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Modules with Injective Cohomology, and Local Duality for a Finite Group New York Journal of Mathematics New York J. Math. 7 (2001) 201–215. Modules with Injective Cohomology, and Local Duality for a Finite Group David Benson Abstract. Let G be a finite group and k a field of characteristic p dividing |G|. Then Greenlees has developed a spectral sequence whose E2 term is the local cohomology of H∗(G, k) with respect to the maximal ideal, and which converges to H∗(G, k). Greenlees and Lyubeznik have used Grothendieck’s dual localization to provide a localized form of this spectral sequence with respect to a homogeneous prime ideal p in H∗(G, k), and converging to the ∗ injective hull Ip of H (G, k)/p. The purpose of this paper is give a representation theoretic interpreta- tion of these local cohomology spectral sequences. We construct a double complex based on Rickard’s idempotent kG-modules, and agreeing with the Greenlees spectral sequence from the E2 page onwards. We do the same for the Greenlees-Lyubeznik spectral sequence, except that we can only prove that the E2 pages are isomorphic, not that the spectral sequences are. Ours converges to the Tate cohomology of the certain modules κp introduced in a paper of Benson, Carlson and Rickard. This leads us to conjecture that ∗ ∼ Hˆ (G, κp ) = Ip , after a suitable shift in degree. We draw some consequences of this conjecture, including the statement that κp is a pure injective module. We are able to prove the conjecture in some cases, including the case where ∗ H (G, k)p is Cohen–Macaulay. Contents 1. Introduction 202 2. Depth 204 3. The Greenlees spectral sequence 205 4. The last survivor 208 5. The kappa modules 210 6. Dual localization 211 7. Comparison 212 References 214 Received February 17, 2001. Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 13D45, 13H10. Key words and phrases. Modular representation, idempotent module, Tate cohomology, local cohomology, injective module, Grothendieck duality, spectral sequence, Cohen-Macaulay ring, Matlis duality, depth, Koszul complex. The author is partly supported by a grant from the NSF. ISSN 1076-9803/01 201 202 David Benson 1. Introduction Let G be a finite group and k be a field of characteristic p. By a theorem of Evens and Venkov [10, 23], the cohomology ring H∗(G, k) is a finitely generated graded commutative k-algebra. Further restrictions on the structure of H∗(G, k) can be expressed in terms of the existence of certain spectral sequences. The first of these was constructed by Benson and Carlson [2] using multiple complexes and related finite Poincar´e duality complexes of projective kG-modules. It follows from the existence of this spectral sequence that if H∗(G, k) is Cohen–Macaulay, then it is Gorenstein. Even if H∗(G, k) is not Cohen–Macaulay, the existence of the spectral sequence gives severe restrictions on the possibilities for the ring structure. An example of an application of this spectral sequence is the statement that if H∗(G, k) is a polynomial ring, then it follows that p = 2, the generators are in degree one, and G/O(G) is an elementary abelian 2-group. Greenlees [11] found a way of using the same techniques coming from multiple complexes to construct a cleaner spectral sequence expressing much of the same information. This spectral sequence can be written as ∗ ∗ Hm H (G, k) ⇒ H∗(G, k). Here, m denotes the maximal ideal generated by the elements of positive degree in ∗ ∗ H (G, k), and Hm denotes local cohomology in the sense of Grothendieck [13, 14]. Greenlees and Lyubeznik [12] showed how to apply dual localization to this spectral sequence, with respect to a homogeneous prime ideal p of H∗(G, k). This leads to a spectral sequence whose E2 page is ∗ ∗ Hp H (G, k)p and which converges to the injective hull, in the graded sense, of H∗(G, k)/p as a H∗(G, k)-module, suitably shifted in degree. The first purpose of this paper is to provide a construction of the Greenlees spectral sequence using Rickard’s idempotent modules [22]. This construction is essentially a stabilized version of a construction which Carlson and Wheeler [8] introduced in the context of complexity quotient categories. The construction is given in Section 3, where the following theorem is made more precise. Theorem 1.1. There is a double complex consisting of homomorphisms from pro- jective kG-modules to Rickard idempotent modules, whose spectral sequence is iso- morphic to the Greenlees spectral sequence, from the E1 page onwards. We give an interpretation