COMMUTATIVE ALGEBRA OF UNSTABLE K - MODULES, LANNES' T - FUNCTOR AND EQUIVARIANT MOD - P COHOMOLOGY by Hans-Werner Henn Abstract Let p be a fixed prime and let K be an unstable algebra over the mod - p Ste* *enrod algebra A such that K is finitely generated as graded Fp- algebra. Let Kfg-U denote the abe* *lian category of finitely generated K - modules with a compatible unstable A - module structure. We st* *udy various concepts of commutative algebra in this setting. The r^ole of the prime ideal spectru* *m of a commutative ring is here taken by a category R(K) which, roughly speaking, consists of the A * *- invariant prime ideals of K together with certain "Galois information"; sheafs will correspond to f* *unctors on this category, and the r^ole of the sheaf associated to a module will be taken by the compo* *nents of Lannes' T - functor. We discuss the notions of support, of a - torsion modules (for an i* *nvariant ideal a of K) and of localization away from the Serre subcategory T ors(a) of a - torsion * *modules in our setting. We show that the category Kfg- U has enough injectives and use these injecti* *ves to study these localizations and their derived functors; they are closely related to the de* *rived functors of the a - torsion functor Fa. Our results are formally analogous to Grothendieck's r* *esults in the classical situation of modules over a noetherian commutative ring R [Gr]. Important for applications is the case K = H*BG, the mod - p cohomology of a* * classifying space of a compact Lie group (or a suitable discrete group), and M = H*GX where X * *is a (suitable) G - CW - complex. In these cases the category R(K) and the functor on R(K) assoc* *iated to H*GX can be described in terms of group theoretic and geometric data, and our theory * *yields a far-reaching generalization of a result of Jackowski and McClure [JM] resp. of Dwyer and * *Wilkerson [DW2]. As a concrete application of our theory we describe the size of the kernel of t* *he restriction map from the unknown mod - 2 cohomology of the S - arithmetic group GL(n; Z[1=2]) to the * *known cohomology of its subgroup Dn of diagonal matrices. 0. Introduction Let p be a prime number and K be an unstable algebra over the mod - p Steenrod * *algebra A [S, SE]. An unstable K - A - module is an unstable A - module together with an * *A - linear map K M -! M which defines on M the structure of a K - module. We will ususall* *y drop the A from the notation and say that M is unstable K - module. The mod - p coho* *mology H*X of a space X is the main source of examples for unstable algebras and examp* *les for unstable H*X - modules are given by the mod - p cohomology of spaces over X, or* * more generally of Thom spaces of vector bundles on spaces over X. 1 We will be mostly concerned with the case that K is finitely generated as an Fp* * - algebra, i.e. noetherian, and that M is finitely generated as K - module. We will call* * such an M an unstable finitely generated K - module and denote the abelian category of su* *ch modules by Kfg - U (cf. section 1). Interesting geometric examples are provided by th* *e case of equivariant mod - p cohomology (cf. Theorem 0.2). Our first result reads as fol* *lows. THEOREM 0.1. Let K be an unstable noetherian algebra. Then the category Kfg - U* * has enough injectives. We will prove this in section 1 by actually constructing enough injectives. If * *V is an elemen- tary abelian p - group (i.e. V ~=(Z=p)n for some natural number n) and if K = H* **BV , the mod - p cohomology of the classifying space of V , such a theorem was proved by* * Lannes and Zarati [LZ2], and in fact, they even determined all indecomposable injectives. * *We would like to point out that there would not be enough injectives if we worked with module* *s which are finitely generated using both the K and A - module structure together, as the c* *ase K = Fp (i.e. the case of ordinary unstable A - modules) shows (cf. [LSc]). Now assume that K is noetherian. We will use these injectives to study localiza* *tion functors on the category Kfg - U away from suitable subcategories. However, before we wi* *ll get to this we need to discuss various concepts of commutative algebra in Kfg - U. The r^ole of the prime ideal spectrum in the classical case will be taken by an* * appropriate category R(K) (cf. 1.3 for a precise definition and section 2 for a discussion * *of R(K)). Here we just recall that the objects of R(K) are morphisms of unstable algebras ' : * *K -! H*BV where V is an elementary abelian p - group such that ' makes H*BV into a finite* *ly generated K - module. (Observe that Rad (Ker '), the radical of the kernel of ' is a prim* *e ideal which is invariant with respect to the action of the Steenrod reduced power operation* *s and, in fact, all such "invariant" prime ideals are obtained in this way (cf. 2.3).) ___ For each unstable finitely generated K - module M we have a functor M from R(K* *) to Kfg - U, which sends (V; ') to the "component" TV_(M; ') where T denotes Lannes* *' functor (cf. 1.4 for precise definitions). The functor M should be considered as the a* *nalogue of the sheaf fM which one associates to M in the classical case of modules over a comm* *utative ring. In fact, if one considers R(K)op as a site equipped with the trivial Grothendie* *ck topology then sheaves on this site are precisely covariant functors on R(K) (cf. [MM, II* *I.2 and III.4]). We will not go into the details of this analogy but we will point out further a* *nalogies when it seems appropriate. For a further justification for this philosophy we refer * *to the comments after Theorem 0.4. The discussion of support in section 2, in particular 2.10, * *suggests also to consider TV (M; ') as an unstable analogue of the appropriate stalk of fM, i.e.