PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP DAVID BENSON AND HENNING KRAUSE 1.Introduction Let G be a finite group and k be a field of characteristic p. The connection * *between the modular representations and the cohomology of G has been a subject of major interest ov* *er the last several years. For example, recent study of the stable category of kG-modules StMod(kG* *) led Rickard [27] to introduce certain idempotent modules and functors. This has led to a th* *eory of varieties for infinitely generated modules [3, 4], and the classification of thick subcat* *egories of the stable category stmod(kG) of finitely generated modules, at least in the case of a p-g* *roup [5]. The cohomology of the group G is intimately related to the properties of the * *trivial kG-module k and its shifted copies nk, n 2 Z. It is therefore natural to ask for a descript* *ion of the kG-modules which arise as a direct summand of a (possibly infinite) direct product of modu* *les of the form nk, n 2 Z. Note that such a module is always pure injective because any module * *over a finite dimensional algebra is pure injective if and only if it is isomorphic to a dire* *ct summand of a direct product of finite dimensional modules. Pure injective modules have been studied for some time in representation theo* *ry of finite di- mensional algebras, mostly because certain infinitely generated pure injectives* * (so-called generic modules) control the representation type of an algebra [11]. For example, gener* *ic modules have been used to show that the representation type of an algebra is an invariant of* * the stable module category [20]. More recently, the notion of a phantom map in StMod(kG) was introduced in [16* *, 6] as an analogue of a classical concept from stable homotopy theory. In this context, * *it became appar- ent as well that pure injective modules play an important role, as the modules * *which receive no phantom maps. In particular, under rather restrictive hypotheses, some of Ricka* *rd's idempotent modules were observed to be pure injective. This paper grew out of an attempt t* *o understand this phenomenon. The purpose of this paper is to investigate a certain functor T from injectiv* *e modules I over the cohomology ring H*(G; k) to pure injective modules in the stable category StMod* *(kG). Most of the arguments work when kG is replaced by any finite dimensional cocommutative * *Hopf algebra A over k, and so this is the generality in which we work, although at some stages* * we need to make an assumption about the behavior of negative Tate cohomology which we only know* * to be true in the finite group context. Namely, either Tate cohomology is periodic, or negati* *ve Tate cohomology is nilpotent, in the sense that there exists an integer n > 0 such that every p* *roduct of at least n elements is zero. We do not know of an example of a finite dimensional cocomm* *utative Hopf algebra for which this fails. ___________ Date: November 1999. The first author is partly supported by a grant from the NSF. 1 2 DAVID BENSON AND HENNING KRAUSE The construction of the functor T is described in Section 3, and is basically* * given by applying Brown representability to obtain an isomorphism Hom H*(A;k)(H^*(A; M); I) ~=Hom_A(M; T (I)); functorial for M in StMod(A). On injective modules I having no simple submodule* *s, the functor T is fully faithful, and this allows us to understand the endomorphism ring of * *T (I). We begin in Section 2 by classifying the injective modules over the Tate coho* *mology ring H^*(A; k). We prove that under the condition that either Tate cohomology is pe* *riodic or nega- tive Tate cohomology is nilpotent, every injective module over Tate cohomology * *is coinduced from an injective module over the ordinary cohomology ring. A module of the form T (* *I) is determined by its Tate cohomology which is the coinduced module ^I= Hom *H*(A;k)(H^*(A; k)* *; I). This explains the relevance of coinduced modules for our study of the functor T . Under the assumption on Tate cohomology mentioned above, the modules of the f* *orm T (I) are precisely the direct summands of direct products of modules of the form nk, n 2* * Z. This is proved in Theorem 4.7. We show in Section 5 that the modules T (I) are pure injective,* * and usually but not always -pure injective. In Section 6, we show how Theorem 4.7 can be interp* *reted in terms of spectral categories. This establishes a complete classification of the modul* *es which arise as a direct summand of a direct product of modules of the form nk, n 2 Z. In Section 7, we specialize to the case of a finite group, and describe the v* *ariety in the sense of Benson, Carlson and Rickard [4] of the module T (I) for every indecomposable in* *jective module I. This is essentially a question of translating a proof from Hovey, Palmieri and * *Strickland [18] into the language of group representation theory. In Section 8, we investigate T as a map from the projective variety Proj(H*(A* *; k)) to the Ziegler spectrum Ind(A) of indecomposable pure injectives, endowed with the Zariski top* *ology. We prove that this map induces a homeomorphism between Proj(H*(A; k)) and its image in I* *nd(A). The image of a point of Proj(H*(A; k)) is a generic module in the sense of Crawley-* *Boevey [11] if and only if the point is a generic point for a component of the variety, or in othe* *r words, it corresponds to a minimal prime in H*(A; k). An interesting consequence is the fact that A is o* *f strongly unbounded representation type whenever ^H*(A; k) is nonperiodic and the field k is infini* *te. This is a special case of the Second Brauer-Thrall Conjecture which asserts that every finite dim* *ensional algebra over an infinite field and of infinite representation type has strongly unbound* *ed representation type. We show in Section 9 that in case A = kG, the generic modules corresponding t* *o minimal primes are, up to translation, the modules investigated in our recent paper [7]. More * *generally, it turns out that for a nonmaximal prime ideal p of H*(G; k), the module obtained by app* *lying T to the injective hull of H*(G; k)=p is closely related to the corresponding Rickard id* *empotent module (V ) introduced in [4]. The proof of this involves a detailed analysis of the Greenl* *ees-Lyubeznik spectral sequence [17], and will be the subject of a separate paper. Conventions and notation. When we talk about modules and maps over a Z-graded r* *ing R, we always mean graded modules unless otherwise specified. If M and N are R-modules* * and n 2 Z, we write M[n] for the graded module with M[n]i= Mi+n. We write Hom R(M; N) for the* * degree zero maps, and Hom *R(M; N) for the graded maps. In other words, Hom nR(M; N) = Hom R(M[n]; N): If R is graded commutative, then Hom *R(M; N) is a graded R-module in an obviou* *s way. PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 3 Acknowledgements. It is a pleasure to thank Jon Carlson and Jeremy Rickard for * *their interest in this subject. Jon Carlson provided the idea for the proof of Proposition 2.4* *, and Jeremy Rickard provided the direct proof of Lemma 4.3. The modules T (I) were first defined by Hovey, Palmieri and Strickland [18], * *definition 6.0.8 (c), but they are not systematically investigated in that paper. 2. Injective modules over the cohomology ring Before introducing and investigating the functor T , we need to study the inj* *ective modules over the ordinary cohomology ring and also over the Tate cohomology ring. In this se* *ction we classify the injective modules over the Tate cohomology ring, under a hypothesis which w* *e know to hold in the case of the group algebra of a finite group. Let A be a finite dimensional cocommutative Hopf algebra over a field k. We d* *enote by k the trivial A-module and the cohomology ring H*(A; k) is by definition Ext*A(k; k).