PRIMITIVES AND CENTRAL DETECTION NUMBERS IN GROUP COHOMOLOGY NICHOLAS J. KUHN Abstract.Fix a prime p. Given a finite group G, let H*(G) denote its mod p cohomology. In the early 1990's, Henn, Lannes, and Schwartz introduced two invariants d0(G) and d1(G) of H*(G) viewed as a module over the mod * *p- Steenrod algebra. They showed that, in a precise sense, H*(G) is respect* *ively detected and determined by Hd(CG(V )) for d d0(G) and d d1(G), with V running through the elementary abelian p-subgroups of G. The main goal of this paper is to study how to calculate these invaria* *nts. We find that a critical role is played by the image of the restriction o* *f H*(G) to H*(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Ho* *pf algebra H*(C) H*(G)Fp, a number that tends to be quite easy to calculat* *e. Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using Benson-Carlson duality, we show tha* *t in this case, d0(G) = d0(P) = e(P), and a similar exact formula holds for d* *1. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomol* *ogy, reproving and sharpening a theorem of Adem and Karagueuzian. In general, we are able to show that d0(G) max{e(CG(V )) | V < G} if certain cases of Benson's Regularity Conjecture hold. In particular,* * this inequality holds for all groups such that the difference between the p-r* *ank of G and the depth of H*(G) is at most 2. When we look at examples with p =* * 2, we learn that d0(G) 14 for all groups with 2-Sylow subgroup of order u* *p to 64, with equality realized when G = SU(3, 4). Enroute we study two objects of independent interest. If C is any cent* *ral elementary abelian p-subgroup of G, then H*(G) is a H*(C)-comodule, and we prove that the subalgebra of H*(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H*(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural resu* *lts. 1.Introduction Fix a prime p, and let H*(G) denote the mod p cohomology ring of a finite gro* *up G. The p-elementary abelian subgroups of G have had a featured role in the study of group cohomology since D.Quillen's famous work [Q ] in the late 1960's. In p* *ar- ticular, these subgroups become the objects in a category A(G) having morphisms the homomorphisms generated by subgroup inclusion and conjugation by elements in G. The inclusions V < G then induce a map Y H*(G) ~0-! lim H*(V ) H*(V ), V 2A(G) V ____________ Date: December 5, 2006. 2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40. This research was partially supported by a grant from the National Science F* *oundation. 1 2 KUHN and ~0 is shown to have kernel and cokernel that are nilpotent in an appropriate sense. Viewing H*(G) as the mod p cohomology of the classifying space BG makes it evident that H*(G) is an object in K and U, the categories of unstable algebras and modules over the mod p Steenrod algebra A. The 1980's and 1990's saw a revolution in our understanding of these categories, and the 1995 paper of H.- W.Henn, J.Lannes, and L.Schwartz [HLS1 ] revisited Quillen's approximation of H*(G) from this new perspective. For each d 0, the group homomorphisms V x CG (V ) ! G induce a map of unstable algebras Y H*(G) ! H*(V ) H d (CG (V )), V where M d denotes the quotient of a graded module M* by all elements of degree more than d. The image of this map lands in an evident subalgebra of `compatibl* *e' elements which Henn, Lannes, and Schwartz show can be naturally identified with LdH*(G), where Ld : U ! U is localization away from the localizing subcategory generated by (d+1)-fold suspensions of unstable modules. Thus Quillen's map can be viewed as the just the bottom of a tower of localizations of H*(G) associate* *d to the nilpotent filtration of U: .. . | | fflffl| L2H*(G)99 ss ssss |p2| ~2 ssss fflffl| ssss iL1H*(G)44 sss iiii ssssii~1iiiii |p1| siiiiiiss~0 fflffl| H*(G) ________________//_L0H*(G), where we have Y H*(G) ~d-!LdH*(G) H*(V ) H d (CG (V )). V This caused the authors of [HLS1 ] to introduce two new invariants of G: d0(G) and d1(G) are the smallest d's such that H*(G) is respectively detected by, and isomorphic to, LdH*(G). Alternatively, d0(G) is the smallest d such that H*(G) contains no (d + 1)-fold suspensions of a nontrivial unstable module, and d1(G)* * is the smallest d such that also Ext1A( d+1N, H*(G)) = 0 for all N 2 U. These invariants satisfy a few easily verified nice properties: d0(GxH) = d0(* *G)+ d0(H), d1(G x H) = max {d1(G) + d0(H), d0(G) + d1(H)}, and di(G) di(P ) if P is a p-Sylow subgroup of G. However, they are not well behaved under taking subgroups, quotient groups, and extensions; e.g., every G embeds in a symmetric group n and d0( n) = 0. Rough upper bounds for d0(G) and d1(G) were found in [HLS1 ]; e.g., d0(G) is bounded by n2 if a p-Sylow subgroup of G admits a faith* *ful n dimensional complex representation. However, in all but a few examples, these bounds seem far from optimal. Up to now, what determines these group invariants GROUP COHOMOLOGY PRIMITIVES 3 has remained mysterious, and they have not been connected to other work in group cohomology. A main goal of this paper is to present a way to calculate the number d0(G), * *and, in some cases, d1(G). Our finding is that these numbers seem to be controlled by the restriction of cohomology to maximal central p-elementary abelian subgroups. Our results are complete when G has a Sylow subgroup that is p-central, i.e. a group in which every element of order p is central. For example, when p = 2, we compute that d0(SU(3, 4)) = 14 and d1(SU(3, 4)) = 18, where, by constrast, the estimates from [HLS1 ] yield only that d0(SU(3, 4)) 64 and d1(SU(3, 4)) 120. Our method is to combine U-technology, in the spirit of [HLS1 ], with duality results as in the work of D.Benson and J.Carlson [BC1 ]. We ultimately connect a conjectured upper bound for d0(G) to Benson's Regularity Conjecture [Be], known to hold if the p-rank of G and the p-rank of Z(G) differ by at most 2. This is * *the case for all 2-groups of order 64 or less, and, using cohomology calculations f* *rom [CTVZ ], we've been able to verify by hand that d0(G) 14 for all such groups. A number of side results of independent interest come up in our investigation* *s. We are led to study carefully the cohomology of central extensions, in partic* *ular the structure of associated algebras of primitives. One outcome of this is a n* *ew proof of A.Adem and D.Karagueuzian's theorem [AK ] that p-central p-groups have nonzero essential cohomology. We show that in an explicit degree there is a non* *zero cohomology class that is simultaneously essential and annihilated by all Steenr* *od operations of positive degree. Deriving our general estimate of d0(G) involves a careful study of the depth essential cohomology of Carlson, et. al. [CTVZ ] in the important special case* * that the depth of H*(G) equals the rank of the center. We prove that then the depth essential cohomology is both nonzero - reproving the main theorem of [G1 ] with* *out D.Green's hypothesis that G be a p-group - and Cohen-Macauley. In the next section we describe our results in more detail. 2.Main Results 2.1. The cohomology of central extensions. Suppose we have a central exten- sion of finite groups C i-!G q-!Q, where C is p-elementary abelian of rank c. We define various objects associated to this situation. The extension corresponds to an element o 2 H2(Q; C). Since H2(Q; C) = Hom (H2(Q), C), the extension can also be considered as corresponding to a homo- morphism o : H2(Q) ! C, or, equivalently, its dual o# : C# ! H2(Q). Let {E*,*r} denote the Serre spectral sequence associated to the extension, c* *on- verging to H*(G), and with E*.*2= H*(Q) H*(C). Under the identification C# = H1(C), it is standard that o# corresponds to d2 : E0,12! E2,02. Let Io H*(Q) be the ideal generated by A . im(o#G), so that H*(Q)=Io is an unstable algebra. It is easy to see that Io is contained in the kernel of infl* *ation q* : H*(Q) ! H*(G). Call a*subalgebra A of H*(G) a (G, C)-Duflot subalgebra, if the composite A H*(G) i-!im(i*) is an isomorphism, where i* : H*(G) ! H*(C) is the restriction. As we will describe more precisely in x2.5, as an algebra, the Hopf algebra im(* *i*) 4 KUHN H*(C) will necessarily be free graded commutative on c polynomial generators, possibly tensored with an exterior algebra on some generators in degree 1, if p* * is odd. It follows that Duflot subalgebras exist and have the same form. Let QA H** *(G) denote the graded algebra1 of A-indecomposables H*(G) A Fp, or, equivalently, the quotient of H*(G) by the ideal generated by the positive degree elements of* * A. H*(C) is a Hopf algebra, and the multiplication map m : C x G ! G induces a map of unstable algebras m* : H*(G) ! H*(C) H*(G) making H*(G) into a H*(C)-comodule. We define the associated algebra of prim- itives to be PC H*(G) = {x 2 H*(G) | m*(x) = 1 x} m* * = Eq {H*(G) -!-!H (C x G)}, ss* where ss : C x G ! G is the projection. It is easy to check that PC H*(G) is an unstable algebra that contains the image of the inflation map. Thus q* : H*(Q) ! H*(G) refines to a map of unstable algebras qo : H*(Q)=Io ! PC H*(G). Theorem 2.1. With the notation as above, the following are true. (a) H*(G) is a free A-module. Moreover {E*,*r} is a spectral sequence of free E* *0,*1- modules, and applying QE0,*1to the spectral sequence yields a spectral sequence* * con- verging to QA H*(G) with E2-term QE0,*1H*(C) H*(Q). (b) The composite PC H*(G) ,! H*(G) i QA H*(G) is monic. (c) Both PC H*(G) and QA H*(G) are finitely generated H*(Q)-modules. (d) The map qo : H*(Q)=Io ! PC H*(G) is an F -isomorphism2, and the rings H*(Q)=Io, im(q*), PC H*(G), and QA H*(G), are all Noetherian of Krull dimen- sion equal to (the p-rank of G) - (the rank of C). Let C(G) < G be the p-elementary abelian part of Z(G). If C = C(G), the first part of statement (a) recovers J. Duflot's result [D ] that the depth of H*(G) * *is at least as great as the rank of C(G).3 We will call a (G, C(G))-Duflot subalgebra* * of H*(G) simply a Duflot subalgebra. The p-rank of G equals the rank of C exactly when C = C(G) and G is p-central, and we have the following corollary. Corollary 2.2. If G is p-central, and C = C(G), then the rings H*(Q)=Io, im(q*), PC H*(G), and QA H*(G) all have Krull dimension zero and so are finite dimen- sional Fp-algebras. ____________ 1QAH*(G) will not necessarily be an unstable algebra, as A need not be close* *d under Steenrod operations. 