* * as an analogue of the classical localization of M at the prime ideal Rad (Ker '). For earlier * *work on relations between Lannes' functor and classical localization we refer to work by Dwyer an* *d Wilkerson [DW1]. For the remainder of the introduction we concentrate on the case of equivariant* * mod - p cohomology. This case is easier to explain and is also particularly interesting* * because here 2 the category R(K) can be understood in group theoretic terms and the components* * of the T - functor are of geometric and group theoretic significance (cf. Theorem 0.2* *). However, we stress that the main results (Theorem 0.4 and Corollary 0.5 below) are true * *for unsta- ble finitely generated K - modules over an unstable noetherian algebra K (Theor* *em 3.9 and Corollary 3.10) and the proof in the case of equivariant cohomology require* *s the same machinery as in the general case. To state Theorem 0.2 we introduce some notation. As usual we denote the classif* *ying space of the topological group G by BG, the total space of the universal principal G * *- bundle over BG by EG and the mod - p cohomology of the Borel construction EG xG X by H*GX. * *Fur- thermore, for a fixed prime p, let A(G) denote the category whose objects are t* *he elementary abelian p - subgroups of G and whose morphism sets consist of those group homom* *orphisms which are induced by conjugation with an element in G [Q]. THEOREM 0.2. Fix a prime p. Assume we are in one of the following cases. a) G is a compact Lie group and X is a G - CW - complex with finitely many G - * *cells. b) G is a discrete group for which there exists a mod - p acyclic G - CW - comp* *lex F with finitely many G - cells and finite isotropy groups, and let X be any G - CW - c* *omplex, again with finitely many G - cells and with finite isotropy groups (e.g. X = F ). c) Let G be a profinite group such that the continuous mod - p cohomology H*cG * *is finitely generated as Fp - algebra. I. Then H*BG is an unstable noetherian algebra and there is a canonical equival* *ence of categories A(G) ! R(H*BG); E 7! (E; resG;E) with resG;Edenoting the restriction homomorphism H*BG -! H*BE. II. Furthermore H*GX is an unstable finitely generated H*BG - module and there * *are iso- morphisms TE (H*GX; resG;E) ~=H*CG(E)(XE ) which are natural in E 2 A(G). (Here CG (E) is the centralizer of E in G, XE denotes the E - fixed points of E* * acting on X, cohomology is with coefficient in Fp, and if G is profinite we assume that X is* * a point and we read H*GX as H*cG.) If X = F (in case b)), then the spaces XE are also mod - p acyclic by Smith the* *ory, and we obtain the following easy but important consequence. COROLLARY 0.3. Assume G satisfies the assumptions of Theorem 0.2.b). Then there* * are isomorphisms TE (H*BG; resG:E) ~=H*BCG (E) ; _ which are natural in E. |_| 3 Statement II of Theorem 0.2 and the Corollary are to a large extent due to Lann* *es. Case (a) is proved in the important and still unpublished preprint [L1] (cf. [L3] for a * *proof in case G is finite). The cases (b) and (c) are consequences of part (a) (for G finite); * *(c) was proved in [H2] and (b) will be proved in the appendix of this paper. In the case that * *G is a group of finite virtual cohomologial dimension (f.v.c.d.) the corollary was also prov* *ed in [L1]. The finite generation part of Statement I is well known in case (a); for part (b) w* *e refer again to the appendix. The equivalence of categories is essentially a folk result (cf. [* *HLS2, I.5.3]). Interesting classes of groups which are covered by (c) are p - adic analytic gr* *oups in the sense of Lazard [Lz], while (b) covers the case of (S) - arithmetic groups [Se]* *, mapping class groups of orientable surfaces [H], outer automorphism groups of free groups [CV* *] (these are all groups of f.v.c.d.) and word-hyperbolic groups in the sense of Gromov [GH]. Now we turn to localizations. We consider an invariant ideal a in H*BG, i.e. a * *is invariant with respect to the action of the Steenrod reduced power operations. Then the * *class of unstable finitely generated H*BG - modules which are annihilated by some power * *of a (we will call such modules unstable a - torsion modules) forms a Serre class and we* * will study localization away from the full subcategory T ors(a) of such modules. These localizations away from T ors(a) are closely related to the right derived* * functors RiFa of the functor which associates to M its largest unstable a - torsion submodule* * FaM. The ideal a determines a subcategory O(a) of A(G), namely the full subcategory of a* *ll objects E for which a 6 Rad (Ker resG;E), i.e. for which a is not contained the radical o* *f the kernel of the restriction map resG;E: H*BG -! H*BE. The category O(a) should be thought o* *f as the analogue of the open complement (in the classical prime ideal spectrum) of * *the subset V (a) which is defined by a. As usual we denote the inverse limit of a functor F defined on a category C by * *limCF and its derived functors by limiCF . Our main result reads now as follows. THEOREM 0.4. Let G and X be as in Theorem 0.2 and let a be an invariant ideal i* *n H*BG. a) Then there is a natural exact sequence 0 -! FaH*GX -! H*GX -ae!limO(a)H*CG(E)(XE ) -! R1FaH*GX -! 0 in which the components of ae are induced by the obvious inclusions on the grou* *p and space level. In particular, the kernel and cokernel of ae are unstable finitely gene* *rated a - torsion modules. Furthermore, ae is localization away from the subcategory T ors(a). b) There are natural isomorphisms limiO(a)H*CG(E)(XE ) ~=Ri+1FaH*GX for all i > 0. In particular, limiO(a)H*CG(E)(XE ) is an unstable finitely gene* *rated a - torsion module for all i > 0. 4 For the generalization of this result to the case where K is noetherian and M i* *s in Kfg - U the reader is referred to 3.9. If one ignores the assertion about finite generation (which is a consequence of* * Theorem 0.1) this result is formally analogous to the classical situation [Gr]. There one co* *nsiders an ideal a in a noetherian commutative ring R and the derived functors of the functor a,* * which associates to an R - module M its a - torsion submodule. These derived funcors * *are identified with the cohomology groups H*V (a)(specR; fM) of spec(R) with support in the cl* *osed set V (a) and coefficients in the sheaf fM. The inverse limit and its derived functors co* *rrespond in this picture to the cohomology of the open complement of the closed set V (a) with c* *oefficients in Mf (The analogy can be made more precise by noting that O(a) is, in fact, an o* *pen subobject in the topos of sheafs on the site A(G)op [AGV,IV.8.4], and by refering to the * *topos theoretic versions of cohomology with support (cf. [AGV,V.6.5]). In the important special case where a is the invariant maximal ideal m of all p* *ositive dimen- sional elements in H*BG, the submodule Fm is the largest unstable finite H*BG -* * submodule of M and we will write F instead of Fm . The category O(m) turns out to be the * *full subcat- egory of A(G) consisting of all non-trivial E. We will write A*(G) for this cat* *egory. Then Theorem 0.4 takes the following form. COROLLARY 0.5. Let G and X be as in Theorem 0.2. a) Then there is a natural exact sequence 0 -! F H*GX -! H*GX -ae!limA*(G)H*CG(E)(XE ) -! R1F H*GX -! 