* * This is a finitely generated graded commutative k-algebra by a theorem of Friedlander and Suslin [* *13]; in particular, it is a Noetherian ring. We write Mod (A) for the category of all A-modules and homomorphisms, and mod* *(A) for the full subcategory whose objects are the finitely generated A-modules. We write S* *tMod(A) for the stable category, which has the same objects as Mod(A), but where the homomorphi* *sms are given by Hom_A (M; N) = Hom A(M; N)=PHom A (M; N) where PHom A(M; N) denotes the subspace of Hom A(M; N) consisting of maps which* * factor through a projective module. This is a triangulated category, in which the triangles co* *me from short exact sequences in Mod (A). The translation in this category is obtained by taking t* *he cokernel of an injective hull, and is written -1. Its inverse, , is obtained by taking the ker* *nel of a projective cover. We write stmod(A) for the full subcategory of finitely generated modules* * in StMod(A). Given an A-module M, the Tate cohomology ^H*(A; M) of M is defined as the coh* *omology of the complex Hom A(P^*; M) where ^P*: . .-.! P1 -! P0 -ffi!P-1 -! P-2 -! . . . is a complete resolution of the trivial module, namely an exact sequence of pro* *jective modules such that Im(ffi) ~=k. The Tate cohomology of M is the same as the graded homomorphi* *sms k ! M in the category StMod(A). More precisely, H^n(A; M) ~=Hom_A(nk; M) for all n 2 Z. The nonnegative part P* of ^P*is a projective resolution of the * *trivial module, and the canonical map ^P*! P* induces a homomorphism H*(A; M) ! ^H*(A; M) since H*(* *A; M) is the cohomology of the complex Hom A(P*; M). This gives an exact sequence of H*(* *A; k)-modules 0 -! PHom A(k; M) -! H*(A; M) -! ^H*(A; M) -! ^H-(A; M) -! 0 (2.* *1) where ^H-(A; M) denotes the negatively graded part of Tate cohomology. In parti* *cular, the coho- mology ring H*(A; k) can be viewed as a subring of the Tate cohomology ring ^H** *(A; k). The elements of positive degree in H*(A; k) form a maximal ideal m = H+ (A; k* *) and we have H*(A; k)=m ~=k. By Tate duality (see for example Cartan and Eilenberg [10], sec* *tion XII.6), the negative cohomology ^H-(A; k) is the k-dual of H*(A; k), and is hence isomorphi* *c to the injective 4 DAVID BENSON AND HENNING KRAUSE hull of k as an H*(A; k)-module. The negative Tate cohomology ^H-(A; M) of any * *A-module M is an m-torsion module, in the sense that given x 2 m, and a 2 ^H-(A; M), there ex* *ists n 1 such that xna = 0. This is because elements from m have positive degree while elemen* *ts from negative Tate cohomology have negative degree. If M is a module over H*(A; k), then we obtain a module over ^H*(A; k) by the* * recipe M^ = Hom *H*(A;k)(H^*(A; k); M): (2.* *2) Note that the coinduction functor which sends M to ^Mis right adjoint to restri* *ction from ^H*(A; k)- modules to H*(A; k)-modules. Therefore, it is left exact and takes injective H** *(A; k)-modules to injective ^H*(A; k)-modules. We denote by oM :M^ ! M the canonical H*(A; k)-mod* *ule homomor- phism given by evaluation on the identity. Lemma 2.1. Let I be an injective H*(A; k)-module and suppose Hom *H*(A;k)(k; I* *) = 0. Then oI:I^! I is an isomorphism of H*(A; k)-modules. If M is another H*(A; k)-modul* *e then the natural map HomH*(A;k)(I; M) -! Hom ^H*(A;k)(^I; ^M) is an isomorphism. Proof.Every quotient of ^H-(A; k) is m-torsion and every nonzero m-torsion modu* *le has a submod- ule isomorphic to k[n] for some n. Therefore we have Hom *H*(A;k)(H^-(A; k); I) = 0: So applying Hom *H*(A;k)(-; I) to the exact sequence (2.1)(with M = k) gives th* *e first part. The second part follows easily because one gets an inverse map HomH^*(A;k)(^I; ^M) -! Hom H*(A;k)(I; M) * * __ by sending a map OE to the composite oM OOE O(oI)-1. * * |__| We denote by Im the injective envelope of the H*(A; k)-module k ~=H*(A; k)=m. Lemma 2.2. Let I = Im[n], n 2 Z. Then ^I~= ^H*(A; k)[n - 1] and End ^H*(A;k)(* *^I) ~= k. In particular, the ring ^H*(A; k) is self-injective. Proof.We have Im[n] ~=Hom *k(H*(A; k); k[n]): So by Tate duality we have I^~=Hom *H*(A;k)(H^*(A; k); Hom*k(H*(A; k); k[n])) ~=Hom *k(H^*(A; k); k[n]) ~=^H*(A; k)[n - 1]: In order to compute the endomorphism ring of ^I, observe that End *^H*(A;k)(^I) = End*^H*(A;k)(H^*(A; k)) ~=^H*(A; k): * * __ Therefore EndH^*(A;k)(^I) = ^H0(A; k) = k. * * |__| PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 5 The classification of injective modules over a graded commutative Noetherian * *ring 1 R is well known, and can be found in Bruns and Herzog [9]. An arbitrary direct sum of inj* *ective modules is injective, and every injective module decomposes essentially uniquely as a dire* *ct sum of injective indecomposables. Given a homogeneous prime ideal p, the injective hull Ip = E(* *R=p) of the quotient R=p is indecomposable, and each injective indecomposable is isomorphic* * to a shifted copy Ip[n] for some prime p and some n 2 Z. A theorem of Matlis [23] describes the e* *ndomorphism ring of Ip[n]. It is isomorphic to the completion R^pof the localization Rp with res* *pect to the powers of the maximal ideal pp. Each element of R^pacts locally nilpotently on Ip[n]. In terms of this classification of injective modules, if an injective H*(A; k* *)-module I has no summand isomorphic to Im[n] for any n, then Hom *H*(A;k)(k; I) = 0, and so by L* *emma 2.1, we have I = ^I. Next we shall investigate the injective modules over the Tate cohomology ring* *. Note that this ring is usually not Noetherian. However, in the case of the group algebra of a * *finite group, negative Tate cohomology is usually nilpotent, and this allows us to classify all inject* *ive modules. Lemma 2.3. If G is a finite group of p-rank greater than one, and H is a subgr* *oup of p-rank one, then the restriction to H of any element of nonzero degree in ^H*(G; k) is nilp* *otent. Proof.Let x 2 Hn(G; k). If resG;H(x) is not nilpotent, then it is invertible, * *since H^*(H; k) is periodic. So there exists y 2 ^H-n(H; k) such that resG;H(xTrH;G(y)) = resG;H(x)y = 1: Thus xTrH;G(y) = 1. But H^*(G; k) is not periodic, so this implies that x is a* * multiple_of the identity, and hence has degree zero. * * |__| Proposition 2.4.Let G be a finite group and k be a field of characteristic p. I* *f the p-rank of G is at least two (i.e., H*(G; k) is not periodic) then there exists n > 0 such t* *hat (H^-(G; k))n = 0. Proof.Restriction to a Sylow p-subgroup is injective in cohomology, so without * *loss of generality, G is a p-group. Since G has p-rank at least two, and the center of a nontrivial* * p-group is necessarily nontrivial, it follows that every maximal elementary abelian p-subgroup of G ha* *s rank at least two. If G is elementary abelian, it follows from theorem 3.1 of [2] that (H^-(G; k))* *2 = 0, so we may assume that G is not elementary abelian. The rest of the proof is based on an a* *rgument of Quillen and Venkov [26]. By induction and Lemma 2.3, we may assume that there exists n * *> 0 such that every proper subgroup H of G has the property that (H^-(H; k))n = 0. Then ever* *y element of (H^-(G; k))2nis divisible by the Bockstein fi(x) for each 0 6= x 2 H1(G; Fp), b* *y the Quillen-Venkov lemma. By a theorem of Serre [28], there exist nonzero elements x1; : :;:xm 2 H* *1(G; Fp) such that * * __ fi(x1) : :f:i(xm ) = 0. Therefore (H^-(G; k))2nm = 0. * * |__| For a ring R we denote by Inj(R) the full subcategory of injective R-modules,* * and we denote by Inj0(R) the full subcategory of injective R-modules I satisfying Hom R(S; I) = * *0 for every simple R-module S. For example, Inj0(H*(A; k)) is the full subcategory of injective H** *(A; k)-modules I satisfying Hom *H*(A;k)(k; I) = 0 since every simple H*(A; k)-module is isomorp* *hic to a shifted copy ___________ 1Note that in a graded commutative ring, ab = (-1)|a||b|ba, so that it is not* *, strictly speaking, commutative. However, it is routine but tedious to check that the standard theorems about st* *rictly commutative graded rings apply equally well to graded commutative rings. 