2In the sense of Quillen [Q ]: ker(~q*) is nilpotent, and for all x 2 PCH*(G* *), there exists a k so that xpk2 im(~q*). 3We are claiming no originality in the proof of this, which is similar to al* *l proofs of Duflot's theorem following [BrH]. The spectral sequence refinement seems to be a new obs* *ervation. GROUP COHOMOLOGY PRIMITIVES 5 2.2. Quillen's category and functors involving primitives. Our algebras of primitives arise in two formulae associated to H*(G), viewed as an object in K.* * To describe these, we need to introduce some notation. Given a small category C, we let C# denote the associated twisted arrow categ* *ory: the objects of C# are the morphisms of C, and a morphism ff _ fi from ff : A1 !* * A2 to fi : B1 ! B2 is a commutative diagram in C A1 __ff_//A2OO | | | | fflffl|fi| B1 _____//B2. The functor assigning H*(V ) to V 2 A(G) is contravariant, while the assignme* *nt of H*(CG (V )) is covariant. Now observe that the assignment of Pff(V1)H*(CG (V* *2)) to ff : V1 ! V2 can be viewed as defining a contravariant functor of A(G)# . Let AC (G) denote the full subcategory of A(G) having as objects the V con- taining C(G). If G is p-central, then AC (G) has a single object and morphism. 2.3. A formula for the locally finite part of H*(G). If M is an unstable A- module, we define MLF , the locally finite part of M, by MLF = {x 2 M |Ax M is finite}. This is again an unstable module, and is an unstable algebra if M is. Theorem 2.3. There is a natural isomorphism of unstable algebras H*(G)LF ' limffPff(V1)H*(CG (V2)), V1-!V2 where the limit is over AC (G)# . Corollary 2.4. If G is p-central, then H*(G)LF = PC(G)H*(G). 2.4. A formula for ~RdH*(G). An unstable module M 2 U has a canonical `nilpo- tent' filtration [S1, K1, HLS1 ]: . . .nil2M nil1M nil0M = M. In general, nildM=nild+1M = dRdM, where RdM is reduced, i.e. has no nontriv- ial submodules that are suspensions. We let ~RdM denote the nilclosure L0RdM of RdM. The module nildM identifies with the kernel of ~d : M ! Ld-1M, and a bit of diagram chasing will show that dR~dM is isomorphic to the kernel of LdM ! Ld-1M: see Proposition 3.1. Thus d0(G) is the length of the filtration of H*(G), and also is the biggest d such that ~RdH*(G) 6= 0. Theorem 2.5. There is a natural isomorphism of unstable modules ~RdH*(G) ' lim H*(V1) Pff(V1)Hd(CG (V2)), V1-ff!V2 where the limit is over AC (G)# . Corollary 2.6. If G is p-central, then there is an isomorphism of unstable modu* *les R~dH*(G) ' H*(C(G)) PC(G)Hd(G). 6 KUHN 2.5. Invariants of restriction to C(G). If i : C < G is a central p-elementary abelian of rank c, then ( H*(C) ' F2[x1, . .,.xc] if p = 2 (x1, . .,.xc) Fp[y1, . .,.yc]if p,is odd where |xi| = 1 and yi= fi(xi), and is a Hopf algebra in the usual way. In x6, we will see that, after a change of basis for H1(C), the image of the restriction homomorphism i* : H*(G) ! H*(C) will be a sub Hopf algebra of H*(C) of the form ( j1 jc F2[x21 , . .,.x2c ] if p = 2 im (i*) = pj1 pjb Fp[y1 , . .,.yb , yb+1, . .,.yc] (xb+1,i.f.,.xc)p,is odd with the ji forming a sequence of nonincreasing nonnegative integers4. Now suppose that C = C(G). We will say that G has type [a1, . .,.ac] where ( j 1, . .,.2jc) if p = 2 (a1, . .,.ac) = (2 (2pj1, . .,.2pjb, 1, .i.,.1)f p.is odd The type of G has the form [1, . .,.1] if and only if G = C x H, where Z(H) h* *as order prime to p. In all other cases, a1 = 2pk for some k 0. Define e(G) and h(G) by Xc e(G) = (ai- 1), i=1 and 8 ><2pk-1 ifa1 = 2pk withk 1 h(G) ' >1 ifa1 = 2 :0 ifa1 = 1. For example, Q8x Z=4 has type [4, 2] when p = 2, so that e(Q8x Z=4) = 4, and h(Q8 x Z=4) = 2. Remark 2.7. The careful reader will observe that the type of G is just the list* * of the degrees of the unstable A-algebra generators of im(i*), listed in decreasing or* *der, e(G) is the top nonzero degree of the finite dimensional Hopf algebra H*(C) H*(G)Fp = H*(C)=(im(i*>0)), and h(G) is the top nonzero degree of the module A . H1(C) projected into this Hopf algebra. 2.6. PC(G)H*(G), d0(G), and d1(G) when G is p-central. If G is p-central with C = C(G), then PC H*(G) is a finite dimensional unstable algebra. We identify i* *ts top degree and more. Theorem 2.8. Let G be p-central, C = C(G), and A be a Duflot subalgebra of H*(G). Then both PC H*(G) and QA H*(G) are zero in degrees greater than e(G), and one dimensional in degree e(G). Furthermore, PC He(G)(G) is annihilated by ____________ 4In the odd prime case, c - b will be the rank of the largest subgroup of C * *splitting off G as a direct summand. GROUP COHOMOLOGY PRIMITIVES 7 all positive degree elements of the Steenrod algebra, and, if G is a p-group, c* *onsists of essential5 cohomology classes. The last statement implies the main result of [AK ]: p-central p-groups have nonzero essential cohomology. This theorem, combined with Corollary 2.6 and related results, leads to the following calculation. Theorem 2.9. Let G be p-central. Then d0(G) = e(G) and d1(G) = e(G) + h(G). Furthermore, if G is a finite group with p-Sylow subgroup P , and P is p-centra* *l, then d0(G) = d0(P ) and d1(G) = d1(P ). Corollary 2.10. If G is p-central, and H < G, then d0(H) d0(G) and d1(H) d1(G). Examples 2.11. (a) As Q8 is 2-central of type [4], d0(Q8) = 3 and d1(Q8) = 3 + 2 = 5, in agreement with [HLS1 , (II.4.6)]. (b) The hypotheses of p-centrality is needed in the last part of the theorem: as observed in [HLS1 , II.4.7], if G = GL2(F3) and P = SD16, then P is the 2- Sylow subgroup of G, but d0(G) = 0 < 2 = d0(P ), and d1(G) = 2 < 4 = d1(P ). Similarly, G needs to be p-central in the corollary: if H = Z=4 < D8 = G, then d0(H) = 1 > 0 = d0(G) and d1(H) = 2 > 0 = d1(G). The example H = Z=4 < Z=8 = G shows that the inequalities of the corollary can be equal- ities, even when H is a proper subgroup of a p-central p-group G. (c) The 2-Sylow subgroup P of the simple group SU(3, 4) is 2-central of type [8* *, 8]. Thus d0(SU(3, 4)) = d0(P ) = 14 and d1(SU(3, 4)) = d1(P ) = 18. Similarly, the 2-Sylow subgroup Q of the simple group Sz(8) is 2-central of type [4, 4, 4]. Th* *us d0(Sz(8)) = d0(Q) = 9 and d1(Sz(8)) = d1(Q) = 11. We will see that P and Q have the largest d0 of all 2-groups of order dividing 64. For more about the SU(3, 4) example, see x9. (d) In [AKM ], the authors associate a 2-central Galois group GF to every field F of characteristic different from 2 that is not formally real. (They call thi* *s the W -group of F because of its connections to the Witt ring W F [MiS ].) From the* *ir construction it is easy to deduce that GF has type [2, . .,.2]. Thus d0(GF) = r* * and d1(GF) = r + 1, where GF has rank r. In particular, the universal group W -group W (n) has d0(W (n)) = n+12 and d1(W (n)) = n+12+ 1. For more about this example, see x9. 2.7. Central essential cohomology. Our calculation of d0(G) when G is p- central relies on Corollary 2.6. To understand d0(G) for general G, one needs to use the more complicated formula given in Theorem 2.5. Using some analysis of this already done by us in our companion paper [K3 ], we are led to a formula6 * *for d0(G) that makes use of the following variant of essential cohomology. ____________ 5Recall that x 2 H*(G) is essential if it restricts to zero on all proper su* *bgroups. 6Thus far, we have not found an analogous formula for d1(G). 8 KUHN We define Cess*(G), the central essential cohomology of G, to be the kernel of the restriction map Y H*(G) ! H*(CG (U)), C(G) 0, nildM = ker~d-1. An unstable module M is called reduced if nil1M = 0. As observed in [K1 , Prop.2.2], nildM=nild+1M = dRdM, where RdM is a reduced unstable module. (See also [S2, Lemma 6.1.4].) Then ~RdM is defined to be the N il1-closure of R* *dM. Thus RdM L0RdM = ~RdM. ____________ 8This section necessarily overlaps with the presentation in our recent prepr* *int [K3]. 9What we are calling Ld here was called Ld+1in [HLS1]. GROUP COHOMOLOGY PRIMITIVES 11 We have the following useful alternative definition of R~dM. (Compare with [HLS1 , I(3.8.1)].) Proposition 3.1. There is a natural isomorphism dR~dM ' ker{LdM ! Ld-1M}. The functors Ld and Ld-1 are left exact, as they are localizations, and thus * *we conclude Corollary 3.2. R~d: U ! U is left exact. Proof of Proposition 3.1.Let cd+1M = coker{~d : M ! LdM}. Then cd+1M 2 N ild+1, and there is an exact sequence 0 ! nild+1M ! M ! LdM ! cd+1M ! 0. Diagram chasing then shows that there is a natural short exact sequence 0 ! nildM=nild+1M ! ker{LdM ! Ld-1M} ! ker{cd+1M ! cdM} ! 0. As the middle module here is N ild+1-closed, and the right module is in N ild+1* *, we see that the left map identifies with ~d. Recalling that nildM=nild+1M = dRdM, this says that there is a natural isomorphism Ld( dRdM) ' ker{LdM ! Ld-1M}. The proof of the proposition is then completed by observing that Ld( dRdM) ' dR~dM, a consequence of the next proposition. Proposition 3.3. There is a natural isomorphism Lc+d( dM) ' dLcM, for all M 2 U. Proof.We need to check that the map d~c : dM ! dLcM satisfies the two properties characterizing localization away from N ilc+d+1. That ker( d~c) and coker( d~c) are both in N ilc+d+1 is clear, as ker(~c) and coker(~c) are both N ilc+1, and the d-fold suspension of a module in N ilc+1 wi* *ll be in N ilc+d+1. To see that the range of d~c is N ilc+d+1-closed, we check that if M 2 U is N ilc+1-closed then dM is N ilc+d+1-closed. This follows from the following characterization of N ilc+1-closed modules: M 2 U is N ilc+1-closed if and only* * if it fits into an exact sequence of the form Y Y 0 ! M ! H*(Vff) Mff! H*(Wfi) Nfi, ff fi with all the modules Mffand Nficoncentrated in degrees between 0 and c. See [BrZ2 , Prop.1.15]. 