THE ESSENTIAL IDEAL IS A COHEN-MACAULAY MODULE DAVID J. GREEN Abstract.Let G be a finite p-group which does not contain a rank two ele- mentary abelian p-group as a direct factor. Then the ideal of essential * *classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in t* *he affirmative a question posed by J. F. Carlson (Question 5.4 in [7]). Introduction Let G be a finite p-group and k a field of characteristic p. By definition, the* * essential ideal Ess(G) in the cohomology ring H*(G, k) consists of those cohomology class* *es whose restriction to every proper subgroup is zero. The essential ideal is invo* *lved in the cohomological characterization of p-groups whose exponent p elements are* * all central [1], and it has featured prominently in investigations concerning the d* *epth of the graded commutative ring H*(G, k) (see [6]). The essential ideal also plays a key role in Carlson's method [8] for computi* *ng H*(G, k). In this paper we shall show that the most important case of a question posed by Carlson (Question 5.4 in [7]) can be answered using existing technique* *s. That is, we shall prove the following theorem: Theorem A. Let k be a field of characteristik p, and let G be a finite p-group which does not have the elementary abelian p-group of order p2 as a direct fact* *or. If the essential ideal Ess(G) in H*(G, k) is nonzero, then it is a Cohen-Macaul* *ay module with Krull dimension equal to the p-rank of the centre of G. Another formulation of the result is as follows: Theorem B. With the assumptions of Theorem A, denote by C the largest central elementary abelian subgroup of G. Suppose that i1, . .i.dare homogeneous elemen* *ts of H*(G, k) such that the restrictions ResC(i1), . .,.ResC(id) form a homogeneo* *us system of parameters for H*(C, k). Then the essential ideal Ess(G) in H*(G, k) * *is free and finitely generated as a module over the polynomial algebra k[i1, . .,.* *id]. Carlson's method for computing H*(G, k) depends upon the essential ideal being free and finitely generated. Up till now the computer program had to check this property for each group it was presented with. Now that the question is a theor* *em, programs based on Carlson's method should run faster. Note that the same questi* *on was also mentioned by H. Mui several years earlier, but in a much weaker form [* *11]. The equivalence of Theorems A and B is demonstrated in Lemma 1.3, which also shows that it suffices to find one sequence i1, . .,.id satisfying the assumpti* *ons and conclusion of Theorem B. Finite generation of Ess(G) is proved in Lemma 1.2 for all finite p-groups G; and freeness is proved for the groups under consideratio* *n in x2, ____________ Date: 26 February 2004. 2000 Mathematics Subject Classification. Primary 20J06; Secondary 13C14. 1 2 DAVID J. GREEN using the fact that the cohomology ring of G is a comodule over the cohomology ring of any central subgroup. From now on we shall write H*(G) for H*(G, k). 1.Group cohomology and commutative algebra We shall demonstrate that Theorems A and B are equivalent by recalling several well documented facts about group cohomology and commutative algebra. The following lemma is certainly not new. Lemma 1.1. Let k be a field of characteristic p and G a finite p-group whose essential ideal Ess(G) is nonzero. Denote by C the largest central elementary a* *belian subgroup of G. Let A, I and J be the following ideals in H*(G): o A is the annihilator ideal Ann H*(G)(Ess(G)); o I is the kernel of restriction H*(G) ! H*(C); and o J is generated by all transfer classes from all proper subgroups. p __ p_ p__ Then A = I = J , and this radical ideal is a prime ideal. Proof.The cohomolgy ring of an elementary abelian p-group is a polynomial algeb* *ra (p = 2) or the tensor productpover_k of a polynomial algebrapwith_an exterior algebra (p odd). So H*(C)= 0 is a polynomial algebra, and I a prime ideal. Quillen proved that x 2 H*(G) is nilpotent if its restrictionpto_every elementa* *ry abelian subgroup is nilpotent (Coroll. 