Z) in the same range is proven. Wediscuss it in sections 16 and 17, the last two sections of this paper. For some of the earlier work on subvarieties of small codimension in n-space the papers by Hartshorne [37] and Fulton and Lazarsfeld [27] may be consulted. In the next section we discuss 6tale cohomology, the main tool used in [63]. 15. ~TALE COHOMOLOGY AND COHOMOLOGICAL DIMENSION A thorough and detailed treatment of dtale cohomologyis given in [31, 22, 32]. Booksby Milne [74] and Freitag and Kiel [26] give a more accessible presentation. For a short (about 75 pages) summaryof the main ideas the subject Deligne’s article "Cohomologie6tale: les points de departs" in [22] is highly recommended. With every scheme X there is associated an object that bears a stri:king resemblance to a topological space. It is called the dtale site of X and denoted X6t. One defines abelian sheaves on X~t and their cohomology groups by analogy with the cohomologyof abelian sheaves on a topological space. ]~tale cohomologyprovides a unified approach to several cohomologytheories, such as Galois cohomology, the cohomologyof quasicoherent sheaves and singular cohomology. Namely, if X is the spectrum of a field k, the cohomologyof sheaves on X6t is basically the Galois cohomology of k. If X is any scheme, the cohomology of a quasicloherent sheaf on X6t is the same as its ordinary (i.e. Zariski) cohomologyon X. And finally, if X is scheme of finite type over the complex numbers, then X6t is a purely algebraic analogue of the analytic space Xan associated with X and one has’, the following comparison theorem. Theorem 15.1. Let X be a scheme of finite type over the complex numbers C, let G be a finite group and let ~ be the associated constant sheaf on X dt. Then Hi(Xdt, ~) "~ H~ing(Xan , G).
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This theorem underlies a purely algebraic approach to the singular cohomologyof algebraic varieties with torsion coefficients (for a non-torsion group G the theorem fails). By analogy with Definition 2.1 we state the following. Definition 15.2. Let X be a scheme over a field k and Y C X a locally closed subscheme. The dtale cohomological dimension of X with support in Y, denoted dcd(X,Y), (resp. the dtale cohomological dimension X, denoted dcd(X)), is the largest i such that there exists a torsion sheaf .~ on Xdt with the local gtale cohomology group H~(Xdt, 2r ) ~ 0 (resp. Hi(Xdt,.~) ~ 0). If X is a d-dimensional schemeof finite type over a separably closed field k, then ~cd(X, Y) _< 2d, [74], VI.I.1, while if, in addition, X is affine, then ~cdX_< d [74], VI,7.2. Let X be affine with coordinate ring R and let Y C X be a closed subscheme defined by an ideal I C R. If I is generated up to radical by r elements, i.e. araI _< r, then X \ Y is covered by r affines which (via ~hech cohomology) implies that cd(X \ Y) _< r - 1 and in a similar 6cd(X \ Y) _< d ÷ r - 1. Thus we get two lower bounds on araI: araI _> 1 + cd(X \ Y) and araI _> 1 + ecd(X \ Y) Which one is better? Conjecture. If U is a schemeof finite type over a separably closed field k, then ~cdU>_ d + cd(U). This conjecture, if proven, would show that ~tale cohomologyprovides a better lower bound on the arithmetical rank. Someexamples with a strict inequality ~cdU > d + cd(U) are found in Bruns and Schw/inzl [15] in characteristic p > 0 and in Ogus[77], 4.6 in characteristic 0. These provide some evidence for the conjecture. 16. TOPOLOGY OF ALGEBRAIC SUBVARIETIES
OF SMALL CODIMENSION IN
n-SPACE
The fact that ~tale cohomologyis defined purely algebraically makes it possible to state and prove (!tale analogues of the upper bounds on the cohomological dimension proven in [48], some of which were discussed in Section 14. If X is a d-dimensional scheme of finite type over a separably closed field k, an dtale analogue of the inequality cd(X, Y) _< a would the inequality ~cd(X, Y) _< d + a. For example, ~cd(X, Y) _< 2d is an ~tale analogue of Grothendieck’s result that cd(X, Y) _< d, while for an affine the ~tale analogue of the well-known fact that cd(X) = 0 is the fact that ~cdX<_ d. In our paper [63] we prove the following 6tale analogue of 14.4. Theorem16.1. Let k be a separably closed field and let V C P~ be a closed subset consisting of irreducible componentsof codimensions at most b. Then i) 6cd(P~ \ V) <_ 2n - [n/b] if no additional assumptions on V are made,
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ii) dcd(I?~ \ V) _< 2n - 1 - [(n - 1)/b] if V is irreducible, iii) dcd(l?~ \ g 2n - [(n+ 1)/( b + 1][n/(b + 1)] if V is a nalytically irreducible. This leads to the following sharpening of 14.5, where Hi(l?~:, V; Z) is the i-th singular cohomologygroup with integer coefficients and ~ri(]?~, V) is the i-th higher homotopygroup of the pair ( c, V). Corollary 16.2. ([63], 10.7) Let V C l~ be an algebraic set consisitng of irreducible components of codimensions <_ b. Then Hi(i?~,V;Z) : 0 and ~i(l?~, V) = 0 provided i) (cf. Peternell [80]) i <_ [n/b] - 1, if no additional assumptions on V are made, ii) i <_[(n - 1)/5], if V is irreducible, iii) i <_[(n + 1)/(b + 1)] [n/(b + 1)] - 1,if V is analytically irr educible. Proof. Assume 6¢d(~ \ V)<_ m. Then Hi(~ \ Vdt; Z/gZ) = 0 for i :> m. By Poincare duality [74], VI, 11, Hc(~(: V~t;Z/~Z ) : 0 fo r i < 2n- m, whereHcZ(l?cin \ V6t; Z/~Z) is the i-th ~tale cohomologywith compactsupi n port [74], p.39. The long exact sequence ... n Hi[]~ ; Z/~Z) -~ Hi(V~t;Z/~Z) -+ Hic+l(I~ V~t; Z/ ~Z) -~... [7 4], p. 94, ~ c~t shows that the natural mapsHi(FC~t,n¯ Z/gZ) --> Hi(Vdt; Z/~Z) are isomorphisms for i < 2n-m- 1 and injective for i = 2n-ra- 1. A slightly more general version of 15.1 shows that the natural mapof the usual singular cohomology groups Hising []pn. (V; Z/gZ) are isomor~ c’ Z/~Z) -~ Hi.s~ng phisms, or injective, for the same values of i. Hence the singular cohomoli tFn V;Z/gZ) = 0 for i < 2n- m. Now the universal ogy groups Hsing~ coefficients theorem shows that the singular homologygroups with integral coefficients Hi(I~, V; Z) vanish for i < 2n - m. This implies the ranis:hAng of Hi(]?~, V; Z) in prescribed ranges considering 16.1. Finally, V is simply connected if V consists of irreducible componentsof codimensions_< n/3 [48], 6.6 or if V is irreducible of codimension< n/2, [27], Cot. 5.3, p. 47, so Hurewi~implies the vanishing of ~ri(]?~:, V) in prescribed ranges. [] ¯ The bounds 16.1i) and 16.2i) are exact for all n,b [63] 10.9 while the bounds 16.1ii) and 16.2ii) are exact for all n, b with b < n and b /~(n-[63], 10.10. The examples proving the exactness of these bounds are very similar to the ones proving the exactness of 14.2i) and 14.2ii). Openquestions. 1. Are 16.1ii) and 16.2ii) exact even if bl(n - 1)? 2. Are 16.1iii) and 16.2iii) exact? For [(n - 1)/b] = n - 1], i.e. b : 1, the bound 16.2ii) is the well-known theorem of Lefschetz, while for [(n - 1)/b] = 2 this bound is an immediate consequenceof a result of Fulton and Lazarsfeld ([27] Cor. 5.3, p. 47). Thus 16.2ii) tells us what happens for the intermediate values of [(n - 1)/b], i.e. for2< [(n-1)/b]
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If V is an algebraic variety of small codimensionin affine space, the bounds on ~tale cohomological dimension imply the vanishing of the cohomologyof V with compact support in a certain range [63], 9.4. 17. THE PICARDGROUPOF ALGEBRAIC VARIt~TIES OF SMALL CODIMENSION IN PROJECTIVESPACE Let V C F~ be a closed algebraic set, where k is an algebraically closed field. We denote the Picard group of V by PicV. One knows that PicV = NSSPic°V,where NS is a finitely generated abelian group, called the NeronSeveri group of V and Pic°V is a divisible abelian group. One knowsthat Pic°Y is isomorphic to the group of closed points of a commutative group schemeof finite type over k. For this and other important facts about the Picard group we refer to [29, 30]. The relationship between ~cd(F~ \ V) PicV is given by the following. Proposition 17.1. Assume dcd(P~ \ V) <_ 2n - 4. Then (i) If chark = 0, then NS is generated by Oy(1) and Pic°V is torsion-free. (ii) If chark = p > O, then YS/{Oy(1)} is a finite abelian p-group (where {Oy(1)} is the cyclic subgroub generated by Oy(1)) and Pic°V has no torsion prime to p. Proof. Let ~ be any integer prime to p. Kummer theory [74], p. 125, leads to the exact sequence HI(V~t,Z/~Z)-~
PicY ~ PicY -~ H2(V~t,Z/~Z).
Since ~cd(P~\V) _< 2n-4, the natural maps Hi(F~St, z/ez)-~ H~(V~t, z/eZ) are isomorphismsfor i _< 2 (see the proof of 16.2). Hence1 (V~t, Z/t~Z) _ HI(P~t,Z/~Z ) = 0 and H2(V~t,Z/~Z) ~_ H2(P~t,Z/~Z ) _ Z/~Z. Since the map Pic(F~) -~ H~(IP~t, Z/gZ) ~ Z/~Z sends O(1) to a generator 2 n H (Fk~t, Z/gZ), the commutative diagram --~ PicF~ PicV n H 2(P~dt’Z/~Z)
-~
H2(V~t, Z/gZ)
shows that the map PicV -~ H2(V~t,Z/~Z) ~_ Z/~Z sends Ov(1) to generator of H2(V~t, Z/gZ). Hence we have the following exact sequence for every g prime to p: 0 -+ PicV -~ PicV ~ Z/~Z ~ 0. This immediately implies that PicV is torsion free (except, possibly, for ptorsion if char k = p > 0). This also shows that NS/{0v(1)} is divisible every prime except, possibly, by p if char k = p > 0. But NS/{Ov(1)} is finitely generated. []
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n Then NS Corollary 17.2. Assume V is irreducible of codimension < -~. is generated by Ov(1) (up to p-torsion if char k = p > O) and Pic°V torsion-free (except, possibily, for p-torsion if char k = p > 0). []
Proof. 16.1ii) and 17.1.
n _ 1. Then PicV Corollary 17.3. Assume V is normal of codimension < -~ is generated by Ov(1) (up to p-torsion, if char k = p > 0). Proof. 16.1iii) and 17.1 show that Pic°V is torsion-free (except for ptorsion, if char k = p > 0). But Pic°V, for a normal V, is an abelian variety. The only torsion-free abelian variety is the zero variety. Hence Pic°V = O. Now(4.1) shows that NS is generated by Ou(1) (up to p-torsion if k=p>0). [] The latter corollary had been knownonly for non-singular V; see [37], 2.2, [78], 4.10 and [90], 3.1. Openquestion. AssumeV is normal of codimension _< ~ - 1. Is the divisor class group of V generated by Ov(1) (up to p-torsion, if chark = p > REFERENCES [1] J. Alvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals, J. Pure Appl. Alg., 150 (2000) 1 - 25. [2] J. Alvarez Montaner, R. Garcia Lopez, S. Zarzuela, Local cohomolgy, arrangement of subspaces and monomial ideals, to appear. [3] H. Bass, On the Ubiquity of Gorenstein Rings, Math. Z., 82 (I963) 8 - 28. [4] R. Belshoff, C. Wickham,A note on local duality, Bull. London Math. Soc. 29 (1997), no. 1, 25-31. [5] R. Belshoff, S. Slattery, C. Wickham,Finiteness properties for Matlis reflexive modules, Comm.Algebra 24 (1996), no. 4, 1371-1376. [6] R. Belshoff, S. Slattery, C. Wickham,The local cohomologymodules of Matlis reflexive modules are almost cofinite, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2649-2654. [7] J.-E. BjSrk, Rings of Differential Operators, North-Holland, 1979. [8] R. Bogvad, Some Results on D-modules on Borel Varieties in Characteristic p > O, J. of Algebra, 1’~3 no. 3, (1995), 638-667. [9] M. Brodmann, Cohomological invariants of coherent sheaves over projective schemes - a survey, this volume. [10] M. Brodmann, A rigidity result for highest order local cohomology modules, Archiv der Math., to appear. [11] M. Brodmann, C. Huneke, A quick proof of the Hartshorne-Lichtenbaum vanishing theorem, in: Algebraic geometry and its applications (West Lafayette, IN, 1990), 305-308, Springer, NewYork, 1994. [12] M. Brodmann, A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128 (2000), 2851 - 2853. [13] M. Brodmann, C. Rotthaus, R. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure Appl. Alg., 153 (2000) 197 - 227. [14] M. Brodmann, R. Sharp, Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998. xvi+416 pp. [15] W. Bruns, R. Schw/inzl, The number of equations defining a determinantal variety, Bull. London Math. Soc., 22 (1990) 439-445.
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[68] G. Lyubeznik, Finiteness Properties of Local CohomologyModules for Regular Local rings of Mixed Characteristic: the Unramified Case, ComIn. in Algebra, 28 no. 12 (2000), 5867-5882. [69] T. Marley, The associated primes of local cohomology modules over rings of small dimension, Marius. Math. 104(4) (2001) 519-525. [70] Z. Mebkhout, Local cohomology of analytic spaces, Proceedings of the Oji Seminar on Algebraic Analysis and the RIMSSymposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976). Publ. P~es. Inst. Math. Sci. 12 (1976/77), supplement, 247-256. [71] Z. Mebkhout, Cohomologie locale d’une hypersurface, Fonctions de plusieurs variables complexes, lII (Sm. Franois Norguet, 1975-1977), pp. 89-119, Lecture Notes in Math., 670, Springer, Berlin, 1978. [72] L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999) no. 3, 417-423. [73] E. Miller, The Alexander duality functors and local duality with monomialsupport, J. Alg., 231 (2000), 180- 234. [74] J. Milne, t~tale cohomology, Princeton Univ. Press, 1980. [75] M. Mustata, Local cohomology at monomial ideals, J. Symb. Comp., 29 (2000), 709 - 720. tensor product, lo[76] T. Oaku, N Takayama, Algprithms for" D-modules - restrictions, calization and algebraic local cohomologygroups, J. of Pure and Appl. Alg., 156 2-3 (2001), 267-308 [77] A. Ogus, Local Cohomological Dimension of Algebraic Varieties, Ann. Math., 98 (1.976) 999- 1013. [78] A. Ogus, On a formal neighbourhood of a subvariety in projective space, Amer. J. Math., 97 (1976) 1085-1107. [79] C. Peskine and L. Szpiro, Dimension Projective Finie et Cohomologie Locale, I.H.E.S. Publ. Math. 42 (1973), 47-119. fiir Schnitte in projektiv algebraischen Mannig[80] M. Peternell, Ein.Lefschetz-Satz faltigkeiten, Math. Annalen, 264, 361-388 (1983). [81] K. Ra8havan, Local-global principle for" annihilation of local cohomology, Contamporary math., 159 (1994), 329 - 331. [82] V. Reiner, V. Welker, K Yanagawa, Local cohomology of Stanley-Reisner rings with support in monomialideals, J. Algebra, to appear. [83] R. Sharp, The Frobenius Homomorphism and Local Cohomology in Re9ular Local Rings of Positive Characteristic, J. Pure Appl. Algebra 71 (1991), 313-317. [84] P. Schenzel, On the use of local cohomology in algebra and 9eometry, Six lectures on commutative algebra (Bellaterra, 1996), 241-292, Progr. Math., 166, Birkh£user, Basel, 1998. [85] A. Singh, p-torsion elements in local cohomologymodules, Math. Res. Lett., 7 (2000), 165- 176. [86] K. Smith, Tight Closure of Parameter" Ideals, Inv. Math., 115 (1994) 41-60. [87] K. Smith, Test Ideals in Local Rings, Transactions of the AMS,347 no. 9, (1995), 3453-3472. Amer. J. of Math., 119 [88] K. Smith, F-rational Rings Have Rational Singularities, (1977) no. 1, 159 - 180. [89] K. Smith, Fujita’s Freeness Conjecture in Terms of Local Cohomology, J. Alg. Geom., 6 no. 3 (1977), 417- 430. [90] R. Speiser, Vanishin9 criteria and the Picard 9roup for projective varieties of low codimension, Comp. Math., 42 (1981) 13-21. [91] N. TerM, Local cohomol9y modules with respect to monomial ideals, Preprint, 1999. [92] U. Walther, Algorithmic computation of local cohomotogy modules and the local cohomological dimension of algebraic varieties J. Pure Appl. Alg., 139 (1999) 303 321.
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[93] U. Walther, On the Lyubeznik numbers of a local ring, Proceedings of the AMS:~129 (6)(2001)1631-1634. [94] K. Yanagawa, Bass numbers of local cohomoloyy modules with support in monomial ideals, Math. Proc. Camb. Phil. Soc., to appear. [95] K. Yanagawa, Sheaves on finite posets and modules over normal semigroup rings, J. Pure Appl. Alg., to appear. [96] A. Yekutiely, An Explicit Construction of the Grothendieck Residue Complex, Ast~risque, 208 (1992), Soc. Math. de France. [97] K.-I. Yoshida, Cofiniteness of local cohomology modules for dimension one ideals, Nagoya Jour. Math., 147 (1997), 179-191. [98] C. Zhou, Functors of the category of abelian sheaves on regular affine scheme, Acta Math. Sinica (N.S.) 12 (1996), no. 4, 413-414. [99] C. Zhou, Higher Derivations and Local Cohomology Modules, J. Algebra 201 no. 2, (1998) 363- 372. DEPARTMENTOF MATHEMATICS,UNIVERSITY OF MINNESOTA,MINNEAPOLIS,MN 55455 E-mail address: gennady¢math.umn.edu
p-TORSION
ELEMENTS IN LOCAL MODULES II
COHOMOLOGY
Anurag K. SinghUniversity of Utah, Salt LakeCity, Utah
1.
INTI~ODUCTION
GennadyLyubeznik conjectured that if R is a regular ring and a is an ideal of R, then the local cohomology modules H~(R) have only finitely manyassociated prime ideals, [Lyl, Remark3.7 (iii)]. While this conjecture remains open in this generality, several results are nowavailable: if the regular ring R contains a field of prime characteristic p > 0, Hunekeand Sharp showedin [HS] that the set of associated prime ideals of HE(R ) is finite. If R is a regular local ring containing a field of characteristic zero, Lyubeznik showedthat H~(R) has only finitely manyassociated prime ideals, see [Lyl] and also [Ly2, Ly3]. Recently Lyubeznik has also proved this result for unramified regular local rings of mixedcharacteristic, [Ly4]. In [Hu] Craig Hunekefirst raised the following question: for a Noetherian ring R, an ideal a C R, and a finitely generated R-moduleM, is the number of associated primes ideals of H~(M)always finite? For some of the work on this problem, we refer the reader to the papers [BL, BRS, He] in addition to those mentioned above. In [Si] we constructed an example of a hypersurface R for which a local cohomology module H3~(R) has p-torsion elements for every prime integer p, and consequently has infinitely manyassociated prime This manuscript is based on work supported in part by the National Science Foundation under Grant No. DMS0070268.
155
156
Singh
ideals. Since this is the only knownsource of infinitely manyassociated prime ideals so far, it is worthwhileto investigate whether similar techniques may yield an example of a regular ring R for which a local cohomology module H~(R) has p-torsion elements for every prime integer p. This leads to some very intriguing questions as we shall see in this paper. Our results thus far support Lyubeznik’s conjecture that local cohomologymodules of all regular rings have only finitely manyassociated prime ideals. Let R be a polynomial ring over the integers and ~, Gi be elements of R for which FiG1 + F2G2 +’-. + FnGn = 0. Consider the ideal a = (G1,..., ~n)R and the local cohomology modute H2(R)= li---~ (a~l, . . ., Gk~) R where the maps in the direct limit system are induced by multiplication by the element GI’" Gn. For a prime integer p and prime power q --- pe, let P Note that has A~has integer coefficients, i.e.,
that A~E R. Consider
R r/~ = [A~+ (Gql .... , Gq~)R] ~ H:(R) -’- ~ (G~,..., G~)R" It is immediately seen that pr/~ = 0 and so if ~/q is a nonzero element of H~(R), then it must be a p-torsion element. Hence if the local cohomology module H~(R) has only finitely many associated prime ideals then, for all but finitely many prime integers p, the elements ~/~ as constructed above must be zero i.e., for some k ~ N, we have A~(GI...Gn) k e (G~+k,..., Gq~+~)R. This motivates the following conjecture: Conjecture 1ol. Let R be a polynomial ring over the integers Fi, G~ ~ R such that F~G~+ ... + F,~G~= O.
and let
Then for every prime integer p and prime power q : pe, there exists k 6 N such that (F~G,) ~ + ...+ (f~G~)q ~...a~)~, . . ., P Weshall say that Conjecture 1.1 holds for G~,..., G~E R, if it holds for n
all relations
~ FiGi = 0 with F1,...,
F~ ~ R.
157
p-Torsion Elementsin Local Cohomoiogy Modules Remark 1.2. The above conjecture is easily this case we have FiG1 + F~G2 = 0 and so (F1G1) q q+ (F~G~)
~ (fla~)~
P
established
if n : 2 since in
+ (-F1G~) :{(oF~G~)q
P
if
p:2,
if
p~2,
which is an element of (G~, G~)R. Example 1.3. The hypersurface example. Conjecture 1.1 is false if the condition that R is a polynomial ring over the integers is replaced by the weaker condition that the ring R is a hypersurface over the integers. To see this, let R = Z[U, V, W, X, Y, Z]/(UX + VY + WZ) and a denote the ideal (~, y, z)R. (We use lowercase letters here to denote the images of the corresponding variables.) Consider the relation u:e + vy + wz = 0 where,
in the notation
of Conjecture
1.1,
F1 = u, F2 = v, F3 = w and
G1 = x~ G2 : y, G3 : z. By means of a multi-grading established in [Si, §4] that
argument,
it
is
(UZ)p + (vY) p + (WZ)P(~yz)k ~_ (xp+k, yp+k, zp+k)R for all k E N. P Consequently Conjecture 1.1 does not hold here even for the choice of the prime power q -- p. This is, of course, the example from [Si] of a local cohomology module with infinitely many associated prime ideals: more precisely, for every prime integer p,
is a nonzero p-torsion
element.
Conjecture 1.4. We record another formulation of Conjecture 1.1. Fi, Gi be elements of a polynomial ring R over the integers such that FIG~ +...+
F,~Gn = O.
Let a denote the ideal (G1,..., Gn) of R. For an arbitrary and prime power q -- pe, we have (Flal) q ~-"’-}-
Let
prime integer
(Fnan) q =- 0 mod p
which is a relation on the elements ~,...,~qn E R/pR, where -- denotes the image of an element of R in the ring R/pR. This relation may be viewed as an element #q ~ H~-I(R/pR). Conjecture 1.1 is equivalent to the conjecture that this element/Q is in the image of the natural homomorphism H~-I(R) --} H~-I(R/pR).
158
Singh
To see the equivalence
of these conjectures,
/Zq E Image (H~-i(R)-~
suppose
H~-I(R/pR)).
Then the relation (~,..., ~-~) on the elements ~,..., ~q~ R/ pR li fts to a relation in H~-I(R), i.e., there exists an integer k and elements ai ~ such that alG~ +k +’" + anGqn +k = 0 o~i =- F~(GI..
"Gi-lGi+l
and
" "Gn) k mod p for
all
1 < i < n.
Hence we have ((F1G1)q
ql_
.,.~_
(F,~Gn)q)(G1 ...G~)~
~ ....a~ al)a~ +~+...+ (F2G~... ~ ~ a~n-1_a~)a~+ : (F~G~
and so
(Fla,) +...+
e
I
P The proof of the converse is similar.
Remark 1.5. We next mention a conjecture due to Mel Hochster. While this was shown to be false in [Si], our entire study of p-torsion elements originates from this conjecture. Consider the polynomial ring over the integers R = Z[u, v, w, ~, y, z] where a is the ideal generated by the size two minors of the matrix
i.e., a = (A1, A2, A3)R where AI = vz - wy, A2 = w~e - uz, and Aa = uy - w. In the ring R we have uA1 + vA2 + wA3 = 0. The relation (uA1)q
-~
(vA2)q
-~-
(wA3)q
~ 0
may be viewed as an element ttq e H~(R/pR). Hochster conjectured for every prime integer there .exists a choice of q = p~ such that tto
~ Image (H~(R)-+
that
H~(R/pR)),
and consequently that the image of #q in H3a(R) is a nonzero p-torsion element of H3~(R). In [Si] we constructed an equational identity which provided us with an element of H2~(R) that maps to #q ~ H2~(R/pR). While we refer the reader to [Si] for the details of the construction, we would like to provide a brief sketch.
p-Torsion
Elements in Local Cohomology Modules
159
Consider the following equational identity: k
n (_1)
xn
A~k+luk.t_ 1
. i
k -t-
i
k -~
n
n=O k
n i=0
k
n
.
n=0
As ig is a relagion on the elements ~+1 this identigy gives us, for every k ~ N, an element 7~ ~ H~ (R). Using k = q- 1 and examining the binomial coe~cients above mod p, we obtain ((~)q
q-~
+(v~)q+(w~a)q)(~a)
Consequengly 7q-~ ~ ~q under the natural
~0 modp.
homomorphism
~(~ ~ ~(a/pal. Proposition 1.6. If Conjecture 1.1 is true for ~he relations
~ NGi = 0
i=0
and ~ FiGi = 0 where N, Fi,Gi e R = g[X1,...,X~],
~hee i~ is also
i=0
true for ~he rel~ion ~(sN+t~)a~= 0 in S = Z[X~,...,X~,~, ~]. More i=0
precisely, gf for a prime power q = pC, ~here e~is~s k~, k~ ~ N such (Gl’’’Gn)k~ ~(G~+kti:o [~(Ei~i)q] [ ~ (~i~i)qi=O
,"",Gq+kt’Rn ] and
P (~l...~)k~
~=0
~
~ (~+k~,...,~q+k~n
~.~(a~... a~)~ e
]~,
,...
for k = max{k~, k2}. Weleave the proof as an elementary exercise. 2. A
SPECIAL
CASE
OF THE
CONJECTURE
In Theorem2.1 below we prove what is perhaps the first of Conjecture 1.1.
interesting
case
Singh
160
Theorem 2.1. Let R be a polynomial ring over the integers and Fi, Gi be elements of t~ such that F1, F2, F3 form a regular sequence in R and F1G1 + F2G2 + F3G3 = O. Let q = pe be a prime power. Then fork=q-1 we have q+k, aq~+k, ~q+k, .. (FIG1)q q- (F2G2)q -t- (F3G3)q (GIG2G3) k E (G1 P
Remark2.2. In Hochster’s conjecture discussed earlier, the relation under consideration is uA1 + vA~ + wA3= 0 and the elements u, v, w certainly form a regular sequence in the polynomial ring R = Z[u, v, w, x, y, z]. Proof of Theorem2.1. Since F3Ga~ (F~, F2)R and F1, F2, F3 form a regular sequence, there exist a, fl ~ R such that G3 = aF~ + flF2. In [Si, §2, 3] we showed that in the polynomial ring Z[A, B, T] (A + B)q + (-A) q + (-B)q kI(A + B)AB] J P is an element of the ideal (A + B)q+~(T, A)k(T, k + Aq+k(T, B) k(T + B,A +~ + Bq+k(T,A)~(T
- A,A + ~
when k = q - 1. We shall use this fact with A = -F2G2, B = -F3G3 and T = ~F2F3. Note that this gives A + B = FIG~, T + B = -~FtF3, and T - A = -FIG1 - ~FIFa. Consequently (F1G1)q -t- (F2G2)q n a (F3G3)q F~G~F~G2F3Ga P is an element of the ideal
The required result follows from this statement.
[]
p-Torsion
161
Elements in Local Cohomology Modules 3.
A PLUCKER
RELATION
Oneof the main goals of [Si] was to settle Conjecture 1.1 for the relation
~(vz- wy)+ v(wx- ~z) + w(~y- vx)
(1)
in the polynomial ring Z[u, v, w, x, y, z]. This was accomplished by establishing that for every prime integer p and q = pe we have, for k = q - 1, 1--[~tq(v
Z __ wy)q + vq(w x __ ~tZ) q ]"~ wq(?ty __ vx)q
P
× [(vz - ~y)(~x- ~z)(~yk
The syzygy of the matrix (vz - wy wx - uz
uy - vx) is
but, by symmetry,Conjecture 1.1 also holds for the relation x(vz - wy) + y(wx - uz) + z(uy - = 0.
(2)
In the light of Proposition 1.6, this gives us the following: Theorem 3.1. Conjecture 1.I holds for A~=vz-wy,
A~=wx-uz,
A~=uy-vx
~ R=Z[u,v,w,x,y,z],
i.e., if F1,F2,F3 ~ R satisfy FIAi+F2A2+F3A3= 0 then for anyprime power q = pe we have, for k = q - 1,
+~, ~+~,A~+k)R. (F~l)q+ (F:/X~)q+ (F~A~)~ (A~/X~A~)~ ~ (Aq~ P By taking a combination of (1) and (2) above, we get the relation
(vz - ~)(~t- ~) + (~ - ~z)(vt- y~) + (~y - ~)(~t in R = Z[s, t, u, v, w, x, y, z]. This is, of course, the Pliicker relation
where Aij is the size two minor formed by picking rows i and j of the matrix
162
Singh
The syzygy of the matrix (ut - xs
vt - ys wt - zs) is
vz - wy
0
-wt + zs
uy - vx
vt - ys
ut - xs
I
vt - ys ~
Since the Koszul relations are easily treated, the following theorem shows that Conjecture 1.1 holds for ut - xs, vt - ys, wt - zs. Theorem 3.2. Consider the Pliicker
relation
(vz - ~y)(~t - xs) + (~x - ~z)(vt - ys) + (~ - vx)(~t in the polynomial ring R = Z[s,t,u,v,w,x,y,z]. Then for k = q - 1, we have 1 [(vz - wy)q(ut - q + ( wx- uz )q(vt - ys q + (uy- vx)q(wt - zs) q] P
x [(~t - xs)(~t- ~)(~t~)]~ q+k,
~ ((ut--xs)
(vt--ys)
q+k,
(wt--zs)q+k)R.
Towards the proof of this theorem, we first record some identities with binomial coefficients. These identities can be proved using Zeilberger’s algorithm (see [PWZ]) and the Maple package EKHAD,but we include proofs for the sake of completeness. Whenthe range of a summationis not specified, it is assumed to extend /L~
over all integers. Weset Lemma 3.3.
(1) ¯ E(-l)n(mm+;-r)n
[ (-1)r-s(kS"~), if ~n_<~
= I(0 -l)r(m+s:k-1)’
n (kk-:)
ifelse.k+l-<m’
(2) r(-1) r
~
=0
ff
l~m~2k-s.
Pro@ (1) Let n
and G(s, n) (- 1)(k + 1 n)F(s, n).
a(~,~+ 1) a(~,~)
It i s e asi ly veri fied that
(-~)(~- n)(~F(~+ ~,~) ~+~+~--r ~+~+~--~’ (k+ 1-~)(~ ~+~+~)’ F(~,~)
p-Torsion Elements in Local CohomologyModules
163
and these can then be used to obtain the relation (s + 1)F(s + 1,n) - (m + s - k)F(s,n) = G(s,n 1)-G(s,n). Summingthe above equation with respect to n gives (s + 1)H(s + - ( m + s- k) H(s) = 0, and using this recurrence relation for H(s) we get H(s) (m- k + s - 1) .. . (m- k)H( 8"’’1
r. The result nowfollows since H(0) -- (-1)
It is a routine verification that G(r) - G(r 1)= m(2 k + 1 - m-s) F(r Consequently if I _< m _< 2k - s then 1 E F(r) = m(2k + l - m - s) ~ (G(r) - G(r =0. r
r
Weuse the above results to establish the following equational identity: Lemma 3.4.
E r----0
n:0
-
2k+l-r n
2k+1
-r
(T - y)n(y _ Z)2k+l-r-nyrzr
(y2k+lzs
_ z2k+ly s) ,
s----O
Proof. Comparingthe coefficients of T~¢-s, we need to show that k
r
(-1)n-r+S(2k+l-r)(~-:)(rns) 2k+l-r n x (Y - Z)2k+l-r-nyn+szr
-
s(-1) 2k+l-s
i.e., that k
x
r
(-1) n-~ 2k+1 -r
(2k+l-r)(:-:)(rns) n
(Y - z)2k+l-r-nynzr-s
1 2k+l-s
(~)
(y2k+l-s
_ z2k+l-s).
164
Singh
The coefficient of ymz2k÷l-s-m in the expression on the left hand side is
E(-1)n+m+l(2k+l-r-n)(2k+l-r)(~-nn)(rns)~-k-~-~:-~ m-n n rift
=~2k+l-r(r-s) + -m ~(-1)n
-n (~:
which we denote by "~m,s. Note that 7m,s sinceE(-1)n(m+s-r)’(kr-:)
:
0 unless m _< k - s or m _> k + 1
=Of°rsuchmbyLemma3"3(1)"\
n
Wenext consider the case m _< k - s. It follows fl’om Lemma3.3 (1) that
r
If m = 0, then the only term that contributes to this sum is when r = s, -1 (~). Wemaynowassumel<_m<_k-sand and we getTo,s2k+l-s then ")’m,s
=0
--
by Lem~na3.3 (2).
For m _> k + 1, Lemma3.3 (1) gives
=E 2k+l-r r
r-s
m
s
By Lemma3.3 (2), this is zero ifm _< 2k-s. Ifm = 2k+l-s the only term that contributes to this sum is whenr = s, and 72k+l-s,s - 2k + 1 - s .s "
This lemmaenables us to establish the following crucial identity:
p-Torsion Elements in Local CohomoiogyModules
165
Lemma 3.5. k
r
k+l
r=0 n=0 X [X2k+I(T
-g)n(g
_ Z)2k+l-r-ngrzr
- Z) n(z
q_y2k+l(T
+z2k+I(T
-
x)n(x
r- x)2k+l-r-nzrx
-
y)2k+l-r-nxryr]
Proof. Using Lemma3.4, the left hand side in the equation equals k
E(-1)s if-
(~) )2k
+k+l1 - Tk-S×
y2k+l(z2k+lxs
[X2k+l(y2k+lzs-z2k+lys
_ X2k+lz s)
+ Z2k+l(x2k+ly
s - y2k+lxs)]
= O. []
Proof of Theorem 3.2. Since k : q - 1, we have (k + 1)/(2k + 1 - r) q/(2q-l-r). For 0 <_ r G q-2, if’q/(2q-l-r) = a/b for relatively prime integers a and b, then p divides a since q + 1 _< 2q - 1 - r _< 2q - 1. Consequently there exists an integer d E Z such that d is relatively prime to p and dq EZ for all 0
8
we get
r=0n=0 2k + 1 - r x
n
~t
8
V
8
166
Singh
2k+l clears denominators, and gives us the equaMultiplying by ds 4k+l (UVW) tional identity (with integer coefficients): E r=O
n=O
2k+l--r
n
+ (~t- ~)~+~(~z)~(~~z)~+~-~-n(~ z~ - )~(~t --~)~ + (wt - Zs)2k+l(vx)n(uy -- VX)2k+l-r-n(ut -- xs)r(vt ys) r] = 0. Note that d(k+l) 2k+l-r
for0
modp,
if
r=kandn=O,
modp, else.
Consequently we get (~t
-- Xs)2k+I(vz
--
wy)k+l(vt -- ys)k(Wt -- k
+ (,~ _ z~)~+~(~_ ~)~+1(~_ ~)~(~ Finally, this shows that
- ~)~(~t - ~)q + (~ - ~)~(~t- u~)q+ (~u ~[(~
q] ZS)
~ [(~t - ~)(~t - ~)(~t -
ACKNOWLEDGMENTS
I am indebted to Mel Hochster for valuable discussions regarding this material. The use of the Maple package EKHAD is gratefully acknowledged. I also want to express xny appreciation to GennadyLyubeznik and Xavier G6mez-MontAvalos for organizing the Conference on Local Cohomology. REFERENCES M. P. Brodmannand A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), 2851-2853;. [BRS] M. P. Brodmann, C. Rotthaus and R. Y. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure Appl. Alg. 153 (2000), 197-227. [BL]
p-T0rsion Elements in Local CohomologyModules
167
[He]
M. Hellus, On the set of associated primes of a local cohomologymodule, J. Algebra 237 (2001), 406-419. C. Huneke, Problems on local cohomology, in: Pree resolutions in commutative algebra and algebraic geometry (Sundance, Utah, 1990), pp. 93-108, Research Notes in Mathematics, 2, Jones and Bartlett Publishers, Boston, MA,1992. [HS] C. Huneke and R. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), 765-779. [Lyl] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41-55. [Ly2] G. Lyubeznik, F-modules: applications to local cohomology and D-modules in characteristic p > 0, J. Reine Angew. Math. 491 (1997), 65-130. [Ly3] G. Lyubeznik, Finiteness properties of local cohomologymodules: a characteristicfree approach, J. Pure Appl. Alg. 151 (2000), 43-50. [Ly4] G. Lyubeznik, Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case, Comm.Alg. 28 (2000), 58675882. [PWZ]M. Petkovgek, H. S. Wilf and D. Zeilberger, A = B, With a foreword by Donald E. Knuth, A K Peters, Ltd., Wellesley, MA,1996. [sil A. Singh, p-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), 165-176. DEPARTMENT OF MATHEMATICS, UNIVERSITYOF UTAH, 155 S. 1400 E., CITY, UT 84112, USA E-MAIL: s±ngh0math.utah.edu
SALT LAKE
ALGORITHMS FOR ASSOCIATED PRIMES, WEYL CLOSURE, AND LOCAL COHOMOLOGY OF D-MODULES Harrison Tsai Cornell University, Ithaca, NewYork
1. INTRODUCTION Let S = K[xl,...,x,~] be the polynomial ring over a field K of characteristic zero, and let D = K{xl,... ,xn,01,... ,0,~) be the n-th Weyl algebra over K. If is a finitely generated D-module, then Mis also an S-module and we can consider its associated primes, Asss(M) = {P C S prime ideal [ P = arms(m) for some m E Although Mis not generally finitely generated over S, its set of associated primes Asss(M)is still finite. In this paper, we will give an algorithm to compute Asss(M) when M is finitely generated left D-module. For instance, this might be interesting when M is a local cohomology module H}(S) for-an ideal I C S. In general, when A is a regular local ring containing a field of characteristic zero, then Lyubeznikhas shownusing the theory of algebraic/)-modules that the set of associated primes of H}(A) is finite [11]. In the case where A = S is a polynomial ring, algorithms compute H}(S) as a D-module have been given by Walther in [23] and by Oaku and Takayama in [14]. Thus, we obtain an algorithm to compute the associated primes of H~(S). Weremark that when I is a monomial ideal, then Mustata has already given an effective characterization of associated primes in terms of simplicial cohomology[12]. Our algorithm for associated primes consists of two parts. First, we identify a set of candidates for the associated primes by using componentsof the characteristic variety or more generally the primary decomposition of the associated graded module (see Theorem 3.1). Second, we discuss algorithms to compute the torsion modules H~(M) for ideals J C S. This gives us a method to test if a candidate prime P is genuinely associated. Torsion algorithms were first developed by Oaku
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in the holonomic case [13] and next by Oaku, Takayama, and Walther in the specializable case [15]. Wewill extend Oaku’s aigorithm slightly to handle the general non-holonomic case (see Algorithm 4.3). Wealso apply the associated primes algorithm towards the problem of computing Weyl closure. The Weyl closure of a left ideal I C D is defined to be the left ideal R-ICl D, where R = K(xl,...,x,~)(O1,...,On) is the ring of differential operators with rational function coefficients. Chyzak and Salvy posed the question of computingWeylclosure in [5], where they use it in one of the steps of a symbolic integration algorithm. Wewill show (see Algorithm 5.11) how Weyl closure can be computed from the singular locus of D/I by applying the localizaton algorithm of [15]. Finally (see Section 6), we discuss Oaku and Takayama’s local cohomologyalgorithm [14]. In particular, we show how it agrees with a local cohomologyconstruction made by Adolphson and Sperber [1]. The main difference is that Adolphson and Sperber consider their construction on the level of A-modules for arbitrary commutative rings A while Oaku and Takayama consider the special case of a Dmodule. Wewould like to thank Bernd Sturmfels and Nobuki Takayamafor their support and help and for suggesting the question of Weyl closure, Gennady Lyubeznik for suggesting the problem of computing associated primes, and Mircea Mustata and Greg Smith for their discussions on characteristic varieties and local cohomology. Finally, we thank Anton Leykin and Mike Stillman, with whomwe implemented many of the algorithms in this paper in the D-module package of the computer algebra system Macaulay2[8]. This article is based on material in our Ph.D. thesis. 2. ASSOCIATED GRADEDMODULE In this section, we recall a numberof standard definitions. The ring D is naturally filtered by the order of operators, i.e. n
D = U~°=oD(j) where D(j):
(~ K[x]O z and
The associated graded ring of D with respect to the order filtration is a polynomial ring gr(D) K[xl,...,xn,~,...,~n] in 2n commuting var iables. For a fi ni tely generated left D-module M, a filtration U~°=oFk(M)of Mis said to be good with respect to the order filtration if Fk(M)is finitely generated over K[x] and if there exists k0 E N such that D(j) Fko(M) -- Fko+j(M) for all j E N.associate d graded module of M with respect to F, denoted grr(M) or gr(M), is then a module over gr(D). Given a presentation D~/Moof Mwith generators (ei}[=~, there is a corresponding standard good filtration defined by Fo(M)= SpanK {ei}[=~ and Fj (M) D(j). Fo(M). The associated graded module with respect to the standard good filtration r can be expressed explicitly as follows. For an element L = ~i=~ ~ ff~,~(x)o ~ ¯ ei in D~, we define its symbol as
E maximal i=~
K[x,V.
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Si~nilarly for the submodule MoC Dr, we define a(Mo) = {a(L) [ L E Mo}. Then the associated graded module is gr(Dr/Mo) K[x, ~]r/a(Mo). Using the associated graded module, we can define the following important invariant of a D-module M. Definition 2.1. TILe characteristic locus of anngr(D) (gr r (M))-
variety of M, denoted char(M), is the zero
Although the associated graded module grr(M) depends on the filtration F, the characteristic variety char(M) is in fact an invariant of Mand does not depend on the filtration. The dimension of a finitely generated D-module M is defined to be the dimension of the characteristic variety. By a famous theorem proved independently by Kashiwara and Gabber, dim(M) _> n for M~ 0. If the dimension of Mis exactly n, then Mis said to be holonomic.If the dimensionis strictly greater than n, then Mis said to be non-holonomic. To compute associated primes of M, we will have occasion to use both the associated graded module (when Mis non-holonomic) and the characteristic variety (when Mis holonomic). Note that the irreducible components of the characteristic variety char(M) correspond to the zero loci of the minimal primes AsSgr(n) (grr(M)) for any good filtration 3.
ASSOCIATED PRIMES
In this section, we give a theorem for obtaining a finite set of candidates for the associated primes of a D-module M. Using this theorem, we sketch an algorithm for computing associated primes of D-modules. Theorem 3.1. Let M be a finitely generated D-module and let F be a good filtration. If P ~ hssK[x](M), then P = Q K[x] fo r so me as sociated pr ime Q E AssK[,,,¢](grr(M)). If M is holonomic, then Q can also be taken to be a minimal prime. Proof: If P C K[x] is an associated prime of M, then by definition there exists m 6 Mwhose annihilator is anng[x](m) = P. Consider the submodule D. m of which is generated by m. The filtration F induces a filtration on D - m. Note that anngr(D) (grr(D.m))~qK[x] = P. Hence there exists a prime Q c K[x, ~] associated to anngr(D)(grr(D ¯ ra)) such that Q f3 K[x] = P. The prime Q may be taken be minimal. The inclusion 0 -~ D ¯ m --~ Mleads to an inclusion of characteristic modules 0 ~ grr(D ¯ m) -~ grr(M ). The associated primes of grr(D ¯ M) are therefore contained inside the associated primes of grr(M), and hence Q is also associated to grr(M). If Mis holonomic as well, then its submoduleD ¯ mis also holonomic. Since the minimal primes of the characteristic module of a holonomic 1nodule all have dimension n, the minimal prime Q of gr r (D. m) is also a minimmal prime of gr r (M). The theorem gives a set of candidates for the associated primes of Mby considering intersections Q v~ K[x] for associated primes Q of gr(M). To test whether a prime P is actually associated to M, one possibility is to compute the P-torsion module H~(M) = {m ~ M [ pim = fo r so me i > 0}. In the next section, we will discuss algorithms to compute P-torsion. Assumingwe have the torsion module H~(M) in hand, then P will be associated to Mas long
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as some element of H~(M)is annihilated by P and by nothing else. This condition can be determined by checking whether v/anngr(D) (gr(H~ (M)))
P.
If yes, then P is associated to M. Wesummarizeour observations into the following algorithm. Algorithm 3.2. (Associated primes of a D-module) INPUT:a presentation M ~_ Dr/Mo of a left D-module. OUTPUT: prime ideals of K[x] associated to M. (1) Compute the symbol a(M) K[x, ~] r of M0usi ng Grhbner bas es. (2) Computethe set of associated primes Q c K[x, ~] of the associated graded module K[x, ~]*’/a(Mo) by primary decomposition. For each prime Q, compute the intersection P -- Q ¢3 K[x]. If Mis holonomic, then do this only for minimal primes Q. (3) For each prime P, compute H~(M) using Algorithm 4.6, and collect into a
set S those primes P such that ~/anngr(D)(gr(H~,(M))) K[x] = P.