3.2. Further properties of Ld, Rd, and ~Rd. We need to recall some notation and terminology. If V is an elementary p-group, TV : U ! U is defined to be the left adjoint to H*(V ) ___, as famously studied by Lannes [L1, L3]. Given* * a Noetherian unstable algebra K 2 K, Kf.g.- U is defined to be the category studi* *ed in [HLS1 , I.4] whose objects are finitely generated K-modules M whose K-module structure map K M ! M is in U, and morphisms are K-module maps in U. 12 KUHN Proposition 3.4. The functor Ld : U ! U satisfies the following properties. (a) There are natural isomorphisms L0(M N) ' L0M L0N. (b) There are natural isomorphisms TV LdM ' LdTV M. (c) If K 2 K, then LdK 2 K, and K ! LdK is a map of unstable algebras. If K is also Noetherian, and M 2 Kf.g.- U, then LdK 2 Kf.g.- U, and thus is Noetherian, and LdM 2 LdKf.g.- U. Property (b) can be deduced from properties of TV as follows. First, to see t* *hat TV LdM is N ild+1-closed, we compute, for s = 0, 1 and N 2 N ild+1: ExtsU(N, TV LdM) = ExtsU(H*(V ) N, LdM) = 0, since H*(V ) N will be in N ild+1 if N is. Second, TV ~d : TV M ! TV LdM is a N ild+1-isomorphism, as the kernel and cokernel are in N ild+1, since TV is exa* *ct and sends N ild+1 to itself. See [HLS1 , I.4] and [BrZ1 ] for more detail about properties (a) and (c). Proposition 3.5. The functors Rd : U ! U satisfy the following properties. (a) There are a natural isomorphisms R*(M N) ' R*M R*N of graded objects in U. (b) There are natural isomorphism TV RdM ' RdTV M. (c) If K 2 K, then R0K 2 K, and K ! R0K is a map of unstable algebras. If K is also Noetherian, and M 2 Kf.g.- U, then R0K is also a Noetherian unstable algebra, and RdM 2 R0Kf.g.- U, for all d. For the first two properties, see [K1 , x3], and the last follows easily from* * the first. Proposition 3.6. The functors ~Rd: U ! U satisfy the following properties. (a) There are natural isomorphisms ~R*(M N) ' ~R*M ~R*N of graded objects in U. (b) There are natural isomorphisms TV ~RdM ' ~RdTV M. (c) If K 2 K, then ~R0K 2 K, and K ! ~R0K is a map of unstable algebras. If K is also Noetherian, and M 2 Kf.g.- U, then ~R0K is also a Noetherian unstable algebra, and ~RdM 2 ~R0Kf.g.- U, for all d. This, of course, follows from the previous two propositions. A Noetherian unstable algebra K has a finite Krull dimension dimK. We have an addendum to Proposition 3.4. Proposition 3.7 ([K3 , Prop.4.10]). If an unstable algebra K is Noetherian, then dimK = dimL0K. Another special property of L0 that we will need goes as follows. GROUP COHOMOLOGY PRIMITIVES 13 Proposition 3.8 ([L1, Lem.4.3.3]). Let f : M ! N be a map in K. Then L0f : L0M ! L0N is an isomorphism if and only if, for all p-elementary abelian groups V , the i* *nduced map f* : Hom K(N, H*(V )) ! Hom K(M, H*(V )) is a bijection. As in the introduction, given M 2 U, MLF denotes the submodule of locally finite elements: x 2 M such that Ax M is finite. Proposition 3.9. There is a natural isomorphism (RdM)0 = (R~dM)0 ' (MLF )d. See [K1 , x3] for a proof. Finally, Henn [H ] proved the following important finiteness result. Proposition 3.10. Let K 2 K be Noetherian, and M 2 Kf.g.- U. Then the M is N ild-local for d >> 0. In particular, the nilpotent filtration of M has finite* * length. 3.3. Properties of d0M and d1M. The authors of [HLS1 ] define d0M and d1M as follows. Definition 3.11. Let M be an unstable module. (a) Let d0M be the smallest d such that ~d is monic, or 1 if no such d exists. Equivalently, d0M is the smallest d such that Hom U(N, M) = 0 for all N 2 N ild* *+1, or the smallest d such that nild+1M = 0. If M is nonzero, d0M is also the large* *st d such that RdM is nonzero, or the largest d such that ~RdM is nonzero. (b) Let d1M be the smallest d such that ~d is an isomorphism, or 1 if no such d exists. Equivalently, d1M is the smallest d such that ExtsU(N, M) = 0 for s = 0* *, 1 and all N 2 N ild+1. As fundamental examples, we have that d0H*(V ) = d1H*(V ) = 0 for all ele- mentary abelian p-groups V . Proposition 3.12. Let M and N be unstable modules. (a) For s = 0, 1, ds(M N) = max{dsM, dsN}. (b) If M and N are nonzero, d0(M N) = d0M + d0N and d1(M N) = max{d1M + d0N, d0M + d1N}. (c) For s = 0, 1, dsTV M = dsM. (d) If M is nonzero, for s = 0, 1, ds( nM) = dsM + n. For properties (a) and (b) see [HLS1 , Prop.I.3.6]. Using the exactness of T* *V , property (c) follows from Proposition 3.4(b). Property (d) follows from Proposi- tion 3.3. 14 KUHN Proposition 3.13. Let 0 ! M1 ! M2 ! M3 ! 0 be a short exact sequence in U. (a) For s = 0, 1, dsM2 max{dsM1, dsM3}. Furthermore, if dsM3 < dsM1, then dsM2 = dsM1. (b) d0M1 d0M2 and d1M1 max{d1M2, d0M3}. Furthermore, if d1M2 < d0M3, then d1M1 = d0M3. This is proved with straightforward use of the long exact Ext* sequence assoc* *i- ated to a short exact sequence. Compare with [HLS1 , Prop.I.3.6]. Corollary 3.14. If M 2 U is reduced, then d1M = d0(L0M=M). This follows by applying Proposition 3.13(b) to 0 ! M ! L0M ! L0M=M ! 0. 3.4. Basic properties of d0(G) and d1(G). By abuse of notation, if G is a finite group, for s = 0, 1, we write ds(G) for dsH*(G). For example, d0(V ) = d1(V ) =* * 0 for all elementary abelian p-groups V . The properties of d0M and d1M presented above have the following immediate consequences for d0(G) and d1(G) Proposition 3.15. Let G and H be finite groups. (a) d0(G x H) = d0(G) + d0(H). (b) d1(G x H) = max{d1(G) + d0(H), d0(G) + d1(H)}. (c) If P is a p-Sylow subgroup of G, then ds(G) ds(P ) for s = 0, 1. (d) If V is a p-elementary abelian subgroup of G, then ds(CG (V )) ds(G) for s = 0, 1. Properties (a) and (b) follow from Proposition 3.12 (b). As the unstable modu* *le H*(G) is a direct summand of H*(P ) if P is a p-Sylow subgroup, property (c) fo* *l- lows from Proposition 3.12 (a). Similarly property (d) follows from Proposition* * 3.12 (c), as H*(CG (V )) is a direct summand of TV H*(G) [L2]10. 4.Formulae for H*(G)LF and ~Rd(H*(G)) In this section we prove the formulae for H*(G)LF and ~RdH*(G) given in x2. 4.1. A formula for LdH*(G). The starting point for all of these are the followi* *ng constructions. Given a morphism ff : V1 ! V2 in A(G), there are maps ff* : H*(V2) ! H*(V1), ff* : H*(CG (V1)) ! H*(CG (V2)), and m*ff: H*(CG (V2)) ! H*(V1) H*(CG (V2)). ____________ 10It is unfortunate that this much referenced elegant 1986 preprint has neve* *r been published. GROUP COHOMOLOGY PRIMITIVES 15 Here ff* is induced by conjugation by g-1 where g 2 G is any element11chosen so that conjugation by g induces ff, and mff: V1 x CG (V2) ! CG (V2) is the homomorphism sending (x, y) to ff(x)y. We also let mV : V x V ! V denote multiplication in an elementary abelian group V . To state one of the formulae from [HLS1 ], we recall two other bits of notati* *on from x2. Given an unstable module M, we let M d denote M modulo degrees greater than d. Given a category C, we let C# denote the associated twisted arr* *ow category: the objects of C# are the morphisms of C, and a morphism ff _ fi from ff : A1 ! A2 to fi : B1 ! B2 is a commutative diagram in C A1 __ff_//A2OO | | | | fflffl|fi| B1 _____//B2. [HLS1 , Formula I(5.5.1)] now reads Theorem 4.1. The homomorphisms V1 x CG (V2) -mff-!CG (V2) G induce an isomorphism of unstable algebras from LdH*(G) to ~(ff) limffEq { H*(V1) H d (CG (V2))___////_H*(V1) (H*(V1) H*(CG (V2)))}d, V1-!V2 (ff) where ~(ff) is induced by 1 m*ff, (ff) is induced by m*V1 1, and the limit * *is over A(G)# . 4.2. A formula for ~RdH*(G). Recall our notation from x2: if W is a central ele- mentary abelian p-subgroup of Q, then PW H*(Q) denotes the algebra of primitives in the H*(W )-comodule H*(Q). Proposition 4.2. As unstable modules, ~RdH*(G) is naturally isomorphic to limffH*(V1) Pff(V1)Hd(CG (V2)), V1-!V2 where the limit is over A(G)# . Proof.Recall that dR~dM is the kernel of LdM ! Ld-1M. As kernels commute with limits and equalizers, it follows from the previous theorem that ~RdH*(G) * *is naturally isomorphic to ~(ff) limffEq{ H*(V1) Hd(CG (V2))____////_H*(V1) (H*(V1) H*(CG (V2)))d}, V1-!V2 (ff) where ~(ff) is induced by 1 m*ffand (ff) is induced by m*V1 1. But now we observe that the equalizer in this formula is precisely H*(V1) Pff(V1)Hd(CG (* *V2)). For (ff) is the composite m*V1 1 H*(V1) Hd(CG (V2))-----!H*(V1) H*(V1) Hd(CG (V2)) truncate-----!H*(V 0 d 1) H (V1) H (CG (V2)), ____________ 11This is well defined as any two choices will differ by an element of CG(V1* *), and so will agree on cohomology. 16 KUHN and this identifies with * H*(V1) Hd(CG (V2)) 1-ss--!H*(V1) (H*(V1) H*(CG (V2)))d, where ss : V1 x CG (V2) ! CG (V2) is the projection. 