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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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
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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].