(4) Return Example 3.3. Wegive here three simple examples to illustrate Theorem 3.1 and Algorithm 3.2. (1) Let D = K(x,O) and let M = DID ¯ {xO - 2,03}, which is holonomic. The associated graded module is gr~(M) K[x, ~] /(x~, ~3) fo r which th asociated primes are Assgtx,~](gr~(M)) = {(~), (x,~)}. Since M nomic, we only need to consider the minimal prime (~), for which we have (~1 f3 K[x] = (0). In fact, the associated primes of Mare AssK[~](M) {(0)} = {anng[~](1)}. (2) Let D g(z, y, 0~, Cgy)and l et M= D/D.{z OO, yO~, ycgv + 1}, whi ch i s also holonomic. Then gr¢(M) K[x,y,~x,~v]/(~,,y~,,y~v), x 2 fo r wh ich ASSK[~,v,f,,~] (gr¢ (M)) = { (x, y), (~, y), (~, ~v) }. All of these minimal, hence our candidates for the associated primes of Mare (x, g[x, y] = (x, y), (~, y) ~ g[x, y] = (y), and (~,, ~v) g[x, y]= ( 0). Eac of these candidates is an associated prime of M, that is Assg[z,v](M) {(x, y), (y), (0)} = {annK[x,v](0~2), annK[~,vl(0~), annK[~M(1)}. M has an embedded associated prime. (3) Let D g(x, y, 0~,0v) and let M = D/D.{0~(0~ - 1), y( Oz - 1)}, which i non-holonomic. Then gr~(M)= K[x, y,~,~y]/(~,y~), for which the associated primes are ASSK[~,~,~.,~](gr¢(M)) = {(~),(y,~)}. Note only (~) is minimal, however since M is non-holonomic we must consider both (~) K[x,y] = (0) an d (y ,~) N K[x,y] = (y ) ascan didates for the associated primes of M. Wefind, AssK[~,v](M) = {(y), (0}} {anng[~,y](c9z-- 1), annK[~,y](1)}. Example 3.4. We now give an example which required the use of a computer. Let M = D/D.{O~-O~Oz,xOz +yO Gelfand-Kapranov-Zelevinsky hypergeometric system. Wefirst compute the associated graded module or more precisely its annihilator. ±1 : (A = OO[x,y,z, dx,dy,dz, ~/eylAigebra=>{x=>dx, y=>dy, z=>dz}] ~ = #,^1 / ±deal(dy^2-dx*dz, x*dx+y*dy+z*dz+:t, y*dy+2*z*dz+2)
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Algorithmsfor D-Modules J = charIdeal(M)) 9_ ol = ideal (dy - dx*dz, y*dy + 2z*dz, x*dx - z*dz) ol : Ideal
of {~Q [x, dx, y, dy, z, dz]
Next we compute the primary decomposition of the associated graded module by calling the computer algebra system Risa-Asir from Macaulay2 using the OpenXM protocol. i2 : (F = oxasir() oxPrimDecomp(F, J) 2 2 o2 = {{ideal(dz ,dy*dz, dx*dz, dy ,y*dy+2z*dz, ideal (dz ,dy,dx)
2 dx*dy, dx ,-x*dx+z*dz),
{ideal(dz, dy, x), ideal (dz, dy, x)}, {ideal(z, dy, dx), ideal (z, dy, dx)}, 2 2 (-dy +dx*dz, y*dy+2z*dz, 2x*dy+y*dz, -y +4x.z, dx*y+2dy*z, -x*dx+z*dz), 2 2 ideal (-dy +dx*dz, y*dy+2z*dz, 2x*dy+y*dz, -y +4x.z, dx*y+2dy*z, -x*dx+z*dz)}} Our candidates for the associated primes of Mare the intersection of the above primes with K[x, y, z], which are (0), (x), (z), (y2 _ 4xz) . Finally, we c ompute the torsion modules of Mwith respect to these candidates. {ideal
i3 : localCohom(0, ideal(x), M, Strategy=>Walther0TW) o3 = cokernel {0} I dz dy x I i4 : localCohom(0, o4 = 0
ideal(z),
M, Strategy=>Walther0TW)
i5 : localCohom(O,ideal(y’2-4*x*z), M, Strategy=>WaltherOTW) 05=0 It follows that the only associated prime of Mis (x). WhenMis holonomic, Theorem3.1 says that the associated primes of Mare the ideals corresponding to projections of irreducible componentsof the characteristic variety. WhenMis not holonomic, the associated primes can also be described in terms of ideals corresponding to irreducible componentsof various characteristic varieties. The general relationship follows from work of Kashiwara [9] and more recently Smith [18] on the relation between submodules and irreducible components of the characteristic variety. Theorem 3.5. (Smith [18]) If M is a finitely generated D-module whose nonzero submodules all have dimension >_ p, then every irreducible component o] the characteristic variety o] Malso has dimension >_ p.
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Proposition 3.6. Let M be a finitely generated D-module. For i between n and 2n - 1, let Mi be the maximal submodule of M of dimension i, and let {Qij}j c K[Xl,..., x,~, ~1,..., ~,~] be the minimal prime ideals associated to gr(M/) ]or some 9ood filtration F. Then the associated primes of Mare contained in the set of prime ideals {Qij v~ K[xl)/,j. Proof: Suppose that P C K[x] is a prime ideal associated to M and m E M is the element such that that anng[x](m) ---= P. As in the proof of Theorem3.1, there is a minimal associated prime Q c K[x, ~] of gr(D. m) such that Q N K[x] P. Equivalently, the zero locus V(Q) is a component of the characteristic variety char(D, m). Wewould like to show that Q is also a minimal associated prime gr(M/) for some i, or equivalently that V(Q) is a component of the characteristic variety char(M/) for some i. Let i be the dimension of Q, and consider the maximum submodule (D. m)i of D ¯ m of dimension i. Wehave an exact sequence, 0 ~ (D.m)/-~
D.m D.m -~ (D.m)i
-~ 0
so that char(D.m) char((D.m)i)Uchar(D.m/(D.m)~). Moreover, D.m/( D. m)~ contains no submodules of dimension < i, hence by Theorem 3.5 the components of its characteristic variety are all strictly bigger than i. It follows that V(Q) is a component of (D ¯ m)i. Now consider the maximum submodule Mi of M dimension i so that (D ¯ m)i C Mi. Since the characteristic variety of M~has dimension i and char((D re)i) C char(Mi), we seethat the d i~nension i v~r ie V(Q) is also a component of char(M/). 4. COMPUTINGTORSION In this section, we discuss algorithms to computethe torsion of a finitely generated D-moduleMwith respect to a polynomial f, or more generally with respect to an ideal I C K[x]. This completes the ingredients of Algorithm 3.2 for computing associated primes. If M is holonomic, or more generally holonomic away from the zero locus of I, then for any f ~ I, the D-module M[f-~] is also holonomic and thus finitely generated. Moreovera presentation for M[f-~] as well as an explicit represent.~tion of the map M -~ M[f-1] can be computed using a localization algorithm due to Oaku, Takayama,and Walther [15]. If {fl,..., fr} is a set of generators for I, then the torsion module H~(M)is equal to the kernel of the map M-~ (~ir__l M[f[I]. Since this is a map of finitely generated D-modules, its kernel can be computed using GrSbner bases. If Mis not holonomic, then M[f-1] is generally no longer finitely generated as a D-module. For instance, D[f-1] is not finitely generated as a D-module. In this case, we need a different algorithm to compute torsion. The first algorithm to compute torsion of a holonomic D-module was given by Oaku in [13] and is based on restriction rather than localization. Wewill give a slight extension of Oaku’s algorithm to the non-holonoxnic case. Let us start by recalling the theoretical basis for Oaku’s algorithm. Theorem 4.1. [13] Let M be a left (4.1)
Hy(M) ~- ker
D-module and f ¯ K[x]. Then ®~¢[~1 M -~ ~ ®tC[x]
M
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where Dn+l:= D(t, Or), J is the ideal Dn+l" {t-f, Oz. + ~,Ot}~_l, and (Dn+l/J)® M is given the structure of a left Dn+l-moduleby xi(L®m) = xiL O,,(L®m) =O~,L®m+L®O,,rn
t(L®m) = tL®m Ot(L®m) =OtL®m.
Proof: Wefirst observe that Dn+i/J is free as a K[x]-module with the decomposition Dn+l/J : ~i=oK[x]Ot, this can be seen from the isomorphism of left D,~+l-modules ~b : (D,~+~/On+~. {t,O~,...,O~,}) -~ (D,~+~/J) given by sendingxi ~ xi, t ~ t-f, 0~, ~ 0~, + (Of/Oxi)O~, andOt ~ 0~. It follows that ~ i ®g M). We now claim that the desired isomor(D,~+~/J) ®K[x] M = ~=0(0t phism (4.1) is given by the map ker
~ HI(M ) i=0
To see this, suppose m = ~i=o O~ ® mi 6 (Dn+~]J®M),where mi = 0 for all but finitely manyi. Then tm
= ~i>o tO[ ® mi
iO~ -~) ®mi
= Ei~o(OttEi_~O(yO[
--
iO/-1)
mi
= Y’~i>_o Oi~ ~ (fmi - (i 1)mi+l). Therefore, tm = 0 if and only if (i + 1)mi+~ fmi for al l i, or equivalently mi = (1/il)fimo for all i. Since also mi = 0 for all but finitely manyi, this last condition also means fimo = 0 for i >> 0, and it follows that ~ is an injective map from the kernel to H~(M). Conversely, if mo ~ H~(M) so that fNmo = 0 for some N > 0, then the element m = E~~ 0t N (1/i])fimo is in the kernel and maps to m0 under ~. Weconclude that (4.1) is an isomorphism. More generally, the complex in (4.1) computes the local cohomologyof Mwith respect to the principal ideal generated by f. This construction can also be extended to a complex which computes the local cohomology modules of M with respect to any ideal of the polynomial ring. Wewill discuss this construction in Section 6. A presentation of the module (D~+~/J) N~[x] M~ppearing in Theorem 4.1 was given by Walther in [23]. Put Oi = 0~ + (Of/Oxi)Ot and for an element P = ~j pj(xx,..., x~, t, 0~,..., On, Ot)ej ~ D~+~,define
~(P):= ~p~(~,... ,~,,t - f,¢,... ,0,, 0~)~e v~+~. J Lemma4.2. [23] Given a presentation M ~ D~/N, then we have the presentation (D~+~/J) Ngi~] (Dr/N) ~ Dn+x/K(N) as le~ Dn+x-modules, where
U(N):= ~,+~.((t - f)~)~=~+ D,+~. Proof: Consider the map ~: D~+a~(D,~+~/J) NK[~] (Dr/N) of left D~+~-modules defined by ~(ej) = 1 ej . Wecla im tha t ¢ i s sur jective wit h ker nel equ al to K(N). To show surjectivity, note that (D~+~/J) N~[~I (D~/N) is spanned as a K-vector space by the images of "monomials" 0~ N x~O~ej. A computation shows
~(o[~~) = o[~-~’¯ (1, ~) = o[. ~-0g~~d thCraore~ is ~urj~ctiv¢.
To determine the kernel of ~, a computation shows that K(N) ker(¢). To obtain the opposite inclusion, let m be an arbitrary element of ker(~). Note that
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the subalgebras K(x, Ox, Or) and K(x, v~, Or} are equal. Thus modulo the relations {(t - f)e J}j=l, r we can write rn ~cij~O[x’~#~ej. Since m E ker(¢), th is im plies ¢(m) ~(cij~ZO~x~#~) ¯ (1 @ ej) = ~ 0~~ cij ~x ~Ox~ej = 0 i n (D,~+~/J) @g[~) (D~/N). Recall that Dn+~/Jis free as a K[x]-modulewith basis {0~}, and therefore we must have ~a,Z,j cij~¢x~O~¢ej ~ N for all i. It follows that ~ cij~ZO~x~#~ej ~ Dn+~ " ¢(N) and hence m ~ K(N). Nowwe may give an adapted version of Oaku’s torsion algorithm by combining and extending the techniques of [13], [14], and [23]. The algorithm computestorsion for an arbitrary finitely generated D-module. One ingredient of the algorithm and its proof is the notion of V-filtration and V-adapted resolution. For the reader’s convenience, we recall the definitions here. The V-filtration Fy of a shifted free module D~+~[N]with respect to the hyperplane Y = {t = 0} is defined by F~(D~+~[~]) Spang{xaO~tJO~et :
a, ~ e Nn,k -j ~ mt+i}
A free resolution X" of the form, X" :""
~r~+l ~ ~.+t [Nj+~]
~
~ r~Dn+~[my]
~""
is said to be V-adapted if i
rj+l
~ . [m,+,])) C
i
rj
for all i and all j, and if a resolution is also induced on the level of associated graded modules, ¯ ~+~ ~
gr(Dn+ l[mj])
~....
The b-function of a module D~+~/Nwith respect to the V-filtration Fg is the monic polynomial b(s) ~ K[s] of least degree such that b(O)gro(D~+~/P) = 0 where 0 = tot. Algorithm 4.3. (Torsion module H}(M)) INPUT:a presentation M ~ Dr/D¯ {T~,..., T~} of a left D-module. OUTPUT:{RI,..., Rb} C Dr whose images in M ~ Dr/D¯ {T~,... ,Ta} generate the submodule H~(M). (1) Compute the generators {t - f, ¢(T~),..., ¢(T,)} g(g). of K(N) with respect to the (2) Compute a Grhbner basis 6 = {gt,...,g~} weight vector (-w,w) where w = (0,...,0, 1). (Here (-w,w) gives, elements {tej}~=~ weight -1, the elements {Otej}~=~ weight 1, and the elements {xiej,Oiej}ij weight 0). Note that ~ generates gr(K(N)) respect to the induced V-filtration. ~ grk(D~+l/K(g)) is (3) Compute d e ~ such that gr~+l(D~+~/K(g)) injective for k > d. If D~/N is holonomic, then d can be taken to be the maximuminteger root of the b-function b(s) of D~+~/K(N) with respect to V-filtration. If D~/Nis not holonomic, then use Algorithm 4.5. left D,+~-modules (4) Let Cg : D~+~[~] ~ Drn+~[~ be the map of (filtered) (with respect to the V-filtration Fy) defined by sending ej ~ gj, and where ~ ~ ~ is such that mj equals the (-w,w)-weight of gj. Express the map ~g: F¢(AS[~]) ~ F~(hr[~]) induced by Cg as a map of finitely generated D-modules, where A := (Dn+~It. D.~+~).
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s Pijej}i=b 1 C Dn+ s 1 whose images in (5) Compute a set of elements {Pi = ~j=l F~n(Ar[{~]) generate the kernel of ~ r b c D,~+~ (6) Compute elements {Qi}i=~ such that tQi = ~=~ Pijgj for all i. (7) Reduce each Qi ~nodulo Dn+~¯ {(t - f)ej}~= 1 to an element Q~ of the subring K[x,0z,0t] C Dn+l. Let Ri E D" be the element obtained by substituting Ot = 0 in Q~. (8) Return {R~,...,Rb}. Proof: Let the kernel of (D~+I/K(N)) --~ (D~+I/K(N)) be denoted by W. From Theorem 4.1 and Lemma4.2 and using the notation there, we have that HI(Dr/N) = qao¢(W). Step 3 implies that WF~(Dr~+I[-~]/K(N)). Tosee why d may be taken to be the maximuminteger root of b(s) when D~/N is holonomic, note that D~+I/K(N) is also holonomic, hence b(s) ~ O. By definition b(Ot + k) grk(D~+~/K(N)) = 0. To show injectivity, suppose m e grk+~ (D~+I/K(N)) with tm=O. Then0= b(Ot+k+l)m= b(Ott+k)m =b(k)m. Thus ifk > d, then b(k) y~ and m = 0. Let now X° be a V-adapted free resolution of D~+I/K(N) extending the map ¢6 of Step 4, and let Z° denote the complex 0 ~ D~+~[-I~ --~ D,~+~ -+ 0. Then the total complex of X° ®D.+, Z° °is quasi-iso~norphic to both A ®D~+,X and Z° ®D~+,(D~+I/K(N)). Moreover, these quasi-isomorphisms induce quasiisomorphisms on the graded level with respect to the V-filtration Fy. Since the kernel W, which is the -1-th cohomology of Z° ®D.+~(D~+~/K(N)), is contained in F~(Dn+I d r [-~]/K(N)), therefore also the -1-th cohomology of A ®D.+~X° may be generated by cycles in F~(M[nh]), or in other words by the kernel of Note that the maps in A ®D.+~X° are ~naps of left D-modules, and moreover since Fyk (A) = q~/k=oD. 0~, both the source and target of ~g are free D-modules finite rank. Thus the kernel of ~ can be computed in Step 5 by Grhbner bases. ~ ~ The cycles (Pi -- }i=~ can nowbe lifted to cycles {Pi @Qi}i=~ which generate the -1th cohomologyof Tot°(X° ®DZ°). These are in turn projected to cycles{Qi}i=~- b which generate the kernel W, which is the process of Step 6. Finally, for each Qi, we need to compute ~ o ¢(Qi). To do this, we can assume that Qi is reduced modulo D,~+~ ¯ {(t - f)ej}~=~. Nowwe should express Qi in the form Qi = ~,~,~ c~x~O~O~ej. Then ¢(~i) = ~a~ O~k ci ~,~x~cO~e~ and ~ o ~(-~) = ~joc~ Cijoc~X’~O~xeJ. Note however that given the standard expression c~ ~ k t ! Qi = ~jkaf~ CijkaBX O~xVOt ej, thena ~jo~ = Cijo~z. Thus it suffices to substitute Ot = 0 in Qi to get qo o ¢(~i), as we do in Step [] To achieve Step 3 of Algorithm 4.3 when D~/N is not holonomic, we shall use the following proposition. Proposition 4.4. Let B C D~+1 be a left submodule, let w = (0,...,0, 1), and let H : in(_~,~)(B)~D[0~] r where 0~ : tc9~. I] (H : 0~ - k) : H, then the ,nap grk+~(D~+i/B ) ~:~ gr~(Dr~+~/B) is injective. Moreover, ]or any left D[O~]submoduleH’ C D[O~]", the condition (H’ : 19~ - k) = H’ is generic. Proof: Weshall prove the contrapositive, that is, if there exists a nonzero element T ~ ker(grk+~(D~+~/B ) ~+ grk(D~+~/B)) , then (H : ~ - k) ~ H. So assume
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the existence of T. Then T lifts to a (-w, w)-homogeneouselement T’ E D~+1 of weight k + 1 such that T’ ~ in(_w,w)(B) while tT ~ E in(_~,w)(B). Assume now that k _> 0. The argument is similar for k < 0 by interchanging the roles of t and Or. Wecan now write T~ = O~t+IP where P ~ D[Ot]r. Then tk+~T’ = Ot’" (Or -- k)P ~ H. Weshall now show tkotkP = Or"" (St -- k + 1)P in(_~,w)(B), which proves that (H : 0t - k) ~ H. In fact, we claim tiO~t P ¢ in(_w,~)(B) for all 0 _< i _< j _< k. This is true for all j when i = 0 because T’ = otk+lp ¢ in(_w,w)(B) by assumption. For general (i,j) with i _> 1, we argue by contradiction.If t i 0~j P ~ in( _ w,~) (B), then (Or + k - j + i).-. (Or + k - j k P = okt-J+itiv~[P e in(_w,w)(B). Wealso know that OtOtkP = tOkt+lP ~ in(_~,~)(B). Since 0t and (Or + k - j + i)... (0t + k - j + 1) are relatively prime, this implies OtkP E in(_~,~)(B), which contradicts the assumption okt+~P ¢ in(_w,~)(B). To prove the statement about submodules HI C D[0t] r, let F denote the standard good filtration of D[0t] r coming from the order filtration of D[Ot], where 0t gets order 0. For each k ~ Z such that (H~ : Or-k) ~ H, there exists an element Tk ¢ H’ such that (Or - k)Tk ~ H’. Wecan further assume that in(0,~)(Tk) in(0,~) ~) while (Or -k )in(0,~)(Tk) ~ in (0,~)(H’). In other words, the re is a non element of K[x,~x,O]r/in(o,e)(H) having torsion with respect to (Or -k). Since K[x,~x,O]~/in(o,e)(H ’) is finitely generated over the polyno~nial ring K[x,~x,0], this can 0nly occur for finitely manyintegers k by primary decomposition. [] Wecan now make Proposition 4.4 algorithmic. Algorithm 4.5. (Injectivity of grk+~ (D~+~/B) ~ gr~ (D~+~/B)) INPUT:generators {S~,... ,Sa} of B C D~+I. OUTPUT:d e Z with grk+l(Dr~+~/B) ~ grk(D~+~/B) injective for all k > d. {L~,...,Lb} C D~+~ of the (~) Compute (-w,w)-homogeneous generators initial ideal in(_~,~)(B) where w = (0,... ,0,1). di be t he (-w, w)weight of L~, and let H be the left D[Ot]-submoduleof D[Ot]~ generated by {~L~,... ,~cL~} where ~i = t ~’ if di >_0 and ~i = Otd’ if di < O. (2) Computegenerators {hi,..., hc} C K[x, ~, Ot]~ of in(0,e)(H) where (0, the weight vector giving the element x~0~0~ej weight I/~[. (3) Computea Grhbner basis 6 = {gl,-.., gu} of the K(A)[x, ~, 0t, s]-module generated by {(1-s)hi, s(Ot-A)ej }i,~ C g(A)[x, ~, Or, s]~ using any elimination order having {sei}[=l > {xej, ~e~, Otej}j= 1. In the process of computing ~, keep all computations in K[A], that is never perform any division in the field K(A)but rather cross-multiply lead coefficients in forming S-pairs. (4) Compute the greatest integer root d which occurs in any of the leading coefficients p~(A)of gi. (5) Return d. Proof: The submodule H computed in Step 1 of the algorithm is the intersection H = in(_w,~) (B) D[O t] r. Step 2 c omputes the symbol in(o ,e) (H). By the p roof Proposition 4.4, if (in(o,~)(H) : 0t - k) = in(0,~)(H), then {H : 0t - k) implies by the same proposition that the map grk+~(D~+~/B) -~ ) grk(D~+t/B is injective. So we are reduced to analyzing the condition (in(0,e)(H) Ot- k l = in(o,~) (H). Step 3 is a Grhbner basis methodto computethe saturation (in(o,~) Ot - k} (see e.g. [6]). Namely,put = {g /(Ot - A) g ~~ VI /( ’( A)[ ~x,tgt] }. Then
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7) is a Grhbner basis for the saturation (in(0,e)(H) Ot- A). Moreover, sinc in(0,e)(g) is defined over K, it follows that (in(0,e)(H) Ot- A) = i n(0,e)(H). Now suppose that pi(A0) ~ 0 for all i. Then the leading monomials of Gl~0 are the same as those of 6, and hence a straightening relation for any S-pair of 61~o can be obtained from the straightening relation of the corresponding S-pair of ~ by substitution A ~ A0. It follows that ~l~x0 remains a Grhbner basis for the submodule it generates. Put 7)~,o = {g/(Ot Ao) : g E 6~o N K[x, ~x , Ot]}. Then 7)~o is nowa Grhbner basis for the saturation (in(o,e)(H) Ot- Ao). Note als o tha 7)~o = 7)l~Xo because the lead term of any element in G \ 7) contains s and lead coefficient p(A) with p(Ao) ¢ 0. Therefore 7)~o also remains a GrSbner for in(o,~)(H), hence (in(o,~)(H) Ot- Ao) = in(o,e)(H). Th is es tablishes th e validity of the integer d of Step 4 and completes the proof. [] This completes the algorithm for computing torsion modules H](M) of an arbitrary D-module M with respect to a polynomial ]. By intersecting, we may similarly compute torsion modules with respect to arbitrary ideals I C K[x]. Algorithm 4.6. (Torsion module H~(M)) INPUT: a presentation M ~_ D~/D ¯ {T~,...,Ta}
and generators
{f~,...,
fm} of
I c g[x].
OUTPUT:{/~1,... ,I~b} C Dr whose images in M ~- D~/D¯ {T~,... ,T~} generate H~(M). (1) Compute the torsion submodules HI, (M) C Musing Algorithm (2) Computethe intersection of all HI, (M) using Grhbner bases. (3) Return generators of the intersection.
Remark 4.7. Algorithm 4.5 combined with Steps 2 to 6 of Algorithm 4.3 computes the kernel of the map M ~ M where M is any finitely generated left D~+~-module.This kernel is equivalently the 1-st restriction module of Mto the hyperplane t = 0. By computing intersections as in Algorithm 4.6, we can similarly compute the i-th restriction of Mto any codimension i subspace. 5. WEYLCLOSURE Let R = K(x~,..., x~)(01,..., c9,~} denote the ring of differential operators with rational function coefficients. Then the Weylclosure operation is defined as follows. Definition 5.1. Let N C D~ be a le]t D-submodule. The Weyl closure denoted CI(N), is the submodule CI(N) = R. N Cl r.
o] N,
In this section, we use the associated primes algorithm to compute the Weyl closure of submodules of D~. Wefirst give an algorithm for arbitrary submodules N. Wethen give an algorithm for submodules N such that D~/N has finite rank, which in practice is the situation where the Weyl closure has applications. Wealso remark that the Weylclosure of ideals in the first Weylalgebra was studied in [22]. The relation between Weyl closure and torsion is easy to observe. Given an element L ~ Cl(N) \ N, then gL ~ N for some g ~ K[x] or in other words, the image of L in Dr/N has torsion with respect to g. Conversely, if the image of an element L ~ D~ in D~/N has torsion with respect to some g ~ K[x], then L ~ Cl(N). In other words, CI(N) = {n ~ D~ [ gL = 0 in D~/N for some g ¯ g[x]}
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Thus, one algorithm for the Weyl closure which follows immediately from the earlier sections is to compute the associated primes P~ of Dr/N, take a polynomial f~ E P~ for each prime, compute generators of Hy(Dr/N) for f : I-[~f~, and lift each generator to DL The relevant property of the polynomial f = [L f~ is that Cl(N) = D[f-1] ¯ N N r D . Since computing the primary decomposition of the associated graded module in order to obtain the factors f~ maybe intensive, let us mention another approach to identifying polynomials f satisfying Cl(N) = D[f-~] ¯ N ~ Dr. Weuse the following basic fact of D-moduletheory (see e.g. [3, Chapter VII, Lemma9.3]). Lemma5.2. Let M be a finitely generated D-module. Then there exists such that M[f-1] is a free g[x][f-X]-module.
f ~ K[x]
Moreover, the polynomial f guaranteed by Lemma5.2 can be computed using Grhbner bases. The only disadvantage to this approach is that f may have more factors than necessary, thus making the computation of f-torsion more intensive. Algorithm 5.3. (Computing f so that M[f-1] is a free K[x][f-1]-module) INPUT:a presentation M ~_ Dr/D¯ {T1,..., T~} of a left D-module. OUTPUT: f e K[x] such that M[f-1] is free over g[x][f-1]. Choose any term order < on Rr and compute (1) Put N = D. {T~,...,T~}. a Grhbner basis G of R - N with respect to < which also has the property that ~ C N. (2) For each element gj ~ ~, suppose that the lead term of gj is equal to
Y~(x)O~e~. (3) Return f = l-[j fJ(x). Wenow summarize the general algorithm to compute Weyl closure. Algorithm 5.4. (Weyl closure of N C r) r. INPUT:{T1,... ,T~}, generators of N C D OUTPUT: {U1 .... , U~} C Dr, generators of Cl(N). (1) Compute a polynomial f such that CI(N) = D[f-1] ¯ N fq Dr using Algorithm 5.3. (2) Compute {R1,..., R~} C Dr whose images in Dr/N generate the torsion module H~(Dr/N) using Algorithm 4.3. (3) Return {TI,..., T,, R~ .... , Rb}. In practice, we will be most interested in computing the Weyl closure for submodules N whose quotients Dr/N have finite rank. Let us recall the definition of rank. Definition 5.5. The rank of a D-module M, denoted rank(M), is the dimension of R . M as a K(x)-vector space. We say M has finite rank if rank(M) < Finite ferential relation was first
rank modules correspond to overdetermined systems of linear partial difequations and are closely related to holonomic modules. The following was established by Kashiwara [10], although the formulation we present made by Takayama[21]. Proposition 5.6. Let N C Dr be a submodule. Then D~/N has finite rank if and only if Dr/CI(N) is holonomic.
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The question of computing the Weyl closure for finite rank ideals of D was posed in [5] by Chyzakand Salvy, whocall this question the "extension-contraction problem" and consider it more generally for left ideals of Ore algebras. Their motivation to compute Weyl closure is for non-commutative elimination and its application to symbolic integration. Namely, given a left ideal I’ in Rn+d Rn (ti,..., td) (Or1,..., Ota) consisting of operators whichannihilate a function f f(xi,..., xn, ti,..., td), then the left ideal of R~defined by
consists of operators which annihilate the integral g(xi,...,Xn)
=/c fdti’"dtd,
where C is a suitable homologycycle (see [17, Theorem5.5.1] for a proof). Unfortunately, the available techniques to computeJ’(I’) all rely on methods in the Weyl algebra D and not in the ring R. To be precise, given a holonomic ideal I in Dn+d,then the intersection !deal, J(I)
=
(I + O~lDn+ d ~-
.’’
-b O~Dn+d)
can be computed by an algorithm due to Takayama [20] and refined in [17, Algorithm 5.5.4]. Thus, given a function f, the first step in Chyzak and Salvy’s technique for symbolic integration is to obtain enough operators annihilating f so that the ideal I they generate is holonomic. Moreover,it is also clear that if we can make the ideal I even larger, then we will obtain an ideal J(I) which is possibly also larger and therefore a better description of the integral of f. Thus ideally, we would like to obtain the full annihilating ideal of f in D, that is I = annD(f). As we shall see, in manysituations it is easy to obtain generators for a slightly different ideal, annR(f), which is the annihilating ideal of f in R. In this case, the annihilating ideal annD(f) is the contraction annR(f)¢~ D, i.e. the Weyl closure (see Proposition 5.14). Example 5.7. We give an example taken from Stanley’s book [19] and which is the subject of recent work of Pemantle [16]. Given a rational function F(x,t) in two variables and a power series expansion F(x,t) = ~r,~e~ ar~x~t~, then the diagonal function ~(x) = ~re~arrxr is known to be an algebraic function. can be computed using residues as fct-iF(t,x/t)dt, where C is an appropriate homologycycle. For instance, consider F(x, t) = (1 - x - t - -1, for which counts the numberof waysto reach (r, s) from (0, 0) using steps of size (1,0), (0, or (1, 1). Then we would like to compute, t - x
- t 2 - xt"
Let us computedifferential equations describing the integral. Namely, first note that the integrand f = (t - x - ~ -xt ) -~ is a s ol ution of thediff erential oper ators L~ = O~.(t - x - ~ - xt ) and L~= O~(t - x - 2 - xt) . The operatorsL~ andL~ generate the annihilatin~ ideal of at in R since they form a rank 1 system. However, the ideal I = D. {L~, L.z} is not the annihilating ideal of f in D and is not even holonomic. In fact, J(I) = (I + O~D)~ K(x,O~)
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hence the ideal I does not produce any information about the integral. On the other hand, using our implementation in Macaulay 2, we can find the annihilator over D by using the Weyl closure. il : (W = QQ[x,t,Dx,Dt,WeylAlgebra=> {x=>Dx,t=>Dt}]; f = t-x-t-2-x,t; I = ideal(Dx*f,Dt*f); ClI = WeylClosure(I)) 2 ol = ideal (x*Dx + 2t*Dx - t*Dt - Dx - Dr, x*t*Dx + t Dx + ... By someexperimentation, we findthatCl(1)= I + D-{(x+ 2t-l)0xi2 : ClI == I + ideal((x+2*t-l)*Dx(t+l)*Dt) 02 = true Moreover, the integration ideal J(CI(I)) i3 : DintegrationIdeal(ClI, {0,i}) 03 = ideal(x
Dx - 6x*Dx + x + Dx - 3)
Here the parameter {0, 1} of DintegrationIdeal indicates that t is to be integrated out as opposed to x. The integration ideal J(Cl(I)) is a rank one ordinary differential equation whose solution space is spanned by the function p(x) (x 2 - 6x + 1) -1/2. Hence the integral fc f(x, t)dt scalar multiple of p(x), and since a1,1 = 3, we conclude that ~(x) = 2 - 6x+ 1) -1/2. In the general Weylclosure algorithm, recall that the first step is to obtain polynomial f such that CI(N) = D[f-l] . N [~ Dr. WhenDr/N has finite rank, we can obtain a natural candidate for f by using the singular locus. Definition 5.8. Let M be a finitely generated D-module and F a good filtration. Then the singular locus of M, denoted Sing(M), is the zero set Y((anngrr(D)(grr(M)) : (~1,... °°) ~q K[xl ,... ,x~]) C K’~. Theorem 5.9. Let N C Dr be a left D-submodule such that Dr/N is finite rank. Then Cl(N) = D[f-1]. N [q Dr for any polynomial f vanishing on the singular locus Sing(Dr~N). Proof: Wehave f 6 %/(anngrD(gr(Dr/N)): (~1,..., ~n)°°) by definition of the; singular locus, or in other words, gr(Dr/N) is annihilated by elements fc(d~,..., for some c, dl,... ,dn 6 N. Thus gr((Dr/N)[f-~]), which is finitely generated over I([x,~][f-1], is annihilated by ~d~,... ,~,, and in particular is finitely generated over K[x][f -1] as welt. It follows that the D[f-1]-module (Dr/N)[f -~] is finitely generated over K[x][f-~]. It is a basic fact of D-moduletheory that a :Dx-module which is also coherent as an (_0x-moduleis locally free over (3x (see e.g. [7, Lernma 5, Lemma6]). Here, we take X to be the nonsingular variety ~ \ V(f), wh ring of differential operators is precisely D[f-i]. In particular, (D~/N)[f-~] is thus torsion-free with respect to K[x][f-1]. From’this fact, it follows that the D-submodule, D.N ~ Dr C D[f_l] r D[f-~] r r, ~--ll~ " _N N
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is torsion-free with respect to K[x]. In other words, let L G CI(N), which means that gL ~ N C D[f -1].Nf]Dr for someg 6 K[x]. Then the image of L in Dr/D[f-1] ¯ N [q Dr has torsion with respect to g, and hence it must already be that L ~ D[f-1] ¯ N [q Dr. Weconclude that CI(N) = D[f-~] ¯ g N Dr. [] To finish the computation of the Weyl closure, it only remains to compute generators for HI(M) and lift them to Dr where M = Dr/g. Recall that HI(M ) is the kernel of the map M -+ M[f-1], which can be computed by the OakuTakayama-Walther algorithm as long as Mis holonomic away from V(f). Wenow wish to show that a finite rank module Mis holonomic away from its singular locus. Let the non-holonomic locus of Mbe the set of points in the cotangent bundle T*Kn = K2~ where the dimension of the characteristic variety is strictly greater than n. Then we wish to show that the projection of the non-holonomic locus to /(n is contained inside the singular locus. Lemma5.10..IfM has finite rank, then the non-holonomic locus, of M is contained inside ~r-l(Sing(M)), the subvariety of the cotangent bundle which projects to the singular locus. Proof: Suppose the polynomial f vanishes on Sing(M). Then there exists m, N 0 such that fm~g ~ anngr(D)(gr(M)) for all i. This implies that char(M) V(f) tO (~1,..., ~, ~), or in other words, the non-holonomic locus is c ontained insi f = 0 on the cotangent bundle. [] We can now summarize the algorithm to compute Weyl closure of finite modules.
rank
Algorithm 5.11, (Weyl closure of finite rank modules) INPUT:generators {T~,..., Ta} of N C Dr such that Dr/N is finite rank. OUTPUT: G, a generating set for Cl(N). (1) Computea polynomial f ~ K[x] vanishing on the singular locus of Dr/N. (2) Computethe localization map ~o : (Dr/N) -+ (Dr/N)[f -~] using the localization algorithm from [15]. (3) Compute generators G for the kernel of Dr __!+ (Dr/N) _~+ (Dr/N)[f-~]. (4) Return G. Example 5.12. The annihilator of e ~/(~a-y~) in R is the rank 1 ideal generated by
=
-
+
_
_
_
_
Then annD(e~/(~-~)) equals the Weyl closure R. ~ ~q D, and using Macaulay 2, we find that it is generated by the elements ~ t3 {yO~ - ZOz, y2z30~ - 2~x4 O~ 2x~zO~- 2}. il : (w = OO[x,y,z,Dx,Dy,Dz, WeylA~gebra=>{x=>Dx,y=>Dy,z=>Dz}]; 2 = (x’3-y’2*z’2); I = ±deal(f^2*Dx+3*x^2, £^2*Dy-2*y*z^2, 2"2*Dz-2*y^2*z); eli = Wey~¢losure(I)) 2 3 2 2 3 2 ol = ideal (y*Dy - z*Dz, y*z Dx + -*x Dy, y z*Dx + -*x Dz .... 2 2 i2 : CII == I + ideal(y*Dy-z*Dz, y^2*z’3*Dz-(2/3).x^4.Dx-... 02 = true
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Let us end this section by mentioning an analytic interpretation of the Weyl closure. From the analytic perspective, the singular locus of a finite rank module generalizes the notion of singular points of a linear ordinary differential equation. In particular, a special case of the famous theorem of Cauchy-Kovalevskii-Kashiwara is the following. Theorem 5.13. (Cauchy-Kovalevskii-Kashiwara, see e.g. [17, Theorem 1.4.19]) Let M = Dr/N be a module of finite rank and let U be a simply connected domain in C~’~ \ Sin9(M). Consider the system of vector-valued linear partial differential equations, L ¯ ff = O, L E N,
for vectors ff of holomorphic functions on U
Then the dimension of the complex vector space of holomorphic solutions on U, denoted Solu(N), is equal to rank(Dr/N). Using Theorem. 5.13, we arrive at the following analytic interpretation Weyl closure. Proposition
5.14.
Let M = Dr/N and U be as in Theorem 5.13.
of the
Then
CI(N) annD(Solv(N)) where annD(Solv(N)) denotes the set of all differential nihilate the functions in Solv(N).
operators r which an-
Proof: Since Solv(N) = Solv(Cl(g)), we have g C Cl(g) annD(Solv(N)). By Theorem 5.13, the ideal annD(Solv(N)) must have the same rank as N. Therefore, the surjection of finite dimensional K(x)-vector spaces, rR r_R R. N R. annD(Solv(N)) is an isomorphism. It follows that R. N = R ¯ annD(Solu(N)) and hence we get annD(Solu(g)) C R. g n D = CI(N). Using Proposition 5.14, the Weyl closure operation of an ideal I in D can be loosely regarded as analogous to the radical operation of an ideal J in K[x]. That is, the Weyl closure of I is the ideal of operators which annihilate the common solutions of I whereas the radical of J is the ideal of functions which vanish on the commonzeroes of J. 6. LOCAL COHOMOLOGY In Section 4, we realized the f-torsion of a finitely generated D-module Mas the cohomology of a complex of finitely generated Dn+~-modules (Theorem 4.1). In this section, we discuss an extension of this construction to computing local cohomology due to Oaku and Takayama [14]. The complex which they use has also been studied by Adolphsonand Sperber [1]. Wewill bring together these points of view here. Let S = K[x~,... ,x,~] denote the polynomial ring and I = (f~,..., fd) C ,~ an ideal. By definition, H}(-) is the i-th derived functor in the category of S-modules of the left exact functor for torsion, H°~(M) = {m~ M : I~m = 0 for some k > 0}.
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The usual method to realize the local cohomologymodules is via the Cech complex, which we now describe. Let C° (f~) denote the complex of S-modules 0 1 Then H}(M) is the i-th S -~ S[~] -~ 0 where the map is defined by sending 1 ~ 7" d cohomology module of the Cech complex C°(M; fl,..., fd) := ~)i=1 C°(fi) ® M, where the tensor products are all over S. WhenM is a holonomic D-module, the Cech complex is additionally a complex of holonomic D-modules. In this situation, Walther has given an algorithm to compute presentations of the cohomology modules as D-modules [23]. In [14], Oaku and Takayamaintroduce another complex, which we shall call the twisted Koszul complex, to compute local cohomology. They prove the following theorem, which specializes to Theorem4.1 of Section 5 when i = 0 and d = 1. Theorem 6.1. (Oaku-Takayama [14]) Let M be a holonomic D-module. Then for any i >_ O, we have an isomorphism of left D-modules, (6.2)
H}(M) ~- Hi-d
®/qx] M
{~,..... ~d=0) whereDn+ddenotes the (n + d)th Weylalgebra D(tl , . . . , td, 0~1, . , 0~1, where the right hand side denotes the (i - d)-th derived restriction moduleof the D,~+d-module (Dn+d/J) ® M to the subspace {h ..... d Cgfkc9 td =t 0}, and where [1 <_ g= Dn+d.{tj-
fj(x),O~i
+ E ~xi
~ j <_d,l
i < n}.
k:l
Oaku and Takayama also provide in [14] an algorithm to compute the cohomology modules of a derived restriction complex, thus obtaining another algorithm to compute local cohomology of holonomic D-modules. The proof of Theorem 6.1 in [14] consists of an elegant equivalence in the derived category based upon ideas of Kashiwara[10]. On the other hand, a proof "in coordinates" follows from an earlier paper of Adolphsonand Sperber [1], who consider the complex of Theorem6.1 on the level of S-modules and establish a quasi-isomorphism to the Cech complex. More generally, Adolphson and Sperber make a similar construction of a twisted Koszul complex for A-modules M, where A is a commutative ring with identity. Weshall provide an exposition here which brings together the work of Oaku and Takayamaand the work of Adolphson and Sperber. Wealso give a presentation of Dn+d/J® M following [23], which facilitates the algorithmic computation of (6.2). Let us first describe the twisted Koszul complex T°(M; fl,..., fd) on the level of S-modules where M is an S-module and f~,..., fd ~ S. Wewill return to the D-modulestructure and Theorem6.1 afterwards. Let S[t] := S[h,..., td]. Definition 6.2. The twisted Koszul module M[Ot~,..., Oral {f~ ..... fd} of.an S-module M with respect to the polynomials fl,..., fd ~ S is the S[t]-module, M[cgt~,... ’ Otd]{f~ ..... fd} := M ®g K[~t~,..., Otd] = (]~aeNaMO~, where the action of S[t] is as follows:
¯
® a;
=
® -
®.
Wewill usually suppress notation and write M[O~]when f~,...,
fd are clear.
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Definition 6.3. The twisted Koszul complex of an S-module M with respect to fl,..., fd E S is the Koszul complex T°(M; fi,..., ld):= K°(M[Ot];tl, ... ,td). Let us consider the 0-th cohomologyof the twisted Koszul complex. ]it equals H°(T ") = {rfi:=Em~®0 ~ e M[O,] ] tyrfi=O v j= l,...,d} -e~ =0Vj} = {r5 I ~afjm~®O~-ajm,®O~
= {~ I I~-~. = (~ + 1)m.+o~ w, vj} ’= {’~ I ms= (1/-!)I~ f~"m~w}. Since rn~ = 0 for all but finitely manyc~, the last line implies also that f]~mg = 0 for N sufficiently large for all j. It follows that the map~o : M[Ot]--~ Msending ff~ ~ m6 induces an isomorphism H°(T°) ~ H~(M). Theorem 6.4. (Adolphson-Sperber [1]) 990 extends to a quasi-isomorphism, qo° : T°(M; ft,. ..,fd) ..~ .,fd) C°(M; fl, between the twisted Koszul complex and the Cech complex,
T" : 0 -~ M[Ot]--~ ~ M[Ot]ffi ~ ~ M[Ot]ffij---+... l<’i
C°:0~
---+M[Ot]ffl...d--~
0
l<_i<j<_d
M----~ l
s, and ~s where ~" = @s~
:=
l<_i<j<_d
(~i~<...
Proof: The twisted Koszul module M[Ot] has incre~ing S[t]-submodules,
Since M[0t] = U~k=0M[0t]’k,the twisted Koszul complexis the limit of its subcomplexes, lim, K’(M[Ot]k; t~,... ,td), where the maps
~ : ff’(M[0~]~; ta,..., t,~) are induced from the inclusions M[Ot]a ~ M[Ot]k+t. Similarly, the Cech complex can be defined as a limit of Koszul complexes, lim~ K’(M; f(+~,..., f~+l), where the maps 7~ : K.(M;f(+~,...,
fk+id
are induced from the maps Mffi~...i. (f~) Wenow construct a family of chain-maps ~ between the above sub-complexes K’(M[~t]k; t~,..., td) and the Koszul complexes K’(M;f(+1, ¯ ¯ ¯, ~drk+~)" Define
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~ko : M[O~]~~ M where ~(~{a°
~ =
(~
~’"~’, "’" ]i,
{c~ i~(~.....
{~ : a~..... ~e~k}
: ai=O for
i~) A computation shows that ~+~ o ~. = 7~+1 o ~, so that the family of chain maps {~} forms a map of towers of complexes. Wewill next show that T~ is also a quasi-isomorphism.Since (see e.g. [24] for the first equality), HiI(M)
= lim_+Hi(K’(M;fk~+~,...,fkd+~)) = lim_~ Hi(K’(M[Ot]k;t~,...,td)) = Hi(K’(M[Ot];t~,...,td)),
it would then follow that the twisted Koszul complex computes local cohomology and also that ~" is a quasi-isomorphism. So it only remains to prove the following claim. Claim 6.5. p~ : K°(M[Ot]k;t~,...,td) quasi-isomorphism.
-~ K°(M;f~+~,...,f~+~d
, is a chain map
Proof: One checks by a computation that ~° is a chain map. To prove that ~° is a quasi-isomorphism, we induct on the length d + 1 of the twisted Koszul complex. The base case d = 1 is outlined by Oaku in [13]. Base case. For d = 1, we are to show that the following chain map is a quasiisomorphism, 0 -~ M[Ot]k ~ M[Ot]k -~ 0
0
~
M
,M~ --~0
where the maps are k k Ea,®O~l ~ao E b’®O~l ~ E i=0 i=0 i=0 Wehave already seen that ~ induces an isomorphism on the level of homology, so we are left to show that ~ does as well. For surjectivity of W~on the level of homology, suppose m ~ Msuch that f~+~rn = 0. Then the element ~-]~i~=t ~.f~m ~ Ot~ e M[Od~is l in the kernel of [t,. and maps under ~ to m. For injectivity of ~ on the level of homology, suppose k ai @Ot~ i ker[t~.]. Then tl~ = ~i~,(ai - (i 1)ai+,) @ 0[ that ~ = ~i=~ ~ = O, or ai = ~ao for all i. Thus if ~(~) = ao = 0, then g = For surjectivity of ~ on the level of homology, given m ~ M, the element m @ 0~ maps under ~ to m. For injectivity of ~ on the level of homology, let =~i=obi @0~, and supposethat ~k(b) ~ ~ e im[f~+l]. This meansthere exists
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m E Msuch that ~’~=o’i!f~ -ibi = fkz+lm" Wewish to show that ~’ E im[tl]. ao = 0, and inductively set ai = -bi-l+fai-1 - for 1 < i < k. Then, i
Set
Nowlet ~ = ~i=ok ai @0,~.i It follows that, tl~=
~(f~ai-(i+
1)a,+l)@0~
i=0
Finally,
= bi @O~ + ft ak@O~ = i=0
i=0
set ~ = g + ~i~0 ~ @0~, so that t~ = ~, as desired.
General case. The key tool for the induction step is a certain long exact sequence of Koszul complexes. In particular, consider the following three double complexes: (1) Rowsare the Koszul complex K’(M[Ot]d,k; t~,... ,td-1) and columns are induced by the multiplication maptd : M[Ot]d,k ~ M[Ot]d,~, where M[Ot]d,k denotes the twisted Koszul module (M[Oh,...
o
0
0 ~M[Ot].,~ ~ M[Ot]~;~ ...