6 DAVID BENSON AND HENNING KRAUSE of the trivial module k. If ^H*(A; k) is periodic then we have Inj0(H^*(A; k)) = 0 = Inj0(H*(A; k)) since ^H*(A; k) and H*(A; k) both are graded Artinian. Next we study Inj0(H^*(A* *; k)) in the non- periodic case. Lemma 2.5. Suppose that H^*(A; k) is not periodic. Then k is an H^*(A; k)-mod* *ule and every simple ^H*(A; k)-module is isomorphic to k[n] for some n 2 Z. Proof.We claim that ^H-(A; k) ^H+(A; k) is an ideal in ^H*(A; k). This is beca* *use otherwise, we have elements x 2 ^H-(A; k) and y 2 ^H+(A; k) such that xy = 1. In this case, ^* *H*(A; k) is periodic. Thus k = ^H*(A; k)=(H^-(A; k) ^H+(A; k)) is an ^H*(A; k)-module. Now suppose that S is a simple H^*(A; k)-module. The shifted copies of Im co* *generate the category of H*(A; k)-modules. Using Lemma 2.2, we get therefore Hom ^H*(A;k)(S; ^H*(A; k)[n - 1]) ~=Hom H*(A;k)(S; Im[n]) 6= 0 for some n 2 Z. The ^H*(A; k)-module ^H*(A; k)[n - 1] is indecomposable injecti* *ve with a simple_ socle which is isomorphic to k[n]. Thus S ~=k[n]. * * |__| Lemma 2.6. Suppose that (H^-(A; k))n = 0 for some n > 0. (i)If M is an H*(A; k)-module such that M^= 0 then M = 0. (ii)If N is an ^H*(A; k)-module such that Hom *^H*(A;k)(k; N) = 0 then Hom *H** *(A;k)(k; N) = 0. Proof.Let r be the ideal in ^H*(A; k) generated by ^H-(A; k) and let s = r \ H** *(A; k). Then r is nilpotent by our assumption since the ring is graded commutative. (i) Suppose M is a nonzero H*(A; k)-module. Let j > 0 be the smallest intege* *r such that sj:M = 0. Then given any homogeneous nonzero element x in sj-1:M, say of degree* * m, there is a nonzero H*(A; k)-module homomorphism ^H*(A; k)[m] ! M taking 1 to x. Thus M^6= * *0. (ii) Suppose OE: k[m] ! N is a nonzero H*(A; k)-module homomorphism. Let j >* * 0 be the smallest integer such that rj:Im(OE) = 0. Then given any homogeneous nonzero e* *lement x in rj-1:Im(OE), say of degree m0, there is a nonzero H^*(A; k)-module homomorphism* * k[m0]_! N taking 1 to x. * * |__| Proposition 2.7.Suppose that (H^-(A; k))n = 0 for some n > 0. Then the functor Inj(H*(A; k)) -! Inj(H^*(A; k)) sending I to ^Iinduces an equivalence between Inj0(H*(A; k)) and Inj0(H^*(A; k)* *). Proof.First observe that I 2 Inj0(H*(A; k)) implies ^I2 Inj0(H^*(A; k)). This * *follows from the adjointness isomorphism Hom *^H*(A;k)(k; ^I) ~=Hom *H*(A;k)(k; I); together with the description of the simple ^H*(A; k)-modules in Lemma 2.5. In * *Lemma 2.1 it is shown that the coinduction functor is fully faithful on Inj0(H*(A; k)). Thus it* * remains to show that PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 7 it is dense. To this end let J be a module in Inj0(H^*(A; k)). Let J ! E(J) be * *an injective envelope over H*(A; k) and observe that E(J) belongs to Inj0(H*(A; k)) by Lemma 2.6. The* * composite J -! ^J-! [E(J) is a split monomorphism so that J ~=Ker(") for some idempotent endomorphism " o* *f [E(J). Using again Lemma 2.1, we get an idempotent endomorphism OE of E(J) such that ^OE= ".* * Clearly, * * __ I = Ker(OE) belongs to Inj0(H*(A; k)) and ^I~=J. * * |__| The following theorem gives a classification of injective modules over the Ta* *te cohomology ring, under hypotheses which are satisfied at least in the case of the group algebra * *of a finite group. Theorem 2.8. Suppose that either (i)(H^-(A; k))n = 0 for some n > 0, or (ii)^H*(A; k) is periodic. Then an ^H*(A; k)-module is injective if and only if it is isomorphic to ^I= Hom *H*(A;k)(H^*(A; k); I) L for some injective H*(A; k)-module I. Moreover, if I = 2 I is a decompositi* *on into indecom- posables over H*(A; k) then each ^Iis an indecomposable ^H*(A; k)-module, and M M I^~= ^I E ^I 20 2\0 where 0 denotes the set of 2 such that Hom *H*(A;k)(k; I ) = 0. Proof.(i) Let J be an injective ^H*(A; k)-module, and let S = Hom *^H*(A;k)(k; * *J). Write EH^(S) for the injective hull of S as an ^H*(A; k)-module, so that we have a decomposition J ~=J0 EH^(S) with Hom *^H*(A;k)(k; J0) = 0. It follows from Proposition 2.7 that J0 is injec* *tive over H*(A; k) and that ^J0~=J0. We claim that the other summand, EH^(S), is isomorphic to "EH (S), where EH (* *S) is the injective hull of S as an H*(A; k)-module. The map S ! EH (S) induces a map ^S! E"H(S). * * Compos- ing with the natural monomorphism S ! ^S, we see that there are maps of H^*(A; * *k)-modules EH^(S) ! "EH (S)and "EH (S)! EH^(S) whose composite is the identity on EH^(S). * *So we obtain a decomposition "EH (S)~=E ^(S) L: H Now Hom *^H*(A;k)(k; L) = 0, since Hom *^H*(A;k)(k; "EH (S)) ~=Hom *H*(A;k)(k; EH (S)) ~=S: It follows from the first part of the proof that L is injective over H*(A; k) a* *nd that ^L~=L. The split monomorphism ^L! "EH (S)is of the form ^fffor some map ff: L ! EH (S) by * *Lemma 2.1. We have "Ker(ff)~=Ker(^ff) = 0 and it follows from Lemma 2.6 that ff is a monomorp* *hism. Thus L = 0 since Hom *H*(A;k)(k; L) = 0 and Hom *H*(A;k)(k; U) 6= 0 for every nonzero subm* *odule U of EH (S). 8 DAVID BENSON AND HENNING KRAUSE We now take I = J0 EH (S) to see that ^I~=J. Furthermore, M "EH (S)~=E ^(S) ~ E ^I; H = 2\0 where the set \ 0 corresponds to a k-basis for the graded vector space S. (ii) If ^H*(A; k) is periodic, then m is the only prime ideal in H*(A; k). In* * this case, ^H*(A; k) is graded Artinian, and by Lemma 2.2 it is self-injective. So every injective modu* *le is a direct sum * * __ of modules of the form [Im[n]= ^H*(A; k)[n - 1]. * * |__| Remark. The hypothesis of the above theorem is only used via Lemma 2.6. We end this section with a result which shows that ordinary and Tate cohomolo* *gy are the same whenever we work away from the maximal prime in H*(A; k). Proposition 2.9.Let I be in Inj0(H*(A; k)) and p be a nonmaximal homogeneous pr* *ime ideal in H*(A; k). Then for every A-module M the canonical map H*(A; M) ! H^*(A; M) * *induces isomorphisms ~= * Hom H*(A;k)(H^*(A; M); I) -! Hom H*(A;k)(H (A; M); I) and ~ H*(A; M)p -=!H^*(A; M)p: If I = Ip[n] for some n 2 Z, this can be rewritten as ~= * Hom H*(A;k)(H^*(A; M); I) -! Hom H*(A;k)p(H (A; M)p; I): Proof.Every quotient of ^H-(A; M) is m-torsion and every nonzero m-torsion modu* *le has a sub- module isomorphic to k[n] for some n. Therefore we have Hom *H*(A;k)(H^-(A; M); I) = 0: The H*(A; k)-module PHom A(k; M) is isomorphic to a direct sum of copies of k a* *nd therefore Hom *H*(A;k)(PHom A (k; M); I) = 0: So applying Hom H*(A;k)(-; I) to the exact sequence (2.1)gives the first part. For the second part, observe that p-localization is exact and annihilates an * *H*(A; k)-module X if and only if Hom *H*(A;k)(X; Ip) = 0. So the isomorphism H*(A; M)p ! ^H*(A; M* *)p follows from * * __ the exact sequence (2.1). * * |__| 3. The construction Let I be an injective module over H*(A; k). Then the contravariant functor fr* *om StMod(A) to the category of abelian groups which takes a module M to Hom H*(A;k)(H^*(A; M); I) takes triangles to exact sequences and coproducts to products. So by the contr* *avariant version of Brown representability (which always holds, see Brown [8], Neeman [24], wher* *eas covariant Brown representability depends on extra assumptions), there exists a representi* *ng A-module T (I) satisfying Hom H*(A;k)(H^*(A; -); I) ~=Hom_A(-; T (I)): (3.* *1) PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 9 The assignment I 7! T (I) extends via Yoneda's lemma to a functor T :Inj(H*(A; k)) -! StMod(A): Because of the special role of the modules Im[n] in the relationship between * *ordinary and Tate cohomology, we shall often work under the hypothesis that Hom *H*(A;k)(k; I) = * *0. This does not hurt much because of the following lemma. Lemma 3.1. Let I = Im[n], n 2 Z. Then T (I) ~=-n+1(k). Proof.If M is an A-module, then using Tate duality, and the fact that Im[n] ~=Hom *k(H*(A; k); k[n]); we have Hom H*(A;k)(H^*(A; M); Im[n])~=Homk(H^*(A; M); k[n]) ~=Hom k(H^-n(A; M); k) ~=dExtn-1A(M; k) ~=Hom_A(M; -n+1(k)): * *|___| Next we study some of the basic properties of the functor T :Inj(H*(A; k)) ! * *StMod(A). Lemma 3.2. There is an isomorpism of ^H*(A; k)-modules * Hom *H*(A;k)(H^*(A; -); I) ~=dExtA(-; T (I)): Proof.For i 2 Z and an A-module M, we have Hom iH*(A;k)(H^*(A; M);~I)=HomH*(A;k)(H^*-i(A; M); I) ~=Hom H*(A;k)(H^*(A; iM); I) ~=Hom A(iM; T (I)) ~=dExtiA(-; T (I)): * This isomorphism is functorial in M, and so it is an isomorphism of dExtA(M; M)* *-modules. But * * * __ the action of ^H*(A; k) on each side factors through the natural map ^H*(A; k) * *! dExtA(M; M). |__| Lemma 3.3. There is a functorial isomorphism ^I~=^H*(A; T (I)) as modules over* * ^H*(A; k). Proof.This follows by applying Lemma 3.2 to the trivial module k and using the * *definition (2.2) * * __ of ^I. * * |__| Lemma 3.4. For any A-module M, Tate cohomology induces an isomorphism ~= * * Hom_A(M; T (I)) -! Hom H^*(A;k)(H^ (A; M); ^H(A; T (I))): Proof.Using equation (3.1)and Lemma 3.3, we have Hom_A(M; T (I))~=HomH*(A;k)(H^*(A; M); I) ~=Hom ^H*(A;k)(H^*(A; M); ^I) ~=Hom ^H*(A;k)(H^*(A; M); ^H*(A; T (I)): To check that this map is induced by Tate cohomology, check first that the iden* *tity map on T (I) * * __ goes to the identity map on ^H*(A; T (I)) and then use naturality. * * |__| 10 DAVID BENSON AND HENNING KRAUSE Lemma 3.5. If I and J are injective H*(A; k)-modules then the map Hom H*(A;k)(I; J) -! Hom_A(T (I); T (J)) induced by T factors as the map o*I:Hom H*(A;k)(I; J) -! Hom H*(A;k)(^I; J) induced by the natural map oI:I^! I, followed by an isomorphism ~= Hom H*(A;k)(^I; J) -! Hom_A(T (I); T (J)): (3.