8.3.4 in [9]). So Iconsists of those c* *lasses with nilpotent restriction to the centre Z(G).pIt_followspfrom_Carlson's theore* *m on varieties and transfer (x10.2 in [9]) that I = J . Let H G be a subgroup. Then restriction makes H*(H) a module over H*(G), and transfer from H*(H) to H*(G) is a homomorphism of H*(G)-modules. Hence transfer classes and essential classes annihilate each other, which means that * *J A. To finish the proof we willpuse_an argument of Broto and Henn [5] to showptha* *t_ every element of H*(G) \ I is regular, from which it follows that A I. Group multiplication ~: G x C ! G is a group homomorphism, and so induces a map of k-algebras ~*: H*(G) ! H*(G) k H*(C). Moreover, (IdG x"C ) O ~* = IdG and ("G IdC) O ~* = ResGC, where IdG: H*(G) ! H*(G) is the identity map and "G :H*(G) ! k the augmentation. So ~* is a split monomorphism; and ~*(x) 2 1 ResC(x) + H>0(G) H*(C) for all x 2 H*(G). So if ResC(x) is non-nilpotent, then it is a regular element of H*(C), and it follows that ~*(x)* * is a regular element of H*(G) k H*(C). As ~* is injective, it follows that x 2 H*(G) is regular. Lemma 1.2. Let G be a finite p-group, and k a field of characteristic p. Denote by C the largest central elementary abelian subgroup of G. Suppose that i1, . .* *i.d are homogeneous elements of H*(G) whose restrictions to C form a homogeneous system of parameters for H*(C). Then Ess(G) is finitely generated as a module over the polynomial algebra R = k[i1, . .,.id]. Proof.The Evens-Venkov theorem (Coroll. 7.4.6 in [9]) states that H*(G) is a finitely generated k-algebra. Let A, I, J be as in Lemma 1.1, and recall from t* *he proof of that lemma that J A. So Ess(G) is a module over H*(G)=J; and it is finitely generated since H*(G) is noetherian. So it would suffice to show t* *hat H*(G)=J is a finitely generated R-module. THE ESSENTIAL IDEAL IS A COHEN-MACAULAY MODULE 3 If S T are finitelypgenerated_(graded-)commutative k-algebras and L T an ideal such that T= L is a finitely generated S-module,pthen_T=L is a finitely * *gen- erated S-module too. So it sufficespto_showpthat_H*(G)= J is a finitely genera* *ted R-module. But we have seen that J = I; and by assumption H*(G)=I is a submodule of the noetherian R-module H*(C). Lemma 1.3. Let G be a finite p-group and k a field of characteristic p. Denote * *by C the largest central elementary abelian subgroup of G. Assuming that Ess(G) is nonzero, the following three statements are equivalent: (1) Ess(G) is a Cohen-Macaulay H*(G)-module. (2) There is a sequence i1, . .i.dof homogeneous elements of H*(G) such that (a) the restrictions ResC(i1), . .,.ResC(id) form a homogeneous system of parameters for H*(C); and (b) the essential ideal Ess(G) is free and finitely generated as a module over the polynomial algebra k[i1, . .,.id]. (3) Every sequence i1, . .i.dwhich satisfies condition (2a) also satisfies c* *ondi- tion (2b). If these equivalent conditions hold, then d and the Krull dimension of the modu* *le Ess(G) both coincide with the p-rank of the centre of G. Proof.Let A, I, J be as in Lemma 1.1. First we shall see that condition (2a) is equivalent to the following condition: (2a0)The images of i1, . .,.id in H*(G)=A are algebraically independent over * *k, and Ess(G) is a finitely generated module over R = k[i1, . .,.ir]. p_ p__ Since I = A , the images of i1, . .,.id are algebraically independent in H*(G* *)=A if and only if they are algebraically independent in H*(G)=I H*(C). Lemma 1.2 shows that (2a) implies that Ess(G) is a finitely generated R-module. Conversel* *y, (2a0) impliespthat_Ess(G)pand_thereforepH*(G)=A_are finitely generated R-module* *s. Since I= A, it follows that H*(G)= I is a finitely generated R-module. So just as in the proof of Lemma 1.2, H*(G)=I is a finitely generated R-module. But recall from Corollary 7.4.7 of [9] that H*(C) is finite over the image H*(G)=I * *of restriction from H*(G). So H*(C) is a finitely generated R-module. Hence (2a) a* *nd (2a0) are indeed equivalent. But the characterisation of Cohen-Macaulay modules in Theorem 4.3.5 of [3] states precisely that (1), (2) and (3) are equivalent if one replaces (2a) by (* *2a0) in (2) and (3). Moreover, d = dim(Ess(G)) would follow too. As H*(C) is polynomial for p = 2 and polynomial tensor exterior for p odd, and the dimension is equal * *to the rank of C in both cases, d is the p-rank of the centre of G. Remark. Assuming that the equivalent conditions (2a), (2a0) hold, apply the same characterisation of Cohen-Macaulay modules to H*(C). Recalling that H*(C) is a polynomial algebra for p = 2, and polynomial tensor exterior for p odd, one ded* *uces that H*(C) is a free and finitely generated R-module. 