4.3. A formula for H*(G)LF . Proposition 4.3. As unstable algebras, H*(G)LF is naturally isomorphic to limffPff(V1)H*(CG (V2)), V1-!V2 where the limit is over A(G)# . Proof.As there are no nonzero locally finite elements in "H*(V1) H*(CG (V2)), t* *he composite H*(G)LF H*(G) ! H*(CG (V2)) has image in Pff(V1)H*(CG (V2)) for any ff : V1 ! V2 in A(G). Thus one gets a natural map of unstable algebras H*(G)LF ! limffPff(V1)H*(CG (V2)). V1-!V2 That this is an isomorphism follows from Proposition 4.2, recalling that Propos* *i- tion 3.9 said that there is a natural isomorphism (R~dM)0 ' (MLF )d. 4.4. Replacing A(G) with AC (G). Recall that C(G) denotes the maximal central p-elementary abelian subgroup of G, and AC (G) denotes the full subcategory of A(G) with objects C(G) V < G. Theorem 4.4. One can take the limit over AC (G)# , rather than A(G)# in The- orem 4.1, Proposition 4.2, and Proposition 4.3. This will follow quite formally from the following simple observations. Let C* * = C(G). Given V < G, let CV < G be the subgroup generated by C and V . This induces an evident functor C : A(P ) ! AC (G). Furthermore, the natural inclusi* *on V ! CV induces an identification CG (CV ) = CG (V ). Given ff : V1 ! V2, let ffC : V1 ! CV2 be the evident map, and then let ffofffoffC__gff//_Cff, morphisms in A(G)# , correspond to the diagram in A(G) V1_______V1_____//_CV1 ff|| |ffC| Cff|| fflffl| fflffl| fflffl| V2_____//CV2_____CV2. Lemma 4.5. Let F : A(G)# ! Fp-vector spacesbe a contravariant functor such that for all ff : V1 ! V2, F (fff) : F (ff) ! F (ffC ) is an isomorphism. Then* * the natural map : lim F (ff) ! lim F (ff) ff2A(G)# ff2AC(G)# is an isomorphism. Note that both F (V1 ff-!V2) = H*(V1) and F (V1 ff-!V2) = H*(CG (V2)) sat- isfy the hypothesis of the lemma. Theorem 4.4 then follows from the lemma, as the relevant F 's are built from these two examples by constructions that prese* *rve isomorphisms. GROUP COHOMOLOGY PRIMITIVES 17 Proof of Lemma 4.5.We define : lim F (ff) ! lim F (ff), an inverse ff2AC(G)# ff2A(P)# to , as follows. Given x = (xfi) 2 lim F (fi), let (x) = ( (x)ff) 2 Q fi2AC(G)# ff2A(G)#F (ff), where (x)ff= F (fff)-1F (gff)(xCff). One then checks that (* *x) 2 lim F (ff), O = 1, and O = 1. ff2A(G)# 4.5. Rewriting the formulae. If C is a small category, and F : C# ! Fp-vector spaces is a contravariant functor, there is a canonical isomorphism ( ) Y ~ Y limF = Eq F (1C ) -!-! F (ff), C# C2obC ff2morC where, given ff : C1 ! C2, the ff-component of ~ and are induced by applying F to the canonical morphisms in C# from ff to 1C1 and 1C2 respectively. Thus, for example, ~RdH*(G) will be naturally isomorphic to ( ) Y ~ Y Eq H*(V ) PV Hd(CG (V )) -!-! H*(V1) Pff(V1)Hd(CG (V2)), V ff:V1!V2 where ~ and are induced by 1 ff* : H*(V1) PV1Hd(CG (V1)) ! H*(V1) Pff(V1)Hd(CG (V2)) and ff* i : H*(V2) PV2Hd(CG (V2)) ! H*(V1) Pff(V1)Hd(CG (V2)) for each ff : V1 ! V2. (i is the evident inclusion.) Morphisms in A(G) factor as inclusions followed by isomorphisms induced by the inner automorphism group Inn(G), so this last formula rewrites as follows. Proposition 4.6. R~dH*(G) is naturally isomorphic to 8" # 9 < Y Inn(G) ~ Y = Eq : H*(V ) PV Hd(CG (V )) !! H*(V1) PV1Hd(CG (V2)), V V1> 0 [E ]. Recall that the extension corresponds to an element o 2 H2(Q; C), or equiva- lently a homomorphism o : H2(Q) ! C. Under the identification C# = H1(C), its dual o# : C# ! H2(Q) corresponds to d2 : E0,12! E2,02. 5.1. H*(C)-comodule structure of the spectral sequence. As C is central, multiplication m : C x G ! G is a group homomorphism. The induced algebra map m* : H*(G) ! H*(C) H*(G), makes H*(G) into a H*(C)-comodule. The restriction i* : H*(G) ! H*(C) is both an algebra and comodule map, and it follows that E0,*1= im(i*) is a subHopf algebra of E0,*2= H*(C). One can strengthen these last observations to statements about the whole spec- tral sequence. A good functorial model for BG, say the reduced bar construction, shows that BC is an abelian topological group, BG is a BC-space equipped with proper free action via Bm : BC x BG ! BG, and BG ! BQ is the associated principal BC-bundle. The Serre spectral sequence arises from the pullback to BG of the skeletal filtration of BQ. This will be a filtration of BG by BC-subspac* *es, and we conclude the following. Lemma 5.1. For all k and r, Ek,*ris an H*(C)-comodule, such that the maps dr : Ek,*r! Ek+r,*r and Ei,*r Ej,*r! Ei+j,*r are maps of H*(C)-comodules. In particular, E0,*ris a subHopf algebra of E0,*2= H*(C). GROUP COHOMOLOGY PRIMITIVES 19 5.2. A handy Hopf algebra lemma. We now digress to state and prove a handy statement about (connected graded) Hopf algebras that we can apply to the situa- tion of the previous subsection. We need some notation. Let H be a graded connected Hopf algebra over a field F. There is a canonical splitting of vector spaces H = F I(H), where I(H) is * *the augmentation ideal. If M is a right H-module, let the module of indecomposables be defined by QH M = M H F = M=MI(H). Dually, if M is a right H-comodule, let the module of primitives be defined by PH M = Eq {M --!!M H} i = ker{ ~: M ! M I(H)}, where : M ! M H is the comodule structure, i is the inclusion induced by the unit F ! H, and ~ is the composite M -! M H ! M I(H). Lemma 5.2. Let K be a subHopf algebra of a Hopf algebra H. Suppose M is si- multaneously an H-comodule and K-module such that the K-module structure map M K ! M is a map of H-comodules. Then (a) M is a free K-module, and (b) the composite PH M ,! M i QK M is monic. Remark 5.3. To put this in perspective, the lemma has long been known if K = H, and, in this case, PH M ' QH M [Sw , Thm.4.1.1]. Our proof is very similar to the proofs of Proposition 1.7 and Theorem 4.4 of Milnor and Moore's classic pap* *er [MM ]. Compare also to Green's lemma [G3 , Lem2.1]. Before proving the lemma, we note the following consequence. Given H and K as in the lemma, let K -H -Mod be the category of M as in the lemma: an object is a vector space M that is simultaneously an H-comodule and K-module such that the K-module structure map M K ! M is a map of H-comodules. Morphisms are linear maps that are both K-module and H-comodule maps. K - H - Mod is an abelian category in the obvious way. Corollary 5.4. (a) Every short exact sequence 0 ! M1 ! M ! M2 ! 0 in K - H - Mod is split as a sequence of K-modules. (b) The functor sending M to QK (M) is exact on K - H - Mod. Proof of Lemma 5.2.Choose a section s : QK M ! M of the quotient ss : M ! QK M, and let ms : QK M I(K) ! MI(K) be the epimorphism given by ms(x, k) = s(x)k. Statement (a) is asserting that ms is an isomorphism. Let K : MI(K) ! M I(H) be the composite ~ MI(K) M -! M I(H), Statement (b) asserts that PH M \ MI(K) = {0}, i.e. that K is monic. Thus both statements will follow from the following claim: K O ms : QK M I(K) ! M I(H) is monic. 20 KUHN To prove this claim, let FnM be the K-submodule of M generated byPelements of degree upPto n. Given x 2 (QK M)n, and k 2 I(K), let (s(x)) = y0 h0, and (k) = k0 k00. Then K (ms(x, k))= ~(s(x)k) s(x) k modulo terms of the form y0k0 h0k00with either |y0| < |s(x)| = n, or k0 2 I(K). Otherwise said, K (ms(x, k)) s(x) k mod (Fn-1M + I(K)M) I(H). Thus ss( K (ms(x, k))) x k mod (QK M)<2pk-1 ifa1 = 2pk withk 1 and we let e(G) = (ai- 1) and h(G) = 1 ifa1 = 2 i=1 >:0 ifa1 = 1. We have the following lemma about products. Lemma 7.1. Suppose G0 and G1 have maximal central p-elementary abelian sub- groups C0 and C1, and Duflot subalgebras A0 and A1. Then the following hold. (a) C0 x C1 = C(G0 x G1), and PC0xC1H*(G0 x G1) = PC0H*(G1) PC1H*(G1). (b) A0 A1 will be a Duflot subalgebra for G0 x G1, and QA0 A1H*(G0 x G1) = QA0H*(G1) QA1H*(G1). (c) e(G0 x G1) = e(G0) + e(G1), and h(G0 x G1) = max{h(G0), h(G1)}. Note that a subgroup H of a p-central group G is again p-central, and C(H) = C(G) \ H. The next lemma is easily deduced. Lemma 7.2. Let G be p-central, and let A H*(G) be a Duflot subalgebra. If j : H < G is a subgroup, then e(H) e(G), h(H) h(G), and j*(A) will be a Duflot subalgebra of H*(H). Thanks to this lemma, Corollary 2.10 immediately follows from Theorem 2.9. Remark 7.3. The example H = Z=4 < Z=8 = G shows that the inequalities of the lemma can be equalities, even when H is a proper subgroup of a p-group G. 7.1. Benson-Carlson duality. If G is p-central, then QA H*(G) will be a finite dimensional Fp-algebra if A is any Duflot subalgebra. Benson and Carlson tell us much more: Theorem 7.4. If G is p-central and A is a Duflot subalgebra of H*(G), then QA H*(G) is a Poincar'e duality algebra with top class in degree e(G). GROUP COHOMOLOGY PRIMITIVES 29 Under the assumption that A is a polynomial algebra (always true if p = 2), t* *his is an immediate application of the main theorem in [BC1 ]. The general case red* *uces to this one: G and A will admit decompositions G = C0 x G1 and A = H*(C0) A1, with C0 p-elementary, G1 having no Z=p summands, and A1 a (necessarily polynomial) Duflot subalgebra of H*(G1). Then QA H*(G) = QA1H*(G1), and e(G) = e(G1). 7.2. Proof of Theorem 2.8. Let G be p-central, C = C(G), and A H*(G) a Duflot subalgebra. We now prove the various parts of Theorem 2.8. Firstly, Theorem 7.4 implies that QA H*(G) is zero in degrees greater than e(* *G), and one dimensional in degree e(G). Now consider the Serre spectral sequence for C ! G ! G=C, as studied in Theorem 2.1. The bigraded algebra QE0,*1E*,*1is the graded object associated to a decreasing filtration of the Poincar'e duality algebra QA H*(G) with top degr* *ee e(G). This forces the following to be true: there is a largest s, s(G), such th* *at Es,*1 is nonzero, QE0,*1Es(G),*1will be one dimensional and concentrated in total deg* *ree e(G), and nonzero classes in Es(G),e(G)-s(G)1 He(G)(G) will be Poincar'e duali* *ty classes. These classes will also be H*(C)-comodule primitives, as Es(G),*1is a sub- H*(C)-comodule of H*(G), and everything in lowest degree must be primitive. As PC H*(G) is contained in QA H*(G), we conclude that PC H*(G) is also zero in degrees greater than e(G), and one dimensional in degree e(G). By Corollary 2.4, PC He(G)H*(G) is also be the top nonzero degree of H*(G)LF , and so consists of classes annihilated by all positive degree Steenrod operatio* *ns. It remains to show that, under the additional assumption that G is a p-group, PC He(G)H*(G) is essential cohomology. This we prove in the next subsection. 