in terms of this spectral sequence of the “last survivor” method of Section 7 of [2]. The result is a slightly stronger theorem, the proof of which can be found in Section 4. ∗ Theorem 1.2. Let ζ1,...,ζr be a homogeneous set of parameters for H (G, k), r and let ai = deg(ζi)(1≤ i ≤ r). Then there is an element of degree i=1(ai − 1) ∗ in H (G, k) which is not in the ideal (ζ1,...,ζr) and is not annihilated by any nonzero element of k[ζ1,...,ζr]. In Section 5, we provide an analagous construction of a spectral sequence with the same E2 page as the Greenlees–Lyubeznik dual localized spectral sequence. We have been unable to prove that these two spectral sequences are isomorphic, Modules with Injective Cohomology, and Local Duality 203 although this seems very likely. Our spectral sequence converges to the Tate coho- ∗ mology Hˆ (G, κp) with coefficients in the module κp introduced in introduced by Benson, Carlson and Rickard [3] (in that paper, the module is called κ(V ), where V is the closed homogeneous irreducible subvariety of the cohomology variety VG corresponding to p). The modules κp are interesting because they are idempotent, in the sense that ∼ κp ⊗ κp = κp in the stable module category, and tensoring with κp picks out the “layer” in the stable module category corresponding to the prime p ⊆ H∗(G, k). If it could be proved that the two spectral sequences are isomorphic, then it ∗ ∗ would follow that Hˆ (G, κp) is an injective H (G, k)-module. This would allow the modules κp to be compared with the modules T (Ip) introduced by Benson and Krause [5]. The second purpose of this paper is to investigate some special cases in which we are able to make this comparison. We give evidence for the following conjecture. Conjecture 1.3. Let p = m be a homogeneous prime ideal of dimension d in ∗ −d H (G, k). Then T (Ip) and Ω κp are stably isomorphic. The modules T (Ip) are the representing objects for the contravariant exact func- tor which takes a module to the Matlis dual of its Tate cohomology, in the sense that there are functorial isomorphisms ∗ ∼ ∗ DpHˆ (G, M) = ExtkG(M,T(Ip)). In particular, the T (Ip) are pure injective modules, and their cohomology is given ∗ ∼ ∗ by Hˆ (G, T (Ip)) = Ip, the injective hull of H (G, k)/p. The reason for the interest in the above conjecture is that what we know about −d the properties of T (Ip) and of Ω κp have little overlap. That a single module should have all these properties seems to have be significant. It seems likely that there are many consequences for the interactions between commutative algebra and modular representation theory. So for example, it would follow from this conjecture that κp is pure injective, so ∗ ∼ that there are no phantom maps into κp. It also follows that Hˆ (G, κp) = Ip[d], and ∗ ∧ that Endk(κp)isakG-module whose Tate cohomology is the completion H (G, k)p of the localized cohomology ring with respect to the powers of its maximal ideal. Another consequence of the conjecture is the formula ∼ −d T (Ip) ⊗k T (Ip) = Ω T (Ip) in the stable module category. ∗ We prove the above conjecture in the case where H (G, k)p is Cohen–Macaulay of Krull dimension r−d, and also in the case where the depth is r−d−1 and the Krull dimension is r − d. In particular, since any localization of a Cohen–Macaulay ring is Cohen–Macaulay, the conjecture holds whenever H∗(G, k) is Cohen–Macaulay. The proof of the following theorem can be found in Section 7. Theorem 1.4. Let G be a finite group of p-rank r, and let k be a field of charac- teristic p. Suppose that p = m is a homogeneous prime ideal in H∗(G, k) of Krull ∗ dimension d, such that the localization H (G, k)p satisfies one of the following con- ditions: 204 David Benson ∗ (∗) H (G, k)p is Cohen–Macaulay of Krull dimension r − d (i.e., its depth is also equal to r − d),or ∗ (∗∗) H (G, k)p has depth r − d − 1 and Krull dimension r − d. Then the following hold: ∗ ∗ (i) The cohomology Hˆ (G, κp) is injective in the category of graded H (G, k)- modules; in particular, it is isomorphic to Ip[d]. −d (ii) The modules T (Ip) and Ω κp are stably isomorphic. (iii) The modules κp are pure injective. (iv) There are no phantom maps into κp. ∗ ∼ ∗ (v) The ring ExtkG(κp,κp) = Hˆ (G, Endk(κp)) is isomorphic to the completion ∗ ∧ ∗ H (G, k)p of the localized cohomology H (G, k)p with respect to the maximal ideal pp. ∼ −d (vi) The formula T (Ip) ⊗ T (Ip) = Ω T (Ip) holds in the stable module category. ∗ ∗ If H (G, k) is Cohen–Macaulay, then (∗) holds for H (G, k)p.
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