0 in which the components of ae are induced by the obvious inclusions on the grou* *p and space level. In particular, the kernel and cokernel of ae are finite. Furthermore, * *ae is localization away from the subcategory of unstable finite H*BG - modules. b) There are natural isomorphisms limiA*(G)H*CG(E)(XE ) ~=Ri+1F H*GX for all i > 0. In particular, limiA*(G)H*CG(E)(XE ) is is finite for all i > 0. If G is compact Lie and X is a point it is a theorem of Jackowski and McClure t* *hat ae is an isomorphism and all higher limits vanish in 0.5 [JM]. An algebraic version of t* *he theorem of Jackowski and McClure was proved by Dwyer and Wilkerson [DW2] and their result * *motivated the investigations in this paper. Applications in the case where ae is an isom* *orphism (in particular in the cases considered in [JM] and [DW2]) were discussed by Mislin * *[M]. For applications of Theorem 0.4 and Corollary 0.5 the reader is referred to [H2* *,H3,H4] where we study the mod - p cohomology of groups of units in maximal orders of certain* * p - adic division algebras, of the general linear groups GL(p - 1; Zp) (with Zp the ring* * of p - adic integers) and the mod - 2 cohomology of SL(3; Z[1=2]) and GL(3; Z[1=2]). Appli* *cations to mapping classs groups will be considered in joint work with F. Cohen and Y. Xia. 5 The higher limits resp. the derived funcors of F in the case H*GX, G elementary* * abelian will be investigated in [HLO]. They yield new invariants for G - complexes which giv* *e obstructions for equivariant embeddings of finite G - complexes into smooth G - manifolds. As another concrete application of our theory in the case of group cohomology w* *e offer the following result. Here Dn is the subgroup of diagonal matrices in the general * *linear group GL(n; Z[1=2]), cohomology is with coefficients in F2 and the size of a graded f* *initely generated module M over a connected finitely generatedPFp - algebra is given by the order* * of the pole at t = 1 of the power series series idimFpMiti. Note that the size measures t* *he growth of the sequence of numbers dimFp Mi. THEOREM 0.6. The kernel of the restriction map aen from H*BGL(n; Z[1=2]) to H** *BDn has size precisely equal to n - n0 + 1 where n0 denotes the smallest natural nu* *mber such that aen0 is not a monomorphism and n n0. In particular, the size of the kernel of * *aen0 itself is 1, i.e. Ker aen0 is periodic in large degrees. By unpublished work of Dwyer it is known that n0 is finite but no element in th* *ese kernels seems to be known explicitly. Here is a brief outline of the paper. In section 1 we review what we need to kn* *ow about Lannes' functor, discuss injectives and prove Theorem 0.1. Section 2 is concerned with * *concepts of commutative algebra in Kfg- U: we discuss invariant ideals and the category R(K* *), torsion modules, the T - support of an unstable finitely generated module over a noethe* *rian algebra K and its relation to the classical support. In section 3 we study localizatio* *n away from subcategories of torsion modules, we prove the generalization of Theorem 0.4 in* * the context of Kfg-U and discuss its consequences. We also show how to derive the theorem o* *f Jackowski and McClure resp. Dwyer and Wilkerson with our methods. Section 4 is devoted to* * the proof of Theorem 0.6 and in an appendix we prove part (b) of Theorem 0.2. Acknowledgements:_______The research in this paper was inspired by the work of * *Jackowski and McClure [JM] and in particular by the algebraic approach to it by Dwyer and Wil* *kerson [DW2]. The paper should also be considered as a sequal to [HLS2]. In fact, my f* *irst proof (in 1990) of the fact that the kernel and cokernel of the map ae in Corollary 0* *.5 (resp. 3.10) were finite used the main result of [HLS2] in an essential way. I had helpful * *discussions with many different people on the subject matter of this paper and I am especia* *lly happy to acknowledge numerous inspiring discussions with Jean Lannes. In particular, he * *first showed me a result like Corollary 0.5 (resp. 3.10) in the case K = H*BV . I would also* * like to thank John Greenlees for a timely conversation on local cohomology and Bob Oliver for* * comments on a preliminary version of this paper. During the research presented in this * *paper I was supported by a Heisenberg fellowship of the DFG. 6 1. Review of Lannes' T - functor; injectives in Kfg - U We begin by recalling some terminology and facts about Lannes' T - functor. As* * general reference for background information we refer to [L2,L3] and [S]. 1.1______Let p be a fixed prime. Let U resp. K denote the category of unstabl* *e modules resp. unstable algebras over the mod - p Steenrod algebra A. The Steenrod algebra is * *actually a Hopf algebra and its diagonal gives rise to a tensor product on the categories * *U resp. K. For a fixed unstable algebra K we consider the following category K -U: its obj* *ects, which we call unstable K - A -modules (or unstable K - modules for short), are unstable * *A - modules M together with A - linear structure maps K M -! M which make M into a K - mod* *ule; its morphisms are all A - linear maps which are also K - linear. The full subc* *ategory of K - U consisting of those objects which are finitely generated as K - modules i* *s denoted by Kfg - U. Its objects will be called unstable finitely generated K - modules. 1.2.