~M[Ot]e,~ O,
and-colnmns (2) ~ows are the Kos~ul complex K"(M[Ot]e_~,~;tl,...,ta_~) are induced by the multiplication map f~+~ : M[Ot]a_~,~ where M[Ot]a-~,~again denotes (M[O~,..., 0te_~] {I~ ..... ]~-~})k, the twisted Koszul module. 0 ---’+M[Ot]d-l,k ~
’’’ -----~M[Ot]d-l,k M[0daa_-l,k--~ 1~] Sk+
0 ~M[Ot]e_~,~
~Mrn~d-~ ~_~,~ ~ ...
S~+l ~M[Ot]d-~,k~
(3) Rows are the Koszul complex K’(M; ]~+~~ " " ]~+~ " ~ d-1 ~k+l duced by the multiplication map ~d : M ~ M, 0~
M
~
k+l fd
0---+ M -~---~
Md-~...~
Jdfk+l
~0
M
)
O, and columnsare in-
~0
Jdfk+]
M d- ~ -~ . . . --~ M ---~0,
The Koszul complexes K’(M[O~]k; h,..., td) and K’(M; f~+l,..., f~+~) can be identified with the total complexesof the double complexes(1) and (3). To complete the induction step, we construct maps of double complexes #’" from (1) to (2)
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189
v’" froIn (2) to (3) such that on the level of total complexes, Tot(tt") and Tot(v") are quasi-isomorphisms and ~ = Tot(#") o Tot(~"). To define #", note that Md,k = (Md-l,k[Ot~]{fd})k. Thus by the base case (d = 1) applied to Md-~,k and the polynomial fd, we get a chain map quasiisomorphism ~ : K’(Md-l,k[Ot~]k;td) ~ K*(Md-I&; d k k k i=0
i=0
i=0
Since the columns of the double complexes (1) and (2) are copies of the Koszul complexes K’(Ma_~,~[Ote]~; ta) and K’(Ma_I,~; f~+~) respectively, ~ induces a map of double complexes ~’" from (1) to (2). To define u", we use the chain maps, v~ : K’(Md-l,k;ti,...,ta-1) ~ K’(M; f~+~,...,f~), which are quasi-isomorphisms by induction. Then using v~ for both the top and bottom rows defines a map of double complexes v’" between (2) and (3). To show that ~’" and v’" are quasi-isomorphisms, we can use the long-exact sequence of Koszul complexes coming from the row-filtration spectral sequence of (1), (2), or (3). Namely,these are long exact sequences of the ¯ ..
i--2
o
+ H (P~j))
i
¯
¯
¯
+ (N0)) + H (S (j)) ~
H (Pi--1
where N" (j) ~re the induced complexes at the top row of j = (1), (2), or the cohomologyof the column maps, P(~.) are similarly the induced complexes the bottom row on the cohomology of the column maps, and S(~) are the total complexes. In particular, we have (j) (1) (2) (3)
N(~) K’(ker(Md,k : td);tl,... ,td-~) K’(ker(Md-Lk : f~+l);t~,...,td_~) K’(ker(M : rk+l~.¢~+~ Jd J, J1 ,-.. , fk+l~ d-l~
P(~) K’(cok(Md,~: td); h,..., td-~) KO(cok(Md_l,kf~+l); tl ,..., td -1) K’(cok(M:¢d ¢k+1~.~,~ ck+l,’",¢d-l~ck+l~
Since p~ is a quasi-isomorphism, it induces isomorphisms ker(Md,k;td) -~ ker(Md_~,k; +~) cok(Mct,k;td) -- ~ cok(Md_~,k; f~ +~). It follows that #°" induces quasi-isomorphisms, K°(ker(Md,k: td); tt,...,
td-~) --~ K’(ker(Md-~,k: fd~+~); t~,...,
td-~)
K°(cok(Md,k : td);t~,... ,td-~) ~ K’(cok(Md-l,k : f~+~);t~,... Similarly, a quick computation shows, ker(Md_~,k:f~+~) = ker(M; f~+l)[ota,...
{fl , 0td_~]
.....
fd-1}
{1~ ..... cok(Md-l,k: fdk+~) = cok(M;f~+i)[Ot,,..., Ot~_~] Hence ~°° induces chain maps having the form qo~. which are quasi-isomorphisms by induction, K°(ker(Md-l,k: f~+~); tl,...,td-1) K°(cok(Md-x,k: fdk+l);
tl,...,
~ 1K°(ker(M: fdk+l); flk+~,...,
td-1) "-~ K’(cok(M: fdk+l); f~k+~,...,
fdk_+l f~_+~).
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Thus, #’" and u’" induce isomorphisms Hi(N?1)) --~ Hi(N~2)) Hi(N(’I))
Hi(p~)) H~(P(:2))
-~ Hi(N(*2)) Hi(P~)) --~ Hi(P~2))
From the long exact sequence and the five-lemma, we conclude that Tot(# °°) and Tot(u °*) are quasi-isomorphisms. Thus, the composition Tot(#**) o Tot(u°°) is a quasi-isomorphism, and a computation shows that it equals 995. [] Remark 6.6.
There is a chain-map quasi-isomorphism ¢~ : KO(M;f~+l,..., fk+ld ) -~ K’(i[Ot]k; f~+l,...,
jd~k+l~
in the opposite direction fl’om qo° defined by,
(~i ..... ~d<_~}
a : ~i
i~(il
i,)
However, these maps do not agree for increasing k, and hence do not define a map in the opposite direction from the Cech complex to the twisted Koszul complex. Nowsuppose Mis a D-module, ~d let D~+a= D{h ,...., ta, Ot~,..., Ore). Then Oaku and Ta~yama define a D~+d-module structure on M[Ot] as follows. Definition 6.7. Let M be a D-module. Then the twisted Koszul ~nodule M[Ot] of M with respect to fl,.. ¯, fd has a D,~+a-modulestructure given by
xi ¯ (m. ~ 0~) = z~m.~ t~ ¯ (m. ~ 0~ ) = Am.~ OV- ~m~ -~ d
&~¯ (m~~ 0~)= Lemma6.8. If M is a D-module, then the twisted Koszul module M[Ot] of M with respect to f~,..., fa is isomorphic as D~+a-moduleto (D~+a/J) N~([~I M, where d
Proofi Given an element
¯ ~ ¯ (0~®-~) 0~,¯ (o~®m)
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This agrees with the previously defined Dn+d-structure of the twisted Koszul inodule. Moreover, the map of left D~+d-modules K[x][Ot] =D,~+a. {tl, Xi
b-~
Xi
Otj
~-~
Otj
¯ ¯¯,
tj
Dn+d ta, a~ }--~
tj
, ... --
fj(2g)
, a~,}~ D,~+aj a Of~ot~ Oxl
~ Oxl
q-E
~
j:-I
is an isomorphism, so that
~ ®K[x] M ~_ K[x][Ot] ®K[x] M = K[Ot] ®g M. []
The maps in the twisted Koszul complex are maps of D-modules although not maps of D~+d-modules. Moreover, a quick computation shows that ~o° is a chain map of D-modules. Proposition 6.9. The chain map ~° : T’(M;tl,...,td) a quasi-isomorphism of complexes of D-modules.
~ C°(M;fl,...
,fd)
Wenow give a presentation of the twisted Koszul module M[Ot] in terms of a presentation of Mby extending Lemma4.2. As corollaries, we see that if M is finitely generated as a D-module, then M[Ot]is finitely generated as a Dn+dmodule. Similarly, if M is holonomic, then M[Ot] is holonomic. As in Section 4, d redefine ~i = Ox, + ~j=l(Ofj/OXi)Otj and put ¢(P) :: EPj(Xl, . ......... ,Xn,tl -- fl, . ,td -- fd, O1,. ,~n,Otl,, COte)ej re Dn+l J for P = Y~i py(xl,..., x~, h,..., td, 01,..., On, Ot~,..., Ota)ej ~ Drn+d, Lemma6.10. Given a presentation M ~- Dr/N, then we have the presentation left D,~+d-modules( D~+d/J) ®tC[×l (Dr/N) ~- Dn+d/K ( N), where g(g) := Dn+d"{(ti
of
- fi)ej}i,j=d,r 1 -~ On+d"¢(N).
Proof: Consider the map ¢ : D~+d----~(D~+d/J) ®KIwi (D~/N) of left D,~+dmodules defined by ¢(ej) = 1 ® ej. By the same arguments as in the proof Lemma4.2, we have that ¢ is surjective with kernel equal to K(N). Let us nowdiscuss how Theorem6.1 can be used algorithmically to compute local cohomology. By Theorem 6.1, the local cohomology of a D-module M = D~/N at the ideal I = (fl,..., fd) is equal to the cohomology of the Koszul complex ° T = K°((D~+d/K(N)); t~,...,td). This Koszul complex equivalently computes the derived restriction of the Dn+d-module D~+d/K(N) to the subspace {t~ = .’. = td = 0}. WhenD~/N and thus D~+d/K(N) are holonomic, an algorithm to compute derived restriction is given in [14]. The methodis to produce a complex of finitely generated D-modules (whose cohomology can be computed using Grhbner bases) which is quasi-isomorphic to °. We h ave s een t hat t he C ech c omplex is itself a complex of finitely generated D-modules which is quasi-isomorphic to T°. However the Cech complex is not the complex constructed by the restriction Mgorithm. In particular, the Mgorithm produces a complex consisting of free Dmodules of finite rank. The maps between them can be quite complicated and are induced from Gr6bner bases coming from GrSbner deformations. In contrast, the modules which appear in the Cech complex have the form MIni and typically have complicated presentations as D-modules whereas the maps between them are
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noweasy to understand. It would be interesting to investigate which methodis more efficient and under what circumstances. These algorithms have been implemented in Macaulay2 as the scripts localCohom( ..., Strategy => 0T) localCohom(..., Strategy=> Walther) Let us end by discussing the case when the zero locus of I is a nonsingular. In general, when j : Y ~-~ X is a closed embedding of nonsingular varieties, a theorem due to Kashiwara states that the category of/)y-inodules is equivalent to the category of T)x-modules supported on Y. For an ideal I C K[x] such that j : V(I) ~-~ n i s n onsingular, o ne consequence of t his e quivalence i s a relationship between the local cohomologyof a D-module Mwith respect to I and the derived restriction of Mto the variety V(I). Proposition 6.11. (see e.g. Bernstein [2]) Let M be a left D-module, I C K[x] an ideal corresponding to a nonsingular variety, and j : V(I) "-~ n t he i nclusion. Then H~(M) = j. ( TOrdi,~(y(1))_i(g[x]/I , M) ®g[x] WK[×]/I) where Tor~i[~]y(1))_,(g[x]/I,~ M)) is also (dim(V(/)) - i) -t h deri ved rest riction o/M to V(I) and j. denotes the direct image. Proof: A formulation of Kashiwara’s equivalence for the inclusion j : V(I) ~ n is that H°~(M) = j. o Homg[x](K[x]/I,M) where Hom~:[xl(K[x]/I,M) has the structure of a left Dyu)-module. The flmctor j, is additionally exact, hence H~(M)~_ j. o Ext~g[×](K[x]/I, NowExt~[xl( g[x]/ I, M) M) where F" is a free resolution of K[x]/I. Since V(I) is nonsingular, the complex Homg[x](F°, K[x]) is exact except in cohomological degree dim(V(/)) its cohomology is isomorphic to the canonical module Wg[x]/l which is also locally isomorphic to K[x]/I. This shows that H~(HomK[~](F’,K[x]) TOr dim(V(i) )_i( K[x]/ , M) ®¢Og[x]/i.
[]
WhenI is a nonsingular complete intersection, then wK[×]H ~-- K[x]/I as Dg[x]/imodules. Moreover, when I is generated by linear forms f~,..., fd, then the derived restriction modules can be computed directly by the algorithm of Oaku and Takayama. Since direct images are easy to compute for inclusions, this gives an algorithm for local cohomologywith respect to a linear subspace. In other words, we do not need to pass to the twisted Koszul complex or the Cech complex, which are computationally more intensive, to compute local cohomologywith respect to The above observation is useful for instance when f~,..., fd generate the raaximal ideal m= (x~,..., xn) and Mis holonomic. In this case, the derived restriction modules of Mto the origin are finite-dimensional K-vector spaces, and the local cohomologymodules H,/, (M) are the direct images of these vector spaces, which Kashiwara’s equivalence are their injective hulls. This gives a method to corapute the Lyubeznik numbers Ai,n_j(S/I), which by definition are the socle dimensions
Algorithms for D-Modules
193
of the local cohomology modules H~m(HJ~(s)) and hence are equal to the dimension of the derived restriction modules of H] (S) to the origin. We remark that algorithm to cotnpute Lyubeznik numbers was first given by Walther in [23]. Algorithm 6.12. (Computing Lyubeznik numbers) INPUT: a left ideal I C K[x]. OUTPUT:the Lyubeznik numbers "~i,n-j (K[x]/I). (1) Compute the local cohomology modules H](K[x]) of [23] or the algorithm of [14]. the dimensionsAi,,~-j = di*ng H=i(g[x]/(x) (2) Compute the derived restriction (3) Return the
using
the
algorithm
®K[xIL H~(K[x]))
modules of H~(M) to the origin.
Example 6.13. As a simple example, let us compute the Lyubeznik numbers of the ring K[x, y, z]/I where I is the ideal (x(y - z), xyz). First, we compute using Walther’s algorithm the local cohomology modules of polynomial ring K[x, y, z] with respect to I. il : (W = QQ [x,y,z,Dx,Dy,Dz,WeylAlgebra=>{x=>Dx,y=>Dy,z=>Dz}] I = ideal (x*(y-z), x*y*z); HI = localCohom I) ol = HashTable{O => 0 }. 1 => cokernel {0} I Dz Dy xDx+2 x2 l 2 => cokernel {0} I y-lz zDy+zDz+2 xDx+2 z2 1 Second,we computethe derivedrestrictions to the originof each nonzerolocal cohomology module. In the above table, H](K[x]) is accessed by HI#j. i2 : Drestriction (HI#I, {1,1,1}) o2 = HashTable{0 => 0 } 2=>0 3=>O 1 1 => Q~ i3 : Drestriction (HI#2, {i,i,i}) 03 = HashTable{O => O} 1=>0 2=>0 3 => 0 It follows that the only nonzero Lyubeznik number is A2.2 = 1. REFERENCES Ill Adolphson, A., Sperber, S. (1998): A remark on local cohomology.Journal of Algebra 206, 555-567. [2] Bernstein, 3. Mimeographedlecture notes on D-modules. [3] Borel, A. (1987): Algebraic D-modules.AcademicPress, Boston. [4] BjSrk, J. (1979): Rings of Differential Operators. North Holland Publishing Company,Amsterdam. [5] Chyzak,F., Salvy, B. (1998): Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation26, 187-227.
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[6] Cox, D., Little, J., O’Shea, D. (1998): Using algebraic geometry. Springer Verlag. [7] Granger, M., Maisonobe, P. (1993): A basic course on differential modules. In D-modules Coherents et Holonomes, eds. Maisonobe, P., Sabbah, C. Hermann editeurs des sciences et des arts, Paris. [8] Graysoa, D., Stillman, M. (1999): Macaulay 2: a computer algebra system for algebraic geometry, www.math, uiu¢. edu/Maoaulay2. [9] Kashiwara, M. (1976): B-functions and holonomic systems. Inventiones Mathematicae 38, 33-53. [10] Kashiwara, M. (1978): On the holonomic systems of linear differential equations, II. Inventiones Mathematicae 49, 121-135. [11] Lyubeznik, G. (1993): Finiteness. properties of local cohomology modules. Inventiones Mathematicae 113, 41-55. [12] MusCatS, M. (2000): Local cohomology at monomial ideals. Journal of Symbolic Computation 29, 709-720. [13] Oaku, T. (1997): Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Advances in Applied Mathematics 19, 61-105. [14] Oaku, T., Takayama, N. (2001): Algorithms for D-modules -- restrictions, tensor product, localization and algebraic local cohomology groups. Journal of Pure and Applied Algebra 156, 267-308. [15] Oaku, T., Takayama, N., Walther, U. (19~9): A localization algorithm for D-modules. Journal of Symbolic Computation 29, 721-728. [16] Pemantle, R., Wilson, M. (1999): Asymptotics of multivariate sequences, part I: smooth points of the singular variety, preprint. [17] Saito, M., Sturmfels, B., Takayama, N. (1999): Gr6bner Deformations of Hypergeometric Differential Equations. Algorithms and Computation in Mathematics, Vol. 6. Springer Verlag, Berlin. [18] Smith, G. (1999): Irreducible components of characteristic varieties. ht~p :/[~.arXiv,org/abs/math. ~G/9912066. [19] Stanley, R. (1999): Enumerative Combinatorics 2. Cambridge studies in advanced mathematics, Vol. 62. Cambridge University Press. [20] Takayama, N. (1990): An algorithm of constructing the integral of a module - an infinite dimensional analog of GrSbner bases. In: Watanabe, S. and Nagata, N., Eds., Symbolic and Algebraic Computation (ISSAC ’90), Kyoto, Japan. ACMand Addison Wesley, 206-211. [21] Takayama, N. (1992): An approach to the zero recognition problem by Buchberger’s algorithm. Journal of Symbolic Computation 14, 265-282. [22] Tsai, H. (2000): Weyl closure of a linear differential operator. Journal of Symbolic Computation 29, 747-775. [23] Walther, U. (1999): Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties. Journal of Pure and Applied Algebra 139, 303321. [24] Weibel, C. (1994): An Introduction to Homological Algebra. Cambridge studies in advanced mathematics, Vol. 38. Cambridge University Press. DEPARTMENT OF MATHEMATICS,CORNELLUNIVERSITY, ITHACA, NY 14853
COMPUTING
LOCAL
COHOMOLOGY
IN
MACAULAY
2
Anton Leykin University of Minnesota, Minneapolis, Minnesota
ABSTRACT.Wereview
two different methods for computing algebraic local cohomologygroups using D-modules.The first one is due to U. Walther and is based on the construction of a certain (~ech complexof holonomic D-modules. The second is due to T. Oaku and N. Takayama and relies on their algorithm for computingthe derived restriction modules of holonomicD-modules. Weprovide a discussion of the advantages and disadvantages of these approaches, accompaniedby examples of computation done in the computer algebra system Macaulay2 using the package"D-modulesfor Macaulay2" developedjointly by iV[. Stillman, H. Tsai and ourselves. Also we comparethe effectiveness of both methodsin terms of machinetime.
1.
INTRODUCTION
Let K be an algebraically closed field of characteristic 0 and let X be the affine space Kn. By Ox and Dx we denote the sheaves on X of the rings of regular functions and of algebraic linear differential operators respectively (see [1]). Consider polynomials fl,..., ~fd ~ KIwi, where is the set of n independent variables, and put Y = V(.fl, ..., fd) X I ]’l(X) -- - f4(~) = 0}. Let M be a coherent Dx-module. Our goal to compute the algebraic local cohomology groups H~(M) with support Y as D-modules. Recall that H{~(M) is defined (by Grothendieck [3]) as the i-th derived functor of the functor
195
196
Leykin
rr(M) = limHomox(Ox/J~;
M),
where Jy is the defining ideal of Y and the inductive limit is taken as m tends to infinity. If Mis a holonomicD-modulethen so is H~ (M) for ,every i (see [4]). Sections 2 and 3 describe algorithms for localization and restriction of a holonomic D-modulerespectively. These are needed in Sections 4 and 5 when we review two different methods for computing local cohomology: one due to Uli Whither [9], the other due to Toshinori Oaku and Nobuki Takayama [6]. In Section 6 several examples of local cohomologycomputation using Dmodule package for Macaulay 2 are given. They are supposed to give an idea of the level of complexity of the problems that can be solved at the present state of hardware and software development. Also these examples are used in Section 7 to test the effectiveness of the two approaches in discussion. The following notation is used throughout the paper: Dn= K (~1, ..., ~n, c9xl, ..., cgxn) stands for the n-th Weylalgebra. When it is clear what n is, D = D~. Also when it comes to algorithms, a Dmodule should be thought of as a left Weyl algebra module rather than a sheaf module. R = R~ = K[~,..., ~n] is a ring of polynomials, R] is R localized by inverting a polynomial f. 2. LOCALIZATION Let M: D/I be a holonomic cyclic D-module, let f E R = K[~I, ..., ~n]. Wewant to compute R/® Mas a holonomic cyclic D-module, i.e. we would like to find an ideal J C D such that R/® M ~- D/J. There are several algorithms known that find such J, we shall mention two. In case when Mis f-saturated, i.e.f, m = 0 iff m : 0 for all m E M, we refer the reader to Walther’s paper [9] for a detailed description of the localization algorithm. Here we point out the main steps of it. 1. First of all one wants to find jI(fs), the ideal of operators in Dis] annihilating fs®~ ~ Ryls]fS®M, where ~ is the cyclic generator ofM = D/I and fs the generator of R/Is]f" (see [1] for the definition). 2. Another component is the Bernstein-Sato polynomial bly(s), which is defined as the monic generator of the ideal formed by all b(s) ~ g[s] such that b(s)f ~ @i = Q(x, O, s)(.f s+l ® ~), holds in the n[s]-module Ry[s]f ~ ® i for some Q(x, O, s) ~ Dis]. This polynomial exists if Mis holonomic. 3. The final step consists of finding the smallest integer root a of b~ (s) and "plugging in" a for s in the generators of jI(fs). The obtained operators generate J C D that we started our discussion with.
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Computing LocalCohomology in Macaulay 2 3. RESTRICTION
Herewe describe an algorithm forcalculating thederived restriction modules neededfor the algorithmof Oaku and Takayamafor computing localcohomology. LetX -- Kn+dwithcoordinates (Xl,...,Xn,tl,...,td)and Y = {tl = ... = td = 0} C X ~_ Kn with coordinates (~,,...,~) and let f : Y ~ X be the inclusion map. Given a Dx-module M, the derived restriction modules, denoted by H~ (M~.~x), are defined to be the left derived functors of inverse image f* in the category of D-modules. In terms of Tot wehave(xl,..., x~, tl, ..., Hi(M~x)
= Hi(Lf*(M))=
Tor~X(A~,M),
where Av is the right Dx-module Dx/{t~, ..., Q}. Dx. The main computational problem here is that if P* is a projective resolution of Mthen the complex Av @ P" is no longer a complex of left Dx-modules. Indeed, we may view its elements as Dy-modules, but these are not finitely generated as such. A way around this misfortune is based on Vy-adapted free resolutions. Definition 3.1. The V-filtration Fy of a shifled ~ee Dx-module D}[~] is defined by
A ~ee resolution P" of M of the form P" : ...
~ D~+’ [~j+l]
~¢~+’ rj Dx[~j] ~.-
is said to be Vy-adaptedif
for all i and j, and if the induced complex rj+l
~gr(~j+l)
rj
is a resolution on the level of associated graded modules. Definition 3.2. Let 0 = tlOt, + ... + tdOta. The b-function of M for the restriction to Y is the monic polynomial b(O) E K[O]of least degree, which satisfies b(O)gr°(M) : 0 with respect to the Vy-filtration. If there is nonzero polynomial satisfying the above, the b-function is set to equal O. WhenMis holonomic then the b-function is nonzero and we use its largest positive integer root as a point of truncation for finding a quasi-isomorphic finitely generated Dy-subcomplex of Ay ®~)x P°" Algorithms for computing Vy-adapted resolutions and b-functions are given by Oaku and Takayamain [6] and also maybe found in [7]. Weare now ready give an outline of the restriction algorithm. 1. Computethe b-function b(O) of Mfor the restriction to Y.
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Leykin
2. Find the largest integer root kl of b(O). If there are no integer return Hi(M~_~x) = for i = 0, ... , d a ndhalt . 3. Compute a Vlz-adapted free resolution of M p° : ...--+
D~/+I [~i~j+t]
4. Compute the truncated Fr~I(Ay
®Dx p*):..._~
induced
---~
~bj+l
roots
Dr~[~j] --~...
complex
Fk,(A~+I[~j+I]
) _~+1 Fk,(Ay
[~j])_~
...
as a complex of finitely generated left Dy-modules. 5. For i = 0, ..., d compute the -i-th cohomology group ker ¢i/im of the above complex in the formDy/Ni s~ . where Ni is a submodule of 6. Return Hi(M~_+x)= v~’/N~.. 4.
FIRST METHOD:~ECH COMPLBX
Let X = Kn with the coordinate ring R = K[x~, ..., xn] and Y - V(I), where I = (fl, ..., f~). To calculate H~(R) consider the following ~ech complex: 0 --~
C0~ C1 --+ ...
-~ Cd -~ O,
where Ck --
~
Rf~1 ...]~
andthe map Ck --~ C~+~ is the sum of maps R]iI...]~
~ R]~1 ""f~+l,
which are zero if (il, ..., ik} ~ (jl, ..., j~+l} and are natural, i.e. send i ~-+ 1, up to sign (one has to alternate signs in such a way that the sequence above is indeed a complex). Proposition 4.1. The local cohomology groups H~(R) are equal to the cohomology groups Hk(C*) of the constructed ~ech complex. The complex C* enables us to compute the local cohomology algorithmically viewing Ck as holonomic D-modules and the maps between them as D-linear maps. An explicit algorithm for local cohomology is written out in [9] and depends on the localization algorithm, which is used to calculate the components Rf~ ...f~ ~. of C We would like to give an example of using this method in Macaulay 2. Start with loading the D-modules package: J.1
: load "Dloadfile.m2" ......
(For more information
and to download the package see [2])
Example 4.2. Let us. compute H~(R), where R : k[X, {X(Y - Z), XYZ).
Y, Z] and := R.
199
Computing Local Cohomology in Macaulay 2
i2 : W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}]; ideal (X*(Y-Z), X*Y*Z); i4 : time h = localCohom(l, Strategy=>Walther) -- used 3.88 seconds o4 = HashTable{0 => subquotient ({0} I dZ dY dX I, {0} I dX dY dZ l) I => subquotient ({0} [ -XdX-I -dXZdZ-dX dXYdY+ ...
{o} I o
o
o
...
2 => cokernel {0} I XY-XZ -XYZ XdX-2/3YdY-2/3ZdZ 04 : HashTable i5 : pruneLocalCohom
h
05 = HashTable{0 => 0 } i => {0} [ dZ dY XdX+2 X2 l 2 => {0} I Y-Z Z2 dYZ+ZdZ+2 XdX+2 I 05 : HashTable
The meaningof these lines is as follows. Weuse function localCohom with ideal I as a parameter and the option Strategy set to Walthor. Line 04 is the output of the algorithm as it is, which is simplified by prunoLocalCohom in o5. The latter contains a hashtable showing that Ha(R ) = O, H)(R)
= coker¢l,
H](R) = coker¢2, where ¢1 and ¢2 are D-homomorphismsfrom D4 to D given by matrices A1 and A2 respectively (one has to ignore "{0}"’s to the left from the matrices in the output) 31 : (Og Cgy A2
= (Y-Z
XCgX-~U2
2)X
2 Z (Oy+Oz)+2 X
OX+2)
The D-modules H}(R)and H~(R)are nonzero, because should a cohomology module be trivial the program outputs 0. The algorithm was coded exactly as it was described in [9]. Wealso may generalize the method by considering an arbitrary holonomic module Min place of R. One just needs to tensor the above ~ech complex C" with Mover R: Ck(M) = C~ ®~ M. This complex is a complex of holonomic D-modules again, thus computers may be used.
200
Leykin
Example 4.3. (Example 7.5 in [6]) Let D : D3 : K (X, Y, Z, OX, OY, OZ) and M= D/D.{cgX, Oy, z3oz÷ Z} and I= R.{XY, YZ}. Here is how we . find HI(M ) i6 : M = cokernelmatrix{{dX,dY, Z~3*dZ+Z}}; i7 : I = ideal (X,Z, Y,Z); i8 : time h = localCohom(l,M, Strategy=>Walther, LocStrategy=>Oaku) -- used 1.57 seconds 08 = HashTable{0=> subquotient({0} ] dY -dX dXdY XdXZ-Z2dZ-I ... 1 => subquotient({0} [ 0 Y -dX dXYZ dXYdY+dX... {0} I dY X 0 Z2dZ+Z+I dYZdZ+dY ... 2 => cokernel{0} I XZ -YZ YdY+I XdX+l XdXZ-Z2dZ-Z-I[ 08 : HashTable i9 : pruneLocalCohom h 09 = HashTable{0=> {0} I Z dY dX I } 1 => 0 2 => {0} I Y X Z2dZ+2Z+l I 09 : HashTable Thisversionof localCohom for an arbitraryD-moduletakesI and .~I as parameters. The code for this version follows the ideas of [9], but gives you options for the way localization is done. The option LocS~;ra~egy tells to use the Oaku’s algorithm in this case. 5. SECONDMETHOD:RESTRICTION VIA V-STItICT
ttESOLUTIONS
Another way to compute local cohomology was proposed by T. Oaku and N. Takayama in [6] and is based on the algorithm for calculating the derived restriction modules of a holonomic D-module. Theorem 5.1. Let M be a holonomic D-module. Then for any i > 0 there is an isomorphism of left D-modules: H~(M) ~- H~-d(((Du+d/J) ®I<[~] M)’{t I .....
t~:o))
where Dn+d denotes the (n + d)-th Weyl algebra D (t~,..., where the right hand side stays ]or the (i- d)-th derived restriction module o] the Dn+d-module (D~+d/J) @KIwi M to the subspace {~ .... = td := 0}, and where dt Ofko J=Dn+d’{tj-fj(z),O~,+~
~ll~j~d, k:l
1
n}.
Computing Local Cohomology in Macaulay2
201
Provided we know how to get the restriction, the only piece missing is the effective presentation of the module (D~+d/J) ®1
D~+ d -~ D~+d
J where {ej}~ is the standard
basis of D~+ d. Lemma 5.2. Let M ~ Dr/N be holonomic, then (Dn+d/J) holonomic Dn+d-module isomorphic to Dn+d/ K ( N), where K(N) = Dn+d" {(ti
®K[z] M is
a
- .ri)ejli,j=
1 + .Dn+d " Example 5.3. Let D = D2 = g (X, Y, OX, OY), M = DID. (OX, OY) I = R. {X 2 + Y2, XY}. Here is how we find H?(M). iii : W = GO[X, dX, Y, dY, WeylAlgebra=>{X=>dX, Y=>dY}]; i12 : I = ideal (X’2+Y’2,X,Y); i13 : time localCohom(I,Strategy=>0aTa, LocStrategy=>0aTaWa) -- used 3.55 seconds o13 = HashTable{0=> 0 ... I => 0 ... 2 => cokernel{0} I dXY+XdYXY XdX+YdY+4X2+Y2 ... o13 : HashTable 6.
MORE EXAMPLES
Example 6.1. (Twisted cubic) Let the ideal I be the defining ideal of the twisted cubic in ~3, which is generated by 2x2 minors of the matriz ~’2 ~3
~et us determine~ to~t ~o~o~oto~u ~(~) Io~ ~ ~[~, ~,~, ~]. i2 : ~ = ~[x_l..x_4, dx_l..dx_4, ~ey~geb~a => toL£s~ (1..4) / (£->x_£=>dx_£)]; iS : I = minors(2,matrix{{x_1,x_2, x_3}, {x_2, x_3, x_4}}); i4 : time h = localCohom({0,1,3},I, W’I/ideal{dx_l,dx_2,dx_3,dx_4}, Strategy=>Walther, LocStrategy=>Oaku);
202
Leykin -- used 54.93 seconds
i5 : pruneLocalCohom h 05 = HashTable{O => O} 1 => 0 3 => 0 05 : HashTable time localCohom (2, I, W^I / ideal {dx_l, dx_2, dx_3, dx_4}, Strategy=>Walther, LocStrategy=>Oaku); -- used 59.68 seconds Although
local
cohomology
in degrees
0,1,3
tually turns out to be zero, one may see compute and has quite a huge representation, the present time. Here is one more challenge that capabilities of computer systems. Example
6.2.
Consider
ideal
We are looking
for local
H](R) takes which cannot
seems to be on the
I generated
~4
that
is computed
~5
by 2z2 minors
easily
more time to be pruned at
edge of the of the
matrix
x2, x3, z4, xs, z6].
i2 : w = QQ[x_l..x_6, dx_l..dx_6, WeylAlgebra => toList(i..6)/(i->x_i =>dx_i)]; i3 : I = minors(2, matrix{{x_l, x_2, x_3}, {x_4, x_5, x_6}}); 03 : Ideal of W i4 : time h = localCohom ({0,1,3}, I, W^I / ideal {dx_l, dx_2, dx_3, dx_4, dx_5, dx_6}, Strategy=>Walther, LocStrategy=>Oaku); -- used 8.18 seconds i5 : pruneLocalCohom h 05 = HashTable{O => 0 1 => 0 3 => {0} I x_4dx_4+x_Sdx_5+x_6dx_6+6...
i6 : time h = localCohom (2, I,
current
~6
cohomology H~(TI) for R = K[xl,
05 : HashTable
and ac-
Computing LocalCohomology in Macaulay 2
203
W’I/ ideal{dx_l,dx_2,dx_3,dx_4,dx_5,dx_6}, Strategy=>Walther, Lo¢Strategy=>Oaku) -- used140.09seconds Degrees 0,1,3 present no problem and H~( R) is non-trivial this time. However H~ (R) keeps the system busy for almost 4 minutes and produces a representation that appears to be impossible to analyze neither by hand, nor by computers. 7. COMPARISON AND BENCHMARKS It canbe proved(see[8])thatthecomplexes produced by thetwoalgorithms described in thepresentpaperarequasi-isomorphic. However, fromthecomputational pointof viewthereis a bigdifference in thetwoapproaches. Thecomplex produced by therestriction algorithm is a complex of freeD-modules, butthemapsin it arequitecomplicated. On thecontrary the 0echcomplexdescribed aboveis a complexof holonomic D-modules, whichmayhaveno particularly nicepresentations, withthemapsthatare easyto grasp. Example 7.1. Let D : D6 and let M = N1°, where N = D/D. {0z~3, 0x~, 0~3, 0z4, 0~5, Since the algorithms are implemented only .for cyclic modules so far, we produce a cyclic presentation of Musing ma~.eCyc~ic routine. i2 : W = QQ[x_l..x_6, dx_l..dx_6, WeylAlgebra => toList(I..8)/ (i -> x_i => dx_i)]; i3 : M = W’lO/ideal{dx_l’3, dx_2,dx_3,dx_4,dx_5,dx_6}; 03 : W - module,quotient of W i4 : M = makeCyclic presentation M 04 = HashTable{Generator => {0} {0} {0} {o} {o} {o} {o} {o} {o} {o}
x_6"9 x_e~s x_6^7 x_6-6 x_S-5 x_~4 x_6-3 x_6~2
3 io AnnG=> ideal(dx, dx , ~x , dx , dx , dx 5 4 3 2 1 6
204
Leykin
04 : HashTable i5 : time h = localCohom (ideal{x_1,x_2,x_3}, W’I/M.AnnG, Strategy=>Walther, LocStrategy=>Oaku); -- used 4.36 seconds i6 : pruneLocalCohom h 06 = HashTable{O => 0 i=> 0 2=>0 3 => {0} l dx_5 dx_4 x_3 x_2 x_l’3 dx_6"lO [ 06 : HashTable i7 : time h = localCohom (ideal{x_l,x_2,x_3}, WAI/M.AnnG, Strategy=>0aTa); -- used 46.7 seconds i8 : pruneLocalCohom h 08 = HashTable{0 => 0 } 1-->0 2=>0 3 => {0} I dx_5 dx_4 x_3 x_2 x_l^3 dx_6-10 I 08 : HashTable As you see restriction
appears
shows especially when the degrees of the D-module get bigger. Finally to the
let
to take
more time than ~ech complex.
of operators
us give you the time characteristics
examples
described
Algorithm\Example
~ech complex Restriction
involved
That
in the description
for both algorithms
applied
in the paper. 4.2
4.3
5.3
6.1
6.2
7.1
4 sec 1.3 sec 2.3 sec 2 min 3 min 4 se 9 sec 36 sec 3.9 sec oo cx3 46 s,
REFERENCES [1] J.-E. BjSrk, Rings of differential operators, North-Holland (1979). [2] A. Leykin, M. Stillman, H. Tsai, D-modules for Macaulay 2, ( http://www.math.cornell.edu/’htsai/Dmodules.html, ht ~p://www.math.umn.edu/’leykin/Dmodules/ ) [3] A. Grothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA2). (French) North-Holland Publishing Co., Amsterdam; Masson&: Cie, Editeur, Paris, (1968).
Computing Local Cohomologyin Macaulay 2
205
[4] M. Kashiwara, On the holonomic systems of linear differential equations, IL Invent. Math. 49 (1978)~ 121-135. [5] D. Grayson, M. Stillman, Macaulay 2 (http://www.math.uiuc.edu/Macaulay~/}. tensor product, [6] T. Oaku, N. Takayama, Algorithms for D-modules - restrictions, localization and algebraic local cohomologygroups, 3. Pure Appl. Alg. (2001) 156 (2-3), 267-308. [7] M. Saito, B. Strumfels, N. Takayama, Gr6bner bases of hypergeometric differential equations, Springer, (1999). [8] H. Tsai, Algorithms for algebraic analysis, Thesis, Univ. of Calif., Berkley (2000). [9] U. Walther, Algorithmic computation o] local cohomoloyy modules and the local cohomological dimension of algebraic varieties, Journal for Pure and Applied Algebra 139, (1999).
Anton Leykin School of Mathematics, University of Minnesota Minneapolis MN55455 e-maih [email protected]
Squarefree Modulesand Local Cohomology Modulesat MonomialIdeals Kohji Yanagawa Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560, Japan yanagawaOmath, s c ±. osaka-u, ac. j p
1.
INTRODUCTION
Stanley-Reisner rings and affine semigroup rings are very important subjects of modern commutative algebra (c.f. [4, 22]). For the study of these Zn-graded rings, combinatorial descriptions of the local cohomologymodules with supports in graded maximal ideals have been a fundamental tool. But recently, local cohomologymodules with supports in general Zn-graded ideals (i.e., monomialideals) are investigated by several authors (c.f. [1, 17, 18, 19, 20, 21, 25, 26, 27]). It might extend the frameworkof the theory of the Zn-graded rings and modules. This article is basically a survey on the author’s papers [24, 25, 26]. But it also contains manynew results. Especially, the results in Section 6 are entirely new. The organization of this article is as follows. In Section 2, we study squarefree modules over a polynomial ring S = k[xl,... ,xn]. The Stanley-Reisner ring S/Izx of a simplicial complex A is always a squarefree module. Moreover, the syzygy module Syzi(S/I/x), the canonical modulecos, and Extis(s/Izx, cos) are also squarefree. So this is an enough inclusive class for the study of Stanley-Reisner rings. On the other hand, the category Sq of squarefree S-modules is equivalent to the category modAof finitely generated right A-~nodules, where A is the incidence algebra of the boolean lattice 2[ n] over k. Since dimk A < c~, Sq is very easy to treat and enjoys manynice (homological) properties.
207
208
Yanagawa
In Section 3, we study the local cohomologymodule H~a (S). Musta~ [19] and Terai [23] gave an easy procedure to construct H~a (S) from its submodule Ext~s(S/Ia, S), which is a familiar object in the Stanley-Reisner ring theory. Wecan simply explain their results using the concept of squarefree modules, although H~a(S) itself is not squarefree. Wealso study a zn-graded minimal injective resolution of H~,~ (S). Wesee that the Bass numbers H~ (S) coincide with certain values of the Zn-graded Hilbert functions double Ext-modules Ext~ (Ext~(S/I~, S), In Section 4, we study squarefree modules over a normal semigroup ring R. As in the polynomial ring case, radical monomialideals of R are squarefree. The category SqRof squarefree modules over R is equivalent to the category modAof finitely generated right A-modules, where A is the incidence algebra of the face lattice of a convex polytope associated with R. Thus, if R is simplicial, we have SqR -~ Sqs for a polynomial ring S with dim S = dim R. In Section 5, we will study H~(WR)for a (radical) monomialideal I of If R is simplicial, every thing is quite similar to the polynomial ring case. For example, the Bass numbers of H~(wR) are finite. Wecan also prove an analogue of "the second vanishing theorem" for a regular local ring (c.f. [12]) in our context. That is, whenR is a simplicial normal semigroup ring of dimension d and I is a monomialideal of R, H/-I(wR) ----- 0 if and only if Spec(R/I) {m} is connected. If R is notsimplicial, the situ ation is v ery different. For instance, the Bass numbers of H~(wl~) can be infinite, as a famous example of Hartshorne [9] shows. Miller [17] defined the canonical ~ech complex d° Ia of a Stanley-Reisner ideal IA C S. As the usual ~ech complex, we have H~a(M) = Hi(d~ ®sM) for an arbitrary S-module M, but ~a is much smaller than the usual ~ech complex in general. In Section 6, we construct the canonical Cech complex of a radical monomialideal I of a normal semigroup ring R from a minimal projective resolution of R/I in SqR. Using this concept, we can treat H~(R) in a combinatorial way even if R is not Gorenstein. The author would like to thank Professors Miller for stimulating discussions.
Gennady Lyubeznik and Ezra
2. SQUAREFREEMODULESAND STRAIGHT MODULES Let S = k[x~,... n, xn] be a polynomial ring over a field k. Consider an N grading S = ~ael~ Sa = ~ael~ k xa, where xa = l-[~’=l xia~ is the monomial with exponent vector a = (a~,... , an). Let Mbe a Zn-graded S-module, that is, M = ~asZ~ Ma as a k-vector space and Sb Ma C Ma+b for all a ~ Zn and b ~ lh -n. For a ~ Zn, ~Y/(a) denotes the shifted module with M(a)b : Ma+b. We denote the category of S-modules by Mod, and the category of zn-graded S-modules by *lVIod. Here a morphism f in *Mod is an S-homomorphism f : M -+ N with
Squarefree Modules and Local CohomologyModules
209
f(Ma) C Na for all a E n. See [ 7] f or i nformation o n * Mod. For M, N E *Mod and a ¯ Zn, set * Horns(M, N)a := HOm,Mod(M, N(a)). *Homs(M,N)
:= ~ *Homs(M,N)a ~ aEZ
has a natural Zn-graded S-module structure. If Mis finitely generated, then * Homs(M,N) is isomorphic to the usual Horns(M, N) as the underlying S-module. Thus, we simply denote * Homs(M, N) by Homs(M, in this case. In the same situation, Ext,(M, N) also has a Zn-grading with Ext~s(M,g)a EXt~Mod(M, g( a)). Set In] := {1,... ,n}. For a e Zn, set supp+(a):= {i[ai > 0} C Inf. Wesay a ~ Zn is squarefree, if ai = 0, 1 for all i. Whena is squarefree, we sometimes identify a with supp+ (a). Definition 2.1. A Zn-graded S-module M = ~aEZ" Ma is called squarefree, if the following two conditions are satisfied. (1) Mis finitely generated and Nn-graded(i.e., Ma = 0 if a ¢ Nn). (2) The multiplication map Ma 9 y ~-+ xby ~ Ma+b is bijective for all a, b ¯ Nn with supp+ (a + b) = supp+ (a). A free modulesS(-a), a ~ n, i s s quarefree i f a nd only i f a is squarefree. In particular, the Zn-graded canonical module ws = S(-1) is squarefree, where1 := (1,... , 1) E n. Let A C 2[’~] be a simplicial complex, i.e., if F ¯ A and G C F then G ¯ A. The elements of A are called faces, and the maximal faces are called facets. The dimension dimA of A is max{IFI - 1 I F ¯ A}. The Stanley-Reisner ideal of A is the radical monomialideal
of S. Any radical monomial ideal is I/x for some A. We say S/I/x is the Stanley-Reisner ring of A. Stanley-Reisner ideals and rings are always squarefree modules. A Stanley-Reisner ring is one of the central concepts of combinatorial commutativealgebra, see [22, 4]. For F C In], PF denotes the monomialprime ideal (xi [ i ¢ F) of S. Any monomialprime ideal is P~, for some F. Recall that Ass(S/I/x) = {PF I is a f acet of A}. Since dim(S/PF)= IFI, we have dim(S/I/x) = dimA + 1. See for example [4, Theorem5.1.4]. Wedenote by Sqs (or simply Sq) the full subcategory of *Modconsisting of squarefree modules. Using the five lemma, we see that Sq is a subcategorY of *Modclosed under kernels, cokernels and extensions. For the study of Sq, the concept of the incidence algebra of a finite partially ordered set (poser, for short) is very useful. For the reader’s convenience, we will recall basic properties of an incidence algebra here. An
210
Yanagawa
incidence algebra A can be seen as the algebra associated with a quiver with relation, so see [2, § III,1] for further information on A. Let P be a finite poset. The incidence algebra A = I(P, k) of P over k is the k-vector space with basis {ex, y I x,y E P with x < y} and the multiplication is defined by ex, y ez, w = 5y, z ex, w. Wewrite ex for ex, z- ThenA is a finite dimensional associative k-algebra with 1 = ~’~xEPex" iNote that ex ey = 5z,y e~. Thus A ~ ~xEp exA as a right A-module. Denote the category of finitely generated right A-modulesby rnodA. Each exA is projective in mOdA,and any projective object is a finite direct sum of copies of exA for various x E P. If M is a right A-module, then we have M = ~x~p Me~ as a k-vector space. Wewrite Mx for Mex. If f : M --~ N is an A-linear map of right A-modules, then f(Mx) C Nx. Note that Mxe~,y C My and Mzey, z = 0 for y ~ x. Since A is a finite dimensional k-algebra, mOdAadmits the JordanHhlder theorem and the Krull-Schmidt theorem. If M~ mOdAis a simple module, then M= Mz = k for some x ~ P. It is easy to see that [exA]y = k if x _< y, and [exA]y = 0 otherwise. For each x ~ P, we can construct an injective object/~(x) in mOdA.Let ~(x) be a k-vector space with basis {~y I Y -< x}. Then we can regard/~(x) as right A-module by _ ey ¯ ez,w : {~w
ify=zandw<_x, otherwise.
Note that [~(X)]y = k~y if y _< x, and [/~(X)]y = 0 otherwise. The following is well-known. Lemma2.2. The category mOdAhas enough projectives and injectives. An indecomposableprojective (resp. injective) object is isomorphic to exA (resp. ~(x)) for some x ~ P. For any M modA, pr oj. di md M and in j. di md M are at most rankP := max{t [ 2 chain xo < x~ < ... < xt in P}. Proposition 2.3 ([27]). Let 2[hI be the boolean lattice, and A = I(2 [n], k) its incidence algebra. Then we have a category equivalence Sq ~- rnOdA. Proof. For N ~ modA, set M--- (~a~r~ Ma to be a k-vector space with Ma --- Nsupp+(a) for each a ~ n ( note t hat s upp+(a) CIn] is an ele ment of 2In]). Then Mhas an S-module structure such that the multiplication map Ma 9 y ~-~ xby ~ Ma+b is induced by NF ~ y ~ y ¯ eF, G ~ NG, where F = supp+(a) and G = supp+(a + b). By an argument similar to proof of [26, Theorem3.2], we can see that Mis squarefree and the functor ¯ : mOdA~ N ~ ME Sq gives an equivalence modA----- Sq. [] By the above equivalence, a projective object eFA ~ In], mOdA,F ~ 2 corresponds to S(-F), and an injective object ~(F) ~ modd corresponds to S/PF. Hence we have the following.