* *2) Proof.By Lemma 3.3, equation (3.1)defines an isomorphism as in (3.2). It is eas* *y to check_that the composite is the map induced by T . * * |__| Proposition 3.6.The functor T , restricted to a functor Inj0(H*(A; k)) ! StMod * *(A), is fully faithful. * * __ Proof.This follows from Lemma 3.5 and Lemma 2.1. * * |__| This allows us to calculate the stable endomorphism ring of T (Ip[n]). It is * *the degree zero part of Tate Ext, which is described as follows. * * Corollary 3.7.Let I be in Inj0(H*(A; k)). Then dExtA(T (I); T (I)) ~=EndH*(A;k)* *(I). In particular, * if p 6= m is a homogeneous prime ideal in H*(A; k) and n 2 Z, then dExtA(T (Ip[* *n]); T (Ip[n])) is isomorphic to the completion of the local ring H*(A; k)p with respect to the po* *wers of its maximal ideal pp. Proof.This follows from Proposition 3.6, together with the analysis by Matlis [* *23] of the endomor-_ phism rings of injective indecomposable modules over Noetherian rings. * * |__| Another consequence of Proposition 3.6 is the following lemma which we includ* *e for further reference. Lemma 3.8. Let I be in injective H*(A; k)-module. Then T (I) is indecomposable* * in StMod(A) if and only if I is indecomposable. Proof.If I decomposes, then obviously T (I) decomposes, because T takes nonzero* * H*(A; k)- modules to nonzero objects in StMod(A). On the other hand, if I is indecomposa* *ble, then I is of the form Ip[n] for some n 2 Z and some prime ideal p. If p 6= m then the* * endomorphism rings of Ip[n] and T (Ip[n]) are isomorphic by Proposition 3.6. In the case whe* *re p_=_m, we have T (Im[n]) ~=-n+1(k) by Lemma 3.1. * * |__| Next we investigate the property of T to preserve direct products and direct * *sums. Proposition 3.9.The functor T :Inj(H*(A; k)) ! StMod(A) preserves arbitrary dir* *ect products. The restriction of T to Inj0(H*(A; k)) preserves arbitrary direct sums. Proof.It follows immediatelyLfrom the defining isomorphism (3.1)that T preserve* *s direct products. Now suppose that IL= I is a direct sum of injectives in Inj0(H*(A; k)). In * *order to show that the canonical map T (I ) ! T (I) is an isomorphism it is sufficient to show * *that for every finitely generated A-module X the induced map M OE: Hom_A(X; T (I )) -! Hom_A(X; T (I)) PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 11 is an isomorphism. We have an isomorphism M M : Hom_A(X; T (I )) -! Hom_A(X; T (I )) since X is finitely generated, and the composite OE O is isomorphic to the can* *onical map M ae: Hom H*(A;k)(H*(A; X); I ) -! Hom H*(A;k)(H*(A; X); I) by the definition of T and Proposition 2.9. However, H*(A; X) is a finitely gen* *erated H*(A; k)- module by the main result of [13]. Thus ae is an isomorphism and we conclude t* *hat_OE is an isomorphism. * * |__| 4. A characterization via Tate cohomology We have already seen that for every injective H*(A; k)-module I the Tate coho* *mology of T (I) is precisely ^I. This property can be used to characterize the modules of the f* *orm T (I). Let N be the localizing subcategory of StMod (A) consisting of all A-modules * *N satisfying H^*(A; N) = 0. A module L is said to be N-local if Hom_A(N; L) = 0 for all modu* *les N in N. Lemma 4.1. If I is an injective H*(A; k)-module then T (I) is N-local. * * __ Proof.This follows immediately from equation (3.1). * * |__| Proposition 4.2.Let I be an injective H*(A; k)-module. Suppose that M is an A-m* *odule satis- fying (i)^H*(A; M) ~=^Ias modules over ^H*(A; k), and (ii)M is N-local. Then M is isomorphic to T (I) in StMod(A). Proof.By equation (3.1)and (i), the composite H^*(A; M) -~=!^IoI-!I gives rise to a map M ! T (I). By Lemma 3.4, this map induces an isomorphism ^H*(A; M) -! ^H*(A; T (I)): Complete M ! T (I) to a triangle in StMod(A), M -! T (I) -! X -! -1M: Then the long exact sequence in cohomology shows that ^H*(A; X) = 0, so X is in* * N. Now for any N in N, Hom_A(N; T (I)) = 0 by Lemma 4.1, and Hom_A(N; M) = 0 by * *(ii). Applying the long exact sequence, we see that Hom_A(N; X) = 0. Applying this to X, we se* *e that the_identity_ map of X is zero, so X ~=0. Thus M ! T (I) is an isomorphism in StMod(A). * * |__| Recall that a functor F :StMod(A) ! StMod(A) is a localization functor with r* *espect to N if F (M) is N-local, and there is a natural transformation from the identity funct* *or to F such that for each module M, the map M ! F (M) induces an isomorphism ~= Hom_A(F (M); L) -! Hom_A(M; L) whenever L is N-local. 12 DAVID BENSON AND HENNING KRAUSE Lemma 4.3. There is a localization functor FN :StMod(A) ! StMod(A) with respec* *t to the local- izing subcategory N. Proof.It is true in general that localization with respect to a homology theory* * exists, see for example Margolis [22], chapter 7. The homology theory in question is just Tate * *cohomology. For the convenience of the reader, we also provide a direct proof, for which * *we are indebted to Jeremy Rickard. Consider the localizing subcategory K of StMod(A) generated * *by the trivial module k and observe that N is the category of K-local objects. There exists a * *localization triangle EK (M) -! M -! FK (M) -! -1EK (M) such that EK (M) is in K and FK (M) is K-local [27]. In the formation of the qu* *otient category of StMod(A) by N, each morphism is represented by one of the form M -~ EK (M) !* * N, so the arrows form a set. This proves the existence of the quotient category, and henc* *e the localization_ functor with respect to N via Brown representability. * * |__| Corollary 4.4.Let I be an injective H*(A; k)-module. Suppose that M is an A-mod* *ule such that H^*(A; M) ~=^Ias modules over ^H*(A; k). Then FN (M) is isomorphic to T (I) in * *StMod(A). |___| We are now in a position to describe the image of the functor T . This leads* * to a complete classification of all direct summands of direct products of shifted copies of t* *he trivial module. Proposition 4.5.Let I be an injective H*(A; k)-module. Then T (I) is isomorphi* *c to a direct summand of a direct product of modules of the form nk, n 2 Z. Proof.Any injective H*(A; k)-module I can be embedded as a direct summand of a * *direct product of modules of the form Im[m], m 2 Z. Now use Lemma 3.1 and the fact that T pres* *erves_direct products. * * |__| Proposition 4.6.Let I be an indecompoable injective H*(A; k)-module. Then T (I* *) is finitely generated if and only if I ~=Im[n] for some n 2 Z. Proof.One direction is Lemma 3.1. Now suppose that T (I) is finitely generated.* * The module T (I) is indecomposable by Lemma 3.8 and a direct summand of a direct product of modu* *les of the form nk, n 2 Z, by Proposition 4.5. It follows from a result of Auslander (see coro* *llary 3.2 in [1]) that T (I) ~=nk for some n 2 Z. Thus T (I) ~=T (J) for J = Im[-n + 1], and ther* *efore ^I~=^Jby_ Lemma 3.3. We conclude from Lemmas 2.1 and 2.2 that I ~=J. * * |__| Theorem 4.7. Suppose that either (H^-(A; k))n = 0 for some n > 0, or H^*(A; k)* * is periodic. Then an A-module is isomorphic in StMod(A) to T (I) for some injective H*(A; k)* *-module I if and only if it is isomorphic to a direct summand of a direct product of modules of * *the form nk, n 2 Z. Proof.One direction is proved in Proposition 4.5. For the other direction, fir* *st note that by Lemma 3.1, nk is in the image of T . By Proposition 3.9, T preserves products, * *so an arbitrary product of modules of the form nk is in the image of T . Now suppose that M is a direct summand of a direct product of modules of the * *form nk, n 2 Z. Then ^H*(A; M) is an injective ^H*(A; k)-module by Lemma 3.3. By Theorem 2.8, ^* *H*(A; M) ~=^I for some injective H*(A; k)-module I. Now M and T (I) have isomorphic Tate coho* *mology, and__ are both N-local. So by Proposition 4.2 they are isomorphic. * * |__| Remark. The only way in which the hypothesis of the above theorem is used in th* *e