______Now let V be an elementary abelian p - group (i.e. V ~=(Z=p)n for som* *e natural number n). Let TV : U -! U be the functor introduced by Lannes [L2,L3]. It is left adj* *oint to tensor- ing with H*BV , so there are natural isomorphisms Hom U(TV M; N) ~=Hom U (M; H** *BV N) for all unstable modules M and N. TV has a number of remarkable properties. In * *particular, TV commutes with tensor products and lifts to a functor from K to itself and th* *e adjunction relation continues to hold in K: Hom K(TV K; L) ~= Hom K(K; H*BV L) for all u* *nstable algebras K and L. Similarly, TV lifts to a functor from K - U to TV K - U. 1.3.______To an unstable algebra K we associate a category S(K) as follows. Its* * objects are the morphisms of unstable algebras ' : K ! H*BV with V an elementary abelian p - * *group; it will be convenient to denote such an object as (V; '). Then the set of morp* *hisms from (V1; '1) to (V2; '2) are all homomorphisms V1 -ff!V2 of abelian groups such tha* *t '1 = ff*'2. The full subcategory of S(K) of objects (V; ') for which H*BV becomes a finitel* *y generated K - module via ' will be denoted by R(K). Note that in this case the homomorphi* *sms ff has to be injective. If K is a noetherian algebra then this category is equival* *ent to a finite category and it (resp. its opposite) was first investigated by Rector [R]. The * *full subcategory of R(K) having as objects all (V; ') with V non-trivial will play an important* * role for us. We will denote it by R*(K). 1.4.______Now consider the unstable algebra TV K. A morphism of unstable algebr* *as ' : K -! H*BV determines a connected component TV (K; ') of TV K: it is defined as TV (K* *; ') := Fp(')T0VKTV K where Fp(') denotes Fp considered as a module over TV0K (the suba* *lgebra of homogeneous elements of degree 0) via the adjoint of '. Similarly, if M is in K* * -U, we have a "component" TV (M; ') := Fp(') T0VKTV M which has an obvious structure of an un* *stable TV (K; ') - module. Furthermore there is a canonical map of unstable algebras * *aeK;(V;'): K -! TV (K; ') which makes TV (M; ') into an unstable K - module; the map aeK;(* *V;')is the 7 composition of the map flK;(V;'): K -! H*BV TV (K; '), which is adjoint to the * *projection map TV K -! TV (K; '), followed by the projection map H*BV TV (K; ') -! TV (K;* * '). Similarly, there are maps aeM;(V;'): M -! TV (M; '). Now it is straightforward to check that the assignment (V; ') 7! TV (M; ') exte* *nds to a functor S(K) ! K - U. If K is noetherian and M 2 Kfg - U, then we obtain a fun* *ctor R(K) -! Kfg - U (cf. 1.8 and 1.12 below). 1.5.______Next we discuss injectives in the category K - U. First we have the * *analogues of the Brown - Gitler modules in the category U, i.e for each natural number n there i* *s an unstable K - module JK (n) representing the functor M 7! (Mn )*. In fact, if F (n) denot* *es the free unstable A - module on a generator in degree n, then we define JK (n)l, the sub* *space of elements of degree l, as ((K F (l))n)* with ( )* denoting the vector space dua* *l. The A - and K - module structure on JK (n) can then be defined by appropriate maps betw* *een the modules K F (.), just as in the case of the modules J(n) (cf. [LZ1]). This description makes it clear that the module JK (n) is trivial in degrees bi* *gger than n and isLof finite type if K is of finite type. Furthermore, in U the module JK (n) i* *s isomorphic to * n-i * iJ(i)(Kn-i) (with (K ) denoting the dual of the subspace of homogeneous * *elements of degree n - i and being considered as unstable module concentrated in degree * *0). To get more injectives we use the following refinement of the adjunction proper* *ty of the T - functor. PROPOSITION 1.6 [LZ2]. Let K and L be two unstable algebras, V an elementary ab* *elian p - group and g : K -! H*BV L a homomorphism of unstable algebras. For any unstab* *le K - module M and unstable L - module N the adjunction Hom U(TV M; N) ~=Hom U (M; H** *BV N) induces an isomorphism Hom TVK-U (TV M; N) ~=Hom K-U (M; H*BV N) : (Here H*BV N is_a K - module via g and N is a TV K - module via eg: TV K -! L,* * the adjoint of g.) |_| Now consider a map ' : K -! H*BV of unstable algebras and apply this propositi* *on to the map flK;(V;'): K -! H*BV TV (K; '). If N is an unstable TV (K; ') - modul* *e, we will also write H*BV (') N for the unstable K - module H*BV N if its K - modu* *le structure is defined via the map flK;(V;'). Because injectives in TV (K; ')-U a* *re also injective in TV K - U, exactness of TV [L2,L3] implies the following result. PROPOSITION 1.7. Let ' : K - ! H*BV be a map of unstable algebras and_I be any injective in TV (K; ') - U. Then H*BV (') I is injective in K - U. |_| In particular, all the objects H*BV (') JTV(K;')(n) are injective. If K is und* *erstood from the context, and if (V; ') is in S(K), we will also write I(V;')(n) for this in* *jective. 8 1.8.______We will be mainly concerned with the case of unstable noetherian alge* *bras and finitely generated unstable modules over them. We recall that in this case TV K is again* * noetherian and TV M is finitely generated over TV K ([DW1], [H1]). Furthermore, the canon* *ical map K -! TV K makes TV K into a finitely generated K - module and hence TV M become* *s a finitely generated K - module (cf. 1.12 below). If K is also connected, then the map flM;(0:ffl): M -! T0(M; ffl) is an isomorp* *hism (cf. [S, Prop. 3.9.7]). Here 0 is the trivial elementary abelian p - group and ffl is th* *e augmentation of the connected algebra K. In particular, we see that in this case the modules* * JK (n) and I(0;ffl)(n) agree. If K is noetherian, then it is easy to check that the modules I(V;')(n) are fin* *itely generated K - modules for any (V; ') 2 R(K). The following result shows that these modul* *es give enough injectives in the category Kfg - U. We will give two proofs of this resu* *lt which both rely crucially on the main result of [H1]. THEOREM 1.9. (Existence of enough injectives). Let K be an unstable noetherian * *algebra and M an unstable finitely generated K - module. Then there is an embedding M -* *! I in the category Kfg - U in which I is isomorphic to a finite direct product of inj* *ective modules I(V;')(n) for suitable (V; ') in R(K) and natural numbers n. First_Proof_of_1.9:____From Theorem I of [H1] we know that there is a finite fi* *ltration 0 = M0 M1 ::: Mn = M such that the successive filtration quotients