211
Squarefree Modules and Local CohomoiogyModules
Corollary 2.4 ([25]). Sq is an abelian category, and has enough projectives and enough injectives. An indecomposableprojective (resp. injective) object in Sq is isomorphic to S(-F) (resp. S/PF) ]or some F C In]. And sup(proj.dimsqM I M E Sq} = sup(inj,
dimsqM [ M E Sq} --- n.
A projective object S(-F) in Sq is also projective in *Mod. Thus a projective resolution of a squarefree module Min Sq is also a projective resolution in *Mod.In particular, a Z~-graded minimalfree resolution T~. of M~ Sq consists of S(-F)’s. Put c : = I n] \ F. S ince Homs(S(-F),ws) ~ S(-FC), Homs(Po, ws) is a complex of squarefree modules again. Hence have the following. Proposition 2.5 ([24]). If M is squarefree, so is Ext,(M, ws) for all i >_ O. It is noteworthy that many commutative algebraic invariants/operations of/on squarefree modules are naturally described using the incidence algebra A = I(2 In), k) and the functor ¯ : mOdA--~ Sq of Proposition 2.3. For example, dim~(N) = max{ If[ I F C In] with NF ¢ 0 }, proj.dim s ~(N) proj. dimA N, and Exti~(~(N),ws) ~( Ext~(N,A)). In the last isomo phism, we regard Ext,(N, A) as ri ght A-module us ing th e is omorphism A°p ~ A defined by A°p ~ (eF,G)°p ~-~ eG¢,Fc~ A. See [27] for detail. Next, wewill define another central concept of this article. Definition 2.6 ([25]). A Zn-graded S-module M = (~asz~Ma is called straight, if the following two conditions are satisfied. (1) dim~¢Ma< ~x) for all a ~. (2) The multiplication map Ma 9 y ~-~ xby ~ Ma+b is bijective for all a e Zn and b E Nn with supp+ (a + b) = supp+ (a). The injective (2.1)
hull *E(S/PF) of S/PF in *Modis straight.
[*E(S/PF)]a:{~ otherwise,ifsupp÷(a)
In fact,
CF’
and the multiplication map *E(S/PF)a 9 y ~-~ xby ~ *E(S/PF)a+b is always surjective. Note that *E(S/PF) is not injective in Modunless PF = m. Denote by Str the full subcategory of *Modconsisting of straight modules. For a Zn-graded S-module M= (~ae:~ Ma, we call the Nn-graded submodule (~ae~= Ma the n-graded part of M, a nd d enote i t b y 2 17/. I f Mis straight, then ~ is squarefree. For example, the N’n-graded part of ¯ E(S/P~) is isomorphic to S/PF. Conversely, for a squarefree module N, there is a unique (up to isomorphism) straight module Z(N) nwhose N graded part is isomorphic to N, see [25, 17, 10]. According to [10, 17], we say Z(N) is the ~ech hull of N (this terminology is justified by Remark3.6 below). Summingup, we have the following. Proposition 2.7 ([25]). The functors Sq ~ N ~-~ Z(N) St r and Str ~ M~ M~ Sq give a category equivalence Sq --- Str.
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Yanagawa
Corollary 2.8 ([25]). Str is an abelian category, and has enough projectires and enough injectives. An indecomposable projective (resp. injective) object is isomorphic to Z(S(-F)) (resp. *E(S/PF)) for some Inf. Here Z(S(-F) ) is the localization of the free module S(-F) at the multiplicatively Ft. closed set generated by x Remark 2.9. Let M be a straight module. If char(k) = 0, M can seen as a holonomic module over the ring of differential operators An(k) S(01,... , On), see [1, 25, 27] (rather than Mitself, it is morenatural to think the shifted module M(1) is an An(k)-~nodule, see [1, 27]). If char(k) > Mhas a natural F-finite module ([16]) structure, see [25, Remark2.13]. this sense, the results in the next section are a monomialideal version of the theory of Lyubeznik [15, 16]. 3.
LOCAL
COHOMOLOGY
MODULES
WITH
SUPPORTS
IN
MONOMIAL
IDEALS Since Str has enough injectives and an injective object in Str is also injective in *Mod,an injective resolution of a straight moduleMin Str is also an injective resolution in *Mod.So we have the following. Proposition 3.1 ([25]). Let M be a straight S-module, and *E" : 0 --~ M -+ *E° --~ *E~ --~ ... a minimal injective resolution of M in the category *Mod. Then *Ei is straight for all i >_ O. More precisely, *Ei ~~Fc[n]*E(S/PF) "~(PF’M) (degree shifting does not occur), and the Bass numbers #i(PF, M) are always finite. And we have inj.
dim.Mod M < inj. dims M= dimSupp(M) = dim/~/, where Supp(M) = {p ¯ Spec(S) ] My ¢ Let M, N be Zn-graded S-modules, and *E° an injective resolution
of Min *Mod(we will say *E" is a zn-graded injective resolution). If N is finitely generated, then gi(Homs(N,*E’)) ~- Ext~(N,M). And H~(M) -~ Hi(FI(*E’))a monomial idea l I, w here FI(- ) := l imHoms(S/I t, -) i s a functor from *Modto itself.
In particular,
Hi~(M)has a natural zn-grading.
Corollary 3.2 ([25]). If Mis straight, so is Hita (M). Proof. Let *E° be a Z~-graded minimal injective resolution of M. By Proposition 3.1, each *Ei is straight and a finite direct sum of copies of *E(S/PF). Since FI~(*Ei) is a direct summandof *Ei, it is straight again. Hence H~x(M) is straight. [] Since the canonical module ws is straight, a Zn-graded minimal injective resolution *D° of ws is a complex of straight modules by Proposition 3.1. Moreover, *D° is of the form (3.1)
° : 0 -- > *Do -- > *D~ -+... --> *Dn -+ 0,
SquarefreeModulesandLocalCohomology Modules
213
and the differential is composedof the maps(-1) "(j’F) ¯ nat : *E(S/PF) --~ ¯ E(S/P~{i}) for j e F, where nat: *E(S/P~) ~ *E(S/P~(i)) is induced by the natural sm~ection S/PF b,a e A,b e B} for A,B C In]. See [4, ~5.7]. Corollary 3.3 (Musta~ [19], Terai [23]). For all i ~ O, the local cohomology module H~(ws) ~ H~a(S)(-1) is a straight module whose ~-graded part is isomorphic to Ext~(S/IA, Proof. Since ws is straight, so is H~a(ws). Observe that the ~-gr~ded part of Homz(S/IA,*E(S/PF)) = {y ~ *E(S/PF) ~ xy = fo r Vx 6 I A} is isomorphic to S/PF if I~ C PF, and 0 if not. Thus the ~-gr~ded part of Homs(S/IA,*D’) is isomorphic to that of F~a(*D"). Recall that Ext~(S/IA,ws) = gi(Hom~(S/I~,*D’)) is ~-graded. So the ~-graded part of H~ (ws) is isomorphic to Ext~(S/IA,ws). Let P. be a ZU-graded minimal free resolution of S/IA. By the argument before Proposition 2.5, Homs(P.,ws) is a cochain complex consisting squarefree modules S(-F). So we have the ~ech hull Z(Hom$(P.,ws)) consisting of flat modules Z(S(-F)). Definition 3.4 (Miller [17]). Wesay is the canonical Cech complex of IA. Since the ith cohomologyof Homs(P., ws) is Ext~(S/IA, ws), we have and hence Hi ~.
~ =
¯
(S).
Moreover, ~a (-1) is a flat resolution of FIa (*D), in other words, ~ and F~a (*D) are isomorphic in the derived category Db(*Mod),see [18, Since the local cohomology module H~a (M) is the ith cohomology of the derived tensor product F~ (*D)(1) @~M, we have the following. Theorem 3.5 (Miller [17]). module M, we have
For an arbitrary (not necessarily
graded)
(M) = Vs M). Remark 3.6. Let W"be the Taylor resolution of S/IA with respect to minimal generators of IA. Then W"consists of squarefree modules S(-F), so we have a fl~t complex Z(Soms(T’, ws))(1), which also computes H~a(M). This is isomorphic to the usual ~ech complexof IA, see [18].
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Yanagawa
In Section 6, we will construct the canonical ~ech complex of a radical monomialideal of a normal semigroup ring. Wehave an explicit
formula for the Bass numbers of straight
modules.
Theorem 3.7 ([25]). n-gr Let aded M be a straight S-module, and ~ its N part. Then, tti(pg, M) dimk[Ext~-i-IFl(/17/, WS )]F. Corollary 3.8 ([25]). For all i,j, #i(pF,H~~(S))
we have
= dimk[Ext~-i-lEl(ExtJs(s/IA,ws), = dimk[Ext~s-i-IFl(ExtJs(s/IA,S),
ws S)]E. []
Wecan also computethe Bass numbers at an arbitrary prime ideal p, since we have #i(p,H~ (S)) = #i-t(p*,H~a (S)) by [7, Theorem1.2.3], where the largest monomial prime ideal contained in p and t := dim(Sp/p*Sp). The next lemmais useful to construct an injective resolution of H~a(S). Lemma3.9 ([25]). If M is a squarefree module, so is the Nn-graded part of Homs(M, *E(S/PF)). Moreover, it is isomorphic to (MR)* ®k (S/PF). Here (ME)*is the dual k-vector space of ME, but we set the degree of (ME)* to be 0 (since (MF)* essentially comes from Hom~(Mg,[S/PF]E)). Let *D" be a zn-graded minimal injective resolution of ws described in (3.1). For a squarefree module M, denote the Nn-graded part of the cochain complex Homs(M,*D’) by L°(M). Then L°(M) is of the form L°(M) : 0 --~ ° - ~ L~ - + . .. - ~ Ln - ~ O,
(3.2)
L~ --= ~[~ (ME)* ®k (S/PF) FC[n]
by Lemma3.9. The differential
¯
is composed of the maps
® nat: (ME)*S/RE
for j ~ F. Here (vj)* is the dual of the multiplication map vj : ME\{j} 9 y ~ xjy ~ ME. Since gi(Homs(M, *D°)) ~ Ext,(M, cos) and Ext,(M, ws) is squarefree, we have Hi(L°(M)) Ext,(M, ws). Inpar ticular, if M is Cohen-Macaulay i and has dimension d, then L = 0 for all i < n - d and Hi(L°(M)) for all i ¢ n - d. Hence L°(M)[n - d] gives a minimal injective resolution of Ext~-d(M, w:~) in Sq. Here "In - d]" represents the shifting of the homological degree of a cochain complex by n - d places. Since Z : Sq -+ Str is an exact functor sending S/PE to *E(S/PF), we have the following. Proposition 3.10 ([25]). If M is a Cohen-Macaulay squarefree module of dimension d, then the Oech hull Z(L’(M)) : 0 --~ °) -+ Z(L~) -~. .. -+ Z(Ln) --~ 0
Squarefree Modules and Local CohomologyModules
215
Z(L ~) = (~ (MF)* ®~¢ *E(S/PF),
of L’(M)is a Z"-graded minimal i iective resolution of Z(Ext- (M, after suitable shifting of the homological degree. Here, as in Lemma3.9, we set the degree of the dual space (ME)*of MFto be Corollary 3.11. Assume that Ext}(S/I~, ws) is a Cohen-Macaulay module of dimension t. Set M := Ext~-~(Ext~(S/IA,ws),ws)). Then Z(L’(M))[n~] gives a Zn-graded minimal injective resolution of H~a(ws). Proof. By local duality, Mis a Cohen-Macaulay module of dimension t again, and Ext~-~(M, ws) ~ Ext~(S/IA,w~). Since Z(Ext~(S/IA,w~)) H~a(ws), the assertion follows from Proposition 3.10. If S/IA is a Cohen-Macaulayring of dimension d, Ext}-d(s/IA, w8) is a d-dimensional Cohen-Macaulay module, and Ext~-~(Ext}-d(s/IA, ws), is isomorphic to S/IA itself. Hence Z(L’(S/IA))[n - d] : ~FeA *E(S/PF) gives a ZU-graded minimal injective resolution of H~d(ws). But this is nothing other than F~a(*D’)[n - ~. Since H~a(ws) = for al l i ~ n - d in this case, FI~ (*D’)In- d] is clearly a Zn-graded minimalinjective resolution of H~d(wS). But, even if S/I~ is not Cohen-Macaulay, Ext~-d(S/I~, can be a Cohen-Macaulay module (of dimension d). And the squarefree module ~2 := Ext~-d(Ext~-d(s/I~, ws), has a combinatorial description for an arbitrary simplicial complexA of dimension d, see [22, p.72]. So we have non-trivial exa~nples for which Corollary 3.11 is effective. Example3.12. Let A C 2{a ..... e} be a simplicial complexwhose facets are {a, b, c} and {c, d, e}, see Figure 1 below. Set S = kin,..., el. The StanleyReisner ring S/I~ has dimension 3 and embedding codimension 2, and it is not Cohen-Macaulay. In fact, it is not connected in codimension one, and does not satisfy Serre’s condition ($2). But Ext~(S/I~, ws) is a CohenMacaulaymoduleof dimension 3, and the modulestructure of ~2 is described by Figure 2. Note that Figure 2 does not represent a simplicial complex(in fact the vertex c is "doubled"). We have dim~[a~]~ =
if F = {c} or ~, ifF ~. Wecan take a ~-basis {c~,c~} of [~](~/ so ~hat c~ corresponds to the "left c" of Pigure 2 and c~ corresponds to the righL Then ac~, bc~, dc~ and ec~
216
Yanagawa
a
d
a
e
b
Figure 1
d
Figure 2
are non-zero. But ac2 = bc2 = dcl = ecl = 0. Similarly, we can take a kbasis {¢1, ¢2} of [f~2]o. Then a¢1, b¢l, d¢2 and e¢2 are non-zero, c¢1 = cl, c¢2 = c~, and a¢2 = b¢~ = d¢~ = e¢1 = 0. By Corollary 3.11, a zh-graded minimalinjective resolution of H~2a(ws) as follows. 0 --> *E(S/P{a,b,c}) @*E(S/P{c,d,e})
*E( S / ~ *E(S/P{c}) ~ --+ *E(S/m)2 -+ 0
A Stanley-Reisner ideal Ia is defined over Z, but some of its invariants maydepend on char(k) (not only A). Tim vanishing of H~a (S) also depends on char(k). For example, let A be a triangulation of the real projective plane F2 with six vertices (c.f. [4, Figure 5.8]). Note that dims = 6 and dim(S/Ia) = 3 in this case. By Corollary 3.2, H~a(S) = 0 if and only if char(k) ¢ 2. On the other hand, H~3a(S) is nonzero over any k, but Hilbert function depends on char(k). Hence the Bass numbers of H~a(S) depend. If char(k) ¢ 2 then #i(p, H~a(S)) = 0 for all p with dim(S/p) ~- 3-i. But, if char(k) = 2, then #2(m, Hzaa (S)) By Corollary 3.7 and the combinatorial description of f~2 ([22, p.72]), have the following. Corollary 3.13 ([25]). Let I A be a Stanley-Reisner ideal with dim(S/IA) d (i.e., dimA= d - 1). Then we have the following. (1) #i(p, g?~d(s)) = 0 for all prime ideal p with dim(S/p) > d (2) #i(p, g?~d(s)) does not depend char(k), if dim(S/p) = d (3) Assume that dim(S/PF) = d - i. Then #i(pF, g~d(s)) 7~ 0 only if there is some G ~ A such that [G[ = d and G D F. (4) #i(p, g~d(s)) = 0 for all i and all p with dim(S/p) ¢ d- i if and’ only if S/IA is Cohen-Macaulay.
SquarefreeModulesandLocalCohomoiogy Modules
217
(5) #i(p,H}~-d(s)) <_ 1 for all i and all p with dim(S/p) d-i if andonly if S/IA satisfies the Serre’s condition ($2). (6) By (3), #d(ra, H~2d(s)) is always non-zero (this is true in more general situation, see [15]). For simplicity, assume that A is pure, that is, all the associated primes of Ih have dimension d. Then #d(m, H~zd(s)) 1 if andonlyif A is connected in co dimension 1. 4. SQUAREFREE MODULESOVER A NORMALSEMIGROUPRING Let C C Zd C ]~d be an affine semigroup (i.e., a finitely generated subsemigroup containing 0), and R := k[C] = k[x c I c ~ C] C k[x~ 1 x+11 its semigroupring over a filed k. A zd-graded ideal of R is called a monomial ideal, since it is generated by monomials. In this article, we always assume that R is normal, and m := (x c I 0 ¢ e ~ C) C R is the graded maximal ideal. Since R is normal, it is Cohen-Macaulayby [11, Theorem1]. Wealso assume that ZC = Zd. Hence we have dimR = d. Set IR>0:= {r ~ 1~ I r _> 0}. Consider the polyhedral cone ]~>0C:--- {Zrici I ri ~ ]~_>0, ei G C} C and its dual cone
v = {Ve
I (V,c)>0 forall e
Let et,... , en be edges of (~>0C)v, aand take the first lattice point ai ~ Z of ei for each i. Note that each ai corresponds to a (d- 1)-face of l~>0C. is well-knownthat ]~>oC= {z fi I~d I (ai, z) >_ for al l i} . For z e ~d, se t supp+(z) := {i [ (ai, z) > 0} C [n]. Assumethat R is Gorenstein, that there is some w ~ C with WR= R(-w). Then (ai, w) = 1 for all Wesay R is simplicial if ~>0 C can be spanned by d vectors as a polyhedral cone. In this case, n = d. A polynomial ring S = k[Xl,... , Xn] = k[Nn] is simplicial semigroup ring. In this case, supp+(z)of z = (Zl,... ,zn)~ ]~n is given by {i I zi > 0} C [n] as in the previous sections. Remark 4.1. If R is simplicial, for any subset X C [n], there is some c C C II~a such that supp+(c) = X. This is not true in the non-simplicial case. In general, both inclusions {supp+(c) I c e C} C {supp+(z) I z e ~4} [n] can be strict. But even in the non-simplicial case, for any z ~ ~d, the closure of {w e ~d [ supp+(z) = supp+(w)} is a polyhedral cone of dimension Wedenote by L the set of non-empty faces of the cone ~>0 C. Here ll~>0 C itself and the origin {0} are elements of L. The order by inclusion ~nakes L a finite poset. The rank of L as a poset is d. Note that R is simplicial if and only if L is isomorphic to the boolean lattice 2 [4] as a poser. For z ~ ~>_oC, there is a unique face s(z) ~ such that z is contained in therelative inte rior rel-int(s(z)) of s(z). For z, z’ ~ R>0C,supp+(z) = supp+(z’) if and only
=
218
Yanagawa
Definition 4.2. A Zd-graded R-moduleMis called squarefree, iif the following two conditions are satisfied. (1) Mis finitely generated and C-graded (i.e., Ma= 0 for all a t~ map Ma ~ y ~ xby E Ma+biS bijective for all (2) The multiplication a, b E C with supp+ (a + b) = supp+ (a). If R is a polynomial ring, the above definition of squarefree modules coincides with the previous one (Definition 2.1). It is obvious that R itself a squarefree module. Since the canonical module wR of R is isomorphic to the ideal (x c [ c e C with supp+(c) : In]), it is also squarefree. Let A be an order ideal (i.e, if F E A, G E L, and G C F, then G ~ A). Set IA :=(X e[c~Cands(c) ~A) to be a radical monomialideal of R. Any radical monomialideal of R is Ia for some A. For c ~ C, [R/IA]e ~ 0 if and only if s(c) ~ A. Both IA and R/IA are squarefree. WhenR is a polynomial ring (i.e., C ~ n and L~ 2I n] as posers), the above definition of Izx coincides with the Stanley-Reisner ideal of a simplicial complex A. Note that an order ideal A C 2In] can be seen as a simplicial complex whose vertex set is (a subset of) [n]. Wealso remark that if Mis a squarefree R-modules then Ma --~ Mb for all a, b E C with supp+ (a) = supp+ (b). In fact, since supp+ (a) = supp+ (a + b) = supp+ we have Ma ~- Ma+b -~ Mb. dWe denote the category of R-modules by ModR, the category of Z graded R-modules by *ModR, and the full subcategory of *Modn consisting of squarefree modules by SqR. As in the polynomial ring case, for dM,N ~ *ModR, *HomR(M,N) := (~aezdHom.Modn(M,N(a)) is a Z graded R-module. If M is finitely generated, *HomR(M,N)is isomorphic to the usual HomR(M,N) as the underlying R-module. So we simply denote *HomR(M,N) by HomR(M,N) in this case. The same is true for Ext,(M, N). The following result was essentially proved in [26]. But in [26], the author used the term "sheaves on a poset", which is equivalent to modules over the incidence algebra of a poset. Proposition 4.3. Let A = I(L,k) be the incidence algebra of the poset over k. Then we have a category equivalence SqR----- modA. Proof. Let eF G A be the idempotent corresponding to a face F E L. For N ~ mOdA,set M = ~)¢ec Mc to be a k-vector space with Mc --- Nes(c). Then M has a squarefree R-module structure such that the multiplication Ma ~ y ~-~ xby ~ Ma+b, a, b ~ C, is induced by NeF 9 y ~-~ y . eF, G ~ Nea, where F = s(a) and G = s(a + b). Then the correspondence modm~ N M~ SQRgives an equivalence modA----- SQR.See [26] for detail. [] For a face F E L, set P~. := (xc [ c ~ C \ F) C R to be a monomialideal. Since R/PF is the normal semigroup ring k[C V~ F], PF is a prime ideal and
219
Squarefree Modules and Local CohomologyModules
RIPE is Cohen-Macaulay. Conversely, any monomial prime ideal of R is of the form PF for some F E L (c.f. [4, Proposition 6.1.1]). Observe that dim(RIPE) = dimF, where dimF is the dimension of F as a polyhedral cone. If Mis a Zd-graded R-module, then any associated prime of Mis PF for some F E L. As in the Stanley-Reisner ring case, we have Ass(R/IA) = {PF [ is a maximal ele ment of A}. Note that the canonical module ~OR/Prof R/PF satisfies a E C C) rel-int(F), and [wn/p~]a = 0 otherwise. By Proposition 4.3, we have the following.
[a~U/pr]a = k if
Proposition 4.4 ([26]). SqR is an abelian category admitting the JordanHblder theorem and the Krull-Schmidt theorem. A simple object in SqR is isomorphic to WR/pFfor some F ~ L. Corollary 4.5 ([26]). .for all i >_O.
If M is a squarefree R-module, so is Ext~(M, wR)
Proof. Since SqRis a subcategory of *Modeclosed under kernels, cokernels and extensions, it suffices to prove that Ext~R(WR/pF,WR) is squarefree for all F E L by Proposition 4.4. But this is easy. [] The category SqRhas enough projectives and enough injectives by Proposition 4.3. For a face F E L, we denote the radical monomialideal (x ¢ [c ~ C and s(c) D by JF. For example, J{0) is R itself. If F is the maximalelement of L (i.e., F = ~>0C), then JF = WR. It is easy to see that JF is a squarefree module corresponding to the A = I(L,k) module eFA. Thus an indecomposable projective object in Sqn is isomorphic to Jg for some F. WhenR is a polynomial ring S, JF is nothing other than the free module S(-F). Similarly, an indecomposable injective object in SqR is isomorphic to RIPE for some F. Moreover, sup( proj. dimsq" MI M~ SqR} ---- sup{ inj. dimsq" M[ M~ SqR} = d. Proposition 4.6. With the above notation, we have the following. (1) JF is a maximal Cohen-Macaulaymodule (i.e., depth R JF = d). (2) /] F ¢ {0}, JF has pure codimension one as an ideal of R, more precisely, JF =
N
PG.
GEL dimG=d-1 G~F
And {[JF] l dimF >_ d- 1} generates the class group cl(R) of R. (3) HomR(JF,wR) is isomorphic to the radical monomial ideal ( c I c e C such th at s( c) VF= ] R>0C), where s(c) V F ~ L is the smallest face containing both s(c) and F.
220
Yanagawa
(4) We have
proj. dimsqn M _> d - depth R M
for all M E SqR. If R is simplicial, the equality holds for all M. But if R is not simplicial, there exists some M for which the above inequality is strict. Proof. Every statements other than (2) have been proved in [26]. The first statement of (2) is easy. The second statement follows from the fact that cl(R) is generated by [PF]] di mF = d - 1}(c. f. [6, §3.3]). By Proposition 4.6 (2), if R is not a polynomial ring, JF is not a free R-module for some F E L with dimF >_ d- 1. By (1), proj. dimnJF = c~ in this case. 5. LOCAL COHOMOLOGY MODULESWITH SUPPORTS IN MONOMIAL IDEALS (NORMALSEMIGROUPRING CASE) Definition 5.1. Wesay a Zd-graded R-module M = ~aez~ Ma is straight, if (1) dimk Ma< c~ for all a E d. (2) The multiplication map Ma ~ y ~-~ xby ~ Ma+biS bijective for all a 6 7/~ d and b ~ C with supp+(a + b) = supp+(a). The injective fact, we have
hull *E(R/PI~) of R/PF in *ModRis a straight
module. In
if supp+ (a) C supp+ (F), otherwise, for all a ~ ~d. Here supp+(F) := supp+(b) for some b E C N rel-int(F) (it does not depend on the choice of b). Remark 5.2. By the last observation of Remark 4.1~ a straight module M is finitely generated if and only if Mis a finite direct sum of copies of WR. Similarly, a straight module Mis *artinian (i.e., any non-empty family of Zd-graded submodules always has a minimal member) if and only if Mis finite direct sum of copies of *E(k). Denote by StrR the full subcategory of *ModRconsisting of straight modules. For M ~ *Modn, we call the submodule/~ := (~)aec Ma the C-graded part of M. If Mis straight then AT/ is squarefree. For example, the C-graded part of *E(R/PF) is isomorphic to R/PF. If R is simplicial, a straight moduleMis a *essential extension of its C-graded part (i.e., for any non-zero homogeneouselement y ~ Ma, there is some xc 6 R such that 0 ~ xCy ~ f/I). This is not true if R is not simplicial. In fact, the C-graded part of a non-zero straight modulecan be 0.
SquarefreeModulesandLocalCohomology Modules
221
Proposition 5.3 ([25]). Suppose that R is simplicial. For a squarefree module M, there is a unique straight module Z(M) whose C-graded part is isomorphic to M. From this correspondence, Sqa and StrR are equivalent. We say Z(M) is the ~ech hull of a squarefree module M. When R is a polynomial ring, the above new definitions of straight modules and (~ech hulls coincide with the previous ones in Section 2. On the other hand, Helm and Miller [10] defined the ~ech hull of a graded moduleover a general "semigroup graded ring". In general case, the behavior of the ~ech hull is very delicate. But, for a squarefree module over a simplicial normal semigroup ring, their ~ech hull coincides with ours. Theorem 3.7 for a polynomial ring S can be generalized to a normal semigroupring R, if R is simplicial. Theorem 5.4. Assume that R is simplicial, and M is a straight R-module. Then a Zd-graded minimal injective resolution of Mconsists of straight modules *E(R/PF). For a face F E L and c ~ C re l-int(F), wehave #i(PF, M)= dimk[Ext dR- i-dim F ( 2tT/, WR)]c. Here ~/I is the C-graded part of M. Even if R is not simplicial, a zd-graded minimalinjective resolution of wR consists of straight modules*E(R/PF). In fact, it is given by 0-~*D °--+*D1
d-+O, ~...~*D
i= *D
*E(R/PF). ~ FEL dimF:d-i By the same argument as the polynomial ring case, we have the following. Theorem 5.5. Let I be a radical monomial ideal of R. Then H~(wR) is straight module whose C-graded part is isomorphic to Ext,(R/i, wR). Corollary 5.6. Suppose that R is simplicial and I is a radical monomial ideal of R. For all F ~ L and c ~ C ~ rel-int(F), we have #i( pF, g~i(WR)) = dimk[ Extd~-i-dimF( ExtJR(R/ I, wR), )]c < ~. In particular,
inj.
dimR H](WR))=dim(Ext~(R/I,
wR))
If R is simplicial, the direct generalizations of Proposition 3.10, Corollaries 3.11 and 3.13 also hold. But if R is not simplicial, the situation is completely different. Example 5.7. If R is not simplicial, Corollary 5.6 is not true. Set R = k[x,y,z,w]/(xz -yw). Then R is a normal semigroup ring, and ll(>0C is the cone over a square. Hence R is not simplicial. For I := (x,y~ R, we have H~(R) ~ byTheorem 5.8 belo w, whil e Ext~R(R/I, wR) = O. Moreover, H](WR)a ¢ if andonly if su pp+(a) = sup p+(xw-1). Hence
222
Yanagawa
#°(m,H~(wR)) = cx~. This example was appeared counter example to a conjecture of Grothendieck.
in Hartshorne
[9],
If R is not simplicial, a minimal injective resolution of Hi~(wn) does not consist of straight modules in general, that is, a shifted *E(R/PF)(a) might appear. The author does not know whether one can control an injective resolution of Hi~ (WR) only by the combinatorial properties of the face lattice L. Let T be a cross-section intersection of 1~>0 C and d - 1. For a non-empty face By this correspondence, the It is easy to see that
of the cone ~_>0C, that is, T is a transversal a hyperplane of ~d. T is a polytope of dimension F E L of I~>0 C,/~ denotes the face T ~ F of T. face lattice of T is isomorphic to L as a poser.
T= U ~ = ~ rel-int(~) F~L
F~L
is a finite regular cell complex (c.[ [4, ~6.2]). If A C L is an order ideal, ~FeA ~ C T is also a regular cell complex, and we denote it by HAll. IfR is simplicial, A can be seen as a simplicial complex, and the topological space ~[A[[ is a geometric realization of a simplicial complex A. In this case, the dimension of F ~ A as a face of a simplicial complex is dim~ = dimF - 1, since dim F denotes the dimension as a polyhedral cone here. For an order ideal A C L and a ~ Zd, set delA(a)
:= {F e A [ supp+(F)
Note that del~ (a) is an order ideal complexes d.(A)
:=
~ k~
and
of L. Consider the augmented chain
d.(delA(a)):=
FEA
In this ~rticle,
~ supp+(a)}.
F~delA
(a)
we use the following convention on the homological degree:
F~ A, dim ~=i
F~A, dim F=i+I
IfF = {0}, then ~ = 0. Thus ~_~(A) = k for all A ¢ O. We denote the ith homology group Hi(~.(A)) of the chain complex ~.(A) by ~i(A; Needless to say that ~i(A; k) is isomorphic to the reduced homology group ~i(HA]~; k) of the topological space H = ~F~A ~ C T. If R i s simplicial, ~i(A; k) is the ith reduced homology group of the simplicial Wealso set ~.(A,
del~(a))
:= ~,(&)/d,(del~(a))
complex A.
~ FEA supp+(F)D supp+ (a)
k~,
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223
and denote H~(~.(A, del/~(a))) by /~(A, deLx(a);k). It is easy that the degree a-part of Fja(*D") is isomorphic to ~d-,-l(A, del/x(a)) complex of k-vector spaces. Hence we get the following formula. Theorem5.8 ([26]).
With the above notation,
we have
i
[Hj~ (b3R)]a: /:/d_i_l (A, del/x(a); for all i > O d. and a E Z If -a e C, then del/x(a) = ~ and [H~I~(WR)]a~d-i-l(][A][; k) . If R i simplicial, we can regard A as a simplicial complex, and for all a E Za there is a face F e L such that supp+(a) = supp+(F). Then ~.(A, del/x(a)) isomorphic to the augmentedchain complexof lkF A, after suitable shifting of the homological degree. Thus, when R is a polynomial ring, the above formula is nothing other than Terai’s formula in [23]. In the polynomial ring case, the next result coincides with a well-known formula of Hochster ([22, II, Theorem4.1]). Corollary 5.9. With the above notation,
we have
[H~(R/LX)]a= {~O~_~(A,deLx(-a);k) i]aE -C andsupp+(-a) E A,otherwise Proof. By the Zd-graded local duality ([7, Theorem2.2.2]), we have [H~(R/Lx )]a = [Extd~-~(R/Lx,wR)]_a. Since ExtdR-i(R/Izx,WR) is the C-graded part of H[a (wn), the assertion follows from Theorem5.8. [] Recall a fundamental result of the theory of Stanley-Reisner rings, which states that the Cohen-Macaulaynessof S/1/x is a topological property of a geometric realization ]A[ of a simplicial complexA. Theorem5.10 (Reisner, Munkres, c.f. [22, Proposition 4.3]). Let In] AC2 be a simplicial complex with a geometric realization X :: IA[, and let IA C S = k[x~,... ,Xn] be the Stanley-Reisner ideal of A. Then the following are equivalent. (1) S/IA is Cohen-Macaulay. (2) For all F ~ A and all i < dim(lk/x F), we have/~i(lkA F; k) = 0. (3) For allp e X and all i < dimX, ~(X;k) H~(X,X \p ;k) = If a simplicial complex A satisfies the conditions of the above theorem, we say A is Cohen-Macaulay (over k). Wewill see that the above result also holds for a normal semigroup ring. Roughly speaking, the CohenMacaulaynessof a radical mono~nialideal is a "very topological" property. Corollary 5.11. With the above notation, the following are equivalent. (1) R/IA is Cohen-Macaulay.
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Yanagawa
(2) For all a E C and all i < d - 1, ’we have [-Ii(A, delzx(a); k) = (3) The topological space IIAI[ = [JFeh ~ C T is Cohen-Macaulay,that is, satisfies the condition (3) of Theorem5.10. Proof. The equivalence between (1) and (2) directly follows from Corollary 5.9. So it suffices to prove the equivalence between(2) and (3). E be the "barycentric subdivision" of A. In other words, E is the order complex of the poset A \ {0} (c.f. [3, §12.4]). Then IEI -~ IIAll. HenceA satisfies the condition (3) if and only if ~ is a Cohen-Macaulaycomplex. Weregard E as an order complex of A\{0}. Then a face a E E corresponds to a chain of A \ (0}. Let F A bethe maximum elem ent of t he chai n a. Set # E E to be a saturated chain of the order ideal AF:= {G ~ A I G C F} containing a. Let EF C ~ be the order complexof AF\ {0}. It is easy to see that lk~a = lk~Y. lk~. a. Since ]2F] -~ ~JGcF~ ’~ ~’, lkp~F (7 is a CohenMacaulay complex. So lkr. ~ satisfies the condition (2) of Theorem5.10 and only if so does lkr~ ~. Let (F) be the k-vector subspace of d spanned by F. ForG ~ L with G D F, set G’ := G+(F)/(F) to be the cone in ~d/(F) ]I ~d-dim F spanned by the image of G. Let T~ be a cross-section of the polyhedral cone Ua_~FG’ in ]i~ d-dim F Then AF := {G t I~1 Tt [ G ~ A and G D F} is a cell co~nplex such that ~o_dimF(A, dell(a)) ------- ~’) for a E C f~rel-i nt(F). Moreover, the barycentric subdivision of a cell complexAF is isomorphic to Ik~ ~. Thus A, delA(a); k) = ~i(lk~ ~; k). We are done. /~i_dimF( Using Theorem 5.8, we can get a combinatorial proof of "HartshorneLichtenbaum vanishing theorem" [8, Theorem 3.1] for a normal semigroup ring and a monomial ideal (when R is Gorenstein). Corollary 5.12 ([26]). Ilia 7~ m (equivalently, A ¢ {{0}}), Hd~a(WR) = Proof. By Theorem5.8, [Hdla(WR)]a = ~_t(A, deIA(a); k). If a ~ -C, /~_~(A, delA(a);k) =/~/_~([[A[[;k) = 0. If a ¢f -C, then d_i(delA(a)) and hence ~-I(A, delA(a)) = 0. We are done. In general, the Hilbert function of H~a (con) maydepend on char(k). that of H~2~ (con) does not by Theorem5.8. Recall the following well-known connectcdness theorem. Theorem 5.13 (Faltings). Let (A,m) be a uoetherian complete local domain of dimension d, and I C A an ideal. If H}(A) = O forM = d- l,d, then Spec(A/I) {m} is connected. For a normal semigroup ring R and a monomial ideal IA, the above connectedness theorem is simple and precise. Note that Spec(R/IA) {m} is connected if and only if IIA[[ = [.J~’eA ~ is connected. Corollary 5.14. For a monomial ideal IA 7£ R, the following conditions are equivalent.
SquarefreeModulesandLocal Cohomology Modules (1) The C-graded part of H}a(wn) is 0 for i = d 1,d. (2) dim IIA[[ _> 1 (/.e., dim(R/IA) >_ 2) and I[AII is connected. If R is simplicial, the condition (1) can be rephrased to ’CHi i = d- 1, d".
225
= 0 for
The last statement of the above corollary means that "the second vanishing theorem" (c.f. [12, Theorem2.9]) for a regular local ring also holds for a simplicial normal semigroup ring if the support ideal is a monomial ideal (see also Corollary 6.11 below). WhenR is not simplicial, can be non-zero even if IIA[[ is connected. R = k[x, y, z, w]/(xz - yw) and IA = (x, y) provide a counter example again. To prove Corollary 5.14, we need the following lemma. For an R-module M, set Assi(M) = {p E Ass(M) I htp = Lemma5.15 (c.f. [5]). Let M be a .finitely generated R-module. If p Ass(Ext~(M, wR)), then htp >_ i. And Assi(M) = Assi(Exti~(M, wR)) all i >_O. Proof. The proof of [5, Theorem1.1] also works here.
[]
Proof of Corollary 5.14. Wewill prove the implication (1) =~ (2). Wefirst showthat dim [JAil _> 1 by contradiction. If I[A[[ = ~ (equivalently, IA = m), then Hd~a(wR)~O. If dim]JAil = 0 (equivalently, dim(R/IA) 1), th en dim ExtdR-I(R/IA,wR) = 1 by Lemma5.15. In particular, ExtdR-I(R/IA,wR) is non-zero. Hence dim[IA]] _> 1. Since [H]~I(wR)]o ~0(A, delA(0); k) =/-)0(llAII; k) = 0, [JAil is connected. Wecan prove (1) by a similar argument. If R is simplicial, H~(wR) is a *essential extension of Ext~(R/IA, wn) and hence Ass(H~a (wR)) = Ass(Ext~R(R/I~,wR)). If R is not simplicial the inclusion Ass(H~a(wR)) ~ Ass(Ext~c(R/IA, can be st ric t. But a weaker result holds. Corollary 5.16.
We have dimSupp(H~(WR)) <_ d - i
Assi(H~ (w~)) Assi(R/IA) = {PF I F is a maximal element of A and dim F = d - i}. Proof. Since IA Hi (wn) is a subquotient of*Di (~dimF=d-i*E(R/PF), the first inequality is clear. The last equality is obvious. So it suffices to prove i Ass i (H}a(wR)) = Assi(R/IA). Since ExtiR(R/IA,wR) is a submodule of g~.(w~.), we have Ass(H~(wR)) ~ Ass(ExtiR(R/I/~, By Lemma 5.15, Ass~(H}a (wR)) D Assi(R/IA). Wewill prove the opposite inclusion. Assume that PF ¯ Assi(g~(wn)). Since H~(wn) dis straight, there is some a ¯ Z such that supp+(a) D supp+(F) [gt~(WR)]a ~ O. By Theorem 5.8,
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Yanagawa
~-Id-i-l(A, del6(a);k) ¢ Assume tha t sup p+(a) ~ s upp+(F). Sin dimF = d - i (i.e., dim/~ = d - i - 1), ~d_i_l(A, delA(a)) = 0. This contradiction. Thus supp+(a) = supp+(E) and hence ~d-i-1 (A, delA (a)) k ¯ F (in particular, F E A). If F is not a maximal element of A, then /~d_i_l(A,delA(a); k) : 0. So we are done. [] The above result is far from true if R is not a normal semigroup ring. We also remark that H~ (WR) can have an embedded associated prime even if R is a polynomial ring, see [19, Example1]. 6. CANONICAL~ECH COMPLEXESFOR NORMALSEMIGROUPRINGS Let Psq be the full subcategory of Sq consisting of all the projective objects. An object in Psq is a direct sum of finitely manycopies of JF for various F E L. Note that Hom~,sq(jG,
jF)={ ~ ifFcG, otherwise.
Let M, N be objects in Psq. Assumethat M(resp. N) is a direct sum of (resp. t) indecomposable projective objects. Then a ~norphism f : N -+ in l=’sq can be represented by an s x t matrix .A whose entries are elements of k. Suppose that the ith row of.A corresponds to a direct summandJF of M and the jth column corresponds to a direct summandJG of N. Then the (i, j)-entry of ,4 can be non-zero only if F C Wedenote by R(F) the localization of R at the multiplicatively closed set {x¢ [ c e C gl F} = { monomialsin R \ PF}. Lemma6.1. For all a ~ Zd, we have [R(F)]a
{; ifsupp+(-a) otherwise.
Csupp+(F),
Proof. Since R(F) is a subring of k[x~¯1," .~ , xd ~, dim~[R(F)]a <_ i for all a ~ Zd. If supp+(-a) q~ supp+(F), it is clear that [/~(F)]a : 0. So it suffices to show that [R(F)] a ¢ 0 for all a ~ Zd with supp+(-a) C supp+(F). prove this, it is enough to find b ~ C and c E C ~’1F such that a = -c + b. Take c~ ~ C ~ rel-int(F). For sufficiently large m ~ N, we have ~ + a ~ C. So b := me~ + a and c := me~ satisfy the expected conditions. [] The above proof shows that if c ~ C ~ rel-int(F) then the localization R at {(xC)m I m E N} is isomorphic to R(F). Moreover, the localization of R at a multiplicatively closed set consisting of monomialsis isomorphic to R(F) for some By Lemma6.1, the zd-graded Matlis dual (R(F)) v = *HomR(R(g),*E(k)) of R(F) is isomorphic to E(R/PF), inparticular, it is a str aight module. If R is a Gorenstein ring with WR~ /~(--w), then the shifted module R(F)(-w)
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Modules and Local Cohomology Modules
is straight. But if R is not Gorenstein, R(F)(-a) is not straight for a E ~d in general. Let flat be the full subcategory of *ModRconsisting of direct sums of finitely manycopies of R(F) for various F E L. It is easy to see that Homaat(R(F), R(a)) =
otherwise.
Let M, N be objects in flat. Assumethat M(resp. N) is a direct sum of (resp. t) indecomposable objects (i.e., R(~)’s). ~hen a morphismf : in flat can be represented by an t x s matrix ~ whose entries are elements of k. Suppose ~hat ~he jth row of N corresponds ~o of N and ~he i th column corresponds ~o a direct summandR(F) of M. Then the (j, /)-entry of Mcan be non-~ero only if F C Wec~n define an additive contravariant functor ¯ : Psq ~ flat so that O(JF) : R(F) and O(f) of f ~ Homps~(N, M) is represented by the transpose of a matrix representing f. The following is a generalization of Miller’s canonical ~ech complexof a Stanley-Reisner ideal of a polynomial ring (Definition 3.4). Definition 6.2. Let I C R be a radicM monomial ideal, and P. a ~ninimal projective resolution of R/I in SqR. Wesay ~ :: ~(P.) is the canonical ¢ech complex of I. Since max{/[C} ¢ 0} = proj. dimsq~ (R/I), we have d- depth(R/I)
~ max{i]C} ¢ 0} ~
by Proposition 4.6 (4). If R is simplicial, the left inequality becomes equality. If I = IA for an order ideal A C L, P~ = ~ { Jg ~ F is a minimal element of L Hence ~ = R and ~ = ~{R(F)[F is a minimal element of L ~ A}. Example 6.3. Consider the case where I = m. A minimal projective lution P. of Rim in SqRis of the form
reso-
with FEL,dimF=i
and the signs in the differential mapsare given by an incidence function of the regular cell complexllLII --- T. Hence the canonical ~ech complexO~of m is give by
Yanagawa
228 with FEL, dimF:i
It is easy to check that ~ coincides with the complex L° defined in [4, §6.2] to compute local cohomology modules Him(M). Theorem 6.4. Let I C R be a radical monomial ideal with the canonical ¢ech complex ~. For an arbitrary (not necessarily graded) R-module we have Hil(M) In particular,
= Hi(~
@R
H~(R)=
To prove the theorem, we will look at R(F) @RM carefully. Note that c M(F) := R(F) @~M is the localization of M at (x ~ c ~ C ~ F}. Lamina 6.5. Let M be an arbitrary following are equivalent.
R-module.
For an element
(1) The natural image of y in the localization M(F) is (2) There is some c ~ C ~ F such that xCy = O. then (xc’)my = fo r su ~ciently la (a) If c ~ ~ (C ~ rel-int(F)),
y ~ M, the
rge m ~ N.
Proof. The implications (1) ~ (2) and (3) ~ (2) ~re clear. So it to prove (2) ~ (3). Under the above notation, a:= ~-c ~ C~F fo r m >> 0. Hence we have (xc’)my = xmC’y = xa+¢y Lemma 6.6. Let p C R be a (not necessarily Zd-g~uded) prime ideal, and E(R/p) the injective hull of RiP in ModR. Let p* be the largest monomial ideal contained in p. Then p* is a prime ideal, and hence there is some G ~ i such that p* = PG (see [4, 7]). Then F C G, otherwise. Proof. It is easy to see that {x c ~ C~F}~PG : {x~]c ~ C~F}~{x ¢ ]c ~ C~G} = {x ¢ I c e C~(F~G)}. Hence F C G if and only if {xc [ c ~ C~ F} ~ p = 0. So it suffices to show that (R/p) if c I ,=e c n F}n p = R(F) ®R E(R/p) otherwise.
{~
Suppose that (x c I c E C DF}np ~ 0. Take x c ~ p with c ~ CnF. For all y E E(R/p), there is some m E N such that (xC)my = 0. Hence R(F) ®RE(R/P) = O. the othe r hand , if { x c [ c E CnF}~p - q), the multiplication map E(R/p) D y ~4 xCy ~ E(R/p) is bijective for all [] c E C D F. Hence the localization R(F) ®R E(R/p) is E(R/p) itself.