proof is in order to ensure that every injective ^H*(A; k)-module is of the form ^Ifor some injec* *tive H*(A; k)-module I. We do not know whether this is true in general. PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 13 Corollary 4.8.Let G be a finite group and k be a field. Then the conclusion of * *the above theorem holds in the case A = kG. * * __ Proof.By Proposition 2.4, A = kG satisfies the hypotheses of the theorem. * * |__| 5.Purity A short exact sequence of modules is said to be pure exact if it gives a shor* *t exact sequence on applying Hom (F; -) with F a finitely presented module. A module M is said to b* *e pure injective if every pure exact sequence beginning with M splits. In other words, Hom (-; M* *) is exact on pure exact sequences. So for example, every injective module is pure injective. Fo* *r a more detailed introduction to purity, see for example Jensen and Lenzing [19]. A module is said to be -pure injective if an arbitrary direct sum of copies o* *f the module is pure injective. So for example, a ring is left Noetherian if and only if every injec* *tive (left) module is -pure injective. Theorem 5.1. Let I be an injective H*(A; k)-module. Then T (I) is a pure injec* *tive A-module. If Hom *H*(A;k)(k; I) is finite dimensional then T (I) is -pure injective. Proof.We give two proofs of the first statement. The first proof consists of o* *bserving that a module over a finite dimensional algebra is pure injective if and only if it is* * isomorphic to a direct summand of a direct product of finite dimensional modules (see exercise 7.10 of* * [19]). So it follows from Theorem 4.7 that T (I) is pure injective. A second, more direct proof goes as follows. Applying ^H*(A; -) to a pure exa* *ct sequence 0 -! M0- ! M -! M00-! 0 gives an exact sequence, and we get from equation (3.1)an exact sequence 0 -! Hom_A(M00; T (I)) -! Hom_A(M; T (I)) -! Hom_A(M0; T (I)) -! 0: Projective A-modules are injective, and therefore the sequence 0 -! Hom A(M00; T (I)) -! Hom A(M; T (I)) -! Hom A(M0; T (I)) -! 0 is exact. It follows that T (I) is pure injective. Since H*(A; k) is a Noetherian ring, an arbitrary direct sum of injective H*(* *A; k)-modules is injective. By Proposition 3.9, T takes direct sums in Inj0(H*(A; k)) to direct * *sums in StMod(A), which are then represented by the corresponding direct sums in Mod(A). It follo* *ws that arbitrary direct sums of copies of such a T (I) are pure injective. Now if Hom *H*(A;k)(k; I) is finite dimensional, it follows that I is isomorp* *hic to a direct sum of a finite set of modules of the form Im[n] and a module I0 in Inj0(H*(A; k)). It n* *ow follows_from_the previous paragraph and Lemma 3.1 that T (I) is -pure injective. * * |__| Recall from [11] that a module M over a ring R is endofinite if M has finite * *length over its endomorphism ring EndR(M). An indecomposable R-module is generic if it is endof* *inite but not of finite length over R. Now suppose that R is commutative Noetherian. Given a * *prime ideal p, it is not hard to see that the injective envelope E(R=p) of R=p is an indecomposable * *endofinite R-module if and only if p is a minimal prime. This observation explains our next theore* *m. However, the corresponding statement about T (I) is slightly different because of the specia* *l role of the maximal ideal m of H*(A; k), and is as follows. 14 DAVID BENSON AND HENNING KRAUSE Theorem 5.2. Let I be an injective H*(A; k)-module. Then T (I) is an indecompo* *sable endofinite module if and only if I is isomorphic to Ip[n] for some n 2 Z, where p is eithe* *r a minimal prime or the maximal prime m. Proof.By Lemma 3.8, I is indecomposable if and only if T (I) is indecomposable.* * So we only need consider modules of the form Ip[n]. If p = m then Lemma 3.1 shows that T (Im[n]) is an endofinite module, because* * it is even finite dimensional. Otherwise, we use the characterization of endofinite modules given in proposi* *tion 2.1 of [7]. Namely, an A-module M is endofinite if and only if for every finitely generated* * A-module X, Hom_A(X; M) has finite length as an End_A(M)-module. Let p 6= m. Then by Proposition 2.9 we have HomH*(A;k)p(H*(A; X)p; Ip[n]) ~=Hom_A(X; T (Ip[n])): By Proposition 3.6, the endomorphism rings of Ip[n] and T (Ip[n]) are isomorphi* *c, and the above is an isomorphism of modules over this ring. So it suffices to consider the length* * of the left hand side as a module over EndH*(A;k)(Ip[n]). Now the degree zero part of H*(A; k)p is a local ring, and is Artinian if and* * only if p is minimal. The degree n part of Ip[n] is equal to the injective hull of the quotient of th* *is local ring by its maximal ideal. If p is not minimal, then consider the case X = nk. Then the left hand side * *of the above isomorphism is equal to the degree n part of Ip[n], and is not of finite length* * over the degree zero part of H*(A; k)p. Therefore, T (Ip[n]) is not endofinite. If p is minimal, then Ip[n] is a finitely generated H*(A; k)p-module, whose e* *ndomorphism ring is the degree zero part of H*(A; k)p. So for any finitely generated A-module X,* * the left hand side__ has finite length over this Artinian ring. Therefore, T (Ip[n]) is endofinite. * * |__| One reason for our interest in indecomposable endofinite modules is the Secon* *d Brauer-Thrall Conjecture, which asserts that if a finite dimensional algebra over an infinite* * field is of infinite representation type, then it has strongly unbounded representation type. Recall* * that an algebra is of strongly unbouded representation type if, for infinitely many d, there are i* *nfinitely many noniso- morphic indecomposable modules of dimension d. A result of Crawley-Boevey shows* * that strongly unbounded representation type is equivalent to the existence of a generic modul* *e. Corollary 5.3.Suppose that ^H*(A; k) is not periodic. Then there exists a gener* *ic A-module and therefore A is of strongly unbounded representation type provided the field k i* *s infinite. Proof.Take T (Ip) for any minimal prime p. It is an indecomposable endofinite * *A-module by Theorem 5.2 and not of finite length by Proposition 4.6, since ^H*(A; k) is not* * periodic. Thus T (Ip) is a generic A-module. Strongly unbounded representation type for A now follows* * from_theorem 7.3 in [12]. * * |__| We end our discussion of pure injectives in StMod(A) with an alternative cons* *truction of the functor T which does not involve Brown representability. To this end we denote * *by (mod (A)op; Ab) the category of additive contravariant functors mod(A) ! Ab into the category o* *f abelian groups. It is an abelian Grothendieck category which as far as we are concerned means t* *hat it has injective envelopes [14]. Define a functor T 0:Mod(H*(A; k)) -! (mod (A)op; Ab) PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 15 via T 0(M) = Hom H*(A;k)(H^*(A; -); M)|mod(A): One should think of T 0as an extension of T because it sends injective H*(A; k)* *-modules to objects which are injective in the full subcategory (stmod(A)op; Ab) consisting of func* *tors mod(A) ! Ab vanishing on projectives, and the functor StMod(A) -! (stmod(A)op; Ab) which takes N to Hom_A(-; N)|mod(A)identifies the full subcategory of pure inje* *ctive A-modules with the full subcategory of injective objects in (stmod(A)op; Ab). In order to compare T 0with T , we begin with a lemma. Lemma 5.4. For every A-module M there is a functorial monomorphism ffM :Hom_A(-; M)|mod(A)-! Hom k(-; k) A M|mod(A) in (mod (A)op; Ab) which is an injective envelope if M is pure injective withou* *t nonzero projective direct summands. Proof.Auslander-Reiten theory gives for every finitely generated A-module X an * *isomorphism Hom_A(-2X; -) ~=Hom_A(Tr(Hom k(X; k)); -) ~=TorA1(Hom k(X; k); -): We combine this with the monomorphism TorA1(-; -1M) ! -A M which is induced by * *an exact sequence 0 -! M -! P -! -1M -! 