Mi=Mi-1 are ki - * *fold suspensions of modules Li which can be embedded (in the category Kfg - U) into * *a finite direct product of modules H*BV (') = I(V;')(0). It is enough to show that the * *theorem holds for the quotients Mi=Mi-1. Now Mi=Mi-1 can be embedded into a finite dire* *ct sum of modules H*BV (') kiFp. The K - module structure on H*BV (') kiFp is clearly pulled back from the obvious H*BV TV (K; ') - module structure via the map flK* *;(V;'): K -! H*BV TV (K; '). Now H*BV kiFp can be embedded (as H*BV TV (K; ') - module) into H*BV JTV(K;')(ki), hence Mi=Mi-1 can be embedded_in Kfg - U int* *o a finite direct product of modules I(V;')(ki) and we are done. |_| The second proof relies on the following result of [HLS2] which again depends o* *n [H1,Thm.I]. THEOREM 1.10. Let K be an unstable noetherian algebra and M an unstable finitel* *y gen- erated K - module. Then there is a natural number n such that the maps flM;(V;'* *)(which are adjoint to the projection maps TV M -! TV (M; ')) induce an embedding Y fl : M -! H*BV TV (M; ') 0} i* *s an unstable K - submodule which we denote by FaM. In fact, this assignment extend* *s to a functor Fa from Kfg - U to itself which we call the a - torsion submodule funct* *or. It is left exact and its right derived functors are denoted by RiFa. If a = K+ we will wri* *te F instead of FK+ and call F the "finite submodule functor". Proof:______The only part which is not obvious is that FaM is closed under Stee* *nrod operations. For simplicity assume that the prime is 2. Let x 2 FaM, a 2 a, so anx = 0 for s* *ome large n. We may assume that n = 2k and kkis large. Then we apply the total Steenrod o* *peration to this equation and obtain Sq(a)2kSq(x) = 0. Now assume that i < 2k and consi* *derkthe homogeneous part of degree |a2 x| + i (|y| denoting the degree of y). We obtain* *_a2 Sqix = 0, i.e. Sqix is again a - torsion. The argument for odd primes is analogous. |_| The final result in this section together with the embedding results of 1.9 and* * 1.10 shows that primary decompositions (cf. [L]) exist in Kfg - U if K is noetherian. We* * leave it to the interested reader to state and prove the appropriate existence and uniquene* *ss results for such decompositions. PROPOSITION 2.15. Assume K is an unstable algebra and M is an unstable K - modu* *le. Assume we have a map M -! H*BV (') F of unstable K - modules with F a finite TV (K; ') - module for some (V; ') 2 R(K). Then the kernel of this map is prim* *ary with respect to the prime ideal Rad (Ker '). Furthermore, if F vanishes in degrees n* * and bigger, then (Ker ')n kills H*BV (') F . Proof:______To see this take an element x 2 K and consider flK;(V;')(x) 2 H*BV * *TV (K; '). This can be written as flK;(V;')(x) = 'x 1 + y with y 2 H*BV TV (K; ')+and TV (K; * *')+ denoting as before the ideal of elements of positive degree. From this formula * *it is clear that x acts nilpotently on H*BV (') F iff x 2 Rad_(Ker '), and if x1; : :;:xn are i* *n Ker ', then the product x1 : :x:nkills H*BV (') F . |_| 3. Localizations in Kfg - U away from torsion modules 3.1.______Throughout this section K will be an unstable noetherian algebra. We * *will study local- izations on the category Kfg - U of unstable finitely generated K - modules awa* *y from the 14 full subcategory T ors(a) of a - torsion modules for some fixed invariant ideal* * a. We begin with some formal definitions. DEFINITION 3.2. Let K be an unstable noetherian algebra and let a be an invaria* *nt ideal of K. Let M be an unstable finitely generated K - module. a) M is called T ors(a) - reduced (or a - reduced) iff Hom K-U (N; M) = 0 for e* *ach a - torsion module N 2 Kfg - U. b) M is called T ors(a) - closed (or a - closed) iff ExtiK-U(N; M) = 0, i = 0; * *1 for all a - torsion modules N 2 Kfg - U. It is clear that M is a - reduced iff it does not contain any non-trivial a - t* *orsion submodules. Furthermore, M is a - closed iff for any morphism ff : A -! B of unstable finit* *ely generated K - modules whose kernel and cokernel is a - torsion, the induced map Hom K-U (* *B; M) -! Hom K-U (A; M) is an isomorphism. The following proposition follows immediately from the definitions (cf. [G], or* * [HLS1,2] where the same concept has been investigated in other settings). PROPOSITION 3.3. Let K be an unstable noetherian algebra and let a be an invari* *ant ideal. a) If 0 -! M -! M1 -! M2 is exact and M1 is a - closed and M2 is a - reduced th* *en M is a - closed. b) Any finite inverse limit of a - closed modules is a - closed. _ c) Any summand of an a - closed module is a - closed. |_| 3.4.______The categories T ors(a) are "Serre subcategories", i.e. their set of * *objects form a Serre class. They are localizing in the sense of Gabriel [G, III.3. Cor. 1] so the* *re are functors La : Kfg - U ! Kfg - U and natural transformations a : 1Kfg-U ! La such that fo* *r each M 2 Kfg - U: o LaM is a - closed and o kernel and cokernel of a;M are a - torsion. We will study the localization away from these subcategories. The following res* *ult provides general examples for a - closed modules. PROPOSITION 3.5. Let K be an unstable noetherian algebra, let a be an invariant* * ideal and (V; ') 2 R(K) be such that Rad (Ker ') does not contain a, in other words (V; '* *) 2 O(a). Then the following assertions hold for each unstable finitely generated K - mod* *ule M. a) ExtiK-U(N; TV (M; ')) = 0 for each a - torsion module N in Kfg - U and each * *i. In particular TV (M; ') is a - closed. b) RiFa(TV (M; ')) = 0 for all i. The proof of the proposition relies on the following key lemma. 15 LEMMA 3.6. Let F be an unstable finite TW (K; ) - module for some (W; ) 2 R(K* *) and let (V; ') be in R(K). Then there is an isomorphism of unstable K - modules Y TV (H*BW ( ) F ; ') ~= H*BW ( ) F : HomR(K)((V;');(W; )) Proof_of_Lemma_3.6:______By definition TV (H*BW ( )F ; ') ~=Fp(')T0VKTV (H*BW (* * )F ). Futhermore, because F is finite and TV commutes with tensor products we have an* * isomor- phism of unstable modules (see [L2,L3]) Y TV (H*BW F ; ') ~= H*BW F ; Hom(V;W) and we have to identify the TV K - module structure, and in particular the TV0K* * - mod- ule structure on this. Now F is bounded above, so the action of TW (K; ) on F* * factors through an action of (TW (K; )) 0. In particular, limiO(a)TV (M; ') is an unstable finitely genera* *ted a - torsion module for all i > 0. The following special case of this theorem needs to be emphasized. COROLLARY 3.10. Let K be an unstable noetherian algebra and let M be an unstabl* *e finitely generated K - module. a) Then there is a natural exact sequence 0 -! F M -! M -ae!limR*(K)TV (M; ') -! R1F M -! 