SquarefreeModulesandLocal Cohomology Modules
229
Proof of Theorem6.4. This result is a generalization of [4, Theorem6.2.5] concerning the case whenI = m. So we follow the pattern of the proof of this theorem. More precisely, we show that Hi(dr ®R-) are the right derived functors of Let A C L be an order ideal with I = IA. Note that ~®RM= R®RM:M and d~ ®RM = O {M(F) I F is a minimal element of L \ Since I = (xc I c E C V~ rel-int(F)
for some minimal element F of L \
the kernel ofd~®RM--~ d~®/~Mis F,(M)= {y e M [ Imy = 0 for m >> 0 } by Lemma6.5. Since ~ is a complex of flat R-modules, the exact sequence of R-modules
O-~M~-~M~-~M3-~O yields the exact sequence 0 --~ ~ ®RM1 --~ ~ ®RM~--+ ~ ®RM3 -+0, from which we obtain the expected long exact sequence of cohomologymodules. Finally, we must show that Hi(d~ ®RM)= fo r al l i > 0 if M i s an injective R-module. Without loss of generality, we may assume that M = E(R/p) for a prime ideal p. Let G e L be the face with p* = PG, and 7). a minimal projective resolution of R/I in SqR. If T)i = ~ j~(i,v), then C}’ = {~ ~(F)~(i’F) by construction ~i~ ®R E (R/p) = (~ Fca E(R/P)/~(i’f) by Lemma6.6. For a ~ C g~ rel-int(G), we have [:Pi]a = kfl(i’f). ~FCG Moreover, where [7).]a is the degree a-component of ). ( hence achain co mplex ofkvector spaces), and ([:P.]a)* is the k-dual of [T’.]a. Since ). i s a n acyclic complex, ([:P.]a)* is also. Hence H~(d~®nE(R/p))= 0 for all i > 0. Remark 6.7. Assume that R is simplicial, then for any F ~ L there is a unique face Fc ~ L such that HomR(JF,wR)= JF~, see Lemma4.6 (3). also assume that R is a Gorenstein ring with wR = R(-w). Then the ~ech hull Z(Jt~) is isomorphic to R(F)(-w). Henceif 7). is a minimal projective resolution of R/I in SqR, then d r ~ Z(Somn(T)o, wn))(w). Thus the new definition of a canonical ~ech complex is a generalization of the old one for a polynomial ring. Proposition 6.8. If I is a monomialideal of R, the Z~-graded Matlis dual v of Hi~(R) is straight. H~(R)
230
Yanagawa
P,vof. Wemay assume that I is a radical monomialideal and has tlhe canonical ~ech complex ~. By Theorem6.4 and the exactness of the Matlis duality functor, H~(R)v is isomorphic to Hi(~) v = Hi((d~)v). Since (~)v a complex of straight modules *E(R/PF), H~(R)v ~ Hi((d~) v) is straight. (Of course, we can use the usual (~ech complexinstead of the canonical (~ech complex, since it also consists of R(F)’S by the remark after Lemma6.1).. [] Corollary 6.9. Assume that I ~ 0 is a monomial ideal of R. If H~(R) finitely generated as an R-module, then Hi~(R) = Proof. IfH}(R) is finitely generated, then the Matlis dual H}(R)v is straight and *artinian. So H~(R) v = *E(k) m for some m >_ 0 by Remark 5.2. Therefore H}(R) = m. Since I ¢ 0, we must hav e rn = 0 . [] For an ideal I C R, set cd(R,I)
:= max{ilH}(M)
¢ fo r SMe ModR } = max{ i[H}(R) ¢
0} .
WhenR is a polynomial ring, the next result was proved by Lyubeznik [14]. Corollary 6.10. For a radical monomial ideal I C R, cd(R, I) = proj. dimsqR (R/I). In particular, if R is simplicial, cd(R,I)
= d - depth(R/I).
Proof. Set t :=The proj. second dimsq equality For thefrom firsttheequality, first oneit byisProposition enoughto show 4.6 (4). that R (R/I). follows HI(R ) = Ht(~) ¢ O. Let f e L be a maximal element such that JF actually appears in ~} as a direct summand.Then it is easy to check that [Ht(~)]_a ¢ 0 for all a e zd ~ rel-int(F). WhenR is not Gorenstein, the next result improves the last statement of Corollary 5.14. Corollary 6.11. Assume that R is simplicial. IA, the following conditions are equivalent. (1) Hdia(n)= H/d21(R)=0 (equivalently, (2) dim IIAII >_ and II AII is connected.
For a radical monomial ideal cd(n, IA)< d- 1).
Proof. Since cd(R, Ih) = proj. dimsq,(R/Ih), cd(R, IA) can be expressed in terms of the incidence algebra A of 2[n], and hence it only depends on the simplicial complex A (not specific R). So we mayassume that R is a polynomial ring. If R is a polynomial ring (more generally, if R is Gorenstein), the assertion is immediate from Corollary 5.14. []
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P~EFEB.ENCES [1] J. Alvarez Montaner, R. Garcia Lopez, S. Zarzuela, Local cohomology, arrangement of subspaces and monomial ideals, preprint, ht~;p://arXiv, org/abs/~aath/0011181. [2] M. Auslander, 1. Reiten, and S.O. Smal0, Representation theory of Artin algebras, corrected reprint of the 1995 original, CambridgeUniversity Press, Cambridge, 1997. [3] A. Bj6rner, Topological methods, Handbookof combinatorics, Vol. 1, 2, 1819-1872, Elsevier, Amsterdam, 1995. [4] W. Bruns and J. Herzog, Cohen-Macaulay rings, revised edition, Cambridge University Press, 1998. [5] D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235. [6] W. Fulton, Introduction to toric varieties, Princeton University Press, 1993. [7] S. Goto and K. Watanabe, On graded rings. II (Z"-graded rings), Tokyo J. Math. 1 (1978), 237-261. Ann. of Ma~h. 88 [8] R. Har~shorne, Cohomological dimension of algebraic varieties, (1968), 403-450. [9] R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164. [10] D. Helm and E. Miller, Bass Numbers of semigroup-graded local cohomology, preprint, ht tp ://arXiv. org/abs/ma~h/O010003. [11] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972), 318-337. [12] C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102 (1990), 73-93. [13] C. Huneke and R. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), 765-779. [14] G. Lyubeznik, On the local cohomology modules HI,(R) for ideals a generated by monomials in an R-sequence. In: "Complete intersections (Acireale, 1983)", Lecture Notes in Math., 1092, Springer, 1984, pp. 214-220. [15] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to Commutative Algebra), Invent. Math. 113 (1993), 41-55. [16] G. Lyubeznik, F-modules: applications to local cohomology and D-modules in characteristic p > 0, J. Reine Angew. Math. 491 (1997), 65-130. [17] E. Miller, The Alexander duality functors and local duality with monomial support, J. Algebra 231 (2000), 180-234. [18] E. Miller, Graded Greenlees-May duality and the ~ech hull, this volume. [19] M. Musta~h, Local cohomology at monomial ideals, J. Symbolic Comput. 29 (2000), 709-720. [20] M. Musta~a, Vanishing theorems on toric varieties, preprint, http://arXiv.org/ abs/math/0001142. [21] V. Reiner, V. Welker and K. Yanagawa, Local cohomology of Stanley-Reisner rings with supports in monomial ideals, J. Algebra (to appear). [22] R. Stanley, Combinatorics and commutative algebra, 2nd ed. (Birkh/iuser, 1996). [23] N. Terai, Local cohomology modules with respect to monomial ideals, preprint. [24] K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree iNn-graded modules, J. Algebra 225 (2000), 630-645. [25] K. Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc. (to appear). [26] K. Yanagawa, Sheaves on finite posers and modules over normal semigroup rings, J. Pure and Appl. Algebra 161 (2001), 341-366. [27] I~. Yanagawa, Derived category of squarefree modules and local cohomology with monomial ideal support, preprint.
GRADED GREENLEES-MAY DUALITY AND THE ~ECH HULL EzraMiller Massachusetts Institute of Technology,Cambridge, Massachusetts
ABSTRACT. The duality theorem of Greenlees and May relating local cohomologywith support on an ideal I and the left derived functors of I-adic completion [GM92]holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local cohomologyand derived functors of Z~-graded completion, when I is a monomialideal in the Z~-graded polynomial ring k[x~,... , x,~]. Greenlees-May duality for this case is a consequence. A key construction is the combinatorially defined ~ech hull operation on Z%graded modules [Mi198, Mil00, Yan00]. A simple self-contained proof of GMduality in the derived category is presented for arbitrarily graded noetherian rings, using methods motivated by the (~ech hull.
1. INTRODUCTION
Let S = k[xl,... ,Xn] be a polynomial ring over a field k, and I = (ml,... ,mr) C S a Stanley-Reisner ideal generated by squarefree monomials. Local cohomology H~(S/I) with support on the maximal ideal m = (xl,... , xn) of the Stanley-Reisner ring S/I has been familiar to combinatorialists and algebraists ever since the fundamental work of Hochster and Stanley (see [Hoc77, Sta96]) as well as Gr/ibe [Gr~84] relating these objects
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to simplicial complexes, and Reisner’s discovery of a simplicial criterion for S/I to be Cohen-Macaulay [Rei76]. Beginning with Lyubeznik [Lyu84] and continuing with a series of recent papers [Ter99, Mus00a, Yan00, Mil00], increasing attention has been paid to properties of the local cohomology HI(M) of modules Mwith support on a Stanley-Reisner ideal I. One consistent feature of the recent investigations into Hi is the presence of somesort of duality. The duality ranges in character, from the topological duality between homology and cohomology, to a more combinatorial duality in posers (Alexander duality, if the poser is boolean), to algebraic dualities such as Matlis duality and local duality. Elsewhere in this volumeis a full treatment of the rather general Greenlees-May duality [GM92, AJL97], an adjointness between Hi and the left derived functors L./ of I-adic completion in commutativerings (or better yet, for sheaves of ideals on sche~nes), that as yet has not appeared explicitly in combinatorial studies. The purpose of this paper is to compute simple formulas for both sides of the Zn-graded Greenlees-May isomorphism over S, and to show how the computation can be rephrased to give an easy proof of GMduality for arbitrary noetherian rings graded by commutative semigroups. The isomorphism between graded local homologyand the left derived functors of graded completion is an easy consequence. The Zn-graded local duality theorem with monomia! support and the Alexander duality between H(~(S/I) and H~-’(S), whose original proofs were independent of GMduality, illustrate the theory as combinatorial examples. The key construction for the Zn-graded case is the ~ech hull, a combinatorial operation on Zn-graded modules. Although it was first defined in the context of Alexander duality quite independently of local cohomological considerations [Mi198], the ~ech hull turns out to be a natural tool for computing left derived functors _L.~ of ZU-graded I-adic completion. More precisely, Theorem4.4 carries out the computation of _L./ in terms of the ~ech hull of the Zn-graded Hommodule Homs(br., ws) for a free resolution ~’. of I. Whenaltered slightly to avoid using the ZU-grading, this method provides a similar approach to the isomorphism between L./ and local homology H.~ in graded noetherian rings (the new proof of this knownisomorphism is the interesting part; the arbitrary grading comesfor free). The organization is as follows. The ~ech hull and Zn-graded construction of Matlis duality are reviewed in Section 2. The relation between the ~ech hull and local cohomologyis demonstrated in Section 3. The left derived functors _L.~ of graded I-adic completion are introduced in Section 4, and computed over the polynomial ring via the (~ech hull. Section 5 treats Zn-graded Greenlees-May duality (and its consequences) over S. Finally, Section 6 covers the reformulation of the methods to give a conceptually easy proof for noetherian graded rings.
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The reader interested only in the (short) proof of Greenlees-Mayduality for graded noetherian rings is advised to read Sections 4.1-4.2 and Definition 4.7 before continuing on to Section 6, which requires no other results. This paper is intended to be accessible to those unfamiliar with GreenleesMayduality as well as to those unfamiliar with the combinatorial side of local cohomologyand Zn-gradings. Therefore manyparts are longer than strictly necessary, and have a decidedly expository feel. This seems to be in the spirit of the "first ever" conference on local cohomology,as well as this workshop proceedings. Acknowledgements. I am grateful to Joe Lipman and John Greenlees for enlightening discussions (and courses) at the Guanajuato conference on local cohomology. Funding during various stages of this project was provided by an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship and a National Science Foundation Postdoctoral Research Fellowship. 2. BASIC CONSTRUCTIONS 2.1. Matlis duality. Let S = k[xl,... ,xu] be a polynomial ring over a field k, and set xa: xlal.., x~n for a E N~. An S-module J is Zn-graded if J = (~bEZa gb and xaJb C_ Ja+b for all monomials xa E S. Thus the polynomial ring S is ~n-graded, as is the localization S[x~1 I i e F] for any subset F C_ {1,... ,n}. Definition 2.1. Define the Matlis dual jv of J by with S-module structure the transpose of xa : J-a-b
(JV)_b = Homk(Jb, determined by letting
xa : (JV)b --> (Jv)a+b
-~ ,]-b.
It is obvious from this definition that Matlis duality is an exact contravariant functor on Zn-graded modules, and that (jv)v = j if dimk Jb < c~ for all b ~ Zn (such a moduleJ is said to be Zn-finite). To orient the reader, this Matlis duality restricts to the usual one between finitely generated and artinian Zn-graded modules (see Lemma2.3), although this fact won’t arise here. Example 2.2. The Matlis dual of an S-module that is expressed J= k[xi [ i e F][x~-l[jeG] is jV = k[x~l [ i e F][xj I JeG]. It is not necessary that F and G be disjoint, or that F _~ G, or that their union be {1,... ,n}; when Jb ~ 0 but Ja+b = 0, it is understood that xaJb = 0. The easiest and most important example along these lines is when J = S, so that J~/= S~/= k[x~l,... ,x~1] is the injective hullofk. [] Let R be any Zn-graded ring (we will use only S and its Zn-graded subring k concentrated in degree 0). A map ¢ : M--~ N of Zn-graded R-modules is called homogeneousof degree b ~ Zn (or just homogeneous when b = 0)
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if ¢(Mc) _C Nb+c. WhenR = S’ and b is fixed, the set of such maps is k-vector space denoted Horns(M, N)b : degree b homogeneous maps M --> As the notation suggests, Homs(M,N )
= ~ H°ms(M,g)b nbEZ
is a zn-graded S-module, with xa¢ defined by (xa¢)(m) = xa(¢m) ¢(xam). Matlis duality can now be expressed withou~ resorting ~o degreeby-degree vector space duals. Lemma2.3.
For any Zn-graded modules J and M, Homs(J ,Mv) = (M ®s j)v = Homs(M’ jr).
In particular, jv = Horns(j, Sv). Proof. jv can be expressed as the Zn-graded module Homk(J, k), because a k-vector space homomorphismJ --~ k that is homogeneousof degree b is the same thing as a vector space mapJ-b --~ k, the k being concentrated in degree 0. The result is a consequence of the adjointness between Homand ® that holds for arbitrary Zn-graded k-algebras S and S-modules J, M: Homk(M® g,k)
= Homs(M, Homk(J,k))
= Homs(M,
That Mand J can be switched is by the symmetry of ®.
[]
nMatlis duality switches flat and injective modules in the category of Z graded modules. Lemma2.4. J E M is fiat if and only if jv is injective. Proof. The functor Horn s (-, jr) on the right nomk(- ®S J, k) = Horns(-, Ho__.~mk(J , k)). is exact <=~ the functor on the left is exact ¢~ - ®s J is, because k is a field. [] Example 2.5. The Zn-graded dualizing complex for S is the Matlis dual of the (~ech complexon x~,... , x~, appropriately shifted homologically. [] See [GW78,Sta96, Mil00, MP01] for more on the category of Zn-graded modules and its relation to combinatorial commutative algebra. 2.2. The ~ech hull. Definition 2.6 ([Mil00]). The (~ech hull of a Zn-graded module M is the Zn-graded module ¢M whose degree b piece is (¢M)D
= Mb+
where b+ = ~’~ b~e~ b~>_0
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and the ~ech Hull
is the positive part of b. Equivalently, ~M = (~ Ub ®t~ k[x~ 1 I b~ = 0]. nbEN
If ei E Zn is the ith standard basis vector, the action of multiplication by xi is if b~ < 0 ¯ xi : (¢M)b --~ (¢M)e~+b= [,-xi :Mb+~ Me,+b+if bi _>
~identity
Note that (ei+b)+ = b+ whenever bi < O, and (ei+b)+ = ei+b+ whenever bi >_0. Heuristically, the first description of ~Min the definition says that if you want to know what ~Mlooks like in degree b E Zn, then check what M looks like in the nonnegative degree closest to b; the second description says that the vector space Ma for a e Nn is copied into all degrees b such that ’~. b+ = a. The ~ech hull "forgets" everything about Mthat isn’t in N The ~ech hull can just as well be applied to a homogeneousmapof degree 0 between two modules, by copying the maps in the l~-graded degrees as prescribed. Someproperties of ¢ are now immediate from checking things degree by degree. Lemma2.7. The ~ech hull is an exact covariant functor modules.
on Zn-graded
The utility of the ~ech hull stems from its ability to localize free modules and take injective hulls of quotients by primes simultaneously. For instance, ¢S = Six,l,... ,x~l], while ~(S/m) = SV; see [Mil00, Example 2.8] for more details. Here, the applications require only the localization property. The notation henceforth is as follows. For b ~ Zn, the Zn-graded shift M(b) is the module satisfying M(b)¢ = Mb+c. Thus, for a n, the free module of rank 1 generated in degree a is S(-a). If F ~ {0, n, th e localization Mix-F] is M®~ S[x~~ [ Fi = 1]. Setting 1 = (1,... , 1), the next lemmais straightforward. Lemma2.8. /f F E {0, 1}n then ~(S(F 1)) -~ S(F- 1) Ix -F]. Lemma 2.10 will clarify the pertinence of the next definition. Definition 2.9. Define the t th Frobenius power of a monomial ideal I = (ml,... ,mr) tO be I[t] = (m~,... ,m~) for 1 _< t ~ Z. Any direct sum 2: = ~j S(-Fj) with each Fj 6 Nn is isomorphic to a direct sum ~j/xF~) of ideals, so 2:[~] = ~j S(-tFj) is sirnilarly defined. The advantage to Frobenius powers is that their free resolutions are all related (no ~ssumption is required on the characteristic of S). Let ~. be free resolution of S/I. Thenthere is an inducedfree resolution ~’[~] of S/I[~]: choose matrices for the differentials in 9L and replace every occurrence of xi with x[, for all i. Equivalently, if ~oIt] is the k-algebra isomorphismmapping
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t for all i, then 9v. can be consideredas a complex~o[t](~’.) of S[t]-modules, and .T’[t] = S ®sltl qo[t](-T-) (the tensor product over S[t] is via the inclusion SIt] ~ S). This implies that .T[t] is indeed acyclic, since S is a free (and hence fiat) S[t]-module. For the Zn-graded shifts of the summandsin ~-. to work out properly, SIt] must be graded by the sublattice tZ For each t >_ 1 there is an inclusion .T[ t+l] --> .T[t] of complexesvia the homogeneous degree 0 maps S(-(t+ 1)F) --> S(-tF) sending 1 ~ xF. This makes {.T[ t+l]} into an inverse system of complexes. Just as with Koszul complexes,setting w.v -- S(1) and applying Ho____~ms(-,ws) to {br[ t]} yields a directed system {.T[[;]}, in which S(tF- 1) --> S((t 1)F- 1) viathe natural f. inclusion 1 ~-~ x Assuming nowthat I is squarefree, ~’. can be chosen so that all of the summandsS(-Fj) of Definition 2.9 are in squarefree degrees Fj E n. {0., 1} (This may not be obvious, but will follow from Lemma3.2.) In this case 9TM = Homs(~’., ws) also has all of its summandsin squarefree degrees, since Horn s(S(-F), ws) = S(F-1). The reason for introducing Frobenius powers is the next lemma,to be applied in Section 4.
s = k[xl,.., to sI,l =
via
Lemma2.10. Let .~. be a .minimal free resolution If ~;] = Homs(~it],ws)then Fl__,mt
of S/I and ws = S(1).
Proof..T’" is composed of maps S(G - 1) -+ S(F - 1) between free modules generated in squarefree degrees. Since li__7mS(tG- 1) = S(G - 1)Ix -G] and li__,m S(tF- 1) = S(F- 1)Ix -f] are the corresponding localizations, it follows that li__,mgV[;] is composed of natural inclusions S(G- 1)Ix -G] --~ S(F1)Ix-F]. Now use Lemma2.8. [] All of the Zn-graded shifts by 1 in the preceding are essential, because taking ~ech hulls rarely commuteswith such shifts. However,the restriction of minimality in Lemma 2.10 is unnecessary, as is clear from the proof: being generated in squarefree degrees wilt do. See Proposition 3.3 and Example3.4 for examples. Remark2.11. The colimit li__,mtJt of a directed system of Zn-graded modules is a quotient of ~[~ Jt by a Zn-graded submodule. It is therefore naturally Zn-graded. Remark 2.12. In the context of Lemma2.8, Yanagawa’s notion of straight hull [Yan00] coincides with the ~ech hull. The ~ech hull is generalized in [HM00]to algebras graded by arbitrary semigroups, where it isn’t exact. Its derived functors yield a spectral sequence of Ext modules converging to local cohomology,and are used to prove the finiteness for Bass numbers of graded local cohomology of finitely generated modules over simplicial semigroup rings. It is also shownthat the local cohomologyof some finitely generated module has infinite Bass numbersif the semigroup is not simplicial.
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3. THE ~ECH HULL AND LOCAL COHOMOLOGY 3.1. Generalized ~ech complexes. This section reviews the connection between the ~ech hull and local cohomology[Mil00, Section 6] from a new point of view. Let ws = S(-1) be the canonical module of Definition 3.1. If .T. is a free resolution of S/I and ~" = Hom(~.,ws), define d;~ = (¢~=’)(1) to be the generalized ¢ech complex determined by .T.. When~’. is minimal, d~: is called the canonical Xech complex for I, and is denoted by ~. The first task, Proposition 3.3, is to identify the usual ~ech cornplex on generators for I as the generalized ~ech complexfor a certain (usually far from minimal) free resolution. Suppose xF1,... , F~ ES ar e monomials (they don’t need to be squarefree, yet). Given a subset X C_ {1,... ,r}, define Fx so that xg× is the least commonmultiple of the monomials xFj for j E X. The Taylor resolution 7:. on xfl,... , x~ is r
O+-- S ~-~]~S(-F~) ~-... j=l
~- ~ S(-Fx) ~-... Ixl;e
r}) ~0,
e- S(-F{~
where the map S(-Fx\j) ~- S(-Fx) is (-1) ~-~ times the natural inclusion if j is the S th element of X. Lemma3.2 ([Tay60]). T. is a free resolution of
S/(xFI,...
,xFr).
Proof. The Zn-degree b piece ofT. is zero unless b ~ Nn, in which case (T.)b is the reduced chain complexof the simplex whosevertices are {j I Fj ~_ b}, with ~ in homological degree 0. This chain complex has no homologyunless x~ ff (xg~,... ,x~’~), whenthe only homologyis k in homological degree Thus 72. is a free resolution of something. The image of the last map(to S) in T. is obviously (xF~,... , F~). [] The next result is needed for Proposition 3.7. It identifies the ~ech complex not just as a limit of Koszul cochain complexes on generators for i[t], whoseduals /£!tl maynot be acyclic, but as the result of applying the ~ech hull and shift by 1 operations, which are exact, to a complex T" = Homs(T., ws) whose dual is a resolution T. of S/I. Proposition 3.3. If I = (xF~,... ,x F~) is generated by squarefree monomials, so Fj ~ {0, 1}~ for all j, then 8;r is the usual Oechcomplex
......,l-0 on the monomials xF~,...
,x F" generating I.
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Proof. Each original summandS(-F) in T. turns into a corresponding summand S(F - 1) = Hom3(S(-F),ws) in Homs(T.,w3 ). Lemma 2.8 says ~(S(F- 1)) ~ S[x-F](F - 1), and this is isomorphic to Six-F](-1) because multiplication by xF is a homogeneousdegree F automorphism. Shifting by 1 yields the result. [] The proof of Proposition 3.3 says how to describe any generalized ~ech complexd~ in more familiar terms: after choosing bases in the complex.T" = Horn s (.T., ws), simply replace each summandS(F - 1) by the localization Example 3.4. The usual ~ech complex ~ = ~’(xl,... ,xn) is the canonical ~ech complex of the maximal Zn-graded ideal m. In other words, the Cech hull of the Koszul complex is the Oechcomplex, up to a Zn-graded shift by 1. This example, or more generally Proposition 3.3, is the reason for the term "(~ech hull". [] Remark 3.5. The canonical ~ech complex of I depends (up to isomorphism of Zn-graded complexes) only on I, not on any system of Zn-graded generators of I. The term "canonical (~ech complex" refers both to this freedom from choices and the fact that it is, up to Zn-graded and homological shift, the ~ech hull of a free resolution of the canonical module~as/I whenS/I is Cohen-Macaulay. 3.2. Local cohomology. If one is convinced that the minimal free resolution of S/I is in any sense "better" than the Taylor resolution (or any other free resolution), then one should be equally convinced that the canonical ~ech complex d~ is similarly "better" than the usual (~ech complex ~- on the minimal generators of I, by Proposition 3.3. Of course, the main use for the usual ~ech complex is in defining the local cohomologymodules H~(M) = Hi(M ®sd~-), and one needs to be convinced that the canonical ~ech complex is just as good at this local cohomology computation. In fact, this holds for any generalized ~ech complex. Theorem 3.6 ([Mil00]). S-module, then
If J:. is a free resolution
of S/I and M is any
HiI(M) = Hi(M ®s ~’~:). Although it is possible to give a concrete proof using Lemma2.10 as in [Mil00, Theorem 6.2], a new and more conceptual proof is appropriate here, in anticipation of Zn-graded Greenlees-May duality in Section 5. The proof below pinpoints the relation between the canonical ~ech complex and the usual ~ech complex as being analogous to--and a consequence of---the relation between the minimalfree resolution of S/I and the Taylor resolution. It rests on Proposition 3.7, which is the observation that propels the rest of
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the paper: Theorem 3.6 and Corollary 4.8 depend on Proposition 3.7, and Proposition 6.1 is modelled upon it. In what follows, a homology isomorphism of complexes (also knownas quasi-isomorphism) is a map inducing an isomorphism on homology. Proposition 3.7. If 6. and ~. are free resolutions of S/I, then there is a homology isomorphism ~ -+ ~= between their corresponding generalized ~ech complexes. Proof. If 9v.~ is a minimal free resolution of S/I, then there are homology isomorphisms ~’. -+ ~’./ and ~’.~ -~ 6. because .T. ~ is a split subcomplex of every free resolution of S/I [Eis95, Theorem20.2]. Composingthese, we have a homology isomorphism ~’. -+ 6.. Applying Homs(-,ws) yields a map 6" -~ 9v" whose induced map on cohomology is an isomorphism: both have cohomology Ext’s(S/I ,wS). The desired homology isomorphism is therefore simply (¢6")(1) -+ (~bv’)(1), since the ~ech hull and the by 1 are both exact. [] It’s worth stating explicitly the following lemma,even though it is standard (so its proof, the Kiinneth spectral sequence, is omitted), because it will be used so often. A bounded below complex is one that has nonzero modules only in positive homological degrees; thus free resolutions of modules are bounded below. Lemma 3.8. If ~... --+ 1: ~. is a homologyisomorphism of boundedbelow complexes of flat modules and M is any module, then M ® ~. ~ M ® Lr. is a homology isomorphism. Proof of Theorem 3.6. Proposition 3.7 produces a homology isomorphism d~-(,) ~ d~-. Since both of the complexes ~-(,~) and ~.~ are bounded below (and above), Le~nma 3.8 says that H~(M) = Hi(M ® ~-(I)) -~ Hi( M ®~) is an isomorphism. [] Remark 3.9. If the module Mis Zn-graded, then the isomorphism in Theorem 3.6 produces the natural Zn-grading on H~(M), since the right-hand side is still zn-graded. 4. THE ~ECH HULL ANDCOMPLETION 4.1. Graded completion. Suppose for the moment that I is an ideal in an ungraded commutative ring A. Just as F~ takes the direct limit of submodules annihilated by powers of I, the I-adic completion functor AI takes the inverse limit of quotients annihilated by powersof I. Moreprecisely, any A-module Mhas a filtration by submodules I~M for 1 _~ t ~ Z, giving rise to an inverse system M/It+IM --~ M/I~M. The I-adic completion A~(M) is then defined as the inverse limit of this system, li~__.m~M/ItM. In general, I-adic completionis neither left exact nor right exact (intuition from finitely generated modulesover noetherian rings fails badly). However,
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one can still define the left derived functors L~.(M) of h~ by taking a free resolution of Mand applying AI to it. The catch is that the natural map from the zeroth homology L~(M) of the resulting complex to A~(M)need not be an isomorphism, as it would be if AI were right exact. Everything in the last two paragraphs (and everything in the next subsection, as well) makes complete (!) sense, mutatis mutandis, in the category of modules graded by some group or monoid over a similarly graded algebra. This includes the case of the Zn-graded polynomial k-algebra S and Zn-graded modules, to which we return for simplicity of exposition. The phrase "mutatis mutandis" here means only that the inverse limits defining the Z’~-graded completion Ax must be taken in the category of Zn-graded objects and homogeneousmaps of degree zero. Recall what this means for (say) an inverse system {~t} of chain complexes of Zn-graded S-modules: the ZU-graded inverse limit *~ is defined by (1)
*limit
=~ ~ ~ b~
where the ordinary inverse limits of the degree b pieces ~ on the right are in the c~tegory of chain complexesof k-vector spaces. Example 4.1. The polynomial ring S is Zn-graded complete with respect to its maximal ideal m = {x~,... ,x~), and hence with respect to every ~graded ideal I. Indeed, for any finitely generated module M, ideal I, and fixed b, the inverse system (M/It+~M)D ~ (M/ItM)D eventually stabilizes to become Mb~Mb, SO ~I(M) M.On the othe r hand , infi nitely generated modules may behave differently: 1. Completionof the injective hull S~ of k yields Sv for all nonzero ideals I t. However, see Example4.6 for ~(Sv). 2. As in the ungraded case, ~:(M) = 0 if IM = M. This can’t happen if M ¢ 0 is finitely generated by Nakayam£sLemmu,but can occur for localizations Mix-F] (the notation is explained before Lemma2.8). Since graded modules behave so much like vector sp~ces, truly bad behavior of completion requires some infinite dimensionality. With one variable x, for instance, let I = (x) k[ x] and M = ~bezk[x](-b). The completion h:(M) has the vector space h~(M)~ = ~
$ ez k[x] =
kiwi)
completion.
For comparison, the ungraded completion AI(~beZ k[[x]]) is even more infinite than £~(M). The former contains any sequence (xd~)b~Z in which, for each fixed d, there are finitely many b satisfying db ~ d, whereas the latter contains only semi-infinite sequences. The goal of this section is the computation in Theorem4.4 of ~ in terms of generalized ~ech complexes.
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4.2. Telescopes and microscopes. This section follows [GM92], but with gradings. Given a homogeneousmap ¢ : Z" -~ Y" of cochain complexes with differentials 5z and 5y, form the associated double complex Y @¢Z with (Y’, 5y) in the 0 th row; (Z’, -~z) in row -1; and vertical differential ¢ : Z" --~ Y’. The cone or cofiber cone(C) is the total complexTot (Y @¢Z), which yi$ Zi+l as its term in cohomologicaldegree i and differential ~(yi, zi+~) = (hyy i + Czi+1, -hzzi+l). Defining Z[t]" by Z[t] i = Zt+i with differential (-1)thz, there is a short exact sequence 0 -+ Y -~ cone(C) -4 Z[1] -~ whose long exact cohomology sequence has connecting homomorphism¢. Any cochain complex Y" is also a chain complex Y. with Yi = Y-i. Define the fiber of ¢ : Y. -~ Z. by fiber(C) = cone(-¢)[-1]. This is designed precisely so that if ¢ : 2" -~ 17" is a mapof cochain complexes of (say) i, k) = ]~, then k-vector spaces and ¢ : Y. -~ Z. is its transpose, so Homk(~ fiber(C) is the transpose of cone(C). Nowsuppose {¢t : 6~ -~ 6;+1}t>1 is a sequence of cochain maps, homogeneous of degree zero, and define ¢: ~]~ 6~" -+ ~ 61 by ¢(x) = x- Ct(x) x E 6~. The homotopycolimit or telescope of the sequence {¢t}t>l is cone(C), and denoted Tel (6~). Althoughthe projection Tel (6[) -> ~]~ 6[ to the target of ¢ (corresponding to Y" above) is not a morphismof cochain complexes, becomes so after moddingout by the image of ¢. The resulting cochain map Tel (6;) -~ (~ 6;)/image ¢ = li.i_~m 6; is a homologyisomorphism,identifying H~Tel(~;) as li__~m~(H~6i). Dual to the telescope is the homotopy limit or microscope Mic(6[) of sequence of homogeneousdegree zero chain maps {¢* : 6. TM-~ 6~}t>~. It is defined as fiber(C) for the map¢ : 1-1" 6! --~ I-I* .~ sending t he element (x*)t>t ~-~ ~ - Ct+~(xt+~))t>l. Th e pr oducts he re ar e in the cate gory of graded modules, and thus do not agree with the usual products. They can be seen as special cases of *li~__m;moreconcretely, l-I* 6.~ c_ I-[ 6.~ is the submodule generated by arbitrary products of elements of the same homological degree and the same graded degree. It is routine to verify the following relation betweenTel and Mi___~c(check that the maps ¢ and ¢ on the direct sum and product are dual, before actually taking the cone and fiber to get Tel and Mic). Lemma4.2. If {6;} is a sequence of cochain complexes and M is any module, then MicHoms(61,M) = Homs(Tel61, M). In particular, if ~ = ~’~’(1) : Homs(.~.., S) for a sequence of free resolutions :7:~. of some ideals It, then Homs(WelgV~’(1),M) = Mic ~ ® M) Note that Mic depends on the grading while Tel doesn’t (hence the underlining). The association of microscopes to inverse sequences constitutes an exact functor on inverse sequences, since I-I* is exact. Furthermore, setting G.~ = Horns(G;, S) for a system of free complexes G~, sending M~+ Mic (6! ® M) is an exact functor of
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The whole point of microscopes is that they compute _L./. ~Ib get the precise statement, say that two chains {It} and {I~} of ideals are cofinal if each It is contained in some I~, and each Is ~, is contained in some Is. For instance, the l~robenius powers{I[t]} of Definition 2.9 are cofinal with the ordinary powers {It}. Lemma4.3 ([GM92, Proposition 1.1]). Suppose {/t} is a chain of ideals cofinal with the powers{It}, and let .~t. be an inverse sequence of free resolutions of Slit, lifting the inverse sequence of surjections S/It+l --~ Slit. If $. is a free resolution of some module M, then the total complex of the double complex Mic ($-.t @$.) is homologyisomorphic to both Mic (hr. t ® M) and_AI($.). In particular, HiMic(.T’. t ® M) = _L[ (M). Proof. For any module J, the short exact sequence for the fiber defining the microscoperuns 0 --~ I-I*(~’[+. ® J) -~ Mi___~c(~, t ® J) -~ I-[*(~, t ® J) -~ O. The final four terms of the resulting long exact sequence are HoMi___~c (~’. t ® J) -+ []*(Slit ® J) --+ 1-I*(S/It ®J) H-1Mic (~’t ® J)-~0. The H_lMic term is zero because every map in the inverse system {Slit@J} is surjective [Wei94, Lemma3.5.3]. If J is free, all of the higher terms vanish because each ~’.~ is a resolution, and the HoMic_term is At(J) by the cofinality assumption. Setting J = ~. this shows that the homology of the double complexMic(gv. t ® ~.) in the v direction i s A~($.). Taking homologyin the £ direction yields Mic (gv. t @M) by exactness of the microscope construction. The standard spectral sequence argument now applies. [] 4.3. Derived functors of completion. The main result, Theorem 4.4, says that for many modules M, the left derived functors of Zn-graded completion can be calculated using the flat complex ~- that is approximated by t], S). (Recall that ~-!t] is the Frobenius power the telescope Tel Homs(~’[ a free resolution of S/I as in Definition 2.9). This theorem is interesting for three reasons. First, ~: is never projective, so it doesn’t seem agile enough to detect derived functors. Second, its finitely many indecomposable summands make d~= muchsmaller than either a microscope or a telescope: ~ is zn-finite whenU. is, meanining that its degree b pieces are finite k-vector spaces for all b E Zn (note that ¢ preserves Zn-finiteness). Finally, it allows the use of any free resolution to make the generalized ~ech complex. This last point is crucial for Corollary 4.8. Theorem 4.4. If .7. is a free resolution of S/I and M = jv for some J (this includes all Zn-finite modules M), then _L//(M) = Hinoms(~, More precisely,
Homs(d}, ) i s h omology i somorphic t o Mic(.7[ t] ®M)
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245
Noteworthyis the case when~’. = 7: is the Taylor resolution (Lemma3.2), because ~- is then the usual ~ech complex on squarefree generators for I, by Proposition 3.3. Proof. Let ~] = Homs(f~!*],ws), as usual. Assuming for the time being that bc. is minimal, consider the following diagram: Homs(~, jr) = (li__,m~[;](1) (Tell;i(1)
@j)v = Horns(Tell;i(1),
Both "=" symbols use Lemma2.3, while the first uses also Lemma2.10 and Definition 3.1. The downwardmap is obtained by tensoring the homology isomorphism Tell;i(1) ~ ~t~;](1) with J, and then taking Matlis duals. That the downwardmap is itself a homology isomorphism is an application of Lemma3.8. The diagram therefore represents a homology isomorphism Homs(~, jr) ~ Mic(~[~] ~ M) by Lemma4.2, ~nd since the homology ~(M) by Lemma4.3, the result is proved when ~. is minimal. When~. isn’t minimal, the top end of the downwardmapis still homology isomorphic to (~@j)v by Lemma2.10, Proposition 3.7, and Lemma3.8. Remark 4.5. The assumption M= jv for some J in Theorem 4.4 is annoying, but it’s unclear how to get rid of it. Anyway,even the restriction of Zn-finiteness isn’t so bad. It allows for a significant numberof infinitely generated modules, including ~ech hulls of finitely generated modules and their localizations. Example 4.6. Using Theorem 4.4, we compute b~(S v) = (Hide)v. Since Hid~ = (¢Ext i(S/I,
ws)) (1), we conclude that
= By definition, ~(Sv) is therefore the ~ech hull of the Alexander dual of Exti(S/I, ws) as defined in [Mil00, Rhm00], and more cleanly denoted by OEx~(S/I, ws)1. No matter the notation, the vanishing and nonvanishing of ~(Sv) follows the same pattern as Ext}(S/I, ws), since the latter can be recovered from the former. In particular, ~(Sv) is nonzero precisely for d = codim(I) if and only if S/I is Cohen-Macaulay,in which case
is the ~ech hull of the Alexander dual of the canonical module. Notice that neither ~n injective hull nor a localization can substitute for the ~ech hull here in relating ~ (Sv) to the finitely generated moduleExt ~ (S/I, ws) 4.4. Local homology. Closely related to the left derived functors of completion ~[ is ~nother collection of functors HI. In the definition to come, is the Koszul cochain complex on m~,... , m~. To be precise about homological and zn-grading, ~ is the tensor product @;=~(S ~ S(degm~)),
246
Miller
S in cohomological degree 0 and S(deg m~), which is generated in Zn-degree -deg(m~), in cohomological degree Definition 4.7 ([GM92,Definition 2.4]). Let I be generated by ml.,... The local homology of M at I is
, mr.
_H/~ (M) = H~Homs(Tel/~;, Of course, this definition works for any finitely generated graded ideal in a commutative ring with any grading when the mj are arbitrary homogeneous elements. For instance, in the ungraded setting, Greenlees and Mayproved that H./ = L. ~ under mild assumptions on I and the ambient ring. In the present Zn-graded context over S, this isomorphism of functors is an easy corollary to Theorem4.4. Corollary 4.8. If M = jv for some J, then _L~.(M) ~- H_~.(M). Proof. Start with the homology is6morphism Tel ~ -~ ~ 1(:~ = d~- to the ~ech complex, tensor with J, and take Matlis duals to get a homology isomorphism (Tel~®J) v +- (~_®j)v, by Lemma3.8. Now apply Lemma 2.3 and Theorem4.4 with .T. = 7: being the Taylor resolution. [] The convenient feature of Theorem4.4 for this corollary is that the microscopes there already compute_L.~, rather than _H.~ as in [GM92].Thus, to get _/-/.~ -~ _L.~, it suffices to comparetelescopes, whichis easier than comparing microscopes because direct limits behave better than inverse limits. This point will resurface in Section 6. 5.
~n-GRADED
GREENLEES-MAY
DUALITY
5.1. The Zn-graded derived category. The theorem of Greenlees and Mayis a remarkable adjointness between local cohomologyand the left derived functors of completion. It greatly generalizes the local duality theorem of Grothendieck, and unifies a number of other dualities after it has been appropriately sheafified; see [AJL97]for a detailed explanation of these assertions. As with the ungraded versions in [AJL97], the Zn-graded version (Theorem5.3) is most naturally stated in terms of the derived category ~_ of Zngraded S-modules. However, since hardly any of the machinery is used, all of the pertinent definitions and facts concerning ~ can be presented from scratch, so this is done in the next few lines (without actually defining _~). If ~ is a bounded complex of Zn-graded S modules--so ~ is nonzero in only finitely many (co)homological degrees and has homogeneous maps degree 0--then ~ represents an object in ~. Every homogeneous degree b homomorphism¢ of such complexes is a morphism of degree b in _~. Moreover, if ¢ is a homology isomorphism then ¢ is an isomorphism (of degree b) in ~. The usual functors Ext, H~, and _L.~ are replaced by their derived categorical versions ]l~Iom, ll~t, and L~_~ as follows.
247
GradedGreenlees-May Duality and the ~ech Hull
Suppose 6 is a fixed complex. The right derived functor ]lCJ-Iom (6,-) calculated on a complex£ by applying Horn (6,-) to an injective resolution of 6. By definition, such an injective resolution is a complexJ of injectives along with a homologyisomorphism 6 ~ J. The result of taking Hom(6, ,~) is a double complexwhosetotal complexis defined to be liCJ-Iom (6, 6). is a fact that ]ICJ-Iom (6, 6) doesn’t depend (up to isomorphismin ~) on ~. choice of injective resolution 3 Alternatively, as with Ex_~t, one also gets ]ICJ-Iom(6, 6) by taking a free resolution of ~ and applying Ho____~m (-, 6) to it. Duallyto an injective resolution, a free resolution of 6 is a homologyisomorphism ~" -~ ~ from a complex of free modules to 6. Again, Horn (9 v, 6) yields a double complex, whose total complexTot Horn (~’, 6) is isomorphic in ~ to ]lCJ-Iom (G, 6), independent the free resolution 5v -~ ~. Observethat if either 6 is already a complexof free modulesor 6 is already a complexof injectives, then Hom(G, 6) represents the right derived functor, and the [¢ may be left off. The relation between Ii~-Iom (~,6) and Ext modules is seen when 6 = G and 6 = E are both modules: the usual notions of free and injective resolutions of modules may be regarded as homology isomorphismsas above, and Ext" (G, E) is the cohomologyof ]ICJ-Iom (G, (calculated either way). The discussion above worksjust as well with IRFI in place of tlCJ-Iom (G, -) (injective resolution of "-" required) and with I inpla ce of lt~ om (-, 6) (free resolution of "-" required), except that no double complexes appear, so there’s no need to take any total complexes. Remark 5.1. Proposition
3.7 says that ~ and ~(I) are isomorphic in
Lemma4.3 says that Mic(~-.t,M) ---_~ LAI(M); in fact, M there can replaced by a bounded complex 6, since Lemma3.8 works with bounded 6 in place of M. All resolutions of bounded complexes over S can be chosen bounded, so we restrict to bounded complexes to avoid technical issues. As in the previous remark, the next lemmasays that certain objects are isomorphic in ~. Its proof uses the standard spectral sequence arguments, and is omitted. The symbols ~ and e~- denote homology isomorphisms, so as to keep track of their directions (and thus avoid getting too steeped in ~, where ~-_~ would suffice for both). Lemma 5.2.
Suppose ~-~ ~ and ~ -~ ~. Then:
1. If ~ is free then Horn(~’, 6) ~ Horn(~’, ,~). 2. If ~ is injective then Horn_(b~, ~) ¢-~ Hom(6, 3"). 5.2. The duality theorem. Theorem 5.3. If 6 and 6 are any bounded complexes of Zn-graded Smodules, then
mtom
(6),
LS_
248
Miller
Proof. Let £ -~ ~7 be an injective resolution, T.It] the Frobenius power of a Taylor resolution for I on squarefree generators, and T[[~] = Hom(T.[t],ws), as usual. Calculate: L~_I(£) ~ Mic (T. It] ® £) ~ Mi.__9.c(T.It] ®
(2)
I-Iom (Tel Hom(d
The first ~ is by Lemma4.3, where Remark 5.1 has been used to justify replacing M by a complex g; the ~-% uses the exactness of formation of microscopes when the inverse system is fiat; the second ~- uses Lemma4.2; and Eq. (2) is by Lemma5.2.2. Observe that Eq. (2) is a complexof injectives (proof: the functor of given by Horn (-, Horn (flat, injective)) =Hom(- ® flat, injective) is exact). Therefore ll¢&Iom_(~, L~_t(g)) ~_~ Horn (~, Horn (~-, ~ Horn (~ ® ~-, J)
Iom ® and the result is a consequenceof the standard isomorphism~®d~-~-~-~II~! (~) (which is proved by applying FI to an injective resolution of ~). Remark 5.4. The restrictions on Mthat appear in Theorem 4.4 and Corollary 4.8 don’t appear in Theorem5.3 because we allowed ourselves to replace Mby an injective resolution ~7: the homologyisomorphism in Eq. (21) follows without the dualization argument used for Theorem4.4. The fact that ~- isn’t free means we can’t a priori replace ,] by g as in Lemma5.2.1 (or Mas in Theorem 4.4). Zn-graded local duality is a special case of Theorem5.3. Corollary 5.5 (Local duality with monomial support [Mil00]). Suppose is a minimal free resolution of S/I, and let d’y: be the generalized Oechcomplex determined by ~.. For arbitrary Zn-graded modules M, H}(M)V HiHoms(M’ (~ )v). In particular, if S/I is Cohen-Macaulayof codimension d and CO~/I ~:8 the Alexander dual of the canonical module a~s/I (Example ~. 6), then H}(M)v --~ Ext~-i(M, Proof. Setting ~ = Sv and G = Min Theorem5.3 yields the first displ~ayed equation by Example4.6 (note that v i s Z’~-finite). B y Lemma .24 ( C~=) is a complex of injectives (decreasing in homological degree). Under the
Graded Greenlees-May Dualityandthe ~echHull Cohen-Macaulay hypothesis,
249
Example 4.6 also says the homology of (~)v
is ~W~ll, in homological degree d.