7.3. p-central p-groups and essential cohomology. Let P be a p-central p- group. We have shown that He(P)(P )LF = PC(P)He(P)(P ) is a one dimensional subspace of He(P)(P ). Proposition 7.5. He(P)(P )LF is essential. Proof.As P is a p-group, maximal proper subgroups have the form j : Q < P , where Q is the kernel of a nonzero homomorphism x : P ! Z=p. We need to show that j*(i) = 0 2 H*(Q) if i 2 He(P)(P )LF is nonzero. The map j* : H*(P ) ! H*(Q) will take He(P)(P )LF to He(P)(Q)LF . If e(Q) < e(P ), we are done: j*(i) will be an element of a zero group. If e(Q) = e(P ), we reason as follows. Let A be a Duflot subalgebra of H*(G), so that j*(A) is a Duflot subalgebra of H*(Q). If j*(i) 6= 0, it will project * *to a nonzero element in Qj*(A)H*(Q). We show that this is impossible. Regard x as an nonzero element in H1(P ). By construction, j*(x) = 0 2 H1(Q). By Poincar'e duality, there exists y 2 H*(P ) such that i = xy 2 QA H*(P ). But th* *en j*(i) = j*(x)j*(y) = 0 2 Qj*(A)H*(Q). Let A(P, P ) be the two sided Burnside ring over Fp: the Fp-algebra with basis given by equivalence classes of diagrams P Q ff-!P , and multiplication defin* *ed using the double coset formula14. If J is the ideal generated by all such diagr* *ams ____________ 14There are more elegant descriptions, but this is better for our purposes. 30 KUHN with ff not an isomorphism, then A(P, P )=J ' Fp[Out(P )], the group ring of the outer automorphism group. Using transfers (a.k.a. induction), A(P, P ) acts on H*(P ), with a basis ele* *ment * TrPQ [P Q ff-!P ] inducing H*(P ) ff--!H*(Q) ---! H*(P ). As these are unstable A- module maps, it follows that He(P)(P )LF is a one dimensional A(P, P )-submodul* *e. Corollary 7.6. The ideal J acts trivially on He(P)(P )LF . Proof.The previous proposition shows that if a homomorphism ff : Q ! P is not onto, then ff*(He(P)(P )LF ) = 0. It follows that the A(P, P )-module He(P)(P )LF is the pullback of a one dime* *n- sional representation of Out(P ) over the prime field Fp. We let !(P ) denote t* *his representation. Clearly !(P ) will be trivial if p = 2, but this need not be th* *e case when p is odd. Example 7.7. Let p = 3. Then !(Z=9) = H1(Z=9) is nontrivial, as -1 : Z=9 ! Z=9 induces multiplication by -1 on H1(Z=9). 7.4. d0(G) when G has a p-central p-Sylow subgroup. We prove the parts of Theorem 2.9 involving d0. Firstly, if G is p-central, then Corollary 2.6 says that R~dH*(G) ' H*(C(G)) PC(G)Hd(G). Since d0(G) is the largest d such that ~RdH*(G) 6= 0, it follows that d0(G) wil* *l equal the top nonzero degree of PC(G)H*(G), which we have computed to be e(G). Now suppose that G is not necessarily p-central, but has a p-central p-Sylow subgroup P . We show that then d0(G) = d0(P ). We need to show that the largest d such that ~RdH*(G) 6= 0 is d = d0(P ) = e(* *P ). Let e1 2 A(P, P ) be an idempotent chosen so that A(P, P )e1 is the projective * *cover of ffl, the trivial Fp[Out(P )]-module, pulled back to A(P, P ). Standard argum* *ents show that there are inclusions e1R~dH*(P ) ~RdH*(G) ~RdH*(P ). Thus it suffices to show that e1R~e(P)H*(P ) 6= 0. Otherwise said, it suffices * *to show that ffl is a composition factor in the A(P, P )-modules ~Re(P)H*(P ). If p = 2, we are done: by Corollary 7.6, ~Re(P)He(P)(P ) = He(P)(P )LF ' ffl.* * As a bonus, we learn that H*(G)LF is one dimensional in degree e(P ). When p is odd, more care (and maybe luck) is needed. Recall that ~Re(P)H*(P )* * = H*(C(P )) He(P)(P )LF . The fact that J acts as 0 on He(P)(P )LF implies that same is true for H*(C(P )) He(P)(P )LF . Thus we just need to show that the trivial Out(P )-module occurs as a composition factor in H*(C(P )) He(P)(P )LF = H*(C(P )) !(P ), or, equivalently, that !(P )-1 occurs as an Out(P )-composit* *ion factor H*(C(P )). We are done with the following lemma15. Lemma 7.8. If P is a p-central p-group with p an odd prime, then every irreduci* *ble Fp[Out(P )]-module occurs as a composition factor of H*(C(P )). The lemma, in a stronger form than stated, follows by combining [K3 , Prop.5.7 and Cor.6.8]. The key point is that, since C(P ) = 1(P ), the kernel of Aut(P * *) ! Aut(C(P )) will be a p-group if p is odd [Gor, Thm.5.3.10]. ____________ 15This lemma is false if p = 2, as the example P = Q8 illustrates. GROUP COHOMOLOGY PRIMITIVES 31 Example 7.9. Let p = 3 and G be the semidirect product Z=9 o Z=2. Then d0(G) = d0(Z=9) = 1, but "H*(G)LF = 0. 7.5. d1(G) when G is p-central. In this subsection, let G be p-central. We show that d1(G) = e(G) + h(G). We get control of d1(G) by working directly with the desuspended composition factors RdH*(G) of H*(G), rather than their N il1-localizations ~RdH*(G), as was done in our calculation of d0(G). To simplify notation, write Rd for RdH*(G), ~Rdfor ~RdH*(G), and C for C(G). We have that ~R0= H*(C), and R0 = im(i*), where i : C ,! G is the inclusion. In the nilpotent filtration of H*(G), the last nonzero submodule, nile(G)H*(G* *) = e(G)Re(G), has been shown to be isomorphic to e(G)R0 as an unstable module. Thus d1 of this submodule of H*(G) equals e(G) + d1(R0). By Lemma 3.13, the next lemma implies that if d < e(G), d1( dRd) is strictly smaller than this. Lemma 7.10. Each Rd with d < e(G) admits a filtration by unstable modules with subquotients all of the form kR0 with d + k < e(G). Again appealing to Lemma 3.13, we then have the next corollary. Corollary 7.11. d1(G) = d1( e(G)Re(G)) = e(G) + d1(R0). Thus we will have proved that, when G is p-central, d1(G) = e(G) + h(G), once we have proved Lemma 7.10, and calculated that d1(R0) = h(G). We begin the proof of Lemma 7.10 here, and then both finish it, and calculate d1(R0), in the* * two subsections that follow, which correspond to the cases p = 2 and p odd. Proof of Lemma 7.10.For all d, we have inclusions R0 PC Hd(G) Rd ~R0 PC Hd(G), where PC Hd(G) is regarded as an unstable module concentrated in degree 0. These are inclusions of unstable modules, enriched with compatible R0-module structur* *es and ~R0-comodule structures. Call the category of such objects R0 - ~R0- U. Say that M 2 R0 - ~R0- U admits a nice filtration if it admits a filtration in R0 - ~R0- U with subquotients all of the form kR0. We will show that each Rd admits a nice filtration. (That the composition factors will then also satisfy d + k < e(G) follows imm* *e- diately from the fact that QR0( dR*) is a graded object associated to QA H*(G), which we know is one dimensional in degree e(G) and zero above that.) We claim that, if N admits a nice filtration, and M N, then M also admits a nice filtration. To see this, suppose F0N F1N . .i.s a filtration of N wi* *th FjN=Fj-1N = kjR0. Let FjM = M \FjN. Then FjM=Fj-1M FjN=Fj-1N = kjR0 will be an inclusion of objects in R0-R~0-U that will be split as R0-modu* *les, thanks to Corollary 5.4. We conclude that FjM=Fj-1M is either 0 or kjR0. Thus to prove Rd has a nice filtration, it suffices to prove that ~R0 PC Hd(* *G) has a nice filtration, or just that ~R0has a nice filtration. We show this in t* *he next two subsections, which separately deal with the cases p = 2 and p is odd. 7.6. A calculation of d1(R0), and a nice filtration of R~0, when p = 2. Suppose that p = 2, and that j1 2jc R0 = F2[x21 , . .,.x1 ] F2[x1, . .,.xc] = ~R0, with j1 . . .jc. We show the following. 32 KUHN Lemma 7.12. R~0has a good filtration as an object in R0 - ~R0- U. Lemma 7.13. If j1 > 0, d1(R0) = 2j1-1. Proof of Lemma 7.12.For all 1 b c and 1 ib jb, the module i1 2ic R(i1, . .,.ic) = F2[x21, . .,.xc ] will be an object in R0 - ~R0- U in the evident way. Clearly R(j1, . .,.jc) = R0 admits a nice filtration. The short exact sequenc* *es ib 0 ! R(i1, . .,.ic) ! R(i1, . .,.ib- 1, . .,.ic) ! 2 R(i1, . .,.ic) ! 0 then shows that if R(i1, . .,.ic) admits a nice filtration, so does R(i1, . .,.* *ib - 1, . .,.ic). By downward induction, we conclude that R(0, . .,.0) = R~0 admits a nice filtration. Proof of Lemma 7.13.By Proposition 3.12(c), it suffices to prove that, if j > 0, j j-1 d1(F2[x2 ]) = 2 . In the short exact sequence j 2j-1 2j-1 2j 0 ! F2[x2 ] ! F2[x ] ! F2[x ] ! 0, d1 of the middle term is strictly less than d0( 2j-1F2[x2j]) = 2j-1: this is cl* *ear when j = 1, and for larger j this follows byjan inductive hypothesis. Thus Prop* *o- sition 3.13(b) applies to say that d1(F2[x2 ]) = 2j-1. 7.7. A calculation of d1(R0), and a nice filtration of R~0, when p is odd. Suppose that p is odd. We can assume that j1 pjc * R0 = Fp[yp1 , . .,.y1 ] (x1, . .,.xc) Fp[y1, . .,.yc] = ~R0, with j1 . . .jc, and we show the following. Lemma 7.14. R~0has a good filtration as an object in R0 - ~R0- U. Lemma 7.15. If j1 = 0, d1(R0) = 1. If j1 > 0, d1(R0) = 2pj1-1. Proof of Lemma 7.14.As a first step, we note that the filtration of ~R0given by* * let- ting FkR~0= k(x1, . .,.xc) Fp[y1, . .,.yc] is a filtration in the category R0* *-R~0- U, and the associated subquotients are direct sums of suspensions of Fp[y1, . .* *,.yc]. It follows that it suffices to prove the lemma with ~R0replaced by Fp[y1, . .,.* *yc]. Our next reduction will allow us to reduce to the case when c = 1. If K is a subHopf algebra of a Hopf algebra H, and both objects and all struc* *ture maps are in U, one has a category K - H - U, analogous to R0 - ~R0- U. One can then say that M 2 K - H - U has a good filtration if it has a filtration wi* *th subquotients that are all suspensions of K. It is easy to see that if M1 2 K1-H* *1-U and M2 2 K2 - H2 - U has a good filtration, then so does M1 M2, viewed as an object in K1 K2 - H1 H2 - U. Applying this observation to the evident tensor decompositions of K = R0 and H = Fp[y1, . .,.yc], we are left just needing to show that Fp[y] has a nice fil* *tration, when viewed as an object in Fp[ypj] - Fp[y] - U. By downwards induction on i, we show that, for 0 i j, Fp[ypi] has a nice filtration, when viewed as an object in Fp[ypj] - Fp[y] - U. The case i = j is * *clear. GROUP COHOMOLOGY PRIMITIVES 33 For the inductive step, we filter Fp[ypi]. For 0 r p - 1, define M(r) to be the span of {ypim | m s modPp, for some0 s r}. Using the formulae Pkyn = nkyn+k(p-1)and (yn) = k nkyk yn-k, one easily checks that each M(r) is an object in Fp[ypj] - Fp[y] - U: if nk6 0 mod p, and n has the form pi(pa + s) with 0 s r p - 1, then both n + k(p - 1) and k also have this * *form. Thus we have a filtration in Fp[ypj] - Fp[y] - U: i+1 pi Fp[yp ] = M(0) M(1) . . .M(p - 1) = Fp[y ], and we are assuming by induction that Fp[ypi+1] has a good filtration. Now one checks that M(r)=M(r - 1) ' 2pirM(0) as objects in Fp[ypj] - Fp[y] - U, so by upwards induction on r we conclude that each M(r) has a good filtration. Proof of Lemma 7.15.By Proposition 3.12(c), it suffices to prove that d1(Fp[y])* * = 1, and, if j > 0, d1(Fp[ypj]) = 2pj-1. Corollary 3.14 (or Proposition 3.13(b)), applied to the short exact sequence 0 ! Fp[y] ! *(x) Fp[y] ! Fp[y] ! 