0 17 in which the components of ae are induced by the maps aeM;(V;'). In particular,* * the kernel and cokernel of ae are finite. Furthermore, ae is localization away from the subca* *tegory of finite unstable K - modules. b) There are natural isomorphisms limiR*(K)TV (M; ') ~=Ri+1F M _ for all i > 0. In particular, limiR*(K)TV (M; ') is finite for all i > 0. |_| 3.11._Remarks.______a) Theorem 0.4 and Corollary 0.5 of the introduction are cl* *early just special cases of 3.9 and 3.10: the two subsets resp. subcategories both labelled O(a)* * (of A(G) and R(H*BG) respectively) clearly correspond under the equivalence of categorie* *s A(G) ~= R(H*BG) of 0.2.I; furthermore for M = H*GX the isomorphisms of 0.2.II are compa* *tible with the maps aeM;(V;')and the maps H*GX -! H*CG(E)(XE ) induced by the inclusions. * *This is obvious from the construction of the isomorphisms (as desribed in the appendix). b) Of course, as in the case of Theorem 0.4 of the introduction we have here th* *e same formal analogy with the classical case considered in [Gr]. c) If a is the ideal of positive dimensional elements of K then there is a vani* *shing result for the higher limits due to Oliver [O]: limiR*(K)TV (M; ') = 0 if i > d(K) where d(K) * *is the maximal rank of an elementary abelian p - group V such that (V; ') 2 R(K) for some ' : * *K -! H*V . This is analogous to the vanishing theorem of Grothendieck [Ha, Theorem III.2.7* *]. However, for more general ideals this analogy breaks down: e.g. if O is the open (!) set* * consisting of all (V; ') with Rad(Ker ') equal to a fixed minimal prime ideal then O = O(a) w* *here a is the intersection of all other invariant primes. In this case O(a) is equivalent to * *the one-object- category associated to the automorphism group Aut R(K)((V; ')) and the higher l* *imits can be identified with group cohomology and can be non-zero in arbitrary high degre* *es. The following result represents the key step in the proof of Theorem 3.9. LEMMA 3.12. Let K, a and O(a) be as in 3.9, let (W; ) be in R(K), F be an unst* *able finite TW (K; ) - module and consider the unstable K - module M = H*BW ( ) F . a) If (W; ) is in O(a) then ae : M -! limO(a)TV (M; ') is an isomorphism and for each i > 0 limiO(a)TV (M; ') = 0 : b) If (W; ) is not in O(a) then for each i 0 limiO(a)TV (M; ') = 0 : 18 Proof:______By Lemma 3.6Qthe functor on O(a) which sends (V; ') to TV (M; ') is* * given by TV (H*BW ( ) F ; ') ~= HomR(K)((V;');(W;())H*BW ( ) F ). In other words it* * is in- duced from the graded vector space H*BW ( ) F if (W; ) 2 O, and trivial other* *wise. (Here we identify the category of graded vector spaces with the category of fun* *ctors from the "trivial category", with one object (W; ) and the_identity morphism only, to t* *he category of graded vector spaces.) The Proposition follows. |_| Proof_of_Theorem_3.9:_____Consider an injective resolution Io of M in Kfg - U a* *s provided by Theorem 1.9 and Corollary 1.11. Because T is exact the complex of functors T- (* *Io; -) is a resolution of the functor T- (M; -). By the previous proposition the higher der* *ived functors of limO(a) vanish on the functors T- (Ik; -). This together with the fact that* * limO(a) is left exact implies that limiO(a)T- (M; -) can be computed as the cohomolgy of t* *he complex lim OT- (Io; -) Again by the previous proposition the complex limOT- (Io; -) is obtained from I* *o by throwing away those I(V;')(n) for which (V; ') is not in O(a) and keeping all the others* *. In other words, we get an exact sequence of complexes 0 -! FaIo -! Io -! limO(a)TV (Io; ') -! 0 : The long exact sequence belonging to this short exact sequence yields the exact* * sequence of a) and the isomorphisms in b). _ Finally the inverse limit is a - closed by 3.3 and 3.5. |_| 3.13.______The proof of 3.9 together with 2.15 gives also information about the* * height of the a - torsion modules RiFaM in terms of a given injective resolution. For example, if* * p = 2 and if ni is the maximum of all n such that for some (V; ') 2 C(a) the injective I(V;'* *)(n) occurs as a summand in Ik , then the height of the a - torsion module RiFaM is at most nk. In the case of equivariant cohomology information about injective resolutions i* *s often available (see [HLS2, II.1 and II.2], [HLO]). In joint work with F.Cohen and Y. Xia we wi* *ll apply this in the case of mapping class groups to get new classes in their cohomology. Theorem 3.9 together with Theorem 0.2 and the obvious generalization of Corolla* *ry 3.8 imply the following result. It contains the theorem of Jackowski and McClure [JM] (se* *e also [DW2]) as a special case. (Take O = A*(G) and C be isomorphic to Z=p!) Note also that * *under the equivalence of categories of Theorem 0.2.I a subset O of A(G) will be called op* *en if E 2 O implies E0 2 O whenever E is subconjugate to E0. COROLLARY 3.14. Let p be a prime. Assume G is a compact Lie group and let C b* *e an elementary abelian p - subgroup of G which is central in some p - Sylow subgrou* *p. Let O be any open subset of A(G) containing C. 19 a) Then the restriction maps H*BG -! H*BCG (E) induce an isomorphism ae : H*BG -! limOH*BCG (E) : _ b) Furthermore limiOH*BCG (E) = 0 for all i > 0. |_| The following change of rings type result gives us some flexibility for computi* *ng the higher limits. In particular it allows a change of groups in the situation of equivari* *ant cohomology. The result is implicit in [DW2]. PROPOSITION 3.15. Let K, L be unstable noetherian algebras and let f : K -! L * *be a homomorphism of unstable algebras which makes L into an unstable finitely gener* *ated K - module. Let M be an object in Lfg - U which we also consider as an object of Kf* *g - U via the map f. Then for any open set O 2 R(K) there are natural isomorphisms for al* *l i > 0 limiOTV (M; ') ~=limif*-1OTV (M; ) : (Of course, f* denotes the induced map R(L) -! R(K) and f*-1 O is the full subc* *ategory of R(L) whose objects are mapped to O under f*.) Proof:______The map f induces a functor f* : f*-1 O -! O and a functor f! : O -* * mod -! f*-1 O - mod . Here we write D - mod for the category of functors from a cate* *gory D to the category of abelian groups. The objects of D - mod are called D - modu* *les. Let fe!: f*-1 O - mod -! O - mod be the right Kan - extension along f!so that we ha* *ve natural isomorphisms Hom O-mod (F; ef!G) ~=Hom f*-1O-mod (f!F; G) for any O - module F and any f*-1 O - module G. We recall that the right Kan - extension of an f*-1 O - module G is given on th* *e object (V; ') 2 O as follows: Let (V; ') # f! be the under category with respect to f* *!. Then (ef!G)(V; ') = lim(V;')#f!G. We will need the following lemma. We omit its proof which is not difficult and * *completely analogous to that of Lemma 2.9 of [H3]. LEMMA 3.16. Let (V; ') be an object in O. Then the under category (V; ') # f! i* *s a disjoint union of categories each of which has an initial object. The components are ind* *exed by those (V; ) 2 f*-1 O which extend (V; '), i.e. for which_' = f holds. These objects* * are at the same time the initial objects of the components. |_| The lemma implies that the Kan extension is given by the following formula: Y (ef!G)(V; ') = G(V; ) : (V; ) 20 The product in this formula is indexed by those (V; ) 2 f*-1 O which extend (V* *; '). In particular, the Kan - extension ef!is exact. The functor f! is clearly exact an* *d we conclude that it carries projective resolutions to projective resolutions. Taking a proj* *ective resolution of the constant functor with value Z we obtain for any R(L) - module G: limiOef!G ~=limif*-1OG : Now let M be an unstable L - module and consider the f*-1 O - module T- (M; -).* * Now let (V; ') 2 O. It is easy to check that (with an obvious abuse of notation) w* *e have (f!T- (M; -))(V; ') ~=TV_(M; '). In other words, we obtain the right Kan - ext* *ension and the result follows. |_| 4. An application to H*BGL(n; Z[1=2]) 4.1.______We give an application of 3.9 to a qualitative study of the mod - 2 c* *ohomology ring H*BGL(n; Z[1=2]). Here GL(n; Z[1=2]) is the general linear group of rank n ove* *r the ring Z[1=2]. Let Dn be the subgroup of diagonal matrices with diagonal entries 1. Th* *e image of the restriction map aen : H*BGL(n; Z[1=2]) -! H*BDn has recently been determ* *ined by Mitchell [Mt]; it is isomorphic to a tensor product F2[w1; : :;:wn]E(e1; : :;:e* *2n-1) of a poly- nomial algebra on generators w1; : :;:wn and an exterior algebra on generators * *e1; : :;:e2n-1, with indices giving the degrees of the elements. Let n0 be minimal such that ae* *n0 is not in- jective. According to Dwyer (private communication) n0 is finite. We will show * *in [H4] that n0 > 3. The theory developed here can be used to get some qualitative informati* *on about the size of Ker aen. To do this we recall (cf. [HLS2,II.5.2]) that the objects of Rn := R(H*BGL(n; * *Z[1=2])) can be identified with the faithful representations of elementary abelian 2 - g* *roups V , i.e. with formal sums OnOO where O runs through the characters of V and the nO are * *non- negative integers such that OnO = n and such that the set of O with nO > 0 span* *s V *, the group of characters of V . (The representation ' is identified with the obj* *ect (V; '*) in R(H*BGL(n; Z[1=2])), if '* denotes the induced map in cohomology, and correspon* *ds to the object Im ae in A(GL(n; Z[1=2])) under the equivalence of categories in 0.2.I.)* * Furthermore, if ' = OnOO (resp. '0 = O0nO0O0) are faithful representations of V (resp. V * *0) then Hom Rn ('; '0)Q= {ff 2 Hom (V; V 0)|' = O0nO0O0ff}. We note that the centraliz* *er of Im ' is isomorphic to OGL(nO; Z[1=2]). Let On Rn be the subset consisting of those representations ' for which nO < n* *0 for all O. This is clearly an open subset and we have On = O(an) where an is the invariant* * ideal given asQthe intersection of all Rad (Ker '*) with (V; '*) =2On. ThenQTV (H*BGL(n; Z[* *1=2]); ') ~= O H*BGL(nO; Z[1=2]) by Theorem 0.2 and hence embedds into OH*BDnO ~= H*BDn for all ' 2 On. Consequently the kernel of the localization map away from T ors* *(an) modules agrees with the kernel of aen. In particular, Ker aen is an - torsion. If ' i* *s a representation 21 of V which is not in On then V has rank at most equal to n - n0 + 1, therefore * *the size (as defined in the introduction) of H*BGL(n; Z[1=2])= Rad(an) is at most equal to n* * - n0 + 1. Clearly this number is an upper bound for the size of the an - torsion module K* *er aen0. In fact, we have the following theorem. THEOREM 4.2. The kernel of the restriction map H*BGL(n; Z[1=2]) -aen!H*BDn is * *the largest an - torsion submodule and for n n0 it has size n - n0 + 1 where n0 de* *notes the smallest natural number such that aen0 is not a monomorphism. In particular, t* *he size of Ker aen0 is one, i.e. Ker aen0 is periodic in large degrees. Proof:______We have already seen that the size is at most n - n0 + 1. Let dn be equal to the size of Ker aen. Consider an embedding of Ker aen0 as i* *n 1.9, and assume it is minimal in the sense that no factor I(V;')(n) can be dropped witho* *ut loosing the embedding property. Then it is clear that the size of Ker aen is equal to * *the maximum of the sizes of the injectives I(V;')(n) involved, and hence exactness of T and* * Lemma 3.6. imply that each component TV (Ker aen; '*) has size at most dn. It suffices the* *refore to find a faithful representation ' : V - ! GL(n; Z[1=2]) such that TV (Ker aen; '*) has * *size n - n0 + 1. Such a representation can be obtained as follows. Let V be elementary abelian* * of rank n-n0+1 and let Oi, 1 i n-n0+1 be a dual basis of V . Then consider ' = n0O1+i* *6=1Oi. Applying TV (-; '*) to the exact sequence 0 -! Keraen -! H*BGL(n; Z[1=2]) -! H*BDn yields an exact sequence Y 0 -! TV (Ker aen; '*) -! H*BGL(n0; Z[1=2]) H*BDn-n0 -! H*BDn where the product is taken over all homomorphisms : V - ! Dn such that its co* *mposition with the inclusion of Dn into GL(n; Z[1=2]) is conjugate to ' (cp. the discussi* *on of the Kan extension in the proof of 3.15.) Now the different maps H*BGL(n0; Z[1=2])H*BDn-* *n0 -! H*BDn differ only by the action of an appropriate element of the symmetric grou* *p Sn on the target, in particular all these maps have the same kernel, which is equal t* *o Ker aen0 H*BDn-n0. This has size n - n0 + dn0 and hence we only have to show that the s* *ize of Ker aen0 is positive. However, Theorem 0.2 and Proposition 3.5 show that H*BGL(* *n; Z[1=2]) does not contain any unstable finite ideals (take for a the ideal of elements i* *n positive_degrees and for ' the restriction map to the central Z=2), so the size must be positive* *. |_| The method used in the discussion above should lead to similar results for gene* *ral linear groups over rings of S - integers in other number fields. 