[]
The usual Zn-graded Grothendieck-Serre local duality theorem (support taken on m = (xl,... , xn)) is a special case of Corollary 5.5, by Example2.5. Another consequence is the theorem in combinatorial commutative algebra relating local cohomologyof S with support on I to local cohomologyof S/I with maximal support. The combinatorial interpretation is as an equality of Zn-graded Hilbert series whose coefficients are Betti numbers of certain simplicial complexesrelated to the Stanley-Reisner simplicial complex of I. Corollary 5.6 ([Mus00a, Ter99, Mil00]). For a squarefree monomial ideal I, v): "~ xt¢E is(SI, ws) ¯ HI i( ws) :"~ ¢(H~-i(s/I) / Proof. The second isomorphismis by usual local duality, and the first is the Matlis dual of Corollary 5.5 with M = ws = S(-1), using Example 4.6. Judging from the relation between the usual local duality theorem and Serre duality for projective schemes,it seemsclear, in view of the connections made in [Cox95, EMS00,Mus00b] between local cohomology with monomial support in polynomial rings and sheaf cohomologyon toric varieties, that graded Greenlees-May duality has something to do with Serre duality on toric varieties. It would be interesting to see exactly howthe details work out. In particular, what will be the role played by the cellular fiat complexes related to toric varieties in [Mil00, Example6.6] and IMP01,Proposition 5.4, Examples 6.3 and 8.14]? 6. GRADEDNOETHERIANRINGS Let I = (~,... , (~r) be a finitely generated graded ideal in a commutative ring A graded by a commutative monoid. The ZU-graded methods of this paper suggest a transparent proof of GMduality in this general graded case, at least when A is noetherian. This is not to say that the graded case doesn’t follow with care from knownproofs for proregular sequences in arbitrary commutative rings [GM92,AJL97]. Rather, the innovation here is the simplicity of the proof; the consideration of a grading is done "for the record", because it requires no extra effort at this point. The interesting part about the proofs in Sections 4 and 5 is that they work with the direct limits associated to telescopes, whereas previous, methodsin [GM92,AJL97] worked with the inverse limits associated to microscopes. The direct limit technique is applied below; it has the advantage that the colimits are exact, and the adjunction between ll¢~F~ and LA1 gets reduced directly to the adjunction between Homand ®. In general, the adjointness of Hi and _H.~ is essentially by definition, while the identification of the latter with _L.~ requires hypotheses. The proof here relies on two facts about noetherian graded rings. The first, whose standard
250
Miller
proof is omitted, is that the (~ech complexC’(O~l,... , O~r) is isomorphic in the derived category to ]t~t A (apply -® C" (al,... , at) to a graded injective resolution). The second is contained in the next subsection. 6.1. General analogue of ~ech hull colimits. The main feature of the ~ech hull that madeit useful earlier was its expression as a direct limit, resulting in the homology isomorphisms of Proposition 3.7. Although the combinatorial construction is lost with general gradings, the direct limit still makes sense, and the homology isomorphism survives, thanks to the next proposition. Let r
K:! = K:.(a~,...
,ate)=
at
(~(A(-degaS)A_~
j=l
be the Koszul chain complex whose tensor factors are in homological degrees 1 and 0. Proposition 6.1. Suppose that A is arbitrary, but .for every t >_ 1 the ideal I [t] = (ate,... ,atr) has a resolution by finite-rank free A-modules.Thenthere is a morphism{pt} : {1~!} __~ {~..t} of inverse systems of complexes~indexed by t >_ 1) in which 1..T t. is a resolution of A/I It] by finite rank free modulesfor all t >_1; 2. the mapset : .T.t+l _~ 9r.t lift the surjections A/I[t+~] ~ A/I[t]; 3. the maps on tensor factors given by the identity in homological degree 0 and multiplication by aj in homological degree 1 determine the maps nt :/~!+I _~ l~!; and ~. each map pt : ~! __~ ITt. induces an isomorphism on homology in degree O. Under these conditions, the transpose direct system {Pt} : {-Tt’} -+ {K:;} obtained by applying HOmA(, A) to {pt} induces a homology isomorphism
lim Proof. All complexesof free modules appearing in this proof will be assumed to have finite rank in each homological degree, by the hypothesis on the ideals I[t]. Conditions1 and 2 can be forced uponany list {fi-.t } of resolutions for the quotients A/IIt]. Moreover,the acyclicity of fi-.t and the freeness of K:! imply that mapst5 t : K:! --~ fi-.t as in condition 4 exist and are unique up to homotopy[Wei94, Porism 2.2.7]. Although {t~t} may not a priori constitute a morphism of inverse systems, the uniqueness up to homotopy can be used to remedythis, by constructing .T. t, et, and pt inductively, starting with a resolution ~.~ of A/I and a chain mappl as in condition 4. Having defined ~-.t and pt, choose a resolution fi-.t+~ of A/I [~+~] and a map~5t+~ : K:!+l --~ 3~.t+l as in condition 4, and let ~t : fi-.t+l __~ ~=.~ be any lift of the surjection A/I It+t] --~ A/I It]. Then ~t+~t+~ is homotopic
Graded Greenlees-May Dualityandthe (~echHull
251
Letting a lowered "t" index denote transpose, so (for instance) Pt : .T’[ ~ /~ is obtained by applying HomA(-,A ) to pt, it follows that fSt+l(~t+l is homotopicto atPt :.T t" --~/C~+1. Any choice of homotopy induces a map Pt+l : cyl(~t) ~ /~+1 from the mapping cylinder of (~t to K:~+1. (See [Wei94, Section 1.5] for definitions and generalities concerning mappingcylinders; only the properties of cyl(~t) required here are presented below.) The mapPt is produced essentially by [Wei94,Exercise 1.5.3], and satisfies: (i) There is an inclusion et : br[ ~ cyl(~t) of complexes, Pt+l et = ~tPt. (ii) 9~’+~injects into cyl(~t), and the composite"~t’+l "-)* cyl(~t) ~-~ ]~+1 is/St+:. (iii) The inclusion ~t’+l ~-~ cyl(~) is a homotopyequivalence, and the composite 5r~" -~ cyl(~~) -> ~’+l is just ~. to ptnt.
The transpose p~+~ : /C! +1 --~ HOmA(cyl(~),A ) =: ~-.t+l satisfies the required conditions, with et : 9v.TM-~ hr. ~ being the transpose of the mapet in (i) and(iii). The induced map on direct limits is a homology isomorphism because Iim, Ext~A( A/ I[~], A) =li___,m Hi( ~[) =Hili__,m (.T~) -~ H~li__,m (1Ci) = is the canonical isomorphism on the local cohomology module H)(A); note that the graded moduleExt ~4 (A/I[t], A) is naturally isomorphic to the usual ungraded module Ext~(A/I[~], A) because A/I It] is finitely presented. [] Remark 6.2. Like the generalized ~ech complexes of Definition 3.1, the object li__7m.T ~" of Proposition 6.1 is a complexof flat modules.This is because the colimits are taken over directed systems of complexes of free modules, so complexes of flat modules result by a theorem of Govorov and Lazard [Eis95, TheoremA6.6]. It is unclear whether the methods of Proposition 6.1 can be madeto apply whenthe only assumption is that (al,... , at) C A is a proregular sequence [GM92,AJL97]. In general, under what conditions will (a~,... ,ate) have finite Betti numbers as an A-module for infinitely many t _> 1? Perhaps characteristic p > 0 criteria are possible. 6.2. Graded noetherian Greenlees-May duality. In the following theorem, the derived category statements concern complexes ~ and Y that are bounded, for simplicity. The proof is nearly the same as that of Theorem5.3, but is repeated in full to be self-contained, and for notation’s sake. Theorem 6.3 ([GM92, AJL97]). Let A be a noetherian ring graded by a monoid and I C A a graded ideal. Then H_I.(M) ~- _LI.(M) for any graded A-module M. If ~ and ~ are bounded complexes of graded A-modules, then
e) 3- om(6,
252
Miller
Proof. Let the ideals i[t], the free resolutions ~’.~, their transposes ft’, the Koszul chain complexes/C~,and their transposes/C~ be as in Proposition 6.1. If ~ -+ 3" is an injective resolution, then (3) (4)
~ Mie (~-.t
® 3")
(5) (6)
~ Hom (~U[,
fl)
(7) (8) (9)
~ Horn (Tel El, ~).
Eq. (3) follows by replacing Mand its free resolution ~. in Lemma 4.3 with complex~ and a free resolution of it (the proof is the same). Eq. (4) uses exactness of formation of microscopeswhenthe inverse system is fiat. Eq. (5) is by Lemma 4.2. Eq. (6) is because fl is injective and Tell:- t" ~ li__,mgVt., and similarly for Eq. (8). Proposition 6.1 implies Eq. (7) (this is the key step!). Finally, Eq. (9) uses the fact that Tel/C~is free. When$ = Mis a module, the conclusion of Eq. (9) implies that _L./ (M) _H./(M), by definition. To get the derived categorical statement, observe first that Eq. (7) is a complex of injectives because the functor given Horn (-, Hom(fiat, injective)) = Horn (- ® fiat, injective) is exact. Therefore I~Hom(G,LAZ($))
~- Hom(~,Uom(li__~mlC~,3")) ~-
Hom(~®~,S)
®
s)
the second isomorphismbeing at the level of complexes while the rest are in the derived category. The last isomorphism is the standard isomorphism for noetherian graded rings, mentioned at the beginning of Section 6. [] The above proof essentially requires ~.t : Hom(Horn (5~.t, A), A) to be its owndouble dual (e.g., in Eq. (5)), thus using again the finite rank conditions. Remark 6.4. Looking back at Theorem 4.4, under what circumstances can the injective resolution 3" in Eq. (5) be replaced by a module M? I.e., when will _L//(M) be isomorphic to HiHomA(li__,mt.~t.,M ) (Remark 6.2) HOmA(li_~mtl~,M), rather than having to replace the colimits by projective approximations in the form of telescopes? REFERENCES [AJL97] Leovigi|doA|onsoTarrfo, AnaJeremias Ldpez,and JosephLipman,Local homologyandcohomology on schemes,Ann.Sci. ]~cole Norm.Sup. (4) 30 (1997), no. 1, 1-39.
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David Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50. [Eis95] David Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, NewYork, 1995. [EMS00] David Eisenbud, Mircea Musta~, and Mike Stillman, Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), no. 4-5, 583-600, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). [GM92] John P. C. Greenlees and J. Peter May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), no. 2, 438-453. [Gr~84]Hans-Gert Grebe, The canonical module of a Stanley-Reisner ring, J. Algebra 86 (1984), 272-281. [GW781Shiro Goto and Keiichi Watanabe, On graded rings, II (Z~-graded rings), Tokyo J. Math. 1 (1978), no. 2, 237-261. [HM00] David Helm and Ezra Miller, Bass numbers of semigroup-graded local cohomology, Preprint (math.AG/0010003), 2000. [Hoc77] Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) (B. R. McDonald and R. Morris, eds.), Lect. Notes in Pure and Appl. Math., no. 26, Dekker, NewYork, 1977, pp. 171-223. Lecture Notes in Pure and Appl. Math., Vol. 26. [Lyu84] Gennady Lyubeznik, On the local cohomology modules H~(R) for ideals a generated by monomials in an R-sequence, Complete intersections (Acireale, 1983), Springer, Berlin, 1984, pp. 214-220. [Mi1981 Ezra Miller, Alexander duality for monomial ideals and their resolutions, Preprint (rnath.AG/9812095), 1998. [Mil00] Ezra Miller, The Alexander duality func~ors and local duality with monomial support, J. Algebra 231 (2000), 180-234. IMP01] Ezra Miller and David Perkinson, Eight lectures on monomial ideals, COCOAVI: Proceedings of the International School (Anthony V. Geramita, ed.), Queens Papers in Pure and Applied Mathematics, no. 120, 2001, pp. 3-105. [Mus00a] Mircea Musta~, Local cohomology at monomial ideals, J. Symbolic Comput. 29 (2000), no. 4-5, 709-720. Duke Math. J. (2000), [Mus00b] Mircea Musta~, Vanishing theorems on toric varieties, to appear. Gerald Allen Reisner, Cohen-Macaulay quotients of polynomial rings, Advances [aei76] in Math. 21 (1976), no. 1, 30-49. Tim RSmer, Generalized Alexander duality and applications, Osaka J. Math. [R~m00] (20{30), to appear. Richard P. Stanley, Combinatorics and commutative algebra, second ed., [Sta96] Progress in Mathematics, vol. 41, Birkh~iuser Boston Inc., Boston, MA,1996. [Tay60] Diana Taylor, Ideals generated by monomials in an R-sequence, Ph.D. thesis, University of Chicago, 1960. Naoki Terai, Alexander duality ~heorem and Stanley-Reisner rings, S~rikai[Ter99] sekikenkyfisho KSkyfiroku (1999), no. 1078, 174-184, Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998). [Wei94] Charles A. Weibel, An introduction to homological algebra, vol. 38, Cambridge University Press, Cambridge, 1994. [YanO0]Kohji Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Phil. Soc. (2000), to appear. [Cox95]
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
E-mail address: ezra~math.mit,
edu
RESIDUE
METHODS
l-Chiau Huang Institute R.O.C.
IN
COMBINATORIAL
of Mathematics, Academia Sinica,
ANALYSIS Nankang, Taipei,
Taiwan,
ABSTRACT. This article studies the use of residues defined on local cohomology classes as a computing tool for combinatorial analysis. Such cohomologyresidues are used globally or locally to prove binomial identities. Invariance of residues under variable change gives rise to more identities and explains phenomena such as Lagrange inversion formula and Riordan array. It also leads to the discovery of Schauder bases, which are used to characterize inverse relations with orthogonal property. Furthermore residues are analyzed to compute the diagonal of a rational power series. Many classical numbers can be represented by residues. Newinsights of certain numbers are discovered via computations on these representations. Finally, we extend ground field so that polynomial sequences can be treated as scalars sequences. The methods of cohomologyresidues for scalars thus carries over.
This article is about applications of concrete aspects of Grothendieck duality to combinatorial analysis, which includes binomial identities, diagonals of rational power series, classical numbers, and polynomial sequences.
255
256
I-][uang
In combinatorial analysis, it is quite often that a unified methodemerges from different problems sharing the same structure. Such a method, which reflects the underlying structure, sometimes has certain analytic meaning. When it has not, the method is regarded as "formal" or "symbolic". A typical example is the method of generating functions. For convergent generating functions, there is the theory of complexvariables, which treats some formal operations as analytic processes. For divergent generating functions, the same formal operations can still be performed to yield combinatorial information. Howeveranalytic meaning is absent from the formal operations. In this article the key point of view is that some operations, which are considered formal in generating functions, have in fact a precise meaningin the language of algebraic geometry. With this viewpoint, our methods of cohomologyresidues provide a natural as well as rigorous frameworkfor the validity of formal computations of generating functions without analytic restrictions. Comparedwith classical residues defined by the analytic method, cohomology residues work in more general context: They are defined for varieties over an arbitrary field - algebraically closed or not with arbitrary characteristic. Even more, cohomology residues form building blocks for Grothendieck duality in relative situation, that is, a scheme over another scheme. The importance of such generality lies in Grothendieck’s philosophy on algebraic geometry, which emphasizes morphisms more than schemes. To appreciate our methods, certain prerequisites of algebraic geometry and commutative algebra are necessary. For algebraic geometry, acquaintance with the Hilbert’s Nullstellensatz and the projective line is enough. For commutative algebra, besides the basic language, local cohomologyand universal finite differential modules are also needed. These algebraic tools are used to set up our framework. After defining some notions and proving their properties, one is then free from technical burden. That means, accepting rules of operations, one is able to prove knownfacts and to discover new results with simplicity as the method of generating functions and with convenience as analytic theory. This work is partially supported by National Science Council, Republic of China. Thanks go to George E. Andrews, Luchezar L. Avramov, and Ko-Wei Lih for comments. Special thanks go to Su-Yun Huang for reading the manuscript and to Peter Jau-Shyong Shiue for his encouragement and suggestions. CONTENTS
1. Introduction 2. Algebraic Background 2.1. Power Series Rings 2.2. Residues 2.3. Residue Theorem 3. Binomial Identities
Residue Methodsin Combinatorial Analysis
257
3.1. Applications of Residues 3.2. Applications of Residue Theorem 4. Variable Change 4.1. Lagrange Inversion Formulae 4.2. Inverse Relations I 4.3. Schauder Bases 4.4. Inverse Relations II 4.5. Riordan Arrays 5. Diagonals of Rational Power Series 5.1. Generalized Power Series 5.2. Algebralcity of Diagonals 6. Classical Numbers 6.1. Bernoulli Numbers 6.2. Fibonacci and Lucas Numbers 6.3. Euler and Genocchi Numbers 6.4. Stirling Numbers 6.5. Catalan Numbers 7. Polynomial Sequences 7.1. Closed Formulae 7.2. Laguerre polynomials 7.3. ttermite polynomials 7.4. Tschebyscheff polynomials 7.5. Lerch Polynomials 7.6. Identities 8. Future Developments 8.1. Congruences Modulo Prime Powers 8.2. Residue Theorem for Plane Curves References
I.
INTRODUCTION
Combinatorial identities have a long history and have scattered in diverse areas of mathematics. While proving these identities case by case with various techniques mayshed light on their combinatorial nature, a unified approach emphasizing their underlying structure is also strongly desirable. Amongunified approaches, the method of generating functions is perhaps one of the earliest successes. Although divergence of series sometimescauses discomfort, the elegance and wide applicability make the method of generating functions popular amonggenerations of mathematicians. See [55, subsection 1.2.9] and [97]. Anothersuccess initiated by Andrews[3], as a result of ideas from H. Bateman(see [30, Introduction]), is the theory of hypergeometric series, which treats most binomial identities as special cases. Along this line, Gosper [29]
258
Huang
gave an algorithm to evaluate indefinite hypergeometric sums. Will and Zeilberger [98] used the methodof WZpairs to evaluate definite hyper.geometric (and other) sums. A third unified approach is Egorychev’s method of integral representation of sums [16]. In his approach, many combinatorial sums are represented as contour integrals. Residues thus arise and provide an analytic framework, where some operations becomeavailable. By taking residues before and after these operations, various combinatorial identities are then obtained. The notion of analytic residues can be generalized to a much more general context. Grothendieck discovered that certain complexes with residual structures play an important role for duality theory in algebraic geometry. His theory of duality announced in [33] was worked out at some length by Hartshorne [37] using the language of derived categories. This beautiful but abstract approach to Grothendieck duality theory has hindered it from being well knownto non-specialists on algebraic geometry. As a result, many applications of analytic residues have not yet been put in the context of Grothendieck duality theory. One such gap was filled up by [45], where cohomologyresidues gave rise to an algebraic framework with operations similar to analytic residues. Many classical results in combinatorial analysis, which can be proved using analytic residues, can also be naturally put in the context of Grothendieck duality. Such results include MacMahon’smaster theorem [45, Example 1] and numerous legend binomial identities. See [45, Section 2] for a few examples. Look at the following elementary identity
(1) Let [Zn] be the operator defined by [Zn]f = ’~, the coefficient of f at Z where f is a Laurent series in Z and n E Z. For a fixed n E N, the generating function for the sequence { (~) }k>0 is (1 + n. Hence
n+ = l[zk+l](1
--~
= [zk+l]{z(1
Z)
~-
Z) n}
= + z) n +
=(:)+
n ~ n[zk+l](1
~-
+n z)
Z)
259
ResidueMethodsin CombinatorialAnalysis In this example, the method of generating alytic language: +
-
2~rl ~
i
Z~+2 i
2~i
Z(lt-
function translates
dZ
2~
1
Z>n.dZ_l_
~+~ Z
~ 1
_ 1 f(l+Z)"dZ + ~+~ -
well into an-
Z
~
I
(1-t-Z>ndZ
~+~ Z
~+~ Z f(l+Z)"dZ
n
=
In this analytic framework, the operator (1/2~ri) f has the same effect as the operator [Z-1]. However it operates on meromorphic differentials instead of power series. What we propose is an algebraic framework in which the meromorphic differential (1 + Z)ndZ/Z ~+~ is replaced by an algebraic object ,
called a generalized fraction. This object is so named as (1 + Z)ndZ acts like an numerator, Zk+l acts like a denominator, and generalized fractions enjoy most properties that we expect from usual fractions. In this algebraic framework, the operator (l/2~rl) ~ is replaced by an algebraic residue map, denoted by res, which satisfies res
Z~+I
= [Z~]f
for any power series f in Z. The proof for (1) by the method of generating functions translates also well into algebraic language:
(7++~) = res[ (I+Z)’~+’dZZ~+~ ] = res = res
:
Zk+2 Zk+~
+ res
Zk+~
+res
n
In the above computation, although the notions have not yet been defined and the rules for operations have not yet been stated, the reader should already see the similarity between the algebraic and analytic methods. The next example illustrates a difference between the algebraic and the analytic methods, and it also illustrates an advantage of our residue maps over the formal aspects of generating functions. Let t = u/d?, where ¢ is an
260
Huang
invertible element in ~[[u]]. For f e ~[[u]], computations involving generalized fractions [~dt] [~du] res [ tn j ---- res
__
we have the following "formal"
[tnCnddJ~udu res [ tnun
j
]
[¢ndd~du ] = res un
for all n _> 1, that is,
n[tn]f= This proves the Lagrange inversion ingfunctionology [97].
formula as stated
in the book generat-
Lagrange Inversion Formula. [97, Theorem 5.1.1] Let f(u) and ¢(u) be power series in u, with dp(O) = 1. Then there is a unique formal power series u : u(t) that satisfies
u : re(u). Further, the value f(u(t)) of f at that root u = u(t), power series in t about t : O, satisfies [tn](f(u(t))}
when expanded
= ~[un-1]{ft(u)¢(U)n}.
The proof of the Lagrange inversion formula given in [97] combines analytic method and the method of generating functions: First a formal computation, similar to (2), is made and then the validity range of the formal computation is checked with the help of analytic theory. This validity consideration is not necessary in our approach, since cohomology residues provide a framework for formal computations similar to those of generating functions without analytic restrictions. In this sense, our algebraic approach enhances the method of generating functions. The original statement of the Lagrange inversion formula in the first edition of generatingfunctionology [96] has an analytic assumption on f and
¢: Lagrange Inversion Formula. [96, Theorem 5.1.1] Let f(u) and ¢(u) functions of u that are analytic in some neighborhood of the origin (in the u-plane) with ~(0) = 1. Then there is a neighborhood of the origin (in t-plane) in which equation
u = re(u), regarded as an equation in the unknown u, has exactly one root. Further, the value f(u(t)) of f at that root u : u(t), when expanded in a power series t about t : O, satisfies [tn]{f(U(t))}
: ~[un-1]{f’(u)¢(u)n}.
To extend the formula to arbitrary power series f and ¢ as in the second edition of generatingfunctionology, one observes that f and ¢ can be replaced by polynomials so that analytic methods can be applied. In our algebraic
261
Residue Methods in Combinatorial Analysis
approach, such reduction is not necessary. Methods for convergent power series once suitably interpreted apply to divergent power series! The purpose of Lagrange inversion formula is to find relationship between representations of a power series in terms of two different sets of variables. Subtler interplay of more than one power series, represented by different sets of variables, provide further combinatorial information. This idea gives an interpretation of the theory of Riordan arrays. Another direction to generalize Lagrange inversion formula is to allow more flexible representations of power series. Wewill introduce $chauder bases, which generalize variables and give a natural interpretation for inverse relations. A natural follow-up question in our algebraic approach is how other concrete aspects of Grothendieck duality theory determines combinatorial identities. In this article, we find applications of the residue theorem for the projective line to combinatorial analysis. Roughlyspeaking, the residue theorem is a global theorem in the sense that it gives a relation of residues of a given meromorphicdifferential at all points 1 on the projective line. Computingresidues of a given meromorphicdifferential, various combinatorial identities are obtained. The approach to combinatorial analysis by the residue theorem for the projective line is a generalization of the usage of partial fractions. Moreprecisely, this methodreduces to the usage of partial fractions if the meromorphicdifferential has vanishing residue at the infinity. The residue theorem for the projective line explains some identities. The meaningof the identity (3)
’~ ~ (2:-
2kk)(2:):4
k=O
has attracted attentions. Manycombinatorial explanations of the identity have been found, see [93]. In our approach, the sum on the left hand side of the identity is the residue of the meromorphicdifferential w :=
(I +
2~+1 X)
O-x)x+ldX
at the origin (see the proof of Identity 15 for details). One mayverify that the residue of w at the infinity also equals the left hand side of identity (3). At X : 1, the residue of w equals -22~+1. These are all the points at which the residue of w does not vanish. The residue theorem gives the following relation of these residues.
lln this article, stated.
points are assumed closed in the language of schemes, except otherwise
262
Huang
Hence identity (3) conveys certain combinatorial information revealed by the meromorphicdifferential w through the residue theorem for the projective line. Diagonals of rational power series is another instance that more than one residue of a meromorphicdifferential have to be considered. In analytic case, the diagonal of a rational power series is a sumof analytic residues over singularities inside certain circle, see [92, p. 182]. Using cohomologyresidues, we will show that the transcendental process for describing diagonals is in fact algebraic. An advantage of our approach is that it works for fields with arbitrary characteristic. The reader will see that our methods for combinatorial analysis sometimes require change of variables. Such procedures do not seem possible for computerized proofs in spite of some recent progresses in computerized proofs for hypergeometric identities, see [70]. Howeverwe are able to handle classical numbers, such as Bernoulli and Stirling numbers, which can not be treated by methods of hypergeometric functions, such as Gopser’s algorithm or WZpairs. Another category of problems to which cohomology residues can apply is special functions. Our treatment of special functions can be seen as a response to G.-C. Rota~s statement [80]: In the field of special .[unctions, as elsewhere in combinatorics, we are confronted with too manynontrivial identities and too few concepts with which to understand them. To illustrate our ideas, many knownresults are included in the article with new proofs by our methods. There are also results which the author is not able to locate in the literature. These results might be new or not. To the author’s opinion, what is important is the methods of arriving at them. With these methods, we wish that the reader is able to discover new results or find interpretations of knownphenomena. 2. ALGEBRAIC BACKGROUND Likemanyalgebraic approaches to combinatorial analysis, our residue methods are alsobasedon formalpowerseriesrings.However ourdefinitionof formalpowerseries ringsdiffers fromthetraditional onein view of variables. Thisis an analogue to thesituation of vector spaces: An ndimensional vectorspaceovera fields is an abelian groupwithan extra algebraic structure. Itselements canbe described by alUl +’..+ anUn
(ai E ~),
once a basis ul,’",Un is chosen. One knows that there is no canonical choice of basis. In our view, a formal power series ring is a commutative ring with some extra algebraic properties free from the notion of variables. Its elements can be described by formal power series y~ ai,...i,X~’ " " Xin" (ai,...i,~
G to),
ResidueMethods in Combinatorial Analysis
263
once variables X1,... , Xn (that is, regular system of parameters, see Section 2.1 for the definition) are chosen. There is neither canonical choice of regular system of parameters! The concept that there is no canonical choice of variables (that is, regular system of parameters) plays an important role the development of our methods. In [45], local cohomologywas introduced to combinatorial analysis. In the context of local cohomology,residue maps provide a device turning algebraic information into combinatorial information. This is made possible by the notion of generalized fractions, which represent elements in some top local cohomology modules. From the viewpoint of algebraic geometry, the tool used in [45] is an object defined on the local ring of a single point on a variety. A global theorem, namely the residue theorem, which relates local algebraic information for all points of the variety, shall provide further combinatorial information. To benefit from this global consideration, we need to render concrete realization of the residue theorem, which is as subtle as other parts of Grothendieck duality. Fortunately, for varieties sitting in low dimensional spaces, this can be done in an elementary way, see [46]. In Section 2.3, we will give a self-contained account for the residue theorem of the projective line. 2.1. Power Series Rings. Given a field ~ and variables X1,... denote by ~[[X1,..’, Xn]] the ring consisting of all power series 2 "’’X,~
~ aili2...~rtX~lX~
(ail./2...i
,X~, we
n ~ ~)
il ,i2 ,’" ,in~_O
in X1,’" , X~(with usual addition and multiplication). One sees easily that a power series of the above form is invertible if and only if a0...0 does not vanish. Non-invertible elements in ~[[X1,... ,X~]] form the unique maximal ideal, denoted by va, of ~[[X~,-.-,Xn]]. If n elements fa,-’", fn in a[[X1,’", Xn]] generate m, that is, every element f E m can be written as f
-~
glfl
-~-’’"
-~
gnfn
(gi
e t~[[Xl,""",
Xn]]),
we call these n elements a regular system of parameters of a[[X1,.", Xn]]. The variables X1,." ,Xn form a regular system of parameters. From a given regular system of parameters, we can form other regular systems of parameters, for instance, X1+ X2, X2, .-- , Xn and X~+ X12,X2, .. ¯ , Xn. If fa, ¯ ¯ ¯ , fn is a regular systemof parameters,every powerseries in X~, ¯ ¯ ¯ , X~ can be represented uniquely as a power series in f~,... , f~. Moreprecisely, this means that there is an isomorphism ~[[Y~,""", Yn]] -~ ~[[X,,... , Xn]] of rings mappingl~ to fi (1 < i < n) with ~ fixed. Thus there are manyways to represent a power series ring depending on choices of regular systems of parameters. One mayeven define formal power series rings without referring to variables (regular system of parameters).
264
Huang
Proposition 2.1. Let n be a field. A commutative ring R is isomorphic to a ring of the type n[[X1, ¯ ¯ ¯, Xn]]/f and only if the .following three conditions are satisfied. ¯ The Krull dimension of R is n. ¯ R is a complete regular local ring. ¯ n is a coefficient field of R (that is, R contains n and the induced map from ~ to the residue field of R is an isomorphism). This proposition should be considered as a part of the Cohen’s structure theorem on complete regular local rings. The reader is referred to [68] for the definitions used in the proposition and more details. Wecall a ring R an n-dimensional formal power series ring over ~ if it is an n-dimensional complete regular local ring with ~ as a coefficient field. Fromnowon, the notation n[[X1,... ,Xn]] means an n-dimensional formal power series ring over n together with a specified regular system of parameters XI,... , Xn. The notation ~[[X~,..., Xn]]= ~[[Y1, "’", Yn]] means an n-dimensional formal power series ring over n together with two specified regular systems of parameters X1,..., X~ and Y1," "", Yn. 2, Another algebraic object we use is universal finite differential modules which in formal power series rings play the role as modules of K~ihler differentials in polynomial rings. Werecall its definition. Let R be a n-algebra and M be an R-module. A ~-derivation of R into Mis a map d : R --~ M satisfying the rules d(f W g) = d(f) + d(fg) : fd(g) + gd(f) d(a) = for all f, g E R and a E n. The universal finite differential module is a finite R-module ~R/~, together with a n-derivation d: R --~ ~R/~, which satisfies the following universal property: for any finite R-moduleM, and any ~-derivation ~: R --~ M, there exists a unique R-linear homomorphism f: f~/~/~ --~ Msuch that ~ = f o d. In general, the universal finite differential module may not exist. In our context, where R is an n-dimensional formal power series ring over ~, the universal finite differential module not only exists but also has a very simple structure. If X1,..’, Xn is a regular system of parameters of R, then elements in ~R/~ can be represented uniquely as fldX1 +... q- fndXn for some f~,... , fn ~ R. Thus elements in the n-th exterior power of [~/~ can be represented uniquely as fdX1 ...dXn
:= fdX1 A...A dX,~
for some f ~ R. 2Theuniversalfinite differentialmodule of a powerseries overa field is also the universal separateddifferential module usedin [45].
Residue Methods in Combinatorial Analysis
265
If R is a formal powerseries ring over a field ~ of characteristic 0, wemay define exponential, logarithmic and trigonometric functions as in [31]. For f in the maximal ideal m of R, we define e f := 1-~f+...+~-.~ log(1
+...,
f2
+ f):=
n
as elements in R. One can check that el°g(l+]) = 1 + f and log(e/) = f for fern. ForgERandf6m, wedefine (1 d- f)g := gl°~(l+y). e :For c~ ~ ~¢ and n > 0, wedefine
Then
For f e m, we define sinf ,
f2n+l := f--~-. f3 +--.+(-1)’~(2n+l)!+...
cos/ ,
:= 1-~.~+...+(-1)n(-~n)~.+...
tan f .SeE f ::
sin f cosf ’ 1 COS/ ’
as elements in R. For f,g,h e R satisfying ffl = h and f ~ 0, we write g = h/f even though f maybe not invertible in the formal power series ring. For instance, ~ n ex - 1 X X X X -1+~.~+-~.~ +...+ (n+l)-----~+.... For f E ra, we define fcotf
:=
fcscf
.-
f COSf
sin f ’
f
sinf
Note that cot f and csc f can not be defined in formal power series rings.
266
Huang
In ~[[Y]], let X : sin Y. Then ~[[Y]] = ~[[X]] and Y can be written as a power series in X, which we denote by arcsinX. If we let X = tanY, then Y can also be written as a power series in X, which we denote by arctan X. One can show that arcsinX = X + ~-~1"-23-: "’’(2n-1) n:l oo
arctanX
= ~-~(-1)
X2"~+’
2n+ 1’
n X2n+ l
~nTi"
n--~0
See Sections 6.1 and 6.3 for expansions of other trigonometric functions¯ 2.2¯ Residues. Let R be a noetherian commutative ring and I be an ideal of R generated by n elements up to radical, (that is, every element of has some power contained in the ideal generated by these n elements). The functor FI on R-modules given by ri(M) = {~ M[IS~ = 0 fo r so me s > 0} is left exact. The i-th right-derived functor of the functor FI is denoted by H}(.). Wecall H}(M)the i-th local cohomology module of Msupported I. If R is an n-dimensional formal power series ring over a field ~ and I is the maximal ideal of R, elements fl,"" , fn generating I up to radical are called a system of parameters of R. In such context (or more generally if R is a Cohen-Macaulaylocal ring), the notion of systems of parameters is equivalent to that of regular sequences. Recall that a sequence al,... ,am of elements of a commutative ring R is a regular sequence if the following conditions hold: ¯ al is R-regular (that is, b E R and atb = 0 implies b = 0). ¯ a2 is (R/alR)-regular (that is, b E R and a2b ~ a~R implies b ~ aiR). ¯ am is (R/(a~,..., a,~_~)R)-regular. ¯ (al,...,am)R ~£ R. ~m for any If al,...,a,~ is a regular sequence of R, then so is a~, ¯¯ .,a m positive integers v~,..., Vm-If R is local, for instance R is a formal power series ring over a field, any permutation of a regular sequence is again a regular sequence. The reader is referred to [68, §16] for details. System of parameters has an analytic meaning. If a is the field of complex numbersand f~,... , fn are holomorphic in a sufficiently small neighborhood of the origin, then one can showthat fl," "", fn form a system of parameters if and only if they have the origin as an isolated commonzero [32, p. 660]. Let R be a noetherian commutative ring and I be an ideal of R generated by fl,-" , f~ up to radical. For an R-moduleM, we define the localization
ResidueMethods in Combinatorial Analysis of Mat the product of fl,"", Mf~...fn
:=
267
fn to be the set {(fl...
f,~)~ I w E M, s >_
modulo the relations Wl
O32
where for some s _> 0. One can show that H~(M)= 0 for i > n. For i = n, there is a canonical map (4)
Mfl...fn
-+ H~(M)
with kernel generated by elements of the form f~w/(f,.., Mand s _> 0. Wedefine a generalized fraction
fn) ~ for some w E
i-i~w/(f~.., fn) i, where i > ij for all j, under to be the image of f~-h ... f~ the map (4). The map(4) is surjective, hence every element of H~(M) generalized fraction. Generalized fractions enjoy the following properties: Linearity Law. Forwl,w2 ~ M, i~,... ,is > O, and a~,a~ ~ R,
Transformation Law. [44, 2.3.ii] or [62, 7.2.b] f~,"" , f~n generating I up to radical,
(5)
[ w ] [ det(rij)w = ,
For w ~ M and elements
,
n
The equation (5) is also called the transition formula in someliterature. Vanishing Law. [44, 2.3.i] or [62, 7.2.a] For w ~ M,
e (f~,(s+~) . . fi,,(s+~)) ifa~do~lyif (f~’...fi~:,)s~ M forsomes > O. In particular, for any 1 _< j _< n,
For this reason, we allow some power ij of fj to be zero by convention that
[
]
268
Hiuang
if some ij : 0. The vanishing law becomes simpler, if R is an n-dimensional formal power series ring over a field n and M = An~R/~. To see this, we need the following lemma. Lemma2.2. Let fl,’" , fn be a regular element h E R satisfies hf~..,
sequence of a local
f~ ~ (f;+l,...
for some s > O, then h ~ (fl,"",
ring R. If an
, f~+l)R
f~)R.
Proof. We prove the lemma by induction on n. For n : 1, the lemma holds, as fl is a non-zero divisor. For n > 1, assume that the lemma holds for all regular sequences gl,’",ge with £ < n. Since f~+l, f2,"", fn is also a regular sequence, modulo f~+l and by the inductive hypothesis, we get hf~ ~ (f~+l, f2"", Since f2,"", fn, f~+l is still a regular by the inductive hypothesis, we get h e (f~,...,
The above lemma implies ishing law.
immediately
f,~)R. sequence,
modulo (f2"",
fn)R and
f~)R.
the following
version
of the van-
Vanishing Law for Power Series Rings. Let R = ~[[X1,.-. ,X~]] and ra be the maximal ideal of R. Given h ~ R and a system of parameters fl,...,f~
of R, fl," ,f~ [ hdXl...dX~
=0 ]
(as elements in I-I~(A’~0R/~)), if and only if h ~ (fl, Still assume that R is an field ~ with the maximal ideal ements in H~n(AnfiR/~) using can be written uniquely as a
--
ii 0 ,...
,in>
" " , f,~)R
n-dimensional formal power series ring over a m. In such case, it is convenient to describe elgeneralized fractions: Elements in H~(An~.R/~) finite sum
...
[ ai,...~.dXx...dX~ ’
]
(ah...i.
I~),
where X~,..., Xn is a regular system of parameters. In other words, the local cohomology module H~n(An~R/~) as a ~-vector space has a basis consisting of elements of the form
where il, ¯ ¯ ¯ , in > 0. Thus we can give the following definition.
ResidueMethods in Combinatorial Analysis Definition
269
of the Residue Map. The residue
map
resxl ,... ,x, : Hm(A is defined to be the a-linear mapsatisfying =in=l; resx1 ,--"
,Xn
"
il otherwise. [ XdXi...dX~, .... 1 ,.. ,X~" ] {1, O,ilia
One of the most important theorems of our residue methods is that the residue mapis independent of the choice of regular system of parameters. Theorem of Independence of Variables. For any two regular systems of parameters X1,’" ,Xn and Y1,"" ,Yn of R, resx1,... ,Xn: resy1,... ,Yn " A proof for a more general case of this theorem can be found in [44, (5.3)]. The reader is also referred to [62, p.63] for some history of residue maps. Fromnowon, we will write the mapresx1,...,x~ simply as res. Using modules with zero dimensional support, the notion of residues can be extended to power series rings over a complete local ring [44, Chapter 5]. In such general context, residue maps are transitive. Here we only state a special case, whichwill be used later. Transitivity Law. For fo, f~, f~"" ~ a[[X]], Xn, ym
= resx
Xn
.
One difference between cohomologyresidues and analytic residues is vanishing law. For analytic residues, the integral 1 vanishes if f is of the form anZnq-an+l Zn+l q-....
But its algebraic analogue
res [ fdZ already vanishes before taking residue, that is,
The next proposition clarifies the concept of vanishing for the case of homogeneous polynomials. Proposition 2.3. Let h, fl,"" , f~ E a[X1,"., XT~] be homogeneous polynomials. Assume that fl,..., fn is a system of parameters considered as elements in ~[[Xb..., (1) If deg h > deg f~ +... + deg fn - n, then fl," ,fn [ hdX~...dX,~
]
=0.
270
Huang
(2) If deg h < deg fl +"" + deg f~ - n, then res
fl,’" , fn -- O. [ hdX~...dX~ ] (3) Ifdeg h = deg fl +’" .+deg fn -- n, the following three conditions are equivalent. (a) h e (fl,"" ,f~)~[X~,... ,X~]; (b)
hdx~’’’dX,~ ] Yl," ,fn
(c) res
:0;
fl," , fn = [ hdX,...dX~ ]
Proof. Choose g _> 0 and homogeneouspolynomials vii in ~[X~,... , Xn] of degree 1 + £ - deg fj such that
:
By transformation
r2n
r21 r22
law,
[ hdX~."dXn ye+l ye+l fl," , fn ] [:" hdet(rij)dXl...dXn (1) If deg h > deg f~ + ... + deg fn - n, then
¯
deg (h det(rij)) : deg h + n(1 + g) - (deg fl +"" + deg fn) >ng. Hence the homogeneouspolynomial h det(rij), considered as a power -.. , Xn]], equivalently series, is contained in (Xle+~,... , Xne+~)~[[X1, : 0.
[ hdet(rij)dX~...dX~ yg+l y/?+l
hi . Hence (2) If degh < degf~ +.-’+degfn-n, then deg (hdet(rii)) the coefficient of h det(rij) at (X1 "’Xn)e is zero, and therefore res
fl, "" , fn
: res
ye+l ~L1
,’’’,’~n
vt+l
--- O.
(3) It is easy to see that (a) implies (b) and that (b) implies (c). that deg h = deg fl + "" + deg f~ - n and res
fl," ,fn [ hdX,...dX~
]
= O.
Then res
yg+l
"’1
,-",~
~g-g+l
= res
fl,"
,f~
= 0.
271
ResidueMethodsin CombinatorialAnalysis As h det(rij)
is homogeneous of degree n/~,
[hdet(rij)dX1...dXn]..1yt?+l ,’’" ,’’n res =
ye+l "~l
ye+l ~’’’
y~+l
,~n
/J dXl.., K~+l
dX,~
"
Therefore [ hdX1...dXn fl," , f~
=
ye+l yg+l ] -[ hdet(r~j)dX1...dXn
0.
Equivalently
h : Ag~ +"’+ for some g~,..., y~ E ~[[X],..., Xn]]. Collecting homogeneous polynomials of the same.degree gives h S (fl,"", f~)~[X1,..’, Xn].
The assumption that fl,"", f~ are homogeneous is needed in the proposition. For instance, for n = 2, h = 1, fi = XI + X21, f2 = X2 + X22, the degree assumption deg h < deg f~ + -.. + deg f~ - n is satisfied. However res
X1’ X2 X1 + X21, X2 + X~
res
= res
X1,X2 ][dXldX2
: 1.
Weremarkalso that the first part of the propositionis just the Macaulay’s theoremconcerningideal membership of homogeneous polynomials. Macaulay’s Theorem. Let f~,..., fn ~ n[X1,’",X~] be homogeneous polynomials which form a system of parametersof ~[[X~,..., Xn]]. If h ~ n[X1,...,
Xa] is a homogeneous polynomial degh > deg fl
then h ~ (fl,-.., Proof. Proposition such that
fn)~[Xl,’"
+’"+ degf~ - n,
, Xn].
2.3 implies that there exist f~g~ + ...
Collecting
with
homogeneous polynomials
gi,"",
gn ~ n[[X~,-..,
Xn]]
+ f~g~ = h. of the same degree gives the theorem.
The next result sets up a base for comparisons between our approach and others to combinatorial analysis. It will not be used in this article. Let Hom~(R, a) be the R-module consisting of all a-linear maps u: R -4 which are continuous for the m-adic topology of R and the discrete topology of ~¢ (i.e. u(mn) = 0 for some n).
272 Local Duality. (6)
Huang The R-linear map ~ nHm(A~R/~)
Hom~(R, a)
defined by sending w E Hm(A~R/.) to the ~-linear continuous map f res(fw) is an isomorphism. A direct proof of this theorem, which gives explicitly the inverse of (6), can be found in [44, (5.9)]. With local duality, one can translate a generalized fraction to an operator, see [47~ Section 3]. 2.3. Residue Theorem. In general, residue theorem asserts that certain maps of graded modules are maps of complexes (see Section 8.2). In this section, we concentrate on a very special case. Let F be the projective line Proj(a[X0, X1]) over an algebraically closed field ~. In this case the residue theorem means that the sum of certain residues is zero. Let Vo (resp. V~) the open subset of F consisting of the points with homogeneouscoordinates (1, a) (resp. (~,1)). The point (1, a) (resp. (~,1)) is identified maximal ideal of a[~] (resp. a[~]) generated by ~ - a (resp.~x° through the Hilbert’s Nullstellensatz. Wecall the point (0, 1) the infinity and denote it by ~. Given a (closed) point p ~ P, let O~,p be the local ring of p 6 F, mp its maximalideal, and ~o~.~/~ be the module of K~hler differentials of O~,v over g. Wedefine M(p) := 1 ~ (~o~,~/~), the first local cohomologymodule of ~o~,v/~ supported at mv. Elements of M(p) can be written as generalized fractions: Choose a ~, say V0, containing p, so p is of the form (1, ~) for some ~ ~ a. Then O?,p is isomorphic to the localization ~[~](~_~) of a[~] at the maximal ideal (~- ~). Elements of ~o~,~/~ can be written uniquely as ~d~ for some k~,k2 ~ a[~] with k:(a) ~ 0. Choose an element f ~ a[~] which generates m~ up to radical. In concrete terms, this means f is a non-zero polynomial and its value at a is zero. Every element of M(p) can be written as a generalized fraction
for some m e N and k,,k: e ~[~] with k:(~) ~ If p is also contained in V~, elements of M(p) can be represented by generalized fractions in terms of ~. A relation of these two representations is easy to describe: Choose homogeneous polynomials K~, K~, and F in ~[X0~ X~] of degrees nl, n2, and n such that kl : g,(1,~), k2 : g2(1, ~), f : F(1, ~) (as elements in the quotient field g(X0, X~) of g[X0, X1]). ~,p, ~ ~X~I
~X~
~)~"
ResidueMethods in Combinatorial Analysis
~ 0 dXX~0-
--
273
~’2-z~1-2¢]
~/X°~
XO
~ Z~(;
~Xl"
Hence
tX~,
!
~X~
!
If f is chosen to be of the form -~. - a for somea 6 n, Chenevery elemen[ of M(p) can be written uniquely as a n-linear combination of the elements of the form
Wedefine resp: M(p) ~ ~ to be the ~-linear map satisfying res~
(~-
~)~
the condition
0, otherwise.
Wealso write
One can check that the mapresp is independent of the choice of ~ containing p. The details are left to the reader. If a = 0, the mapresp is essentially the same as the residue map defined in Section 2.2. To see this, we need to identify M(p) with H~m~(~o~p/~)., Note that the completion ^~,~ of O~,~ with respect to m~ is an one dimensional formal power series ring over n. The required identification is given by the n-linear isomorphism 1
which maps
TM
,
to
] If Pn is the generic point of I~, the local ring O~,p~is isomorphic to the fields n(~0) and n(~). Wedefine M([~) := f~Or,~/~
274
Huang
and elements vector in M(F) meromorphic differentials. Note that M(F) is one the dimensional space over O~,p~. d~0 ° ancall x1 and d~ x0 are both basis of M(F). They satisfy the relation
:
(8) We would like
to define ~-linear e:
~ X0 / ~X~ "
maps
M(~)
and tr: $ M(p) ~ called trace, where the direct sums range over all (closed) points of ~. Given p ~ P, we choose a ~, say V0, containing p and ~ ~ ~ such that ~ - ~ generates mo. We define a ~-linear map %: M(~) ~ M(p) by the formula ~ r~d~ [ ~P~ Xo I :
kd~ g
] P
where k ~ ~[~] and g is a non-zero element in ~[~]. Using properties of generalized fractions and relations (7) and (8), one may check that the map ep is well-defined, that is, independent of the choice of ~ containing p and representations by generalized fractions. Given a meromorphic differential w and a point p of P, we call
:= the residue of w at p. Note that, for a meromorphic differential w, ep(W) vanishes except at finitely many points. Thus ~ ep ranging over all points of ~ has the image in SM(p), which is defined to be the map ~. We define tr := ~ resp, where the direct sum ranges over all points of ~. To justify the name for trace, we work on its restrictions to ~psv~f(P), i : 0, 1, which determine the trace. As these two restrictions have the same form, we consider only i = 0. Denote X = X~/Xo. Lemma 2.4.