0; and this gives ffM . If M is pure injective then the functor FM = Hom k(-; k) A M|mod(A) is an injective object in (mod (A)op; Ab) and every direct summand of FM is of* * the form FN for some direct summand N of M (see theorem 7.12 of [19]). We conclude from the fun* *ctoriality of__ ffM that the map ffM is an injective envelope provided that M is minimal. * * |__| Theorem 5.5. Let I be an injective H*(A; k)-module and denote by E(T 0(I)) the* * injective envelope of T 0(I) in (mod (A)op; Ab). Then T (I) ~=(E(T 0(I))(A)) in StMod(A). Proof.The defining isomorphism (3.1)for T (I) gives an isomorphism T 0(I) ~=Hom_A(-; T (I))|mod(A) and Lemma 5.4 provides an explicit recipe for the injective envelope of T 0(I) * *since T (I) is pure injective by Theorem 5.1. This enables us to get T (I) from the functor T 0(I) * *by_evaluating_E(T 0(I)) at A. * * |__| 6. Spectral categories Let T(A; k) be the full subcategory of StMod(A) consisting of A-modules which* * are isomorphic to a direct summand of a direct product of modules of the form nk, n 2 Z. We ar* *e interested in a classification of the objects in this category. The results of Section 4 show t* *hat the category T(A; k) is closely related to Inj(H*(A; k)) via the functor T . Spectral categories pro* *vide the appropriate framework to express this relationship. For example, the spectral category of a* * ring controls the injective modules and the invertible homomorphisms between them. In this sectio* *n we show that the functor T induces an equivalence between the spectral category of H*(A; k) * *and the spectral 16 DAVID BENSON AND HENNING KRAUSE category of T(A; k) under our assumption on the negative Tate cohomology. We ob* *tain therefore a complete classification of the objects in T(A; k) since we have a classificat* *ion of the injective H*(A; k)-modules. Let C be an additive category. We define the spectral category Spec(C) of C a* *s follows: the objects are the same as in C, but the morphisms are given by Hom (X; Y )=radHom (X; Y ) where radHom (X; Y )denotes the subgroup of Hom (X; Y ) consisting of maps OE: * *X ! Y such that for every :Y ! X the composite OOE belongs to the Jacobson radical rad* *(End (X)) of the endomorphism ring of X. Note that every object X in C has End (X)=rad(End (X)) as endomorphism ring in Spec(C). The spectral category Sp(R) of a ring R is by* * definition the spectral category Spec(Inj(R)). In [15], Gabriel and Oberst studied the spectra* *l category of a ring R: it is an abelian Grothendieck category where every object is projective and * *injective. Moreover, two injective R-modules are isomorphic if and only if they are isomorphic as ob* *jects in Sp(R). Theorem 6.1. Suppose that either (H^-(A; k))n = 0 for some n > 0, or H^*(A; k)* * is periodic. Then the functor T induces an equivalence Sp(H*(A; k)) ! Spec(T(A; k)). The proof of this theorem is based on the fact that we can write T :Inj(H*(A;* * k)) ! StMod(A) as the composite of two functors ^T Inj(H*(A; k)) -H!Inj(H^*(A; k)) -! StMod (A): The first functor is H = Hom *H*(A;k)(H^*(A; k); -), so it sends I to ^I. Given* * an injective ^H*(A; k)- module J, the second functor is defined via Brown representability by the isomo* *rphism Hom ^H*(A;k)(H^*(A; -); J) ~=Hom_A(-; ^T(J)): The adjointness isomorphism Hom H*(A;k)(M; I) ~=Hom ^H*(A;k)(M; ^I) for every ^H*(A; k)-module M implies that T (I) ~=^T(^I) for every injective H** *(A; k)-module I. We proceed by showing that H and ^Tinduce equivalences on the level of the sp* *ectral categories. To this end we need the following lemma about the spectral category of a ring R* *. We denote by S :Inj(R) ! Sp(R) the canonical projection. Lemma 6.2. Let R be a ring. Then the indecomposable injective R-modules are p* *recisely the simple objects in Sp(R). If (I )2 is a family of injective R-modules then the* * canonical map M M S(I ) -! S(E( I )) is an isomorphism in Sp(R). Therefore the following conditions are equivalent: (i)Every object in Sp(R) is semisimple. (ii)Every injective R-module is the injective envelope of a direct sum of inde* *composable injective R-modules. * * __ Proof.This follows from the discussion of spectral categories in [15]. * * |__| PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 17 Note that every Noetherian ring R satisfies the equivalent conditions in Lemm* *a 6.2. In particular, this holds for H*(A; k). Lemma 6.3. Suppose that either (H^-(A; k))n = 0 for some n > 0, or ^H*(A; k) i* *s periodic. Then the functor H :Inj(H*(A; k)) ! Inj(H^*(A; k)) sending I to ^Iinduces an equival* *ence Sp(H*(A; k)) ! Sp(H^*(A; k)). Proof.The lemma is a consequence of Theorem 2.8 and Lemma 6.2. In deed, H induc* *es a bijection between the isomorphism classes of indecomposable injectives, and for every fam* *ily (I )2 of indecomposable injective ^H*(A; k)-modules, we have M M H( I ) ~=E( H(I )) and therefore M M M S(H( I )) ~=S(E( H(I ))) ~= S(H(I )): It remains to observe that H induces an isomorphism ~= EndH*(A;k)(I)=rad(End H*(A;k)(I))-!End^H*(A;k)(^I)=rad(End ^H*(A;k)(* *^I)) * * __ for every indecomposable injective I. * * |__| Lemma 6.4. The functor ^T:Inj(H^*(A; k)) ! StMod(A) is fully faithful; it indu* *ces an equivalence between Inj(H^*(A; k)) and T(A; k). Proof.It follows from Lemma 3.3 and Lemma 3.4 that ^Tis fully faithful. The arg* *ument given in Proposition 4.5 shows that the image of ^Tis contained in T(A; k). Conversely, * *given any module M in T(A; k), we have ^T(I) ~=M for I = H^*(A; M). Therefore Inj(H^*(A; k)) an* *d T(A; k) are * * __ equivalent via ^T. * * |__| Proof of Theorem 6.1.We combine Lemma 6.3 and Lemma 6.4. The functor H induces * *an equiva- lence Sp(H*(A; k)) ! Sp(H^*(A; k)), and ^Tinduces an equivalence Sp(H^*(A; k)) * *! Spec(T(A; k)). It follows that T induces an equivalence Sp(H*(A; k)) ! Spec(T(A; k)) since T i* *s the composite of * * __ H and ^T. * * |__| Corollary 6.5.Let I and J be injective H*(A; k)-modules. Then T (I) and T (J) a* *re isomorphic in StMod(A) if and only if I and J are isomorphic. Proof.Suppose that T (I) and T (J) are isomorphic. Then I and J are isomorphic * *as objects in Sp(H*(A; k)) by Theorem 6.1. We get homomorphisms ff: I ! J and fi :J ! I sucht* * that idI- fi Off 2 rad(End H*(A;k)(I)) and idJ- ff Ofi 2 rad(End H*(A;k)* *(J)): It follows that Ker(idI- fi Off) and Ker(idJ- ff Ofi) are essential submodules * *of I and J. We conlude that fi Off and ff Ofi are isomorphisms, and therefore I and J are isom* *orphic_as H*(A; k)- modules. * * |__| We obtain the following classification of the objects in T(A; k) from the cla* *ssification of the injective H*(A; k)-modules. 18 DAVID BENSON AND HENNING KRAUSE Corollary 6.6.The indecomposable objects in T(A; k) are, up to isomorphism, pre* *cisely the A- modules of the form T (Ip[n]) for some homogeneous prime ideal p in H*(A; k) an* *d some n 2 Z. An arbitraryLA-module M belongs to T(A; k) if and only if M is a pure injective en* *velope of a direct sum 2 M of indecomposable objectsLin T(A;Lk). If (N )2K is a second family of* * indecomposable objects in T(A; k) such that M and N have the same pure injective envel* *ope, then there exists a bijection o : ! K such that M and No() are isomorphic. To prove this result we need the following lemma about the full subcategory P* *inj(A) of A-modules which are pure injective. We denote by S :Pinj(A) ! Spec(Pinj(A)) the canonical* * projection. Lemma 6.7. Let (M )2 L be a family of pure injectiveLA-modules and denote by M* * the pure injective envelope of M Then the canonical map S(M ) ! S(M) is an isomor* *phism. * * __ Proof.See theorem 8.25 of [19]. * * |__| Proof of Corollary 6.6.We apply Theorem 6.1. It follows from this result that T* * induces a bijection between the isomorphism classes of indecomposable objects in Inj(H*(A; k) and T* *(A; k). Therefore the indecomposables in T(A; k) are precisely the modules of the form T (Ip[n]). To prove the second part of the assertion letLM be an arbitrary A-module. Sup* *pose first that MLis a pureQinjective envelope of a direct sum M of objects in T(A; k). The* * canonical map Q M ! M is a pure monomorphism and thereforeQM is isomorphic to a direct* * summand of M . It follows that M belongs to T(A; k) since M belongs to T(A; k). Now suppose that M belongs to T(A; k). We consider Spec(T(A; k)) as a full s* *ubcategory of Spec(Pinj(A)). It is closed under taking arbitrary direct sums by the argument * *given in the previous paragraphLin combination with Lemma 6.7. In Spec(T(A; k)), we get a decomposit* *ion S(M) = 2 S(M ) into simple objects by Lemma 6.2. It followsLfrom Lemma 6.7 that M * *is isomorphic to the pure injective envelope of the direct sum M . The uniqueness of the f* *amily (M )2 follows from the uniqueness