22 Appendix. Lannes' T - functor and Borel constructions of discrete groups Let p be a fixed prime. As before we surpress the coefficients from our notatio* *n. In this appendix we prove part (b) of Theorem 0.4. We will use freely results a* *nd terminology of [HLS1]. Let V be an elementary abelian p - group and ae a homomorphism from V to the (d* *iscrete) group . Let X be a - space. Denote the centralizer of the image of ae in by a* *eand the fixed point set with respect to the image of ae by Xae. Then the homomorphism V* * x ae! , (v; g) 7! ae(v)g induces a map BV x (Eaexae Xae) -! E x X which we denote by c* *ae. Passing to cohomology and using adjointness we obtain a map ad(c*ae) : TV H*X -* *! H*aeXae. THEOREM A.1. Let be a discrete group and X a finite dimensional - CW - comple* *x of finite orbit type whose isotropy groups are all finite. Then the natural map Y TV (H*X) -! H*ae(Xae) ae2Rep(V;) with components ad(c*ae) is an isomorphism for each elementary abelian p - grou* *p V . (Here Rep (V; ) denotes the set of - conjugacy classes of homomorphisms from V to * *and we have chosen a representative ae from each conjugacy class. ) Proof:______Because TV is exact and commutes with direct sums, it is enough to * *do the case of an orbit, i.e. X = G=H with H finite. In this case we have natural isomorphisms Y TV H*X ~=TV H*BH ~= H*BHae ae2Rep(V;H) where the second isomorphism comes from Lannes' Theorem [L1,L3] and has compone* *nts ad(c*ae) as in the statement of the theorem. Now it follows from [H3, Lemma 2.8* *] that there is a natural isomorphism Y Y (A:1:1:) H*BCH (ae) ~= H*ae(Xae) : ae2Rep(V;H) ae2Rep(V;) It is straightforward_to check that this string of isomorphisms is given by the* * natural map in question. |_| If X can also be chosen mod - p acyclic then by Smith theory the fixed point se* *ts Xaeare mod - p acyclic as well. Hence we have H*X ~=H*B and H*aeXae~=H*Bae. Consequent* *ly we obtain the following result. 23 COROLLARY A.2. Let be a discrete group which admits an action on a finite dime* *nsional mod - p acyclic - CW - complex with finite orbit type and with finite isotropy* * groups. Then the natural map Y TV (H*B ) -! H*(Bae) ae2Rep(V;) _ is an isomorphism for each elementary abelian p - group V . |_| THEOREM A.3. Let be a discrete group and X be a - CW - complex with finitely * *many equivariant cells and finite isotropy groups. Then H*(X) is a finitely generate* *d algebra. Proof:______We will first prove that H*(X) is noetherian up to F - isomorphism,* * or more precisely, that its Nil - closure is noetherian. Consider the contravariant functor g(H*X) from elementary abelian p - groups to* * sets which associates to an elementary abelian p - group V the set Hom K (H*X; H*BV ). T* *his set is given by the spectrum of the p - Boolean algebra TV0H*X [L2,L3] which by Theore* *m A.1 above can be identified with the disjoint union a a ss0(Eaexae Xae) ~= ss0(Xae)=ae: ae2Rep(V;) ae2Rep(V;) It follows from (A.1.1) together with our finiteness assumptions that this set * *is finite (although the set Rep (V; ) need not be finite). The functor g(H*X) has finite transcenda* *nce degree d in the sense of [HLS1,II.5] and, in fact, d is equal to the maximal rank of a* *n elementary abelian subgroup which occurs as isotropy subgroup in X. Furthermore the End (* *V ) - set g(H*X)(V ) is noetherian in the sense of [HLS1], which means that the Nil - clo* *sure of H*X (which is given, up to isomorphism, by Quillen's inverse limit) is noetherian [* *HLS1,II.7]. We can now pick an unstable noetherian subalgebra K of H*X which is F - isomorp* *hic to the Nil - closure of H*X. The spectral sequence of the map E x X -! \X is a sp* *ectral sequence of H*X - modules and hence one of K - modules. Because K is noetherian* * and this spectral sequence has only finitely many columns it is enough toLshow that its * *E1 - term is a finitely generated K - module. The E1 - term is given as Es;*1= oeH*(Boe) whe* *re oe runs over the set of s - cells in \X and oedenotes the isotropy group of a chosen re* *presentative of the set of cells in X which project to the cell oe in \X. It is clearly enou* *gh to consider a single cell, i.e. to show that H*(Boe) is a finitely_generated K - module. Ho* *wever, this follows immediately from [HLS1, II. Prop. 7.8]. |_| In particular, if X can also be chosen mod - p acyclic we obtain. COROLLARY A.4. Let be a discrete group which acts on a mod - p acyclic - CW - complex with finitely_many - cells and with finite isotropy groups. Then H*B i* *s a finitely generated algebra. |_| 24 We repeat that interesting examples of such groups are (S)-arithmetic groups, m* *apping class groups, outer automorphism groups of free groups and word-hyperbolic groups in * *the sense of Gromov. Proof_of_Theorem_0.2.b_____: The functor g(H*B) which by A.2 sends V to Rep(V; * *) determines the category R(H*B) (see [HLS1,II.7]); as in the other cases of Theorem 0.2 its* * objects can be identified with the conjugacy classes of monomorphisms from elementary abeli* *an p - groups to . This category is (for any group ) equivalent to the category A() and part * *I follows. _ Part II follows immediately from Theorem A.1. |_| A.5_Remark._______In [H3] we use that Corollary 0.5 of this paper holds under t* *he assumptions of Theorem A.3 which are a bit more general than those of 0.5 because it is not* * assumed that H*B is noetherian. 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