Every element
of ~pevoM(P) can be written peV0
as
g
for some f, g e ~[X]. Proof. Every element of Spey0M(P) can be written
i=1
gi
where p~ e V0 is given by X - ~, fi 6 ~[X] and gi = (X - ai) m’ for some ai 6 ~ and mi ~ N. One can check that m
y= i=1
275
Residue Methodsin Combinatorial Analysis
and g : g1""gmare the required elements. Lernma 2.5.
If g f
: Xn+alXn-l~-...+an : bl Xn-1 + b2Xn-2 ~r’"+ bn
then
(9)
peYo
g
P
Proof. In ~(X), write (bij , oq E ~).
Then -1 _~_...,+mbim~)dX pEVo
"q
(X
P
-
’
’
and hence : bll
p~Vo
+-"+bel
: hi.
g
Weremark that the formalism in (9) resembles Tate trace [94]. Our trace mapcertainly also relates to usual trace. Given a non-zero element g E ~[X], the multiplication by an f e ~[X] is a x-linear map,~[X]/(g) -+ a[X]/(g). Taking trace, we get an element in ~, which we denoted by traceg f. For an element G ~ ~[g] of the form n + ang G : ao + atg + a2g~ +’" (ai E ~), wedefine consta G :-- a0. Proposition 2.6. Given f,g ~ a[X], tr ~-~ [ f dg ] =tracegf. p~y0 g P
(10)
Proof. Wemay assume g is of the form g = Xn +alX~-1 +...+a,~ (a{ ~ ,~). Then the images of the elements Uo = 1 Ul
=
X -t- al
u2
:
X2 +alX A-a~
Un-1
: Xn-1 "q- al Xn-2 "-b "’" -b a=_~
276
Huang
in ~[X]/(g) form a basis for the n-vector space ~[X]/(g). Both sides of (10) vanish if f is a multiple of g. Furthermore the maps considered are ~-linear with respect to f. So we may assume f = Xm for somern < n. In such case, by Lemma2.5,
p~Vo = tr
g
~ [ (nXm+u-i p~Vo 1 0 0 : 0 n
al 1 0 :
+ (n-
¯ ¯¯ ¯ ¯¯
a2 al 1 :
0 (n-1)al
ara-1 am-2 am-3 :
... ".. ’ .,"
0 (n-2)a2
To compute trace 9 Xm, it suffices tion
an-lXm)dX
1)a~Xm+U-~m+". g
]
am-1 am-2 :
1 (n-m+l)am_~
al
to know each const 9 hii in the representa-
xmui : hiouo +"" "~ hi(n_l)un-1
(hij
~[g]).
"" ¯ ¯¯
am-1 am-2 am-3
For i < m, 1 al 0 1 0 0 constg hii =
:
:
a2 al 1
"" ¯¯¯ "’’
ai ai-1 ai-2
ai+l ai ai-1
"’"
am ara-[ am-2
:
0 0 0 1 a~ a2
:
0 ai
... ."
0 0
1 0
... ...
al 0
For/
1
al 0 1 0 0 constg hii : : :
a2 al 1
¯¯¯
0 0 1 al
0 a2
... ¯¯ ¯
"" "’" "..
am-1 am-2 am-3 1 am-1
am am-1 am-2 :
=0.
am
Therefore traceg
Xm = E, constg hii = i=o
For further reading on residues and traces, [84] and [102].
pEVo[
g
P
see [50], [51], [60], [63], [83],
Residue Methods in Combinatorial Analysis
277
The Residue Theorem for the Projective Line. the maps M(?) 2_~ ~M(p) ~2~
The composition
of
is zero. That is, the total sum of residues of any meromorphicdifferentials is zero. Proof. Meromorphic differentials ~ can be written as
are of the form ~ d~, where p e n(~).
i,j
for
some~0 ~a[~],flij
are a-linear,
~ Xo
~,ai~a,
andnij ~N. Since the mapseandtr we mayassume that ~ = ~ ~n Xo ~for some n > -0 or assume that
~=l/(~-~)Uforsomen>Oand~ea. If ~ = (~)~ for some n E O, it is trivial to see that ep(~d~) vanishes for any point p ~ V0. At the infinity ~ d~ has residue zero, since it can be written as Xo ] ~X0
~ X1 ]
Therefore the total sum of residues of (~)nd~ is zero if n ~ O. If ~ = 1/(~ - ~)n for some n > 0 and ~ ~ g~ then there are only two points at which ~ d~ mayhave non-zero residues, namely the point q in V0 defined bY ~ - ~ and the point of infinity. ~ d~ can be written as 1 ~
d~ _ XO
’ x1 ’ (l_~)n
d~"
/(1- ,~)n is contained in Oe,~, and hence ~d~ has zero residue at ~ as well as at q. Ifn = 1, then ~d~ has residue 1 at q and has residue -1 at the infinity since
Ifn>
1,
then
(xi)
Therefore the total sum of residues of~(~x0 - ~)-~d~ is zero. In practice, it occurs often that the residue of a meromorphicdifferential at the infinity is zero. The next proposition shows a situation where such case happens. Proposition 2.7. Let f,g ~ a[~]. If degg ~ degf + 2, then the residue x~ of the meromorphicdifferential ~£d~ at the infinity is zero. Pro@Let m and n be ~he degrees of f and g, respecgively. f=amtxo)
+’"
+a~(~)
+a0
Write
278
Huang
and
~ ¢ X_~n g =vntXo, +’"+bl(-~a)+bo for some am,’" , ao, bn,"" , bo E ~. Then :_f dx1 = _ am +’" + al(~[)x° m-1 5’ aot :~ )"X° wnC x° ~n-m-2dX-~ j’.l’rt,
T
OkXI]
T cJl l,’~l
Since n > m + 2, eo~(
dx~)=-
[,~m~x~
’~-a +’"+al(~-~)
b~+... + bl(~)n-1+ ~0(~)n
¯
Sinceb~+...+~tx~, + b0(~)~is invertiblein ~[[~]], e~(~d~) This implies that Zd~ has residue zero at the infinity. g Xo A consequence of Proposition 2.7 and the residue theorem is that pevo
g
if f, g ~ ~[~] s~tisfying deg g > deg f + 2. Note that if ~ is a simple zero of LXoJ 9, then
~_~ ~’(~)"
Xo
Therefore we recover a well known result of Euler and Abel which states that if deg g E deg f + 2 and all zeros of g are simple, then
V Y(~)~(~)=0
~’(~)
3. BINOMIALIDENTITIES In this chapter, we introduce two residue methods for binomial identities. Section 3.1 explains systematically the local methoddeveloped in [45], which uses residues of a single point. ~ansformation law is emphasized by Identities 10, 11, and 12. Constraints are discussed. Section 3.2 consists of a global methodwhich uses residues of all points on the projective line. 3.1. Applications of Residues. Let ~ be a field and R be an n-dimensional formal power series ring over n with maximal ideal m. Wework on generalized fractions in H~(An~R/~). For n = 1, we assume that X is a regular system of parameter of R. For n = 2, we assume that X, Y is a regular system of parameters of R. In this section, some arguments involve exponentiM, logarithmic, or trigonometric functions. For these cases, we assume that ~ has characteristic zero, otherwise, we do not make any assumption on
N.
Our first residue methodconsists of three steps:
Residue Methodsin Combinatorial Analysis
279
Step 1: WRITE ONE SIDE OF THE IDENTITY AS RESIDUES OF GENERALIZED FRACTIONS.
Step 2: SIMPLIFY GENERALIZED FRACTIONS. Step 3: TAKE RESIDUE TO GET THE OTHER SIDE OF THE IDENTITY. Below is an example of this method. Identity 1 (Chu-Vandermonde convolution). ~=o(nk)(nm_k)
= (m+nn)"
Pwof. Write the terms in the left hand side as residues of generalized fractions: xn_k+ 1
n - k = res Simplify the generalized fractions:
.
k=O
=
Xn+i
(transformation
law)
k=0
=
(linearity law)
X~+~
:
xn+l m+ ][ ndX (l+X) Taking residue, we get (m+nn).
¯
[]
Wegive some tips for this method. (1) In step 1, we usually work on the more complicated side of the identity. For instance, the left hand side of Identity 1 looks more complicated than the right hand side. Thus we write the left hand side of the identity as residues of generalized fractions. (2) In step 1, we need to write combinatorial numbers as residues of generalized fractions. The following are some useful formulae. (a)
= res
n
= res
;
xk+l ~+~ (1 - X) xn+l
= res xk+l
1 (d) 2k_ l (2;)
;
=res [ -%4+X1 dX
280
Huang (e) ~ = res (f)
~.
xk+l
= res
xk+l
;
for
a E n.
(3) In step 1, in order to get generalized fractions of simple forms, we usually leave some combinatorial numbers, which we know their generating functions, untouched. For instance, in the proof of Identity 1, the binomial coefficient (~) is untouched. (4) In step 1, sometimes it is more convenient to use the following variants of binomial coefficients for writing one side of the identity as residues of generalized fractions.
n
(c)
=
k
~kk]
;
2k+ikk+l ].
(5) In step 1, sometimes we need to write co~nbinatorial numbers easier forms before writing them as residues of generalized fractions. In the following list, k is a running index of a summation and m, n, p, q are fixed numbers. The leh hand sides of the identities consist of product or quotient of two combinatorial numbers involving k. The right hand sides are easier than the left hand sides in the sense that they are linear combinations of combinatorial numbers involving k.
+
1(m-k)=--1(m-k)~(m-k-1)
k-
;
~ kn-k]
law, transformation law, and vanishing (6) In step 2~ besides linearity law, we also use generating functions for combinatorial numbers to simplify generalized fractions. The following are some useful formulae.
(b)
k=0
n+ k Xk
k=0
=
k=0
.
(1 - n+l’ X)
n
xk:
(c)
1
¯ (1 - X)
ResidueMethodsin CombinatorialAnalysis
281
(7) In step 3, sometimeswe changevariables before taking residues. The followingare someuseful variables change.
(a) T=-X; x (b)
(c) The above available also that properties
T=I_X; X+XY Y+XY and V 1 - XY 1 -XY"
tips consist of five lists of formulae. Of course, the more formulae in the lists, the more convenient it is in applications. But note some formulae in the above lists can be obtained from others using of our algebraic tools. For instance, the formula (2kk
=res[-
/
Xk+l
can be proved by the formula (n:k)
=res[
(1 -- X)~+IdX xnq-1
and the variable changeY = (1 - v/~ - 4X)/2 as follows: res[
~dX
]
(2kk)=res [
~d
Belowwe further demonstrate our methodthrough examples. Identity 2. [30, (3.101)] E(--1)k\/~/ k:r
\k-r]
ifn --r is odd.
I O~
Proof. Wewrite the terms in more complicated side as residues of generalized fractions. The binomial coefficient (~) and powers of -1, 2 are untouched:
(-1)k
k
r
2n-k
:
res (-1)
(k)2 xk-r+l (1
¯
282
Hu~ng
Simplify the generalized fractions:
(1+ x)2kdX
( 1 k n,]2n-k xk-r+l
J
+
]
xn-r+l =
~-~ [ (--1)k(~)2n-k(1w Xn_r+ 1 k=0 ~-~k:o(-)(k)2n-k(
=
xn_r+ 1
=
k [ (2X-Xn-r+l(1 + X)2)ndX
--
n-r+l [ (-1)n(x2 1) ndX X ~
x)2kxn-kdX
(vanishing
1 + 2kXn-kdx X)
(linearity
n - r is odd, the above generalized fraction
3 (Graham and Riordan).
E\
law)
law)
law)
j" has residue (-1) n ((n-~)/2)"
If n- r is even, the above generalized fraction
Identity
(transformation
has residue zero.
For m >_ n,
~2n+l)(m2:k) 2k
= (2rn+l) \ 2n
Proof. Wewrite the terms in more complicated side as residues of generalized fractions. The binomial coefficient ~2n] (m+~ is untouched:
\(2n+l)2k
(m2+nk)
\2n/2n+l_2k+l)(m2:k) = res
~
2n ]x2n_2k+2
.
Simplify the generalized fractions:
k>0 = =
~, 2n ]x2n_2k+2 [ (m+k’(l+X)2n+ldX
E ~ 2n
!
x2n+2m+2
(1 + X)2n÷l Ek>0 k(m+k~x2k+2md 2n X2fi-+2m+ 2
(transformation
(linearity
law)
law)
Residue Methodsin Combinatorial Analysis
283
Taking residue, we get ~>0 \ 2k
m+ :
=
res
x2n~:2m+ 2
(~-x)~+~
res
x2m_2n+2
\ 2n
Note that, in the above proof, we use the representation 2k
X2n_2k+2
\2n- 2k + 1 = res
in step 1 to get a simple generalized fraction in step 2. To evaluate E ~ 2k
2n
k=O
’
we use the representation
The same procedure gives the following identity. Identity
4. Porm ~ ~, k=O ~ 2k
"
Nowwe use residues for more th~n one variables. Identity 5. [30, (6.21)] n
~(_l)k(;
. n 2
) (2n: ~
k)(~)=(_1)
Proof. Wewrite the left-handed side as residues of generalized fractions. The binomial coefficient (~) and the power of -1 are untouched: (_l)k
(:)(2n:k)(;):
(-1) k(~)(1
+X)2nxn+l,yj+l -k(1 +Y)kd
J¯
284
Huang
Simplify the residues of generalized fractions: Eres
(-
+X)2n-k(1 xn+l, yj+l Y) kdXdY
(x,(1
= res ~a=j(-1)
(~¢)(1
+ X)2n-k(1
Xn+I,
Y) kdXdY
yj+l
lkn
1 + X)~n-k(1 + y)kdXdY ] = res ~-]-k:O(-- ) (k)(Xn+I, yJ+~ [ n =
res
Using transitivity
¯
xn+l, yj+l nd ][ XdY (I+X)n(X-Y) law, we get
res [ (1 + j+l X)n(Z - Y)ndXdY ] xn+I,Y
[ [
(-1)J(y)
= res = res
(1 X)nXn-JdX n+l X
+ X)ndX j+~ X 2
Identity 6. [30, (6.28)] (n:j)(n+J)" n
k=O
Proo]. ~(~)2( k=o
k ) = ~res[ n - j :
(~)(1
Eres
+ x)n(1 xk+l,
+ x)n(1
Y) kdXdY ] yn-j+l
xn+l,
Y) kXn-kdXdY ] yn-j+l
k=O
= res [ Y]~=o (~)(1 zn+l, + x)n(1 yn-j+lY) kxn-kdXdY ] =
res
xn+l~ yn-j+l nd ][ XdY (I+X)n(I+X+Y) xn+l
=
Residue Methods in Combinatorial Analysis
285
The next identity occurs in computations of a canonical moduleof a set of points. Identity 7. [8, Lemma7.2]
~-.,~(_l)~(n+i-jn j=O
n+o~-I a+j
~-l+j’~ a - 1
(n+a+i) Proof.
Wecan handle identities
involving elements in the underlying field.
Identity 8. [30, (3.65)] Given a E ~,
k:O
k:O
286
Huang
Proof.
k=0
= ~ (~)(2n
n- k)(a_l)k"
k=0
Sometimes, we need to rewrite a combinatorial sum before writing it as residues of generalized fractions. Identity 9. [97, p. 134] For n > 0 and a E ~ with a ~ -1,... ,-n, n
n
k=O
t k }
Proof. n (n+a’~ E(__I) k ’n-k! k=O (a+nn)
n
a+k
1
(°:")
-
(--1)k(1
n
1
(°+."):
res
res
1 (~+~ res
X)n+aXkdX ] xn+ l
L[ (--1)k(1+Xn+~X)n+aX~dX]
(1 + X)U+a-*dX n+l X
]
(~÷~-1~ (o÷~ n
]
The proofs of the above identities are basically routine once wechoose suitable generalized fractions whose residues represent the combinatorial sums.
Residue Methods in Combinatorial Analysis
287
However, there are examples which require clever choices of new variables as shownin the next three identities. Identity 10. [16, (2.26)] I
1
E(_llk_ 1 j=l 3
k=l
n
Proof. [45, Example6]
k-: (-1)
-. j=l 3
k=l
Nowwe use a new variable
T = X/(1 - X). One can check that 1 I+T
1-Xand dT -
1 dX. (1 - 2X)
Hence u+:+T)dT] = [ (-1)U+’log(1T [ (X-1)" log(1xn+: (1-- X)dX and E(__l)k_ 1 k----1
_1 = res j=l j
Identity 11 (Saalschiitz’s
(-1)n+l
log(1 + T) dT ]
~+: T
]
theorem).
k>~0 (m a- k) (n b k) (a + ~ + k) : (an~n)(b+nm).
=~.
288
Huang
Proof. (cf. [24]) +k) E(ma_k)(nb_k)(a+bk = Eres k>0
( k )(1 xm-k+l +x)a(l+y)bdXdY ~ yn-k+l
=
k ~k>0k -
res
xm+l ~ ynT1
b[ ] (I+X)a(I+Y) xm+l ~ yn+l Let u := check that
(x + xy)/(~ - xy) ~ndV := (~ + XY)/(~- XY). One I+U-
I+X -1 -XY’
I+V=
I+Y -1 - XY’
can
and dUdV = (1 + X)(1 Y) dxdY. 3(1 - XY) Hence (l+X)a(l+y)b ’" -~ -f----~-~---~l-~d4-5-dXa~ ( -- X xm+l~ yn+l
(1 + U)~+’~(1 v) b+mdUdV ] Urn+l, vn+l
and res
Identity
(1 + U)a+n(1 + v)b+mdUdV um+l ~ yn+l
12 (Dixon’s theorem).
=
1)m-r(n+r](g-r)’,
ifg+m--n=2r.
289
Residue Methodsin Combinatorial Analysis
Proof. (cf. [24]) m
Eres[ (--1)k(~) "
k>0 res
(1 + u)m(1 V)edUdV
ue-n+k+l~
1, e(V1vm+V) ndUdV Ug+V) [ (1 + u)m(1
Let X and Y be new variables defined V = (Y + XY)/(1 - XY). Then res[ =
reS
= res
vm-k+l
J
j"
by U = (X + XY)/(1 - XY) and
[ (1 + u)m(1Ue+~ V)t(U - V)"dUdV ] ’ vm+~ (1-XY) ~+~ ..... X~+I, ym+l l~n-k{n+h~ ~ I
[
{n~xk+hyn-k+h~g~y
zg+l~
] ~
ym+l
which does not vanish only if the system of equations
-~+h=m in k and h has integral solutions. The above system of equations is equivalen~ to ~+m-~=2h k = n m.+ h,
{
which has integral solutions if and only if ~+m-~is an even number. In such case, the above residue equals (-1) m-~(~+~,n , (~-~), where e+m-~= 2r. In particular, n
(~)3
,
nn
=
) if
~t
_l)n/2 ((3
=
is
odd
2), ifn is even.
To apply the method in this section, we need to recognize generalized fractions whose residues represent combinatorial sums. In most cases, this step is not too difficult. Howeverthere are examples of combinatorial sums, which can be represented as residues of generalized fractions of a simple form, but we can not do much about it. For example, ~ (~)a k=O
=[ res(1
+X)n(1
+Y)n(l xn+l~
+XY)ndXdY]
yn+l
290
Huang
But no closed form for ~-:~;:0 (~)3 seemsto exist, at least no hypergeometric closed form exists [70, Theorem8.8.1]. See also [p. 159, ibia.l for the difference between having a closed form and having a pretty formula. The method described in this section has not use the full strength of the residues. In next section, we will introduce another method to extend the sphere of applications of residues to combinatorial identities. The method in this section can be used to evaluate combinatorial sums without knowing the identity in advance. Let’s look at an example: In [53], we find an identity
One might wonder what ~ closed formula for
i=k
looks like. Werewrite the sum using residues: ~
i2(:)
= ~ res
. Xk+~+X)idX 1 ] ] i=k i=k i. We need to evaluate the sum ~i~k i2(1 + X) This is done by using the formula ~ yn+~ 2 2~ - 1 (11) ~ i~Y i - Y + Y~ [i2(
n
Xk+l +X)idX
= res
[~ir~ki2(1
(
)
which is obtained by applying ~he operator Y(d/dY) twice on both sides of
~ yi
= 1 - Y~+~
i=0 No~ethat identity (11) can be also proved by using induction on ~. Nowset Y = 1 + X~ we ge~
~ + (~ - ~lx + ~ ~ i~x~(~ + xl ~ = (~ + xl~+~(~x (~ + x~((~- ~)~x~ 4 (~ - ~)x +~). ¯ aking residue of the generalized fraction
[ (1 + X)n+*(n~X~ + (1 - 2n)X + 2)dX =[
k+4 X
k+4 JX
J
--
Residue Methods in Combinatorial Analysis
291
we get an identity: Identity
13. 2= n
2 /~i
+ +
+2
+ (1 - 2n)
+
.
D. Foata has remarked that the following identity, conjectured by Dyson [15] and proved independently by Gunson [34] and Wilson [99], is closed related to MacMahon’smaster theorem (see Chapter 4). Our point of view that not only Dyson’s conjecture and MacMahon’smaster theorem but also manyof their consequences (for instance [26]) result from residue cMculus. Here we include a short proof of Dyson’s conjecture, which is a modification of Good’s approach [28]. Identity 14 (Dyson-Gunson-Wilson). Let ai,... gers. Then the constant term of
,an be non-negative inte-
1-I (1 is equal to (al + "" + an)! a~! . . . an! Proof. Wework on the formal power series ring Q[[X~,..- , Xn]]. Let D(a~,...
= H (Xj - a’.
,an)
The Dyson-Gunson-Wilsonidentity is equivalent to the identity (12)
res
,an)dXl"" yal.~-,,.~_an~_ 1
dXn
(al
+"- + an)!
X~l+...+an+ 1
~1
As a consequence of Lagrange interpolation, n
we have
n
i=~ j=~ X]Z Xi" If all a~ are positive, multiply the above identity by D(a;,... , an), we get n
D(al,’" ,an) = ~ XI "" ~i’" XnD(al,"" ,ai_l,ai
- 1,ai+l,...
,an).
i=1
Nowwe prove (12) by induction (12) can be &ecked directly. Assume lal
>1
and 02) hdds fo~ smaller ~a}.
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Huang
We may assume that all res
:
=
ai are positive,
X~ZlT...+an+l, . . " ,1 XnalT...Tan+ [X~l’"X~’~D(al,"’,an)dXl"’dXn res
~
then
[
X~ 1
¥-al .+’"’+an ...X’~i-1...X,~’~O(al,
~’al "+’"’+an
...
,ai-
1,.-.
,a,~)dXl.
"dXn]
+:::_+ ~.= al!."
ai-l!(ai
- 1)[ai+l!...an!
(al+’"Tan)] all" " an!
3.2. Applications of Residue Theorem. In this section, we work on the projective line Proj(~[X0, X1]) over an algebraically closed field n. Wewrite X1/Xo as X. Given a point p in V0 described by X - a (a E ~), we use the following notation for the residue of a meromorphic differential w at p. resa w := resp Wealso use the following
notation
to describe elements in M(p).
If (~ = 0, we drop the index: f := f 0 Weuse the residue theorem for the projective identities. Our method goes as follows: Step
1:
WRITE
ONE SIDE
line to prove combinatorial
OF THE IDENTITY
AS RESIDUES
OF GEN-
ERALIZED FRACTIONS. Step 2: SIMPLIFY THE GENERALIZED FRACTIONS. ASSUME THAT WE GET
f
FOR SOME POLYNOMIALS AND g. Step 3: COMPUTE THE RESIDUES OF THE MEROMORPHIC TIAL
(fdX)/(gX
THEOREM GIVES
n)
AT ZEROS OF g AND AT THE INFINITY.
AN IDENTITY.
Below we demonstrate this method through Identity 15. [70, Foreword] n. ~ (2:-2klg)(2:)=4 k=0
some examples.
DIFFERENRESIDUE
ResidueMethods in Combinatorial Analysis
293
Proof. Write the terms in the left hand side of the identity as residues of generalized fractions and simplify them: = E res (2kk)(1 k=O
X)2n-2kdX xn-k+l
-- res(1 +X)2u
i
- res
= res
[2~k Z..,k=OX-’¢~ xn+l ~ ] (1 +X)-2kX~dX
(1 + X)2’~ dX l - 4x xn+l 1 - X dX xn+l
.
The residue of the meromorphicdifferential ~+~ (1 + X) (l - X)Xn+l dX at
Y:=X-lis res
-(2 + y)2,~+~ ] = _22n+~. (1 +Y)n+~ dY Y ~
In terms of the infinity Z := l/X, (1 + ½)2n+l dl (1 + ~"+x Z) (1 - ±~(±~n+l Z = (1 - Z)Z n+l dZ. Hence res0 w = res~ w. 0, 1 and cx~ are the only points at which w mayhave non-zero residues. By the residue theorem, res0 w + res~ w - 2 2n+l -- 0~ from which the required identity follows immediately.
[]
The above proof, providing an algebraic interpretation of the identity, is not the simplest one. A simple proof without residue theorem goes as follows:
~-~(2:-2kk)(2:) k:o The next identity
= re8 /2~’~Xk J" ] [ xn+l ~res[ ~_--~dX k:o is drawn to the author’s attention by Hung-WuWu.
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Huang
Identity
16. [16, §2.2,
Example 2] Assume char ~ % 2. Then
2-k=2n’n ~(n+k) k=0 Proof. Write the terms in the left hand side of the identity generalized fractions and simplify them: n + k 2_ k = n
k=0
~res
res
2(1+x)~ (~ - (~+x X ~)
1 -
)
dX
Xn+l
= res
]
n+l 1 X
-
Using the residue theorem as in the proof of the Identity
.
15, we get
n. _2
res l~(nX+~ The residue
of
XU+~
k=0
=
as residues
dX _
of the meromorphic differential 2(1 ÷ X)" w := (1 - X)X ~+~ dX
at
Y:=X-lis res
-2(2+y)ndy (1 + y)n+l Y
1
n+l. = --2
1 By Proposition 2.7 of Chapter 1, the residue of w at the infinity the residue theorem, res
1 - X Xn+~ [ 2(l+X)ndX
is zero. By
= ] 2n+1.
Hence ~ (n + k) 2-k = 2n+l - 2n = 2n. n k=0
Identity 17. [16, §2.2, root of -l,~. Then
Example 1] Assume char~ ¢ 2. Let i be a square
~(_11~(2n + 1 (1+ i 2~÷1+ (1- il
295
Residue Methodsin Combinatorial Analysis
Proof. n 2n÷l E(_I)~(
=
k----O
E
res
[ (_l)k(1
k=O n
x2k+ 1 +x)2n+ldX]
---- E res i] (_l)k(1..l_x)2n+lx2n_2kdX x2n+ 1 k=0
dX
21 + X
= res
x2n+l 2n [ +1 (-l)n(l+X)
The residues of the meromorphicdifferential (-1)n(1 + 2n+: w :=
(1 + X )X
]
dX
at i, -i, and oo are -(1 + i)2n+1/2, -(1 - i)2n+1/2, and O, respectively. Since these are the only points at which w mayhave non-zero residues, the required identity follows from the residue theorem. [] Identity
18 (Hardy’s identity). E (_l)k
(m k - k)m- l_k 2(-l)m/m, m (-1)m-I/m, if otherwise.
k=0
(--1)k((mk
-k)
res
res (_l)m+l
m
Assume char ~ = O. Then
res [
(_l)m+l
res
+ (re;k-l) + res
X~+I
[ (1 + x)m-k-l(1
2X)dX ]
~-]~=_~
(1
m-t (_l)m-k-1 l+2X Alz]
(1 q- x)m-k-ldX
(1 + 2x)xm-k-ldX xm x)m_k-t
.
xm m [ Let u be a primitive cubic root of 1~ and
(-1)m+1 (1 + 2X) w = m(1 mdX" + X + X2)X
]
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Huang
The residues of w at u, u2, and cx~ are (-1)m+:/(mpm), (-1)m+l/(m~2m), and 0, respectively. By the residue theorem,
1 = m÷
[m/2]
m- k
k=0
mu
=
2m 77tlY
f2(-1)m/m, [(-1)m-1/m,
ifm = 3n; otherwise.
Note that we may also use the representation [m/2] _ res k k) m - k
E (_l)k(mk=0
m(1
(I X_~_X2 +X)m(2+ dX [-[-xm+l ) ]
as [16, §1.5.1] to prove the identity. Identity 19 (Ramus’ identity). Assume char n = 0 and q < n. Let u be a primitive n-th root of 1~. Then E(~ nk m :--l~-~u-qk(l+uk)’~" n k=l + q Proof. E nk + q k
m-nk-q+l X
k
"=
res
The residue of the meromorphicdifferential (X n _ 1)Xm_q+l dX ~is atY:=X-u res = res
_(I+X),~ x’~-~+’(x-u...(x-:~,~-~)(x-~+’ )...(x-u~-~ dX ] kX - u (Y+uk) rn
q+l
-O+v+~’~)’~ kq-1 (Y+uk -1)...(Y+uk -uk -1 )(Y+uk -u" )...(Y+uk -un Y -(1 ÷
pkm-kq+k(l~k -- 1)’’" (/2 k -- /~k-1)(/~k __ /]k+l)...
(/]k __ /~n-1)
Since (pk _ 1)... (uk -- ~-l)(~k is the value of j~(X d -1) resuk w
_ pk+l)...
(l~k
-(1 ÷ .k)m m" -_ _l__r,
kq(1
--
izn--1)
at + t~-k)
1
] ~k
297
Residue Methodsin Combinatorial Analysis
Since q < n, the ineroinorphic differential w has residue zero at the infinity. By the residue theorem, ln_1
k nk +
m. q =-~vkq(l+v-k) k=O
Replacing k by n - k, we get = _1 ~ p-qk(1 + p~c)m. ~
nk+q
n
k=l
This application is suggested by Peter Jau-Shyong Shiue. See [57] for another proof, which uses the recurrence relation S(m,n,q)
- (1+ (-1)n-1)S(m-n,n,q),
E(-i)J-l(~)S(m-j,n,q)-t j---1
where S(m,n,q)
nk + q "
k
This recurrence relation can be verified using cohomologyresidues with the representation S(m,
n
n, q) = res I_Xxm_q+~ .
The details are left to the reader. If g equals the field of real numbers, then Ramus’identity can be also written as
k
~+q
~
4.
~=1
~
VARIABLE CHANGE
In Chapter 3, variable change is used together with properties of residue mapsand generalized fractions to prove binomial identities. In this chapter, we emphasize algebraic meaning of binomial identities and other combinatorial results revealed by variable change. In combinatorics, bijective proofs of identities are preferred in manysituations because they shed light on combinatorial nature of the identities. (They are also called combinatorial proofs!) A bijective proof exhibits explicit bijection between two finite sets. If we identify these two sets via this bijection, we get another viewpoint on the bijective proof: A bijective proof stands for two ways to count the cardinality of a set. In this sense, a proof of an identity by variable change can be viewed as an abstract version
298
Huang
of bijective proofs: A proof by variable change stands for two ways to calculate the residue of a generalized fraction. Such proofs shed light on algebraic nature of the identities. For example, Saalschiitz’s theorem
stands for two ways to calculate
the residue of the generalized
fraction
(1 +X)a(1 +Y)b xm+l,
yn+l
]
one way by variables X, Y and another way by variables (X+XY)/(1-XY), (Y + XY)/(1 - XY). See Identity 11 for details. MacMahon’s master theorem can be also interpreted by variable change: Let n be a field, let (aijE~,
X~=Ea~jxj
i=1,2,...
,n),
and let ml, m2,"- ,mn be non-negative integers. MacMahon’s master theorem asserts that the coefficient of x~nl.-. x~n~ in X~1..- Xumn is equal to the coefficient of w~nl --. w~n~ in the inverse of 1 -- all~Vl --a21w2
[ --anl~On
--al2Wl .... -a22w 2 ....
1
... 1
--an2W n
aln~Ol a2nW2
-- annq2)n
To see the algebraic meaning of MacMahon’s master theorem, we consider the power series ring n[[Xl,... , xn]] and polynomials Y~ = 1 + Xi. One way ~... to equate the coefficient of x’~ x~" in X~nl... X~an is to calculate the residue of
¯ Another way is to calculate the residue of the above generalized fraction using new variables wi = xi/Yi (i -= 1,..- , n). The theorem follows from the straightforward computations: y[n, ... ynm,~ dxl"" dxn (YI""Yn) -ldxl"’’dxn ]
:
=
[
D-ldwt’"dwn w,~+~,... ,w~.+ ~] ¯
The above approach to MacMahon’s master theorem in analytic form was first discovered in [27] and was translated to algebraic form in [45, Example 1].
Residue Methods in Combinatorial Analysis
299
A commonfeature of manyidentities which can be interpreted by variable changes is that they all can be deduced from Lagrange inversion formula. The usage of Lagrange inversion formula in such context is quite natural as the formula itself is nothing but an interplay of representations of a given power series by different sets of variables. Whatsurprising is that, if we allow more flexible representations of power series (by what we call Schauder bases), not only combinatorial identities but also all inverse relations with orthogonal property have a clear interpretation. Furthermore, with local duality, an operator method [59] for computing inverse relations have also a natural interpretation. See [42] for details. In this chapter, besides introducing these concepts, we will also reinterpret the theory of Riordan arrays using our approach. 4.1. Lagrange Inversion Formulae. Lagrange inversion formula is one of the.most important tools in enumerative combinatorics. It is also important in some approaches to the Jacobian conjecture, see [7]. The formula investigate the interplay of representations of a given power series in terms of different sets of variables. Assumethat ~[[XI," ¯ ¯ , Xn]] = ~[[YI," ¯ ¯ , Yn]] is an n-dimensionalformal power series ring over n with variables X1,’" ,Xn and Y~,... , Yn. Given a power series f represented as azl...z,~X1 " " " Xn
(ail...i,~
,
Lagrange inversion formulae seek new representations
f = bil...ioYP-.-Yg° (bi...inE-)
in terms of the coefficients c~!..i, ~
of ,.,il...inX~_
l ¯
Such task is accomplishedif we can write the coefficients
of
O) in terms of ci~ -..i,~" Most work in this theme is restricted to the characteristic zero case, see [7], [21], [23], [25], [39], [61], [54], [73], [101], and their references. Cohomology residues provide a convenient characteristic-free approach to Lagrange inversion formula [47], cf. [6] and [10]. Here are someexamples. Lagrange Inversion Formula 1. ([25]) If Yi = Xi/(pi for an invertible element ¢Pi (i = 1,... ,n), then O~°i d},...i, ~ = res (j) .... Yi’~k )dXl " " dXn . [ Xj99~X~oinn det(Sik ] "+1 X~’+I," , X~
Huang
300
Lagrange Inversion Formula 2. If Y1 = X1 + G for an element G in the ideal generated by X2,... ,Xn and Y2 = X2, ... , Y,~ = X,~, then
where i2+’"+in
. (X1 -~- G)i2+"+i’~-k(--x1)k"
k
k=0
Lagrange Inversion Formula 3. ([7, III Corollary 2.5]) If Y~ = Xe - Ge for an element Ge in the square of the mazimal ideal of ~[[X~,..-,X,]] ( ~ = 1,... , n ) , then (J)
di~...i,, =
~
res
ml~*..~mn~O
, ml ~ ’’" (m~+i’’ [
{m~+i~’X’~ml k mn] ~1 "’" X~I +ii +1 ~
O(Y~’’’’’~) ~n ~m~ 0(X1,...,X~)UA1 xmn +in+l ~
"’"
dXn
]
n
where, by convention, G~ = 1 even g G~ = 0 (g = 1,...
,n).
The reader is referred to [47] for proofs. Weremark that Lagrange inversion formula 1 is proved in [47] by an operator method. But the proof lop. cit.] can be translated to a residue proof by local duality. It is shown in [44, Lemma5.2] that any regular system of parameters of the formal power series ring g[[X~,. ¯ ¯ , Xn]] is obtained from X~,... , Xn by a sequence of the following operations: ¯ Permute a regular system of parameters; ¯ Add to one of the elements in a regular system of parameters by an element in the ideal generated by the rest of the elements in the regular system of parameters; ¯ Multiply one of the elements in a regular system of parameters by an invertible power series. Therefore Lagrange inversion formula 1 and 2 together are enough to corn(J) pute dii...i, in the general case. 4.2. Inverse Relations I. Let ~ be a field and R = r~[[X]] = g[[Y]] be an one-dimensional power series ring. If X as a power series in Y and Y as a power series in X have the same form, then coefficients of a given ele~nent in R as power series in X and in Y have interesting relations. For example, cX Y =--and 1 -bX
X-
c 1+ ~-Y c
301
Residue Methods in Combinatorial Analysis
have the same form, where b and c are non-zero elements in n. Note that the above two identities imply each other. Let g be a positive integer and and B--E biXi
A = E aiYi (ai e n) i=O
(bi
E n)
i=0
be power series with the relations and A = (1 - bX)eB.
B = (1 + !Y)gA
Note that the above two identities imply each other, as (1+ ~Y)(1 -bX) = 1. Using the relation Y2dX = cX2dY, we can write bn in terms of ak’s: bn -- res ~
(1 + Y) AdX --~ res (1
c ~zn+l
xn+l
;
or more concretely (13)
bn:k(g’+n-1) k=0
b n k( 1 n - k (’~) - -~)-’*ak,
cf. [20]. Similarly, an can be written in terms of bk’s: (14)
an = ~ (~ + n - l (-b)’~-k c -’~bk’n -
Note that the relations (13) and (14) imply each other and are known pair of inverse relations. Moregenerally, a pair of inverse relations is a pair of relations of the form n
bn
an
:
E Cnkak’
k~°=dnk bk,
where ai, bi, cji,dji e t~ (0 < i < j). Usually, we assume that {ak} and {bk} satisfy the following orthogonal property E Cmkdkn
:
~mn,
k=O
where ~mn is the Kronecker delta function and, by convention, c.~i = dji = 0 for i > j. One mayverify directly the orthogonal property k(g+mm-1)(~’+k-1)(-1)k-nbm-n=5mn-k of the pair of inverse relations (13) and (14). See also the end of Section Inverse relations are frequently encountered in co~nbinatorial problems and have been extensively studied by Riordan [75]. Systematic and unified
302
Huang
methods were developed by Egorychev [16] using contour integrals and by Krattenthaler [58] using operator methods. See also [91] for Mbbiusinversion over partially ordered sets. 4.3. Schauder Bases. Let R be an one-dimensionM formal power series ring over a field n. A Schauder basis for R is a sequence (fi) in R, with which every element in R can be written uniquely as aofo + alfl + a2f~ +’" for someai E ~. Wecall ai the coefficient of ~ aifi at fi with respect to the Schauder basis (fi). Let m be the maximal ideal of R. If fi~ mi \ mi+1, we call (fi) a strictly monotoneSchauder basis. For n = n[[X]], we give some examples. ¯ The sequence (Xi) is a Schauder basis, which we name an ordinary Schauder basis, or an ordinary basis for short. ¯ Given p,q ~ Z, the sequence (xi(1-I-X) p+iq) is a Schauder basis, which we name a Gould-Schauder basis, or a Gould basis tbr short. If the characteristic of n is zero, more Schauder bases are available: x is a Schauder basis, which we namean exponential ¯ The sequence (~.) Schauder basis, or an exponential basis for short. ¯ Given p, q ~ n, the sequence (Xie (p+iq)x) is a Schauder basis, which we namean Abel-Schauder basis, or an Abel basis for short. ¯ Given p,q ~ Z, the sequence (Xi(ex-~_~) p+iq) is a Schauder basis, which we name a Bernoulli-Schauder basis, or a Bernoulli basis for short. Let (fi) and (gi) be two Schauder bases with the following relation: (15)
fi
Given h E R represented as (16) (17)
h = aofo + a~f~ + a2f2 + a3f3 +"" = bogo + big1 + b292 + b393 +"" (aj,bj
~ ~),
it is easy to check the identity n
(18)
bn = ~ Cnkak k=0
by comparingthe coefficients. For some specific h and Schauder bases (fi) and (gi), identity (18) is particular interest. Identity 20. Let p,q,r be non-negative integers and ~o E ~[[X]] be an invertible formal power series. Assume that q does not divide r. Write O0
=
x i=0
.
Residue Methodsin Combinatorial Analysis
303
Then n
r
~(r_qk)l~(pTqk ) _ b(np+r).
r -- qk %
Un-k
--
Proof. Let (fi) be the Schauder basis (xi99p+iq), (gi) be the ordinary Schauder basis (Xi) and h = TP+". Weuse the notation as in (15), (16) and (17). easy to check that bn = b(np+r) and cnk =b(P+qk) n-k . The identity follows from (18) and the computation: ak : res
9rd(X99q) ] (x99q)k+ 1
=
res
99r-qk-q(99q
=
res xk+l + ~ b~~-q~) ~ +.r_--~ i qk b(r_q~ )
= __
-~ qX99q-l ~x )dX
xk+l
r_ b(r-qk)
r-qk k
res
xk+l
¯
Let T = 1 +X. Then (fi) is the Gould basis (Xi(1-~-X)p+iq). As a special case of Identity 20, we get the following identity. Identity 21 (Gould). Let p,q,r be non-negative integers. Assume that q does not divide r. Then
E For ~ = 1/(1 - X), we get the following identity. Identity 22. Let p,q,r be non-negative integers. Assume that q does not divide r. Then n~ ()( n )(= r r-qk+k-1 p+qk+n-k-1 p+r+n-1 " k -k ~=0 -q~r
~
Let ~ be a field of characteristic zero and ~ = ex. Then (]i) is the Abel basis (xie(P+iq)X). As a special case of Identity 20, we get the following identity. Identity 23 (Abel). Let p, q, r be non-negative integers. Assumethat q does not divide r. Then ~ r (r - qk) k (p + qk) ~-k (p + ~ r) ~ r - qk k~ (n - k)~ k=0
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Huang
Let a be a field of characteristic zero and ~ = X/(e X - 1). Then (f/) the Bernoulli basis (xi(ex-~_~)p+iq). As a special case of Identity 20, we get the following identity. Identity 24. [42] Let p,q,r be non-negative integers. Assume that q does not divide r. Then n r B(k r-qk) (p+qk) B p+r) B(n ~r-qk k~ (n-k)’ nl ’ where B~n) is the i-th Bernoulli numberof order n. (See Section 6.1 for the definition of Bernoulli numbers.) Abel identity and Gould identity were generalized in [89, Theorem3.1]. The generalization has a clear interpretation by Schauder bases: Assume that R = ~[[X]] = a[[Y]] and g, f ~ R are invertible, then (f-~Yig) is a Schauder basis for R. The generalization is nothing but an explicit description of the coefficients of g with respect to the Schauderbases (X J) in terms of the coefficients of g with respect to the Schauder basis (f-lyjg) and the coefficients of f-~yig with respect to the Schauder basis (XJ). Identity 25. [89, Theorem 3.1] Let g g f-lyig
~ = =
2+baX3+..., bo+b~X+b~X aof-~g + a~f-lyg coi+c~iX
+ a~f-ly2g + a3f-ly3g +c2iX 2 +c3iX3+...,
+...
,
whereaj, bj, cji E ~. Then n
bn
:~akCnk. k=0
Proof.
--- res
[ gdX
Xn+l ]
.
= res
~-]k=O akf Y gdX xn+l
=
~’~ res k=0
akf-lykgdX xn+l
n
¯-~" ~ akCnk k=O
4.4. Inverse Relations II. Let R be an one-dimensional formal power series ring over a field g. Let (fi) and (gi) be two strictly monotoneSchauder bases for R with the following inter-relations: fi gi
= coigo + cligl + c2ig2 + c3ig3 + "’" (cji ~ ~), = doifo + dlifl + d2if2 + d3ifa +"" (dji ~ ~).
305
Residue Methodsin Combinatorial Analysis
One can check that the following orthogonal relation holds, see [42]. O~
E Cmkdkn k=0
= ~mn.
Given h E R represented as h = aofo +all1 +a2f2 +a3f3 +"" = bogo + big1 + b292 + b393 + ""
(aj, bj ~ ~),
there is a pair of inverse relations [ibid.]: bn ---
E Cnkak’ k=0 n
an = E dnkbk. k=0
For example, ifp ~ 0, (gi) is the ordinary basis (Xi) and (fi) is the basis iX
((~-x)p+~+~), weget the following pair of inverse relations. Inverse Relation 1. [75, Table 2.1, class 4] W k ak~
k=O an :
k=0
--1) n+k
+ k
bk.
Assumethat n has characteristic zero. If p ~ N, (gi) is the ordinary basis (Xi) ~nd (fi) is the Abel basis (Xie(p+OX), we get the following pair of inverse relations. Inverse Relation 2. bn=~
~--_~. k=O
an=
p+n k=O
ak,
+ k (-n - p)n~n-_--~)~. bk.
Assumethat ~ has characteristic zero. If p ~ N, (gi) is the ordinary basis (Xi) and (fi) is the Bernoulli basis ~X i[ ~ x ~p+i~ ~, we get the following pair of inverse relations. Inverse Relation 3. b~
~ a~" (n - k~ B(P+k) n-k
k=0
~ ~(-n-p)
Huang
306 For details of proofs of the above three pairs of inverse relations, is referred to [42]. Conversely, given a pair of inverse relations
the reader
bn : E Cnkak’ a~o an : E dnkbk with the orthogonal
property
there exist strictly element h E R satisfying fi gi
E Cmkdkn : ~mn, k=O monotone Schauder bases (fi)
= coigo + cligl
and (gi)
for R and
+ c2ig2 + c3ig3 + "" ,
= doifo + d~if~ + d2if~ + d3if~ + "’" ,
and h = aofo
+alfl
+a2f2
+a3f3
+’"
= bogo + big1 + b2g2 --~ b393 + ...
,
see [42]. For example, the pair of inverse relations
k:0 an:~-~(~-~n-l)(--b)n-kc-nbk n -- k k=O with the orthogonal
property
nm- k k ~ (~ + m-1)(~ + k - (-1)k-nbm-~ =Sm k:0 in Section 4.2 comes from the Schauder bases (fi) (X i) and the power series h = B.
- n = ((1 ~y)eyi), (g
i) =
4.5. Riordan Arrays. Identity 25 was first discovered using the theory of the Lagrange group and the Riordan group. We interpret this theory in terms of our language. Given g E R and f in the maximal ideal rn of R, write
g(x)
ao + aiX + a2X~ +...
(ai ~ ,~),
f(X)
2+.. biX .+ b2X
(bi ~ ~).