of the decomposition of S(M) into simple objects. T* *his_finishes the proof. * * |__| 7. Subgroups and varieties If H is a subgroup of G, then restriction is a ring homomorphism from H*(G; k* *) to H*(H; k). This makes H*(H; k) into a finitely generated H*(G; k)-module. If I is an injective * *H*(G; k)-module, we write rG;H(I) for the injective H*(H; k)-module rG;H(I) = Hom H*(G;k)(H*(H; k); I): Thus for example rG;H(Ip[n]) ~=Ires-1G;H(p)[n]: Proposition 7.1.Let H G, and write TG and TH for the functor T with respect to* * kG and kH respectively. If I is an injective H*(G; k)-module, we have TG(I)#H ~=TH (rG;H(I)): PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 19 Proof.For any kH-module M, we have Hom_kH(M; TG(I)#H )~=Hom_kG(M "G; TG(I)) ~=Hom H*(G;k)(H^*(G; M "G); I) ~=Hom H*(H;k)(H^*(G; M "G); rG;H(I)) ~=Hom H*(H;k)(H^*(H; M); rG;H(I)) ~=Hom_kH(M; TH (rG;H(I))): * *|___| Next, we calculate the variety of the module T (Ip[n]), in the sense of Benso* *n, Carlson and Rickard [4]. This is essentially done in theorem 6.1.8 (b) ) (e00) of Hovey, Pa* *lmieri and Strickland [18]. For the convenience of the reader, we translate their argument into the * *language of group cohomology. Recall that for every finitely generated kG-module M, the variety VG(M) of M * *is a closed homogeneous subvariety of the cohomology variety VG. For an arbitrary kG-module* * M, the variety of M in the sense of [4] is a collection of closed homogeneous subvarieties of * *VG which is denoted by VG(M). Next, recall from Rickard [27] that for each closed homogeneous subva* *riety V of VG, there is a triangle EV -! k -! FV -! -1EV in StMod(kG) such that EV is a colimit of finitely generated modules with varie* *ty contained in V , and FV is local with respect to such modules. The modules EV and FV are id* *empotent in StMod (kG). Lemma 7.2. Let p be a prime ideal of H*(G; k), corresponding to a closed homog* *eneous irreducible subvariety V of VG. Suppose that M is a kG-module such that H^*(G; M) is p-to* *rsion. Then H^*(G; FV k M) = 0. Proof.Write C for the localizing subcategory of StMod(kG) generated by M. Every* * module in C has p-torsion cohomology. This applies in particular to FV k M. If 0 6= i 2 Hn(G; k) is represented by a cocycle ^i:nk ! k, we write Li for t* *he kernel, so that there is a short exact sequence ^i 0 -! Li -! nk -! k -! 0: Choose elements i1; : :;:it in degrees n1; : :;:nt, generating an ideal whose r* *adical is p; in other words VG \ . .\.VG = V: Then Li1k. .k.LitkFV is projective. Assume, by way of contradiction, that ^H*(G* *; FV kM) 6= 0. Set Xi= Li1k . .k.Liik FV k M; so that ^H*(G; X0) 6= 0 but ^H*(G; Xt) = 0. Choose i > 0 so that ^H*(G; Xi-1) 6* *= 0 but ^H*(G; Xi) = 0. Then the short exact sequence ^ii 0 -! Xi-! nik Xi-1-! Xi-1! 0 20 DAVID BENSON AND HENNING KRAUSE shows that ii acts isomorphically on H^*(G; Xi-1). But Xi-1is in C, and ii is * *in p, so ii acts nilpotently on ^H*(G; Xi-1). So ^H*(G; Xi-1) = 0. This contradiction completes * *the_proof of the lemma. * *|__| Theorem 7.3. Let p be a nonmaximal prime ideal in H*(G; k), corresponding to a* * closed homo- geneous irreducible subvariety V of VG. Then VG(T (Ip[n])) = {V }. Proof.Using Proposition 7.1 with H a Sylow p-subgroup of G, we may assume that * *G is a p-group. By Lemmas 2.1 and 3.3, H^*(G; T (Ip[n])) ~=[Ip[n]~=Ip[n] is a p-torsion H*(G; k* *)-module. So by Lemma 7.2, ^H*(G; FV k T (Ip[n])) = 0. Since G is a p-group, this implies that * *FV k T (Ip[n]) is projective. So only subvarieties of V can occur as elements of VG(T (Ip[n])). B* *ut if W is a proper subvariety of V then for any finitely generated kG-module M with VG(M) W , equ* *ation (3.1) shows that Hom_kG(M; T (Ip[n])) = 0. So W is not an element of VG(T (Ip[n])). S* *ince T (Ip[n])_is not projective, this implies that VG(T (Ip[n])) = {V }. * * |__| 8.Zariski topology The isomorphism classes of indecomposable pure injective A-modules form a set* * which we denote by Ind(A). Recall that a functor F :Mod(A) ! Ab into the category of abelian gr* *oups is coherent if there exists a presentation Hom A(Y; -) -! Hom A(X; -) -! F -! 0 which is induced by a map X ! Y between finitely generated A-modules. Given a c* *oherent functor F , we define D(F ) = {M 2 Ind(A) | F (M) = 0}: The sets of the form D(F ) for some coherent F form a basis for the open subset* *s of the Zariski topology on Ind(A). This topology on Ind(A) has been introduced by Prest [25]; * *it is the analogue of the Zariski topology on the set of isomorphism classes of indecomposable inj* *ective quasi-coherent sheaves on a Noetherian scheme which has been considered by Gabriel (chapitre V* *I of [14]). We need to work in the stable category. We write Ind_(A) for the set of nonpr* *ojective elements of Ind(A), or equivalently the pure injective indecomposables in the stable mod* *ule category. The topology on this is determined in the stable category by the following two lemm* *as. Lemma 8.1. Let P be an indecomposable projective A-module. Then there exist co* *herent functors F and G such that D(F ) = {P } = Ind(A) \ D(G): Proof.We use the fact that a projective A-module is injective. A nonzero map A* *=rad(A) ! P induces the following exact sequence of coherent functors 0 -! Hom A(P=soc(P ); -) -! Hom A(P; -) -! Hom A(A=rad(A); -) -ss!F -! 0: * * __ It is easily checked that D(F ) = {P } and Ind(A) \ D(G) = {P } for G = Ker(ss)* *. |__| Lemma 8.2. Given any coherent functor F on Mod (A), there exists a coherent fu* *nctor G on Mod (A) such that the following hold. (i)G vanishes on projective modules, and (ii)D(G) \ Ind_(A) = D(F ) \ Ind_(A). PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 21 Proof.The sets of the form V (F ) = Ind(A) \ D(F ) for some coherent F form a b* *asis for the open subsets of the Ziegler topology on Ind(A) which has been introduced by Ziegler * *in [29]; see [21] for the definition of this topology in terms of functors. A Ziegler-open set is qua* *si-compact if and only if it is of the form V (F ) for some coherent functor F (cf. corollary 4.5 in * *[21]). Now let P be the finite set of projectives in Ind(A). Note that {P } is Ziegler-open and Zie* *gler-closed for each P in P by Lemma 8.1. Given a coherent functor F , it follows that V (F ) \ P is* * a quasi-compact Ziegler-open subset of Ind(A), hence of the form V (G) = V (F ) \ P for some co* *herent functor G._ By construction, G vanishes on every projective module and D(G) \ Ind_(A) = D(F* * ) \ Ind_(A). |__| Note that a coherent functor F vanishes on projectives if and only if it has * *a presentation of the form Hom_A(Y; -) -! Hom_A(X; -) -! F -! 0 with X and Y finitely generated. Projective modules are separated by coherent functors, so it follows from Lem* *ma 8.1 that the space Ind(A) with the Zariski topology is just the disjoint union of the closed* * set Ind_(A) and a closed point for each projective indecomposable. We denote by Proj(H*(A; k)) the set of homogeneous prime ideals in H*(A; k) w* *hich are different from the maximal ideal m. The assignment p 7! Ip allows us to identify Proj(H*(* *A; k)) with a set of indecomposable injective H*(A; k)-modules. Given a finitely generated H*(A; * *k)-module K, we define D+(K) = {p 2 Proj(H*(A; k)) | Hom H*(A;k)(K; Ip) = 0}: This gives a basis of open sets for the Zariski topology on Proj(H*(A; k)). Theorem 8.3. The map Proj(H*(A; k)) ! Ind(A) which sends a prime p to T (Ip) i* *nduces a homeomorphism between Proj(H*(A; k)) and its image in Ind(A). Proof.The assignment p 7! T (Ip) defines an injective map Proj(H*(A; k)) ! Ind(* *A) by Proposi- tion 3.6 and Theorem 5.1. Given a coherent functor F on StMod(A) determined by a map of finitely genera* *ted modules X ! Y , let K denote the kernel of the induced map H*(A; X) ! H*(A; Y ). For I * *2 Ind0(A), it follows from Proposition 2.9 that F (T (I)) ~=Hom H*(A;k)(K; I): So the inverse image of D(F ) is the open set D+(K). Conversely, if K is a finitely generated module over H*(A; k), then a finite * *presentation gives rise to a map from a finite sum X of modules of the form nk to another such mod* *ule Y . In Tate cohomology, we then get a sequence H^*(A; X) -! ^H*(A; Y ) -! ^H*(A; k) H*(A;k)K -! 