0, shows that d1(Fp[y]) = d0( Fp[y]) = 1. If j > 1, we consider the short exact sequence: j pj-1 pj-1 pj 0 ! Fp[yp ] ! Fp[y ] ! Fp[y ]=Fp[y ] ! 0. We claim that d0 of the last term is 2pj-1, which, by induction, will be strict* *ly more than d1 of the middle term. Thus Proposition 3.13(b) applies to say that d1(Fp[ypj]) = 2pj-1. To verify the claim, one checks that the map Fp[ypj-1] ! 2pj-1Fp[ypj-1] send* *ing ypj-1nto the 2pj-1th suspension of nypj-1(n-1)isja-map1ofjunstablejA-modules,-1* *j-1 and thus induces an embedding Fp[yp ]=Fp[yp ] ,! 2p Fp[yp ] in U. Since the range of this embedding is the 2pj-1th suspension of a reduced module, the same is true of the domain, which thus has d0 = 2pj-1. 7.8. d1(G) when G has a p-central p-Sylow subgroup. Now suppose that G is not necessarily p-central, but has a p-central p-Sylow subgroup P . Here we * *show that then d1(G) = d1(P ). As d1(G) d1(P ) is always true, the point is to show that d1(G) is as big as it could be. Let e! 2 Fp[Out(P )] be an idempotent chosen so that Fp[Out(P )]e! is the projective cover of the one dimensional module !(P )-1. Lemma 7.16. d1(G) = d1(P ) if and only if d1(Re(P)H*(G)) = h(P ), and either of these equalities are implied by d1(e!R0H*(P )) = h(P ). Proof.As we proved that d1(P ) = e(P )+h(P ), we showed that d1(nile(P)H*(P )) = e(P ) + h(P ) and d1(H*(P )=nile(P)H*(P )) < e(P ) + h(P ). This second fact i* *m- plies that d1(H*(G)=nile(P)H*(G)) < e(P ) + h(P ) also holds, as H*(G) is a di- rect summand of H*(P ) in U. We conclude that d1(G) = d1(P ) if and only if d1(nile(P)H*(G)) = e(P ) + h(P ). As d1(nile(P)H*(G)) = e(P ) + d1(Re(P)H*(G)), we deduce that d1(G) = d1(P ) if and only if d1(Re(P)H*(G)) = h(P ). Now reasoning as in x7.4, this last equality would follow if one could show t* *hat d1(e!R0H*(P )) = h(P ). 34 KUHN We now sketch a proof that d1(e!R0H*(P )) = h(P ). This involves redoing the calculation that d1(R0H*(P )) = h(P ) in a way that allows one to keep track of* * the Out(P )-action. Let C = C(P ). The 0-line of the spectral sequence associated to C ! P ! P=C is natural with respect to the action of Out(P ). Thus the filtration studied in x6, H*(C) = E0,*2 E0,*3 E0,*2p+1 . . .E0,*2pk-1+1 E0,*2pk+1= E0,*1= R0H*(P ), is a filtration by unstable modules with an Out(P )-action. Our work above shows that 8 ><2pj-1 ifr = 2pj+ 1 withj 1 d1(E0,*r) = >1 ifr = 3 :0 ifr = 2. We now suppose that p > 2 and k 1: the cases when p = 2 or when E0,*1 equals E0,*2or E0,*3are similar and easier. Recall that then h(P ) = 2pk-1. Usi* *ng Proposition 3.13 in the usual way, we conclude that d1(e!R0H*(P )) = h(P ) if a* *nd only if d0(e!B) = 2pk-1, where B = E0,*2pk-1+1=E0,*2pk+1. From x6, we see that (7.1) B = *(C#0) S*(fi(C#1) + fi(C#2) + . .+. k-1fi(C#k)) S*( k-1fi(C# =C#k))=S*( kfi(C# =C#k)), where C#0 C#1 . . .C#k C# is a filtration of C# as an Out(P )-module. As an unstable module, B thus has the form M (S*( k-1fi(V ))=S*( kfi(V ))), where M is reduced. Now one observes that S*(fi(V ))=S*( fi(V )) = 2N where N is reduced. Thus S*( k-1fi(V ))=S*( kfi(V=)) k-1(S*(fi(V ))=S*( fi(V ))) = k-1( 2N) k-1 k-1 = 2p ( N), which is the 2pk-1st suspension of a reduced module. We conclude that d0(e!B) = 2pk-1 if and only if e!B is nonzero. The image of Out(P ) ! GL(C) lands in the parabolic subgroup GL(C, P ) re- specting the filtration of C, and the idempotent e! will project to a nonzero i* *dem- potent in Fp[GL(C, P )]. We claim that if e 2 Fp[GL(C, P )] is any nonzero idempotent, then e acts non- trivially on B as described in (7.1). Equivalently, we claim that all irreduci* *ble Fp[GL(C, P )]-modules occur as composition factors in B. To prove the claim, we note that all irreducible GL(C, P ) modules will be pu* *ll- backs from the associated Levi factor (i.e. the product of `block diagonal' GL(* *Vj)'s), as the projection from the one to the other has kernel which is a p-group. This reduces us quickly to verifying the following lemma. Lemma 7.17. Every irreducible Fp[GL(V )]-module occurs as a composition factor in S*(fi(V ))=S*( fi(V )). GROUP COHOMOLOGY PRIMITIVES 35 Proof.It is well known that every such irreducible S occurs in S*(fi(V )). Choo* *sing an occurrence of lowest polynomial degree, it is clear that it will remain nonz* *ero in the quotient S*(fi(V ))=S*( fi(V )). 8.Central essential cohomology Recall that Cess*(G) is defined to be the kernel of the restriction map Y H*(G) ! H*(CG (U)). C(G) r - c), and let ae be the regular repres* *en- tation of G=C. For 1 i r - c, let ~~i2 H2(pn-pn-i)(G=C) be the (pn - pn-i)th Chern class of ae, and then let ~i = InfGG=C(~~i) 2 H*(G). It is easy to check that ,1, . .,.,c, ~1, . .,.~r-c is a polarized system of parameters in the sens* *e of [G1 , Def.2.2]. It follows that the element , = ~1 satisfies the conclusion of the le* *mma. Proposition 8.9. Suppose c < r and A = Fp[,1, . .,.,c] is a Duflot algebra of H*(G). The following are equivalent for a fixed integer e 0. (a) e0(G) < e. (b) With , as in the lemma, the kernel of multiplication by ,, ker{,. : Hd(G)=(,1, . .,.c) ! Hd+|,|(G)=(,1, . .,.c)}, is zero for all d e. " (c) ker{,. : Hd(G)=(,1, . .,.c) ! Hd+|,|(G)=(,1, . .,.c)} ,2H"*(G) is zero for all d e. Proof.For each d and , 2 H*(G), we have a commutative diagram f(d) L d (8.1) Hd(G)=(,1, . .,.c)_____// C d e. Given 0 6= ~ 2 Hd(G)=(,1, . .,.c), we need to show * *that f(d)(~) 6= 0. By (c), there exists , 2 "H*(G) such that , . ~ 6= 0. As f(d + |,* *|) is monic by inductive assumption, f(d + |,|)(, . ~) 6= 0. But this equals , . f(d)* *(~), and so f(d)(~) 6= 0. Proof of Theorem 2.13.With notation as in the proposition just proved, we wish to prove that Cess*(G) = 0 if and only if the depth of H*(G) is greater than c. Thanks to Theorem 2.12, Cess*(G) = 0 if and only if statement (a) of the last proposition holds when e = 0. But then statement (b) is true with e = 0, and th* *us the depth of H*(G) is at least c + 1. Conversely, if the depth of H*(G) is at least c + 1, there exists a , such th* *at ,1, . .,.,c, , is a regular sequence on H*(G), and so statement (c) certainly h* *olds with e = 0. Thus statement (a) does as well. Now we study statements (b) and (c) of the last proposition, using work by Carlson and Benson. Proposition 8.10. If r - c = 1, then QA Cess*(G) satisfies Poincar'e dual- ity with duality degree equal to e(G). In other words, the Poincar'e polynomial pQACess*(G)(t) satisfies pQACess*(G)(t) = te(G)pQACess*(G)(1=t). Proof.The conclusion of the proposition is obvious if Cess*(G) = 0, so we can assume that the depth of H*(G) is precisely c. Let ,1, . .,.,c be as in Propos* *i- tion 8.9, and choose , as in the lemma. Replacing , by a large power of itself,* * if necessary, we can assume that, in diagram (8.1), f(d + |,|) is monic for all d.* * Thus QA Cess*(G), the kernel of the top map in (8.1), identifies with the kernel of * *multi- plication by ,, the left map in (8.1). But a careful reading of [BC2 , Lemma 3.* *2 and its proof] reveals that the Poincar'e series of this kernel is precisely the po* *lynomial called `pr(t)' there, and then [BC2 , Theorem 3.9] says that the functional equ* *ation of the proposition holds. Corollary 8.11. If r - c = 1, then e0(G) = e(G) - min{d | Cessd(G) 6= 0} < e(G). To state what we know about the situation when r -c > 1, we need to introduce local cohomology. If I is a homogeneous ideal in a graded ring R, and M is a gr* *aded R-module, H0,*I(M) is defined to be the I-torsion in M, i.e. the set of x 2 M s* *uch that Ikx = 0 for some k. This is a left exact functor of M, and Hd,*I(M) is def* *ined to be the associated dth right derived functor. In [Be], Benson conjectured Conjecture 8.12 (Strong Regularity Conjecture). ( Hi,j"H*(G)(H*(G)) = 0 for j -i ifc i < r j > -i ifi = r. Proof of Proposition 2.19.This proposition asserted that, for a fixed finite gr* *oup G, Conjecture 2.18 is implied by the Strong Regularity Conjecture. GROUP COHOMOLOGY PRIMITIVES 41 q | | 2 | 1 | u, vxv _0_|1__x,_y_________ |0 1 2 p Figure 1. Ep,q3= Ep,q1modulo (a2, b2, c2) In the terminology of [Be], if the Strong Regularity Conjecture holds, then, * *by [Be, Thm.4.5], every filter regular sequence is of type beginning with the sequ* *ence (-1, -2, . .,.-(c + 1)). In particular, with , as in statement (b) of Propositi* *on 8.9, the sequence ,1, . .,.,c, , is the beginning of such a sequence. From the defin* *ition of filter regular, we see that statement (b) of Proposition 8.9 thus holds with* * e = e(G). As mentioned in the introduction, in [Be], Benson shows that his conjecture is true if r - c 2. I have my own `heuristic' proof of statement (c) with e = e(* *G) under the same condition, and the failure of the method to go beyond r - c 2 makes one wonder if a counterexample to both of our conjectures is lurking among the groups of order 128 or 256. Remark 8.13. Slightly milder than the Strong Regularity Conjecture is Benson's Regularity Conjecture, which asserts that the Castelnuovo-Mumford regularity of H*(G) is precisely 0. In terms of local cohomology, this is the statement that Hi,j"H*(G)(H*(G)) = 0 for j > -i. For our purposes, this is enough to deduce th* *at e0(G) e(G). 