ResidueMethods in Combinatorial Analysis
307
Wedefine the value g(f(X)) of g at f with respect to X by g(f(X)) = ao + al(blX 2 +’") + a2( bl X + b2X2 + "" )2+.. .. Let R0 be the subset of R consisting of all invertible elements. With respect to X, for f, g E R0, we define f ¯ g = f(X)g(Xf(X)). It is easy to check f , l~ = l~ , f = f f, (g, h) = (f , g) for all f,g, h E R0. Given f ~ R0, let Y = X f, we represent 1If as power series in Y: 1 -- = ao + alY + a2Y2 +’" + (aie n).
f
In the theory of the Lagrange groups and the Riordan groups, X and Y are treated as dummyvariables. Replacing Y by X on the left hand side of the above equation, we define (with respect to X) ] = ao + alX + a2X2 + ." +. It was shown that f*f=f*f=l,~. Hence (R0, *) is a group, which we call the Lagrange group. Given f E R0 and g E _R, let Y = X f, we represent g as power series in Y: g = bo + b~Y + b2Y2 + ... + (hi ~ ~). Replacing Y by X, we define 8(f) : bo ÷ b~X ÷ b2X2 ÷... ÷. As one of the main theorems in [89], it was proved that (19)
8(~)(Xf(X))
= g(X)
and (20)
g(Xy(X)) = 8(:)(X).
In our language, where X and Y are not dummyvariables but satisfy the relation Y = Xf(X), identities (19) and (20) are conceptually trivial: Identity (19) simply states that the same elements 8(f)(Y) g(X)are r epresented in different ways. Identity (20) should be stated in the equivalent form (21)
g(Y-f(Y)) = 8(:)(Y).
Note that X = Xf(X)y(Y) = Y-f Henceidentity (21) also states that the same elements 8(f)(Y) g(X)are represented in different ways.
Huang
308
A Riordan array is a pair (g, h), where g, h E Ro. Given two Riordan arrays (gl, hi) and (g2, h2), we define (22)
(gl, h~) ¯ (g2, h2) (g~(X)g2(Xh~), h~h2)
It is straightforward to check that (g, h) ¯ (1~, ix) = (1~, 1~) (g, h) = (h) ((gi, hi) * (g2, h2)) * (g3, h3) = (gi, hi) * ((92, h2) (g, h), (~(~1), ~) = (~(-~1), ~). (g, h) for all Riordan arrays (g,h), (gl,hl), (g2, h2) and (g3, h3). Hence the of Riordan arrays together with the operation (22) forms a group, which we call the Riordan group. Were~nark that our definition of the Riordan group, following [89], is essentially the sameas the original definition in [85], although in slightly different guise. Weremark also that the group structure of Riordan arrays was found in [77, p. 43] in terms of Sheffer operators using the umbral calculus. A Riordan array (g, h) determines an infinite lower triangular matrix (cn~), where c~k ~ ~, by the relation ~3
Xn Cnk
¯
n=0
For instance, the Riordan array (1/(1- X), 1/(1- X)) determines the triangle
o 0 o
l!
100... 2 1 0 ... 3 3 1 ...
The product of the Riordan group is in fact comingfl’om product of matrices: Given two infinite lower triangular matrices (Crake) and (Cn2k2) determined by Riordan arrays (gl,hl) and (g2, h2), the product (Cnik~)(c,~2k2) is determined by the product (gl, hi) * (g2, h2). Given an infinite lower triangular matrix (Cnk) defined by the Riordan array (g, h) and a power series f = ao + al X -b a2X2 + ""
(ai ~ n),
the generating functions of the column of Clo cH 0 c20 c2~ c22
0 0
C3.0. 531. C32.
C33.
... ...
al a2 3
"’’. ".
ResidueMethods in Combinatorial Analysis
309
is g(X)f(Xh(X)). From the viewpoint of the Riordan group, this fact with a suitable choice of f gives rise Identity 25, see [89, Theorem3.1]. 5. DIAGONALSOF RATIONALPOWERSERIES The residue methods in previous chapters are restricted to power series rings. In this chapter, we extend the use of cohomologyresidues to Laurent series. Recall that the abelian group of Laurent series with coefficient in an abelian group R, denoted by R[[X, X-1]], consists of all elements of the form
aiX
e
together with termwise addition. Even though R is a ring, multiplication in R[[X, X-1]] in general can not be defined. For instance, we can not multiply ¯ .- +X-2 +X-1 + 1 +X +X2 +.-. to itself. For Laurent series obtained from diagonals of rational powerseries, we will show how to apply residue methods. Let a be a field and f = ~’~=oaijXiY j e g[[X,Y]]. We define the diagonal diag(f) of f to diag(f) := ai i Tie a[ [T]]. i=0 If we embeda[[X,Y]] into the Laurent series a[[T]][[X,X-1]] by fixing g, sending X to X and Y to TX-~, then diag(f) is just the constant term f regarded as a Laurent series in X whosecoefficients are power series in T. The quotient field a(X, Y) of g[X, Y] is contained in the quotient field a((X,Y)) of a[[X,Y]]. A power series f in a[[X,Y]] is called rational if it, as an element in g((X, Y)), is contained in a(X, Y). A power series in is called algebraic if it is algebraic over g(T). It can be shownthat diag(f) is algebraic, if f is rational. For ~ equal the field of complexnumbers, this can be seen by a contour integration. For we have 1 / T.dX (23) diag(/) f( X, -~) --~ under suitable analytic conditions. By the analytic residue theorem, T 1 = ~ resx f(X,-~)-~, X=X(T) where resx denotes the analytic residue and the summation is taken over those singularities of f(X,T/X)/X with the condition limT-~O X(T) = Since f(X, T/X)/X is rational, all these residues are algebraic, hence so is diag(f) [19]. See also [38] for more details. A complete different proof, using power series in noncommutingvariables, can be found [18]. There is also a proof, related to factorizations for Laurent series, given in [22]. (24)
diag(f)
310
Huang
If ~ has positive characteristic, although the diagonal of a rational power series (in fact in manyvariables) is algebraic, the above proof can not carried over, since the integral (23) is no longer available. The purpose this chapter is to replace (24) by a sum of cohomologyresidues in arbitrary characteristic. In this sense, the above analytic process becomesalgebraic. For co~nbinatorial significant of diagonals of rational powerseries, the reader is referred to examplesin [92, §6.3]. 5.1. Generalized Power Series. group
Let ~ be a field.
~{T~} := {E aiTilai i~Q
Wedefine an abelian
e ~}
with termwise addition. For an element f = ~-]~ieQ ai Ti in a{TQ}, we define the support of f to be suppf := {i E Q]ai ~ 0}. Wedefine ~[[TQ]] :: (f e ~(TQ} [supp f is well-ordered}. Recall that a set is well-ordered if every subset of which has a smallest element. Wecall an element in ~[[TQ]] a generalized power series in T with coefficients in ~. Werecall some well-known facts about generalized power series, all of whichare due to Hahn[35], see [36, p. 445-499]. Lemma5.1. Let I and J be well-ordered subsets of Q and k ~ Q. Then (1) I U J is well-ordered. (2) I + J = {i + j l i e I, j ~ J} is well-ordered. (3) i + j : k has only finitely manysolutions (i, j) with i ~ I and j E J. Weleave the proof of the lemmato the reader as an exercise. The reader mayalso find a proof in a more general context in [69, Part 3, 13, Lemma 2.9]. With this lemma, we maydefine addition and multiplication of generalized power series so that n[[TQ]] becomesa ring: E a~T~ + EbiTi i~Q iEQ (EaiTi)(Eb~T~) j~Q ieQ
= E(ai + bi)Ti’ i~Q )T : k" E( E aibi k~Qi+j=k
There is a valuation v defined on ~[[TQ]]: v(E hit i) = minimal/with i~Q (v(0) = oc by convention). Proposition 5.2. ~[[TQ]] is a field.
ai ~ 0
311
ResidueMethods in Combinatorial Analysis
Proof. Weneed to show that any non-zero generalized power series has an inverse. A non-zero generalized power series can be written as aTi(1 - f) where 0 5~ a E ~,i E Q, and v(f) > 0. For each rational number j, the coefficieuts of TJ in 1, f, f2,.., are eventually zero, so 1 + f + f2 + ... can be defined termwisely. One can check that the support of 1 + f + f2 +... is well-ordered [69, Part 3, 13, Lemma 2.10]. It is noweasy to verify that the generalized power series a-iT-i(1 + f + f2 + ... ) is the inverse of aTi(1 f). [] The field of generalized power series is quite large. If ~ is algebraically closed, r~[[TQ]] contains an algebraic closure of ~((T)). For ~ with characteristic zero, this follows from the Newton-Puiseuxtheorem, which says that the algebraic closure of ~((T)) is isomorphic o~
U If the characteristic of ~ is a positive numberp, the Artin-Schreier polynomial Zp -Z -T-1 has no root in the field [.Jn~__l ~((T~)), [11, p. 64]. In Men-Fon Huang’s unpublished PhDthesis (written under Abhyankar) and also in [74], it is shownthat the generalized power series with support S satisfying the following property form an algebraically closed field. ¯ There exists an m ~ N such that every element of mS has denominator a power of p. For instance, the Artin-Schreier polyno~nial has a solution X : T-1/p + T-1/p2 "~ T-1/p3 "~ ’’" , [1]. Therefore, independent of the characteristic, g[[TQ]] contains an algebraic closure of ~((T)). This implies the following proposition. Proposition 5.3. If ~ is algebraically closed, then ~[[TQ]] is a spitting field of~((T)). 5.2. Algebraicity of Diagonals. Theorem5.4. Let ~ be an algebraically closed field and f(X, Y) be a rational power series with coefficients in ~. Write T g f ( g, -~ ) = -~ , where g, h ~ ~[X, T]. Then diag(f)=E ]
p
res [ Xh gdX
’
P
where the summation is taken over those points p with positive (including infinite) valuation.
312
Huang
Proof. (cf. [22, Theorem 6.1]) We work on the abelian series ,~{TQ}[[X, X-ill and its subsets: ~[[TQ]](X)
Laurent
~{TQ}[[X,X-’]]
C
C ~((X))[[TQ]]
group c,f
,~[[T]][[X, X-I ]]. Here the inclusion ~[[TQ]](X) C r;((X))[[TQ]] underlying fields of generalized power series
is defined by first
extending
~[[TQ]]C ~((X))[[TQ]] and then maps X to X. As an elementin ~[[TQ]](X), g
fl
-~ = ~o+ yoX+ x~+ ~ (x :~)~’ k=l
where f0 e a[[TQ]][X], f~ e a[[TQ]][X] has degree less than e0, ~ ~ a[[TQ]], ek e N and 0 ~ ~ e n[[TQ]] ({~ is not necessarily distinct). Rename the as hi,... ,am, ill,"" ,fin, where v(ai) > 0 and v(~j) ~ O. As an element in a((X))[[TQ]], m n g bj fl aiX_C~
~ = ~o+ foX+xe°
i=1 j:l
where hi, bj e a[[TQ]] and ~, dje N. As an element in n{TQ}[[X,X-~]], ~ = (~o + b~ +... h
n
+ bn) + (foX + ~ bjdjfl~lx ~=~
+(fix -e° + ~ aiX -c~ + negative
positive po
wer te rms)
power terms).
i:1
As diag(f) is the constant term of f regarded a[[T]][[X,X-i]] or in n{TQ}[[X,X-~]], we have diag(f) Nowwe calculate valuation. res
either
as an element
= ~o + bi +... + bn.
the residues
of gdX/(Xh) at those points with positive
gdX ] X h o
res [ gdX
:
-c~. --ai(--o~i)
Therefore diag(f)
= res
gdX + Xh o i=,
res
Xh
~i ’
ResidueMethods in Combinatorial Analysis
313
that is, the summation of the residues of gdX/(Xh) at the points with positive valuation. [] We remark that besides the embedding a[[TQ]](X) C ~{TQ}[[X,X-1]] used in the above proof, there is another natural embedding described as follows. ~[[TQ]](X) -+ ~[[TQ]]((X)) ~[[TQ]][[X,X-I]]--~ ~{TQ}[[X,X-1]]. For this embedding, 1/(1 - T/X) maps to X (_~)2
_
(_~)3
T ’ while, for the embedding used in the proof, 1/(1 - T/X) maps to 1+-~+(
)2+....
Corollary 5.5. Let ~ be a field and f(X, Y) be a rational power series with coefficients in ~. Then diag(f) is algebraic. Proof. Wemayassume that ~ is algebraically closed, since diag(f) is algebraic over ~(T) if and only if it is algebraic over Z(T). Write f(X,
T X)
g
where g, h E a(T)[X]. To prove the corollary, it suffices to show that the residue of gdX/(Xh) is algebraic over a(T) at any zero a of Xh. This is clearly true, since this residue is contained in a(T)(a) and a is algebraic ~(T). Example 5.6.
Let 1 1 -X-Y"
Then T 1 -1 f(X,-~)-~ = X2 Z--~ + Let a and fl be the roots of X2 - X + T (in ~[[TQ]]). Since aft = T, the equality v(a) + v(f~) = 1 holds. Without loss of generality, we mayassume that v(a) >_v(fl). Then v(a) 1/2. Using th e re lation a + fl = 1, it is easy to see that v(fl) = and th e constant te rm offl is 1. Now we compute the residue of the point with positive valuation: res
X~_X+T
~=res
~_~
~-
~_~.
The relation (~ - ~)~ = 1 - 4T verifies the algebraicity of diag(f). If ~ characteristic ~ero, ghen fl-~=~-dT
314
Huang
and diag(f)-
fl
a n=O
Note that the above equality holds for ~ with arbitrary characteristic, Weremark that the roots of X2 - X + T have different 2nforms depending ~x~ T on the characteristic of n. If char n = 2, the roots are ~-~u=0 and 1 + ~-~n°~=0T2n. If char n ~ 2, the roots are (1 ± x/q- - 4T)/2. Example 5.7. Let 1 /= 1 -X -Y-XY" Then T 1 -1 f(X, 2+(T-1)X+T" ~)X= Let a and/3 be the roots ofX2 + (T- 1)X +T. Using the relations a+fl = 1 - T and aft = T, it is easy to derive (as in the last example) that one the roots, say a, has positive valuation, the other root fl has zero valuation, and the constant term of fl is 1. Hence 2+(T-1)X+T
diag(f)=res
~-fl-a"
If n has characteristic zero, oo 1 1 2fl - a --x,/1 - 6T + T n-- E D(n, n)T n:O for integers D(n, n) called the central Delannoy number. Independent of the characteristic, diag(/)
= E D(n’n)Tn" n:0
For rational power series of more than two variables, the diagonal maybe not algebraic. For instance, in characteristic zero, the diagonal
of 1
1 __ (
-X-Y-Z
E i+j+k i,j,k>_O i, j, k xiyj
zk
is not algebraic. See [100, Theorem3.8] for a more general result. However, for fields with positive characteristic, muchmore is true. It has been shown that the diagonal of algebraic powerseries of manyvariables is still algebraic, if the underlying field has positive characteristic [13]. See also [86].
ResidueMethods in Combinatorial Analysis
315
6. CLASSICAL NUMBERS Manyclassical numbers can be represented by residues. Newinsights of certain numbers are discovered via computations on these representations. For instance, using the idea of complete sums and other classical results, explanations of various identities involving the Bernoulli numbersand polynomials were found in [49]. See Section 6.1 for somedetails. The phenomenonof complete sums also occurs in the Fibonacci and Lucas numbers. In Section 6.2, we use the idea of complete sums to explain an identity involving the Fibonacci and Lucas numbers. In this chapter, we also treat the Euler and Genocchi numbers (Section 6.3), the Stifling numbers (Section 6.4), and the Catalan numbers (Section 6.5). Webelieve that many knownresults can be proved via our approach and some new results may also come out from there. In this chapter, we assume char a = 0 unless otherwise stated. 6.1. Bernoulli Numbers. The Bernoulli numbers B~n) of order n are defined by B~n): res [ i!(~T-~-~ )UdT TM T ]" Wewrite B~1) simply as Bi. The Bernoulli numbers originate from evaluating sums of powers of consecutive integers. The following well-known identity can be proved using residues [49, Identity 1]. Identity 26. For n >_ 1 and m >_ 2, 1n + 2 n T"" -t-
(m - 1) n : E n + 1 Bn+l_imi. n+l i=1 i
The Bernoulli numbersalso occur in special functions (see Subsection 7.5) and trigonometric functions. As in calculus, the following identities hold in formal power series rings. tanX
:
~-~( -1 )n ~(1-~_ 4n)4nx2n-i, n----1 (2o
XcotX
-4) n=O
XcscX
=
-1
~ t -
¯
Identities involving the Bernoulli numbershave attracted attention of generations of mathematicians. In [49], the following theorem is used to explain various identities involving the Bernoulli numbers.
316
Huang
Theorem of Complete
il
Sum for
Bernoulli
Numbers.
[49,
Theorem
1]
il,"" ,im>_O +...Tim :n
= res[n’fjl(T’e-r-~-l)Tn’+flJ,~(T’eT-~-l)dT], where polynomials f j E Z[U, V] are defined inductively
as follows:
fo:=V and, for j > O, 5 := u--O-~
+ ( V - UV - v )
The main idea of [49] regarding the Bernoulli (1)
(2)
SUMS
WHICH
FROM
COMPLETE
COMPLETE
WE
THE IN
SUMS
To illustrate Identity
ARE OF
BERNOULLI TERMS
INTERESTED
IN
ARE
HANDLE,
AS
THE
EASY
THE
TO
BERNOULLI
NUMBERS
OF
TOO
OF
BERNOULLI
THEY
NUMBERS
HIGHER
OF
ORDERS
NUMBERS
this idea, we prove some classical
27 (Euler).
NOT
below. FAR
OF
ARE
INTEGRAL
HIGHER
ORDERS.
CAN
LOWER
BE
WRITTEN
ORDERS.
results.
For n >_ 2, rt--1
i=1
Proof. 2n
Recall [64, Equation (15)] that, for n, i _>
Bi(n+l)
(25) So we can lower orders
= (1 i
- )Bi - o-i-1.
of the Bernoulli
numbers:
B~2~) - 2B2,~ = -(2n + 1)B2~.
Identity
28 (Sitaramachandrarao
and Davis).
! B2a B2b B2c _ (n + 1)(2n E (2n).(-~a)! (2b)! a,b,c>0 a+b+c:n
AWAY
SUMS.
COMBINATIONS (3)
ARE
numbers is listed
[87] For n >_ 3, 1 - 1) B2n-2. + 1)B2n ~n(2n
317
Residue Methods in Combinatorial Analysis
Proof. By the inclusion-exclusion principle,
a,b,c>0 a+b+c=n
(2n)!(2a)!(2b)!(2c)! B2oB2bB2c
~
i,j,k
i,j,k>_O i+j+k--n
(
~,b~O
2n
)BiBjBk_3
2n( 2~ 2,1,1)B2n_2B1B1
2n \ ]B2a B25 Bo + 3B2,~BoBo. 2a, 2b, 0]
By theorem of complete sum for Bernoulli numbers, 3 ~ B2a B2b B2c E (2n). (~~a)! (2b)! - B~) --~n(2n - 1 )B2n-2 a,b,c>O aTbTc:n
) - 3B~2n
+ 3B2n.
Loweringorders of Bernoulli numbers(see identity (25)), we ~ B2a B2b B2c a,b,c>O aTb+c=n
(2b)!
3 = -(n + 2)B~2n) - 2n~(2)~’2n-I -- -~n(2n- 1)B2n-2 + 3/~2n 1 = (n + 1)(2n + 1)’2n -~n( 2n - 1)B2n-2. Identities 27 and 28 can be generalized to sum of products of more Bernoulli numbers, see [14], [81], [87], and [104]. Our methodapplies not only to these generalizations but also to identities found by .other sophisticated methods. For instance, we are able to prove the following identity which was proved by H. Rademacher[72] using Eisenstein series and by M. Eie [17, Proposition 1] using zeta functions, see [49, Identity 3]. Identity 29. For n > 4, ~-2 ~ (2p2~ - 2)! 2pB2p 2n(2n-_~)~p B2n_2p = (2n + l)(n6n 3) p----2 The idea employed to the Bernoulli numbers in this section can be extended to the Bernoulli polynomials. See [49] for details. 6.2. Fibonacci and Lucas Numbers. Let a be a field of arbitrary characteristic. The numbers defined by a recurrence of the r-th order can be described by residues. Supposethat the sequence a0, a~, a2,.-- in ~ satisfies the recurrence of the r-th order: an = O~lan-1 -~ o~2an-2 ~- " " " ~- O~ran-r
318
Huang
for n _> r, where al,O~2,...
,a r E ~ and ar ~ 0. Let
f = ao + alX + a2X2 +... be the generating relation
function
of the sequence ao, al,
(1 - aiX - a2X 2 .....
Then there
is a
oq_lao for i > 0. The numbers
Xr-1r (~AJ bo + biX + ... + br_l olrX .. ] X2 1 - a~X - 0~2 -z .... xn+~
an ---- res
2.7, the residue
of the meromorphic differential
bo + b~X + ...
:= (1
a2,....
= bo + blX + "" + r-~, br-lX
arX")f
where bo = ao and bi = ai - cqai_~ ..... an can be recovered by residues:
By Proposition
~ a:[[X]]
--
oqX
-
0~2 X2
+ r-1 br_lX .....
n+ldX o~rXr)X
at the infinity vanishes. By the residue theorem, au can be expressed in terms of the residues of Wn at the zeros of 1 - a~X - a2X2 xr. ..... Olr Now we concentrate on the case r = 2 and al = a2 = 1. Let u and v be the zeros of 1 - X - X2 in an algebraic closure of n. Note that u and v are distinct from zero, moreover uv = u + v = -1, (u - v) 2 = 5. If charn ¢ 5, then u ~ v. If char n = 5, then u = v = 2. The Fibonacci numbers Fn and the Lucas numbers Ln defined below are of this type.
ao =0 andal
= 1,
ao = 2 and al = 1. Identity
30. 1
= (,,
+ -
1
if char ~ ~ 5; if char ~ = 5.
Residue Methodsin Combinatorial Analysis
319
Proof. In this case, Wn = dX/(X n-Xn+l-Xn+2). char a ~ 5, then the residue of Wnat u is res
(U+u-v)(U+ U
-- (u- v)u ~ res -1 ~res - (u-v)u -
(u-v)u -1
nres
Let U := X-u. If
1 ] u
dU
[
( u-v
+ 1)(~- + ~ U ,~ U2 UU2 (1-~_~+(~z-/) +...)(1-~-+(~) U [
+..-
)ndU ] u
g u
Similarly, the residue of con at X- v is -1/(v n+l -uvn). Besides X, these are the only points at which con mayhave non-zero residues. Thus the residue theoremgives rise to the required identity. If char ~ = 5, the residue Of wn at u is res (U +r’~
(-~ + 2U1)~
= -anres --3 n res
2 U
=
-3n res
dU U2 U
3n+ln.
The residue theorem gives rise again to the required formula. Note that, if the characteristic Fibonacci numbers as
of ~ is neither 5 nor 2, we can write the
Using the residue theorem, one can also interpret the Lucas numbers: Identity 31. Ln =
2-u (u -- V)it n+l, -3
2-v n+l q- n+l (V ’-- It)V
if
char ~ ~ 5;
if char t~ = 5.
320
Huang
If the characteristic bers as
of n is different
Compare Identities 30 for Fibonacci numbers, we see easily that (26)
from 2, we can write the Lucas num-
numbers with Identity
31 for Lucas
Ln = 2Fn+l - Fn
for n with arbitrary
characteristic.
Theorem of Complete Sum for n >_ m-1 and champ5. Then
Fibonacci
Numbers.
Assume
that
il ,... ,i,n>_o il
~-.’"~im=n
[(m-I)/2] n-m
i=o
~_, i=0
n-m+2i+ n -- rn
~ --n-m+2i+l
-1 l
o," o .(
\ rn --
n-m+2i+l
+ Fn-m+2i+3).
Proof¯
~ il
res [
1-X-X. "Z XZm
~1,’" ,imp0 T’"~-im:n
(FqXq)’"(Fi’~-l Xim-~ ) dXl_x_X2xn ]
Q,...,im_~>_0
= res
The residue
(l_X_X2) m dX
of the meromorphic differential w := Xn_m+l(1
1 _ X - m X2)
dX
Residue Methods in Combinatorial Analysis
at
321
U:=X-uis
(-l)m (u +-~)~-~+: (u +,~-~,)~
res
rn U
[ =
(-1)rn
un-rn+:(u~ - v)
drr] u res
U
m--1
- uu-m+l(u - v)rn ~ -- a=0
n-
:
n-m+k n - m
2
m-~- 1
m
-2 1
Iu~(u- v)
-1 5m-~-:un-m:+:(u
~+1" - v)
The residue of w at X - v is
~
n-m+k
-1
~- m
k=O
k m- 1
~-~-~v~-~+~+~(v- ~)~+~"
If ~ = 2i, then 1 un-m+k+l(u
1 --
v)k~ 1 + vn-rn+k+l(v
__ u)k+l
5i
If k = 2i + 1, then 1
1 + ~-m+~+~(~_ ~)~+~~n-~+~+~(~_ ~)~+~ =
+ 5i+lun-m+k+l(u
--
V)
5i+lvn-m+k+l(v
1
= g+~(F~_~+~+ F~-m+~+~). Since the residue of w at the infinity is zero, the required identity follows form the residue theorem. Since any Fibonacci numbers can be written as linear combinations of two given consecutive Fibonacci numbers, for small m, we may find a and b in terms of n such that E Fi:...
Fire = agn-m+: + bgn-m.
il ,"" ~im~_O il+"’+im:n
See [105] for more details. The followings are special cases. Identity 32. [55, subsection 1.2.8. (17)] Assume that char~ ¢ 5. For n_>l, n ~-~ n- 1 2n F~Fn_~ 5 Fn+~-Fn-~. i=0
322
Huang
Identity
33. [105, Corollary 1.(ii)] E
Assume that charn ¢ 5. For n >_ 2,
5n2 - 9n - 2 Fn_l + 5n2 - 3n - 2 Fn_2" 50 50
FiFiF~=
iTjTk=n
Identity
34. [105, Corollary 1.(iii)] Assume that char~ ¢ 5. For n >_ 3, 4n3 - 12n 2 - 4n + 12 Fn-2 + na - 2n2 - n + 2 Fn-3. i+j+k+£=n
=
If char ~ ¢ 5, we may compute sums which are not too far &way from being complete. Identity
35. [41]
(27)
E Li+IFn-i i:0
: nFn+~.
Proof. If char a = 5, n
n
E ni+lFn-i
= E 3n+3(n - i) ----
i=O
Now assume char~ ¢ 5. By (26), terms of the Fibonacci numbers:
Therefore sums:
3n+4n(n + 1) = nFn+ 1.
i=O
i=0 we see that
we can write
the summation in (27)
i=0 i=0 the summation is not too far away from the complete
n
n+2
n+l
E Li÷lFn-i :2 E fiFn+2-ii=o i=O By Identity 32, n 4n+8_ 2n + 2F, EL~+IFn-~ = g - ,~+2 + ----~-~/~+~ i=0
E FiFn+l-ii=0
2Fn+l"
n - gF,~+,
2n + 2Fn -2Fn+~. 5
Simplify it, we get the required identity. 6.3. by
Euler
and Genocchi
Numbers. k2ex
[]
The Euler ~
numbers
Ek are
defined
X
+ 1 k=O - Ek . The trigonometric numbers:
function
sec X can be represented
secX= ~=0
in terms of the Euler
(2k)!"
323
Residue Methodsin Combinatorial Analysis
All odd Euler numbers are zero. One way to see this is by the variable change Y = -X: ,~.e~--x-~+~,~.~ = res xn+l
En = res
dY = (_l)nEn. n!(-1)n~ yn+l
HenceE2n+I = 0 for n _> 0. It should be noted that, in some literature, Euler numbers are defined differently by k~ X tan X + sec X = ~ Ek k!
the
k=0
The Genocchi numbers Gk are defined by k2X
ex+l-
~
ak
X
Mos~odd Genoechi numbers are zero. This can be also interpreted variable change Y = -X: Gn
:
=
by the
res
xn [ n,~dX ] ~ ~ e~ + ~ dY res g~
res
yn
- res
= ~2(-1) ~-~ res
yn
y~ + (-1)~a~.
Therefore G~n+~= 0 for ~ 2 1 as the case for the Bernoulli numbers. Using another variable Z = 2X, we see furthermore a relation between the ~enocchi and Bernoulli numbers: B2n
=
res
~
r~8
~2n+1 ~2nX2n+2 2n+l
1 -
1
(2~)
2~ 1~ (B~G~,~+ B~nG1).
As B~ = -1/2 and G~ : 1, we get G2n = 2(1 - 22n)B2n.
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Huang
This relation certainly holds for all Genocchi and Bernoulli numbers, that is On = 2(1 - 2n)Bn for all n _> 0. This result will be used in Identity 38 in the next section. 6.4. Stirling defined by
Numbers. The Stirling (log(1
numbers of the first
kind s(n, k) is
+ X)) k ~
~ : ~ s(n,k)n~ X.
k!
n=0
The Stifling numbers of the second kind S(n, k) is defined by yn
(eY~l"- 1)k -7. ~, ~ s(n,~) n=O
It should be noted that, in some literature, kind is defined differently by
the Stifling numbersof the first
x~ (_~o~0 _ x))~ ~~(n,~) = ~ ~. ~ n:0
The Stifling numbers of the first and the second kinds are related by the relations Y = log(1 + X) and X = y - 1 in~[[ X]] = ~[[ Y]]. The se rel ations give rise to the following two fundamental identities of the Stirling numbers. Identity 36. m
~S(~,~)~(n,~)=~. Proof. 5kin
:
res -~.YkdY ~m+l
: Eres
m!s(n,k)-~ ym+~"
n:0
n:0
Using the same method, we get Identity 37. m
~ s(m,~)S(n,~)= nm0
The Stifling numbersare closely related to the Bernoulli numbers as the following identity indicated. Identity 38. [56, p. 585] E S(n’k)k~(-~
k=0
)k -n +2 1(1-2n+~’B ] n+l.
325
Residue Methodsin Combinatorial Analysis
Proof. n
]
)~(~x n!(-½ - 1)~dX F~k s(n,~)k!(-~)= ~res Xn+l k=0 k=0
-
res
xn+l
Gn+l
_
n+l 2 (1 -- ¯2n+l)Bn+l n+l
The following is an analogue of Identity 26. Identity 39. For m >_ 1, lm+2m+’"+nm=E
~,(n
"\k+
S(~n,~).
Proof.
k=l
+ n)dX
This application is suggested by Peter Jau-Shyong Shiue. 6.5. Catalan Numbers. The Catalan numbers Ci are defined as follows: Co := 1 and Cn := CoOn-1 + C1Cn-2 q- "" ~- Cn-lCo
forn>O.
Let
Y = CoX + C1 X2 + C2X3 -~- .... Then X = Y- y2 and hence Y equals (1 + ~/]--L-~-~)/2 or (1- vf~ - 4X)/2. Since the constant term of Y as a power series in X is zero, one sees that Y = (1 - v/~ - 4X)/2. Hence Cn =res [ (1-v~-4X)dX2xn+2 ]= [ ].res
-v~2;2+X2dX
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Huang
Identity
40.
~,i~,... ,ira_>0 Proof.
Let Y be as above. Then
X~ c~, c~ ... i~ ,i2,"",im>_O il +i2+’"+im=n res
E
--
[
il ,i2,"" ,im- l ~_O il+i2+’"+im-1 <_n
E il ,i2,""
:
res
X -il-i2 ..... Ci~512"’’Cim,
i-m- 1+2 1Ydx]
[
i’~+1’’’ CQXQ+Ici2x Cim ~xi’~-~+IydX " xn+m+l -
res ,ira-1 ~0
xn+m+l
]
.
Since dX = (1 - 2Y)dY, res
ymdX
xn+m+l
]
= res
(i’Ly)n+m+l ~-yn+m+l
n --n
The well-known
n-1
+ mm (2n+m-l).n
formula n+l
is a special case. The following identity the Catalan numbers. Identity
41 (Touchard’s
gives another recurrence relation
formula). [n/2~ 2n-2k ( ) 2k k=O
C~= C,~+~.
for
Residue Methodsin Combinatorial Analysis
327
Proof. [~/~]
[,~/~] 2k
k=O
C~ =
~ res
X~_~+~
k=O
res [2~+~(I+X)~_o~(~)~+ldX]X -- ~+~ res[
~
re8
.
xn+3
Use ~he new variable Z = XI(2 + 2X). Then 1 - 2Z = 1/(1 + X), 1 ~ =
(1 - 4z)/(1 - 2z)~ ~d~Z= ~X/(~(1+ x)~). res
Xn+a
=
res
---
res ~ Cn+l.
n+a 2Z [ -~4ZdZ n+3 2Z
]
Another recurrence relation can be also obtained fi’om our method: Identity 42. [52] [(n-1)/2] E k=0
2n-2k-lCk:Cn+l’k+2 nT3(n-1) 2k
Proof. [(,~-l)/~] E k+2 k=O
2k
n+3(n--1)2n-2k-lCk
[(n-1)/2] n+3(1_ ’ ~ +x) n-12n-2k-lCkdX
:
~
res
xn-2k
k:0
=
res[
(n+3)(l+X)n-12n+3~-]f~:oCk(’~’-~)k+2dX] Xn+4 k÷2
To compute
~:o Y75()~+2’
.
328
Huang
we first compute ~ Ck Xk+2. k=O
Note that
: ~x
21-(1
k=O
-
x/~
-
4X)"
Therefore
f = a + ~X + (1 - 4X)~ for somea E n. As the constant term of f as a powerseries in X is zero, a = -1/12 and hence o~ Ck (X2)k+2 1 2 ~2 (1 X2 )~ E ~’--~ T" =-1-~ + + " k=O
Let Z = X/(2 + 2X) as in the proof of the last identity. Then [(n--I)/2]
2n ~ -2k-~Ckk+2 n+3(n;1) k=O ~
res
n+4
X [ (n+3)(I+X)n-12n+s(-~+~X~+~(1-X2)~)dX]
~ t~ +a)(1
-
res
+X)~-~(1-X~)~dX Xn+4
:
res[n+4 ~(n+3)(1-4Z)~dZz
-
Zn+4
_ -_
] res
½(n + 3)v~-4ZdZ n+3 Z
]
n + 3 Cn+~+ + 3) Cn+l 6 3 1(2n+4~ 2(n + 3) (2n + 6\n+2]+3(n+2)\n+l] l (2n + 2~ n+2\n+l]
~- Cn + l .
7.
POLYNOMIAL SEQUENCES
Let n0 be an algebraically charn0 = 0 unless otherwise
closed field. In this chapter, stated. Cohomology residues
we assume that work not only
329
ResidueMethods in Combinatorial Analysis
for identities involving elements in ~0 but also for identities involving polynomials or even power series over ~0. The idea for extending applications of cohomologyresidues from scalars to power series is simple: Let t~ be an algebraic closure of the quotient field of ~0[[X]]. The residue map, which is defined on the top local cohomologymodule of an exterior product of the universal finite differential moduleof ~[[Y]] over ~, treats power series as scalars. The methods of cohomologyresidues for scalars thus carries over to powerseries. In Section 7.1, we derive closed formulae for special polynomials defined by generating functions. Examples include some Sheffer-type polynomials and some Gegenbauer-Humbert-type polynomials. Sheffer-type polynomials have been treated by Roman and Rota using the method of the umbral calculus [77], [78], [79] and by Hsuand Shiue using cycle indicator [41]. Cycle indicator is also used to study some Gegenbauer-Humbert-type polynomials [ibid.]. Being a new computational tool in the theory of special polynomials, we give a new closed formula for the Lerch polynomials in terms of the Bernoulli numbers. Section 7.6 consists of identities involving powerseries. Examplesinclude addition theorems for the Laguerre and Hermite polynomials, sum formula for the Laguerre polynomials, and an identity for power series. 7.1. Closed Formulae. Sheffer-type polynomials Sn E ~0[[X]] are defined by f eXg=ooE-~.Sk yk k=0 for somef, g E ao[[Y]] satisfying res
y
¢ O,
res
y = O,
res
y2
¢ O.
Simplify the representation Sk = res
][eX~dY k’fyk+l using properties of generalized fractions and residues, we get closed formulae for Sheffer-type polynomials. See Subsection 7.2 for the case of the Laguerre polynomials and Subsection 7.3 for the Hermite polynomials. Another type of power series sequences in a0[[X]] that we are interested in is Pn(f ; m, X, o~, ~) defined by (1 - ’~, mXY + aYm)-~ f = E Pn(I;m,X,a,~3)Y n----O
where f ~ ao[[X]][Y], m ~ N, and o~,~ e t~0. Note that Pn(1;m,X,o~,fl) are known as Gegenbauer-Humbert type polynomials. We remark that Gegenbauer-Humbert type polynomials with small m have been treated by
330
Huang
Hsu and Shiue using cycle indicator [41]. Cohomology residues tool for representing Pn(f ; m, X, a, 1~). For instance, Pn(1;m,X,O,/~)
=
res
provide a
yn+l -~ ][ dY (1-mXY)
n+l- mXY)-~d(mXY) = res L [ (rex)n(1(mXY)
"n = (mX)n(n+fl-1) Ifa ~ 0 and fl E N, then P,~(f;m,X,a,~) w :=
is the residue
(1 - n+l mXY + o~ym)~Y
of
dY
at Y. Using the residue theorem, Pn(f; rn, X, a, fl) can be represented in terms of the residues of w at the infinity and at the roots of the equation ym
mXy
1
See Subsections 7.4 for the case of the Tschebyscheff polynomials. Besides Sheffer-type and Gegenbauer-Humbert-type polynomials, treat Lerch polynomials in Subsection 7.5. 7.2. Laguerre polynomials. Let ~ be an element ~) Laguerre polynomials L~ E a0[X] by ~o L(C~) -xY/(1-Y) e
E k=0 -~-’~ Y k
~ L~ a)rk ~=0
a) L~
= res :
res
We define
the
k_
(1 -- y)a+l
a) It should be noted that, in some literature, are defined differently by
Using Z = Y/(1 - Y), a closed obtained:
in a0.
we also
the Laguerre polynomials L(~
-X - Y/O-Y) e(1 - y)a+l
formula
for the Laguerre
~-xv/(~-~),~v a+l (l-Y) yn+l
[
[ n’e-XZ(l zn+l + Z)n+adZ
polynomials
is
Residue Methods in Combinatorial Analysis
331
7.3. Hermite polynomials. Let ~ be an element in r~0. We define the Hermite polynomials H~~) E r;0[X] by ~x~
k
Z..,~k k=0
k! -
It should be noted that, in some literature, are defined differently by oo
the Hermite polynomials ~(a)
k
K-" ~(a) Y’~ ~ k~ -
_
e2XY-ay2 "
k=0
There is a closed formula for the Hermite polynomials obtained by residues: ~) H~
=
res
n~eXY-~ dy yn+~
=
res
~=o
y~+~
~ dY
(x
-
yn-~+~
~ res
=
-
¯
7.4. Tschebyscheff polynomials. In this subsection, the field ~0 has arbitrary characteristic. Let Tn be the Tschebyscheff polynomials of the first kind defined by X’-’ nTnY n:0
1 - XY 1 - 2XY + y2"
There is a closed formula for the Tschebyscheff polynomials obtained by residues and the residue theorem: Tn is the residue of the meromorphic differential 1 -XY wx := yn+l(1 _ 2XY + y2) dY at O. Let u be a solution of the equation 1 - 2XY + Y~ = 0 in Y. Then v = 1/u is also a solution. If the characteristic of ~0 is not 2, then u ~ v. The residues of wa at u and v are (1 - Xu)/(u n+2 - un+lv) and (1 - Xv)/(v n+2 - uvn+l), respectively. Since the residue of Wl at infinity is
332
Huang
zero, by the residue theorem, we get Xu- 1 Xv - 1 + it n+:(~ -- V) Vn÷l(V -U2 -- 1 v~ - 1 + n+l n+l 2U (U -- V) 2v (v -- U)
Tn -
= Let f E n be a solution of the equation y2 _ X2 + 1 = 0 in Y. Then [n/2](nlxn-2~(X2_l)g" 2
k even
~=0
If the characteristic of no is 0, the equation y2 _
2Y cos X + 1 = 0
has solutions eiX and e-IX. Hence 1. niX Tn(cosX) + e-niX) = cos nX. If the characteristic of n0 is 2, then ~T~y~_ k=O
I --+ --XY (1 + XY)(1 +y2 +y4 -I----). 1 +Y~
Hence 1, if n is even; Tn= X, ifnisodd. Nowassume the characteristic of n0 is 0. Let Uu be the Tschebyscheff polynomials of the second kind defined by ~ YV~ ~ - X ~--~ UnYn : 1 - 2XY + y2" n:0
Un is the residue of the meromorphicdifferential
vff- x2
w2 := yn(1 _ 2XY + y2) dY at 0. Let u v
2, = X+iv/~-X 2= X-iv/~-X
Residue Methods in Combinatorial Analysis
333
be the solutions of the equation 1 - 2XY + y2. The residues of w2 at u and v are v~ - X2/(u n+l - unv) and x/~ - X2/(v n+l - uvn), respectively. Since the residue of w2 at infinity is zero, by the residue theorem, we get ~ -X2 ~ - X 2 1 Un - ?.tn(v _ ~t) q- vn(~t - v) 2i (un- vn)"
In more concrete terms,
k odd [(n-l)~2]
= vfi--X2
( ?~ )xn-2g-I(x2
Y~
2g+l
g=O
-
Note also that 1 .
niX
Vn(cosX) = i(e
e-niX)
=
sinnX.
7.5. Lerch Polynomials. Let Pn be the Lerch polynomials defined by ~-~ p~ yn = n=O
(1 - X In(1 + Y))~"
Let Z := X ln(1 +Y). For n _> 1, a closed formula for the Lerch polynomials is obtained by residues:
where B[n) are Bernoulli numbers defined in Section 6.1.
334
Huang Given a, fl E a0, the value of L(~~) at/~ is
7.6. Identities. i(~cO(fl)
= ~ k=O
Identity 43 (Addition Given al,’" ,a~,~,...
(_fl)k
O_y).+~ yn+l
= res
dY
Theorem for the Laguerre Polynomials). ,~ ~ ~o,
~
il~"" i~,... ,i~0 ~i~ il +’"+ik=n
~
*~
’ [67, p. 8g]
"~ "
= r~(~+".+~+~-~(~ +... ~) Proof.
e-BkY/(1-Y)
O-y),~+l
We remark that the above formula is called and a2 = -1 [77, p. 31]. Identity
the Sheffer identity
44 (Sum Formula for the Laguerre Polynomials).
E+ L}~-n)yn’n~= (1 + Y)ae-xz.
Proof. It suffices to show that [(1+
Y)ae-xYdy yn+l
if k = 2
[67, p. 85] Given
~,~0
a-n) L(n n! - res
dY
¯
Residue Methods in Combinatorial Analysis
335
Let Z = Y/(1 + Y), Then res
[ (1 + Y)ae-XYdy yn+l
e_xz/(~_z) (~_Z)~+~_~dZ ]
] = res
= L(na-n)
Givena,/~ e n0, the value of H(na) at fl is H(~a)(fl):=
n-n,(k) k (--~ n
= res
an_k2k_
n!eZY--~yn+l
dy "
It is easy to see that, for a,~,7 e ~0, eZ~Y-~ yn+l
dY ] ’
Identity 45 (Addition Theorem for the Hermite Polynomials). [67, p. 82] Givena, at,... ,O~k,fll,’" ,flk,"~,"Y1,’’" ,’~1¢ E nOsatisfying
and 7 # O, then
Proof.
il,... ,i~>_0 il +’.’+ik=n
n ) 7~ .... i~,...,ik
The special case for k = 2, 3’
ik H(a~)~1 ~) H}: ,~ i~ ~’)""
= 71 : 72 :
1 can be found in [77, (4.2.1)].
336
Huang
Identity 46. [16, §2.2, Example3] Let ~o be an algebraically closed field with characteristic different from 3 and ~ be a primitive cubic root of 0. 1~ The identity
E
n 3k
X3k=
1
1 1 + + n+~ n+l 3(1 - X) 3(1 - ~X) 3(1 - ~2X)n+J
k=O holds in ~o[[X]]. Proof.
= res
1- X3(yn++lY)3dY
The residues of the meromorphicdifferential n(1 + r) (1 - X3(1 7~+IdY Y)3)Y
1,
at (i/x)-
1, and are -1/(3(1-X) --1/(3(1--
tJ2X)n+x), -1/(3(1 - ~X)n+x), and 0, respectively. follows from the residue theorem.
The required identity []
8. FUTUREDEVELOPMENTS This article arises in finding applications of concrete realizations of the theory of Grothendieck duality. We have used the residue map for power series ring over a field and the residue theorem for the projective line in combinatorial analysis. It would be interesting to investigate how other concrete aspects of Grothendieck duality can do in combinatorial analysis. In this chapter, we give somepossible directions for future research. 8.1. Congruences Modulo Prime Powers. In [44], the residue map for power series ring over a field is generalized as follows: Let A be a complete Noetherian local ring, M be an A-module with zero dimensional support (that is, every element of Mis annihilated by some power of the maximal ideal of A), B be a power series ring of n variables over A, and n be the ~naximal ideal of B. There is an A-linear map (28)
res:
H, (A ~/A @A
which in terms of a regular system of parameters X~,... fying res
,Xn of B/A satisi~=1;
yil yin [dX~...dX,~]
{~,
ifi~
.....
337
Residue Methodsin Combinatorial Analysis
for all A E M. It is shownin [48], for constructing residual complexes in a very general context, the residue mapsof the above type are sufficient. Consider the case where A is the ring of p-adic integers Zp~zand Mis an injective hull of the residue field of A given by Q/ZpZ.Elements of A are of the form ao + alp + a2p2 + a3p3 + "" , (0 <_ai < p). For ql E Z and 0 ¢ q2 ~ Z, we denote by q2 the equivalence class of Mcontaining ql/q~. It is easy to see that elements of Mare of the form
The module structure of .4 on Mis given by
Congruences modulo prime powers can be interpreted involving elements in M:
in terms of identities
a = b (rood pn) if and only if pn =
¯
Since Mis an .4-module with zero-dimensional support, one may use the residue map (28) to prove congruences modulo prime powers. Interesting congruences might occur as the residue map (28) is independent of choices of variables. One may also take advantage of the residue theorem to obtain global information. 8.2. Residue Theorem for Plane Curves. Let A~ be a locally Noetherian scheme and ¢: y -~ 2¢’ be a morphism of finite type whose fibers have bounded dimensions. For each point p in X, we assign it an injective hull M(p) to the residue field of the local ring Op,x of p on A’. Given a residual complex A/t" on ,~’ - with additional information that how A4" is built up by the M(p)’s, we can construct explicitly - with additional information that how¢!.t/[" is built up by some injective hulls canonically constructed from the M(p)’s [48]. Furthermore we can construct explicitly a morphism of Ox-modules try/z:
¢.¢!M"
[43]. Wecan also prove the residue theorem, which asserts that try/x is a morphismof complexesif ¢ is proper [ibid.]. As the case of the projective line, the residue theore~n in this general context should contain combinatorial information. However the structure of ¢!A/[° is subtle, see [46, Example3]. A simple case which still contains
338
Huang
wealthy data is plane curves. An elementary theory of residues on the projective plane is given in [46], which includes a canonical residual complex, a trace map, the residue theorem, and intersection numbers. A residual complex and the residue theorem for a plane curve can be derived ea~sily from this theory. Some meromorphic differentials on plane curves, for instance on elliptic curves, might give interesting identities through the residue theorem and residue maps.
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