0: Now recall from Lemma 2.1 that ^I~=I, so that we have Hom ^H*(A;k)(H^*(A; k) H*(A;k)K; I) = Hom H*(A;k)(K; I): So by Proposition 2.9 we have an exact sequence 0 -! Hom H*(A;k)(K; I) -! Hom H*(A;k)(H*(A; Y ); I) -! Hom H*(A;k)(H*(A; * *X); I): 22 DAVID BENSON AND HENNING KRAUSE Complete X ! Y to a triangle X ! Y ! Z ! -1X, and define a coherent functor F* * by Hom_A(-1X; -) -! Hom_A(Z; -) -! F -! 0: The long exact sequence induced by the triangle gives the following exact seque* *nce 0 -! F -! Hom_A(Y; -) -! Hom_A(X; -): Again applying Proposition 2.9, we get F (T (I)) ~=Hom H*(A;k)(K; I): * * __ It follows that the image of D+(K) is equal to D(F ). * * |__| 9.Minimal primes In this section we compute the modules T (Ip[n]) for every minimal prime idea* *l p in H*(A; k) and every n 2 Z. Our analysis is based on a discussion of various localization func* *tors with respect to a prime p. We fix a homogeneous prime ideal p in H*(A; k) and denote by S the set* * of homogeneous elements in H*(A; k) \ p. Given an H*(A; k)-module M, the localization of M wit* *h respect to S is denoted by Mp = S-1M: Lemma 9.1. Let p 6= m and denote by ^Sthe set of homogeneous elements in H*(A;* * k) \ p, viewed as a subset of H^*(A; k). Then the localization of an H^*(A; k)-module M with * *respect to ^Sis isomorphic to Mp as a module over H*(A; k). In particular, ^S-1^H*(A; N) ~=H*(A* *; N)p for every A-module N. Proof.Localization with respect to S and ^Sgives universal maps M ! Mp and M ! * *^S-1M. We need to show that there are maps Mp ! ^S-1M and ^S-1M ! Mp whose composite is t* *he identity on Mp and ^S-1M. Clearly, (S^-1M)p = ^S-1M; and this gives the map Mp ! ^S-1M. On the other hand, Mp is in a canonical way * *an ^H*(A; k)- module. In fact, the injectives Ip[n], n 2 Z, cogenerate the category of H*(A;* * k)p-modules and therefore Mp fits into an exact sequence Y Y 0 -! Mp -! Ip[n ] -! Ip[n ] of H*(A; k)-modules. Applying the functor Hom *H*(A;k)(H^*(A; k); -) and using * *Lemma 2.1, we get Mcp~=Mp since p 6= m. This gives the map ^S-1M ! Mp, and we conclude that the l* *ocalization of M with respect to ^Sis isomorphic to Mp. Finally, we get for every A-module N ^S-1^H*(A; N) ~=^H*(A; N)p ~=H*(A; N)p * * __ where the second isomorphism is taken from Proposition 2.9. * * |__| Recall that a localization functor F :StMod(A) ! StMod(A) with respect to a l* *ocalizing sub- category L of StMod(A) is finite if L is generated (as a localizing subcategory* *) by a class of finitely generated A-modules. PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP* * 23 Proposition 9.2.Let p 6= m be a homogeneous prime ideal in H*(A; k). Then there* * exists a finite localization functor Fp:StMod (A) ! StMod(A) such that ^H*(A; Fp(M)) ~=H*(A; M)p for every A-module M. Proof.Suppose that U is a set of homogeneous elements in ^H*(A; k) such that lo* *calization with respect to U is an exact functor on ^H*(A; k)-modules. Then it is shown in theo* *rem 3.3.7 of [18] that there exists a localization functor F :StMod(A) ! StMod(A) such that ^H*(A; F (M)) ~=U-1H^*(A; M) for every M in StMod(A). In fact, letting Li denote the kernel of a homomorphis* *m nk ! k rep- resenting i, one then takes the finite localization functor with respect to the* * localizing subcategory generated by {Li | i 2 U}. Taking for U the set ^Scorresponding to the prime p,* * the_assertion_ follows from Lemma 9.1. * * |__| Remark. In case A = kG for some finite group G, the functor Fp can be expressed* * as a Rickard idempotent functor [27] as follows. Let U be the set of subvarieties of the coh* *omology variety VG which do not contain the variety determined by p. Let CU be the thick subcatego* *ry of stmod(kG) consisting of finitely generated modules X satisfying VG(X) 2 U. Then Fp is the* * F -idempotent module for CU. From now on we assume that A is the group algebra of a finite group G. In [17* *], Greenlees and Lyubeznik have shown that the localization of the cohomology ring H*(G; k) at a* * minimal prime p is Gorenstein. This allows us to identify for every n 2 Z the corresponding m* *odule T (Ip[n]). Let us recall that FN denotes the localization functor with respect to the localizi* *ng subcategory N of StMod (A) consisting of all kG-modules M satisfying ^H*(G; M) = 0. Theorem 9.3. Let G be a finite group of p-rank r and k be a field of character* *istic p. If p 6= m is a minimal homogeneous prime ideal in H*(G; k) and n 2 Z, then T (Ip[n]) ~=-n+rFN (Fp(k)) as modules in StMod(kG). Proof.In section 7 of Greenlees and Lyubeznik [17], it is shown that H*(G; k)p * *~=Ip[r]. Thus we have ^H*(G; Fp(k)) ~=H*(G; k)p ~=Ip[r] ~=dIp[r] by Proposition 9.2 and Lemma 2.1. In Corollary 4.4 it is shown that Tate cohomo* *logy determines the modules of the form T (I) if one localizes away from N. Therefore T (Ip[r])* * ~=FN (Fp(k)), and_ the assertion of the theorem follows by applying the appropriate shift. * * |__| References 1.M. Auslander, Large modules over artin algebras, in: Algebra, topology and c* *ategories, Academic Press (1976), 1-17. 2.D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure & A* *pplied Algebra 82 (1992), 107-129. 3.D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for in* *finitely generated modules, I, Math. Proc. Camb. Phil. Soc. 118 (1995), 223-243. 4._____, Complexity and varieties for infinitely generated modules, II, Math. * *Proc. Camb. Phil. Soc. 120 (1996), 597-615. 5._____, Thick subcategories of the stable module category, Fundamenta Mathema* *ticae 153 (1997), 59-80. 24 DAVID BENSON AND HENNING KRAUSE 6.D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular represe* *ntation theory, I, Fundamenta Mathematicae 161 (1999), 37-91. 7.D. J. Benson and H. Krause, Generic idempotent modules for a finite group, T* *o appear in Algebras and Repre- sentation Theory, 1999. 8.E. H. Brown, Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 7* *9-85. 9.W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced * *Mathematics, vol. 39, Cam- bridge University Press, 1993. 10.H. Cartan and S. Eilenberg, Homological algebra, Princeton Mathematical Seri* *es, no. 19, Princeton Univ. Press, 1956. 11.W. W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. * *Soc. 63 (1991), 241-264. 12._____, Modules of finite length over their endomorphism ring, in: Representa* *tions of algebras and related topics, eds. S. Brenner and H. Tachikawa, London Math. Soc. Lec. Note Series 168 (19* *92), 127-184. 13.E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a f* *ield, Invent. Math. 127 (1997), 209-270. 14.P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 32* *3-448. 15.P. Gabriel and U. Oberst, Spektralkategorien und regul"are Ringe im von Neum* *annschen Sinn, Math. Zeit. 92 (1966), 389-395. 16.G. Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 * *(1998), 686-702. 17.J. P. C. Greenlees and G. Lyubeznik, Rings with a local cohomology theorem a* *nd applications to cohomology rings of groups, to appear. 18.M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy th* *eory, Mem. Amer. Math. Soc., vol. 128, American Math. Society, 1997. 19.C. U. Jensen and H. Lenzing, Model theoretic algebra, Gordon and Breach, 198* *9. 20.H. Krause, Stable equivalence preserves representation type, Comment. Math. * *Helvetici 72 (1997), 266-284. 21._____, The spectrum of a locally coherent category, J. Pure & Applied Algebr* *a 114 (1997), 259-271. 22.H. R. Margolis, Spectra and the Steenrod algebra, North Holland, Amsterdam, * *1983. 23.E. Matlis, Injective modules over Noetherian rings, Pacific Journal of Math.* * 8 (1958), 511-528. 24.A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and B* *rown representability, J. Amer. Math. Soc. 9 (1996), 205-236. 25.M. Prest, Remarks on elementary duality, Annals of Pure & Applied Logic 62 (* *1993), 185-205. 26.D. G. Quillen and B. B. Venkov, Cohomology of finite groups and elementary a* *beilan subgroups, Topology 11 (1972), 317-318. 27.J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. * *178 (1997), 149-170. 28.J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3* * (1965), 413-420. 29.M. Ziegler, Model theory of modules, Annals of Pure & Applied Logic 26 (1984* *), 149-213. Department of Mathematics, University of Georgia, Athens GA 30602, USA E-mail address: djb@byrd.math.uga.edu Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany E-mail address: henning@mathematik.uni-bielefeld.de