9.Examples Example 9.1. Let W (2) be the universal 2-central group whose quotient by its center C is V2 = (Z=2)2. Thus there is a central extension H2(V2; F2) i-!W (2) q-!V2, where C = H2(V2; F2) ' (Z=2)3. In terms of Hall-Senior numbering, and thus also the numbering in [CTVZ ], W (2) is 32#18. In the associated spectral sequence, one has that E*,*2= F2[x, y, a, b, c], with a, b, c 2 E0,12and x, y 2 E1,02, and d2(a) = x2, d2(b) = xy, and d2(c) = y* *2. As E*,03= F2[x, y]=(x2, xy, y2), it follows that a2, b2, and c2 must be perma- nent cycles. We conclude that W (2) will have type [2, 2, 2] so that d0(W (2))* * = e(W (2)) = 3 and d1(W (2)) = 4. With a bit more work, one can show that E*,*3=(a2, b2, c2) is six dimensional* * with generators as indicated in Figure 1, where u and v are respectively represented* * by bx + ay and cx + by. This is a Poincar'e duality algebra with relations x2 = xy* * = y2 = u2 = v2 = uv = xu = yv = xv + yu = 0. It follows that E*,*3= E*,*1, and then that H*(W (2)) ' F2[ff, fi, fl, x, y, u, v]=(x2, xy, y2, u2, uv, v2, xu, yv, xv* * + yu), 42 KUHN with x, y 2 H1 and ff, fi, fl, u, v 2 H2. Here ff, fi, and fl are represented b* *y a2, b2, and c2 in the spectral sequence. The polynomial subalgebra A = F2[ff, fi, fl] is a Duflot subalgebra. With res* *pect to the H*(C) = F2[a, b, c] comodule structure, the elements 1, x, y are in the * *image of the inflation map q* and so are primitive. The top class xv is not in the im* *age of inflation, but is primitive, by our general theory. The elements u and v are* * not primitive, as m*(u) = 1 u + b x + a y and m*(v) = 1 v + c x + b y in H*(C) E*,*1. Thus each of the inclusions im(q*) ,! PC H*(W (2)) ,! QA H*(W (2)) is proper. The nilpotent filtration works as follows. R0 = H*(W (2))=(x, y, u, v) ' F2[ff, fi, fl] and ~R0= F2[a, b, c]. The embedd* *ing R0 ~R0sends ff to a2, fi to b2, and fl to c2. R1 is the free F2[ff, fi, fl]-module on generators ~x, ~y, ~u, ~vof respectiv* *e degrees 0,0,1,1, and R~1is the free F2[a, b, c]-module on ~x, ~y. The embedding R1 R* *~1 sends ~uto b~x+ a~yand ~vto c~x+ b~y. Thus Sq1(~u) = fi~x+ ff~yand Sq1(~v) = fl* *~x+ fi~y. R2 and ~R2are both 0, as PC H2(W (2)) = 0. R3 is the free F2[ff, fi, fl]-module on a single generator ~x~vof degree 0, a* *nd ~R3is the free F2[a, b, c]-module on this same element. Finally, H*(W (2))LF H*(W (2)) is the algebra spanned by 1, x, y, xv. All nontrivial products and Steenrod operations are zero. Example 9.2. Let G be the group of order 64 with Hall-Senior number #108. Using information from [CTVZ ], we analyzed H*(G) in detail for other purposes* * in [K3 ]. Here we summarize relevant bits to illustrate how one can calculate H*(G* *)LF and ~RdH*(G) by using Proposition 4.7 and Proposition 4.6. The commutator subgroup Z = [G, G] has order 2. The center C is elementary abelian of rank 2, and C = (G), so Z < C and G=C is elementary abelian of rank 4. There is a unique maximal elementary abelian group V of rank 3, and its centralizer K has order 32, so that NG (V )=CG (V ) = G=K ' Z=2. More precisely, K is isomorphic to (Z=2)2 x Q8, with Q8 embedded so that V \ Q8 = Z. We have the following picture of AC (G): __________________________________________* *_____________ C _____//VffZ=2_____________________________________* *___________________________ and from this it is already clear that ~R0H*(G) = H*(V )Z=2. We have maps of unstable algebras equipped with Aut(G) action: * PV H*(K) ,! PC H*(K) -j PC H*(G), where j : K ! G is the inclusion. It is easily checked that j* is onto in degre* *e 1. The maps of pairs (Q8, Z) ! ((Z=2)2 x Q8, (Z=2)2 x Z) = (K, V ) induces an isomorphism of algebras: PV H*(K) ' PZH*(Q8). The algebra PZH*(Q8) is familiar: the calculation of H*(Q8) using the Serre spectral sequence associated to Z ! Q8 ! Q8=Z reveals that PZH*(Q8) = Im {H*(Q8=Z) ! H*(Q8)} = B*, where B* is the Poincar'e duality algebra F2[x, w]=(x2 + xw + w2, x2w + xw2), where x and w both have degree 1. B* has dimension 1,2,2,1 in degrees 0,1,2,3. GROUP COHOMOLOGY PRIMITIVES 43 From this we learn that PC H*(K) ' B*[y] where y is also in degree 1, and thus is generated by elements in degree 1. It follows that j* : PC H*(G) ! PC H*(K) * *is onto, and then that Inn(G) acts trivially on both PV H*(K) and PC H*(K). Proposition 4.7 then tells us that there is a pullback diagram of unstable al* *gebras: H*(G)LF _____//_PC H*(G) | |* | j| fflffl| fflffl| PV Hd(K) _____//PC H*(K). Similarly, Proposition 4.6 tells us that, for all d, there is a pullback diag* *ram of unstable modules: R~dH*(G) __________//_H*(C) PC Hd(G) | | * | |1 j fflffl| fflffl| H*(V )Z=2 PV Hd(K)_____//H*(C) PC Hd(K). Note that the kernel of j* : H*(G) ! H*(K) is precisely Cess*(G), which is described in [CTVZ ]. In our terminology, we learn that a Duflot subalgebra A * *is polynomial on classes of degree 2 and 8 (so G has type [8, 2]), and QA Cess*(G)* * is a graded vector space of dimension 1,3,5,6,5,3,1 in degrees 1,2,3,4,5,6,7. Note* * that this evident Poincar'e duality is predicted by Proposition 8.10. QA Cess*(G) has a basis in which every element is a product of 1 dimensional classes, and thus PC Cess*(G) = QA Cess*(G). In [K3 , Prop.10.2], we further showed that PC Cess*(G) ' B*[y]=(y4) as unstable modules. It follows that there are short exact sequences in U: 0 ! B*[y]=(y4) ! H*(G)LF ! B* ! 0, and 0 ! H*(C) [ B*[y]=(y4)]d ! ~RdH*(G) ! H*(V )Z=2 Bd ! 0. Furthermore, d0(G) = e00(G) = e0(G) = 7, and e(G) = 8. Example 9.3. One can often determine e(G) using minimal information about the extension class o* : C* ! H2(G=C) (where C = C(G)), and in situations where H*(G) has yet to be calculated. For example, suppose that p = 2 and G has no Z=2 direct summands (so that o* is monic). If the image of o* has a basis consisting of products of 1 dimens* *ional classes, then G has type [2, . .,.2] and so e(G) equals the rank of C. To see t* *his, we note that, if d2(a) = xy, then d3(a2) = d3(Sq1a) = Sq1(d2(a)) = Sq1(xy) = x2y + xy2 0 mod (xy). This criterion holds for the important family of groups studied in [AKM ]. T* *here the authors associate a 2-central Galois group GF to every field F of character* *istic different from 2 that is not formally real. They call this group a W -group due to its connections to the Witt ring W F [MiS ]. Thus d0(GF) = e(GF) = r and 44 KUHN q | | 8 | 2 7 | 6 | 6 8 8 8 4 1 5 | 4 | 8 12 8 7 4 3 | 2 | 4 7 8 8 8 1 | _0_|1__4__8__10__8_6_______________ |0 1 2 3 4 5 6 7 8 p Figure 2. The dimension of Ep,q1modulo (a8, b8) d1(GF) = r+1, where GF has rank r. Included among these groups are the universal W -groups W (n), the 2-central group with extension sequence H2((Z=2)n; F2) ! W (n) ! (Z=2)n. n+1 n+1 Thus d0(W (n)) = 2 and d1(W (n)) = 2 + 1. At odd primes p, analogous criteria exist, ensuring that G is p-central of ty* *pe [2, . .,.2]. Interesting families of such groups were studied by Browder-Pakian* *athan [BP ] and Adem-Pakianathan [AP ]. Included among these are the universal groups W (n, p), with extension sequence H2((Z=p)n; Fp) ! W (n, p) ! (Z=p)n. n+1 n+1 Thus d0(W (n, p)) = 2 and d1(W (n, p)) = 2 + 1. Example 9.4. Compared to the families in the last example, at the other extreme among 2-central 2-groups is the 2-Sylow subgroup P of the simple group SU(3, 4). This group has order 64 and Hall-Senior number #187. Its center C is elementary abelian of rank 2. In [G2 ], Green analyzed the associated spectral sequence17.* * In particular, P has type [8, 8], so that d0(P ) = e(P ) = 14 and d1(P ) = 18, and* * the analogue of Figure 1 is the impressively complex Figure 2 (reproduced from [G2 * *]). In spite of this complexity, it is interesting to note that one can get the b* *ound e(P ) 14 quite easily, by using representation theory and characteristic clas* *ses. We thank David Green for the following description of some complex represen- tations of P . Let H*(C) = F2[a, b], and then let aea and aeb be the 1-dimensio* *nal complex representations of C with respective total Stiefel-Whitney classes w(ae* *a) = 1 + a2, w(aeb) = 1 + b2. These representations extend to 1-dimensional represen- tations "aeaand "aebof subgroups Qa and Qb of index 4 in P . Let !a and !b be t* *he 4 dimensional representations one gets by inducing "aeaand "aebup to P : these * *turn out to be irreducible. By construction ResPC(!a) = 4aea and ResPC(!b) = 4aeb. It follows that the to* *tal Stiefel-Whitney classes of !a and !b restrict to (1 + a2)4 = 1 + a8 and (1 + b2* *)4 = 1 + b8 in H*(C). Thus imResPCcontains F2[a8, b8] and so e(P ) 14 must hold. ____________ 17A key simplification comes by computing with F4 coefficients rather than F* *2 coefficients. GROUP COHOMOLOGY PRIMITIVES 45 Alternatively, one can just use the single 8 dimensional representation !a * *!b. This is faithful, as it is faithful when restricted to C, the subgroup of all e* *lements of order 2. It has characteristic classes that restrict to a8+b8 and a8b8 in H*(C)* *. From this, one can formally deduce that the special Hopf algebra imResPCmust contain F2[a8, b8], so that d0(P ) 14 and d1(P ) 18. By contrast, the estimate of H* *enn, Lannes, and Schwartz in [HLS1 ] just lets one conclude that d0(P ) 64 and d1(* *P ) 120 if one knows that P has a faithful 8 dimensional complex representation. Th* *is suggests that there might be some general bounds for ds(G) for an arbitrary gro* *up G, determined by the dimensions of its faithful representations, that are much better than those in [HLS1 ]. 46 KUHN Appendix A. Tables of group invariants Here are various tables of some of our invariants for 2-groups of order divid* *ing 64. The tables were compiled by hand using the calculations in [CTVZ ] and the website version [Ca2 ]. The type of a group G, and thus e(G) and h(G), can be deduced by inspecting the description of restriction to maximal elementary abel* *ian subgroups; this is particularly easy when G is 2-central. If G is not 2-central* *, one can immediately determine if Cess*(G) 6= 0, since both the rank of Z(G) and the depth of H*(G) are given, and then read off the number e0(G) from the descripti* *on of depth essential cohomology. The website source allows one to identify centra* *lizers of elementary abelian subgroups as needed. We say a group is indecomposable if it cannot be written as a nontrivial dire* *ct product of two subgroups. The numbering of groups is as in [CTVZ ] which follo* *ws the Hall-Senior numbering [HS ]. In Tables 1 and 2, recall that, since G is 2-central, d0(G) = e(G) = e0(G) = e00(G), and d1(G) = e(G) + h(G). In Table 3, `2' means Z=2, etc. To compute d0(G), we needed to observe that, in all cases covered by this table, e00(G) = e0(G). Except when G is 32#41, this can be checked by noticing that elements in the top degree in QA Cess*(G) are represented by classes in the image of InfGG=C, and so are primitive. When G is 32#41, elements in the QA Cess5(G) are represented by essential classes of lowe* *st degree, and so are primitive. Table 1: Indecomposable, 2-central, 2-groups of order 32 _______________________________________ ||_Order||#||Type_||d0(G)||d1(G)||Note|s_| ||___2__||1|_|[1]_|_|0___||_0___||Z=2_||_ ||___4__||2|_|[2]_|_|1___||_2___||Z=4_||_ ||___8__||3|_|[2]_|_|1___||_2___||Z=8_||_ ||______||5|_|[4]_|_|3___||_5___||Q8__||_ ||___16_|5|||[2]__||_1___||_2___||Z=16||_ ||______||1|4[|4]_|_|3___||_5___||Q16_||_ ||___32_|1||8[|2,2|,2]3|_||_4___||____||_ ||______||1|9[|2,2|]_2|__||_3___||____||_ ||______||2|1[|2,2|]_2|__||_3___||____||_ ||______||2|8[|4,2|]_4|__||_6___||____||_ ||______||2|9[|2,2|]_2|__||_3___||____||_ ||______||3|0[|2,2|]_2|__||_3___||____||_ ||______||3|5[|4,2|]_4|__||_6___||____||_ ||______||4|0[|4,4|]_6|__||_8___||____||_ ||______||5|1[|4]_|_|3___||_5___||Q32_||_ GROUP COHOMOLOGY PRIMITIVES 47 Table 2: Indecomposable, 2-central, groups of order 64 ___________________________________________ ||__#_||Type|d|0(G)||d1(G)||____Notes______|| ||__11|_|[2]|__|1__||_2___||____Z=64_______|| |__30_[|2,2,2]_|3___|_4___|________________| |__37_[|2,2,2]_|3___|_4___|________________| |__38__|[2,2]__|2___|_3___|________________| |__39__|[2,2]__|2___|_3___|________________| |__41__|[2,2]__|2___|_3___|________________| |__59_[|2,2,2]_|3___|_4___|________________| |__63__|[4,2]__|4___|_6___|________________| |__64__|[2,2]__|2___|_3___|________________| |__65__|[2,2]__|2___|_3___|________________| |__82_[|2,2,2]_|3___|_4___|________________| |__87_[|4,2,2]_|5___|_7___|________________| |__88_[|2,2,2]_|3___|_4___|________________| |__90_[|2,2,2]_|3___|_4___|________________| |__92_[|4,2,2]_|5___|_7___|________________| |__93_[|2,2,2]_|3___|_4___|________________| |_101_|[4,4]__|6___|__8___|________________| |_119_|[4,2]__|4___|__6___|________________| |_139_|[4,2]__|4___|__6___|________________| |_140_|[2,2]__|2___|__3___|________________| |_141_|[2,2]__|2___|__3___|________________| |_145_|[4,2,2]_|5___|_7___|________________| |_149_|[4,2,2]_|5___|_7___|________________| |_152_|[4,2,2]_|5___|_7___|________________| ||_153||[4,4|,4]9|_||11___||2-Sylow_of_Sz(8|)_ | |_162_|[4,4]__|6___|__8___|________________| ||_187||[8,8|]14|__||18___||2-Sylow_of_U3(F|4)_| |_190_|[4,2]__|4___|__6___|________________| |_191_|[4,4]__|6___|__8___|________________| |_192_|[4,2]__|4___|__6___|________________| |_194_|[4,4]__|6___|__8___|________________| |_199_|[4,4]__|6___|__8___|________________| |_210_|[4,4]__|6___|__8___|________________| |_211_|[4,2]__|4___|__6___|________________| |_212_|[4,4]__|6___|__8___|________________| |_222_|[4,4]__|6___|__8___|________________| |_227_|[4,4]__|6___|__8___|________________| |_233_|[4,4]__|6___|__8___|________________| |_235_|[4,2]__|4___|__6___|________________| |_236_|[4,2]__|4___|__6___|________________| |_240_|[4,4]__|6___|__8___|________________| ||_267||[4]_|_|3___||_5___||_____Q64_______|| 48 KUHN Table 3: Indecomposable, non 2-central, 2-groups of order 32 ________________________________________________________________________ ||_Order||#||Type||Depth||Rank_||e(G)||e0(G|)d|0(G|)_|CG_(V_)'s|_N|otes_|| ||___8__||4|_|[2]|_|2___||_2___||1___||-1__||_0___||____22_____||_D8___||_ ||___16_|8|||[4]_||_1___||_2___||3___||-1__||_1___||__4_x_2____||AES16_||_ ||______||9|[|2,2|]_2|__||_3___||2___||1___||_1___||____22_____||_______|| ||______||1|1[|4]|_|1___||_2___||3___||2___||_2___||__4_x_2____||_______|| ||______||1|2[|2]|_|1___||_2___||1___||-1__||_0___||____22_____||_D16___|| ||______||1|3[|4]|_|1___||_2___||3___||2___||_2___||____22_____||SD16__||_ ||___32_|1||6[|4,|2]2_|_||_3___||4___||3___||_3___||__4_x_22___||_______|| ||______||1|7[|4]|_|2___||_2___||3___||-1__||_1___||__8_x_2____||_______|| ||______||2|0[|2,|2]2_|_||_3___||2___||1___||_1___||__4_x_22___||_______|| ||______||2|2[|4]|_|1___||_2___||3___||2___||_2___||__8_x_2____||_______|| ||______||2|6[|4]|_|2___||_2___||3___||-1__||_1___||8_x_2,_4_x_|2__|____|| ||______||2|7[|2,|2]2_|_||_3___||2___||1___||_1___||____23_____||_______|| ||______||3|1[|4]|_|2___||_2___||3___||-1__||_2___||4_x_4,_4_x_|2__|____|| ||______||3|2[|4]|_|1___||_2___||3___||2___||_2___||__8_x_2____||_______|| ||______||3|3[|2,|2]3_|_||_4___||2___||-1__||_0___||__24,_23___||_______|| ||______||3|4[|2,|2]3_|_||_3___||2___||-1__||_0___||____23_____||_______|| ||______||3|6[|2,|2]3_|_||_3___||2___||-1__||_1___||4_x_22,_23_|_|______|| ||______||3|7[|4,|2]2_|_||_3___||4___||3___||_3___||__4_x_22___||_______|| ||______||3|8[|4,|2]2_|_||_3___||4___||2___||_2___||4_x_22,_23_|_|______|| ||______||3|9[|4,|2]2_|_||_3___||4___||3___||_3___||____23_____||_______|| ||______||4|1[|4,|4]2_|_||_3___||6___||5___||_5___||____23_____||_______|| ||______||4|2[|4]|_|3___||_3___||3___||-1__||_0___||____23_____||D8_*_D8||_ ||______||4|3[|8]|_|2___||_2___||7___||-1__||_3___||__Q8_x_2___||D8_*_Q8||_ ||______||4|4[|4]|_|2___||_3___||3___||-1__||_1___||_4_x_2,_23_|_|______|| ||______||4|5[|8]|_|1___||_2___||7___||4___||_4___||Q8_x_2,_4_x|2_|_____|| ||______||4|6[|4]|_|2___||_3___||3___||-1__||_3___||Q8_x_2,_23_||_______|| ||______||4|7[|4]|_|1___||_3___||3___||1___||_1___||D8_x_2,_23_||_______|| ||______||4|8[|8]|_|1___||_2___||7___||6___||_6___||__Q8_x_2___||_______|| ||______||4|9[|2]|_|2___||_2___||1___||-1__||_0___||____22_____||_D32___|| ||______||5|0[|4]|_|1___||_2___||3___||2___||_2___||____22_____||SD32__||_ GROUP COHOMOLOGY PRIMITIVES 49 Table 4: Indecomposable, non 2-central, order 64, with Cess*(G) 6= 0 __________________________________________________________________________ ||__#_||Type|R|ank||e0(G)||||#|T|ype||Rank_||e0(G|)||#|||Type||Rank||e0(G|)_| |__42__|[4]___|2___|_2___|1|73_|[4,4]__|4___|3___|1|93_|[4,2]__|3___|_3___| |__67__|[4]___|2___|_2___|1|75_|[4,4]__|4___|3___|1|96_|[4,2]__|3___|_2___| |_143_|_[4]___|2___|_2___|1|83_|[4,4]__|4___|5___|1|97_|[4,2]__|3___|_3___| |_182_|_[4]___|2___|_2___|2|02_|[4,2]__|4___|2___|1|98_|[4,2]__|3___|_3___| |_245_|_[8]___|2___|_4___|_|__|_____|______|_____|2|00_|[4,4]__|3___|_5___| |_246_|_[4]___|2___|_2___|_|32_[|4,2]__|3___|3___|2|04_|[4,2]__|3___|_3___| |_249_|_[8]___|2___|_6___|_|33_[|4,2]__|3___|3___|2|06_|[4,2]__|3___|_3___| |_255_|_[8]___|2___|_6___|_|40_[|2,2]__|3___|1___|2|07_|[4,2]__|3___|_3___| |_258_|_[8]___|2___|_4___|_|54_[|4,2]__|3___|3___|2|08_|[4,2]__|3___|_3___| |_266_|_[4]___|2___|_2___|_|60_[|4,2]__|3___|3___|2|09_|[4,2]__|3___|_3___| |_____|_____|_____|______|_|61_[|4,2]__|3___|3___|2|13_|[4,2]__|3___|_2___| |_121_|_[8]___|3___|_3___|_|62_[|2,2]__|3___|1___|2|14_|[4,2]__|3___|_2___| |_130_|_[8]___|3___|_3___|_|79_[|4,4]__|3___|5___|2|15_|[4,4]__|3___|_5___| |_133_|_[8]___|3___|_4___|_|80_[|4,4]__|3___|4___|2|16_|[4,4]__|3___|_5___| |_180_|_[8]___|3___|_3___|_|95_[|4,2]__|3___|3___|2|18_|[4,2]__|3___|_3___| |_181_|_[8]___|3___|_3___|_|97_[|4,2]__|3___|3___|2|19_|[4,2]__|3___|_2___| |_247_|_[4]___|3___|_1___|_|98_[|4,2]__|3___|3___|2|20_|[4,2]__|3___|_3___| |_251_|_[8]___|3___|_4___|_|99_[|4,2]__|3___|2___|2|21_|[4,4]__|3___|_5___| |_253_|_[8]___|3___|_3___|1|00_|[4,2]__|3___|3___|2|23_|[4,4]__|3___|_5___| |_254_|_[8]___|3___|_4___|1|02_|[4,4]__|3___|5___|2|24_|[4,4]__|3___|_5___| |_257_|_[8]___|3___|_3___|1|08_|[8,2]__|3___|7___|2|25_|[4,2]__|3___|_2___| |_262_|_[8]___|3___|_3___|1|15_|[8,2]__|3___|7___|2|26_|[4,2]__|3___|_3___| |_____|_____|_____|______|1|16_|[4,2]__|3___|3___|2|28_|[4,2]__|3___|_2___| |__81_[|2,2,2]_|5___|1___|1|18_|[4,2]__|3___|3___|2|29_|[4,2]__|3___|_3___| |_____|_____|_____|______|1|29_|[4,2]__|3___|3___|2|30_|[4,2]__|3___|_3___| |__83_[|2,2,2]_|4___|2___|1|32_|[4,2]__|3___|3___|2|31_|[4,4]__|3___|_5___| |__85_[|2,2,2]_|4___|2___|1|38_|[2,2]__|3___|1___|2|32_|[4,4]__|3___|_5___| |__86_[|2,2,2]_|4___|2___|1|61_|[4,4]__|3___|4___|2|34_|[2,2]__|3___|_1___| |__89_[|4,2,2]_|4___|4___|1|65_|[4,4]__|3___|4___|2|38_|[4,4]__|3___|_5___| |__91_[|2,2,2]_|4___|2___|1|66_|[4,4]__|3___|4___|2|39_|[4,2]__|3___|_3___| |_146_|[2,2,2]_|4___|2___|1|67_|[4,4]__|3___|4___|_|__|______|_____|_____|_ |_147_|[4,2,2]_|4___|4___|1|68_|[4,4]__|3___|5___|_|__|______|_____|_____|_ |_148_|[2,2,2]_|4___|2___|1|72_|[8,2]__|3___|5___|_|__|______|_____|_____|_ |_150_|[2,2,2]_|4___|2___|1|74_|[4,4]__|3___|5___|_|__|______|_____|_____|_ |_151_|[2,2,2]_|4___|2___|1|77_|[4,4]__|3___|3___|_|__|______|_____|_____|_ |_____|_____|_____|______|1|78_|[4,4]__|3___|4___|_|__|______|_____|_____|_ |__94__|[4,2]__|4___|2___|1|79_|[4,4]__|3___|5___|_|__|______|_____|_____|_ |_113_|[4,2]__|4___|_2___|1|85_|[4,4]__|3___|3___|_|__|______|_____|_____|_ |_131_|[4,2]__|4___|_2___|1|86_|[4,4]__|3___|4___|_|__|______|_____|_____|_ |_163_|[4,4]__|4___|_3___|1|89_|[4,2]__|3___|3___|_|__|______|_____|_____|_ 50 KUHN References [AK]A.Adem, and D.Karagueuzian, Essential cohomology of finite groups, Comment.* * Math. 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