2.Freeness As in [5], observe that group multiplication ~: G x C ! G turns H*(G) into a comodule over the coalgebra H*(C). As was noted above, the comodule structure map ~*: H*(G) ! H*(G) k H*(C) is simultaneously a map of k-algebras. 4 DAVID J. GREEN The following lemma is the key to this paper. It is perhaps the natural level of generality for a result that has been known in various special cases for some time (Lemma 3.1 in [2], Proposition 5.2 in [7] and Lemma 1.2 in [10]). The lemma bears a striking resemblance to a classical fact about Hopf algebras (Theorem 4* *.1.1 in [12]), but I do not know see to derive the the former as a corollary of the * *latter. L Lemma 2.1. Let G, C, R be as in Lemma 1.2, and let F = i i0Fi be the free graded H*(G)-module on generators e1, . .,.es (not necessarily in degree zero). Hence F is a comodule over H*(C), with structure map _ given by _(aej) = ~*(a) . (ej 1) 2 F k H*(C). Let N M be graded submodules of F which are simultaneously subcomodules. Then M=N is a free R-module. Proof.Define ~_:M=N ! M=N k H*(C) by ~_(a + N) = _(a) + N k H*(C). By the assumptions, (M=N, ~_) is an H*(C)-comodule. Moreover ~_is a map of H*(G)- modules if one gives M=N kH*(C) the H*(G)-module structure induced from the obvious H*(G) k H*(C)-module structure by ~*. In fact, ~_is a split monomor- phism of H*(G)-modules, the splitting map being IdM=N ", where ": H*(C) ! k is the augmentation map. Inclusion of R in H*(G) induces an R-module structure on M=N kH*(C). We shall show that this R-module is free. It then follows that i1, . .,.iz is a re* *gular sequence for this R-module, and therefore for the R-module M=N too. Hence M=N is a free R-module.L For i i0 set Si:= j i(M=N)i, and let Ti be the R-submodule Si k H*(C) ofTM=N H*(C). Then M=N k H*(C) = Ti0 Ti0+1 Ti0+2 . . .and i i0Ti = {0}. Now, Ti=Ti+1 is H*(C)-module, and as such it is free of rank dimk(M=N)i. Moreover the R-module structure is induced from this H*(C)-module structure by inclusion R ,! H*(G) followed by restriction to H*(C), since for a* *ll æ 2 R, x 2 Siand y 2 H*(C) one has æ.(x y +Ti+1) = x Res C(æ).y +Ti+1. We observed above that H*(C) is a free R-module, and so Ti=Ti+1is a free R-module. By induction, Ti0=Tiis a free R-module for all i. Since Tiis confined to degree* * i, this means that Ti0= M=N k H*(C) is itself a free R-module. Lemma 2.2. In Lemma 1.3 there is always a sequence i1, . .,.id satisfying condi- tion (2a). Proof.Restriction turns H*(C) into an H*(G)-module; recall from Corollary 7.4.7 of [9] that this module is finitely generated. The result follows by the graded* * case of Noether Normalization (Theorem 2.2.7 in [3]) applied to the H*(G)-module H*(C): recall that the annihilator ideal of this H*(G)-module is the kernel of the res* *triction map. One explicit sequence is ii= mth Chern class of the regular representation of* * G, for m = 2(pn - pn-i), where |G| = pn. This construction features in Venkov's topological proof of the Evens-Venkov theorem (x3.10 in [4]). Proof of Theorems A and B.Lemma 1.3 shows that the two theorems are equiva- lent, and that to prove them it suffices to prove the existence of a sequence i* *1, . .,.id satisfying conditions (2a) and (2b). Lemma 2.2 demonstrates that there is always a sequence satisfying (2a), and by Lemma 1.2 the finite generation part of (2b)* * is an automatic consequence of (2a). We shall adopt the notation of Lemma 1.2. Assume first that the cyclic group of order p is not a direct factor of G. Th* *en every maximal subgroup of G contains C, and so Ess(G) is a subcomodule of the THE ESSENTIAL IDEAL IS A COHEN-MACAULAY MODULE 5 H*(C)-comodule H*(G). Hence Ess(G) is a free R-module by Lemma 2.1, so the theorems are proved for this G. Now suppose that G = H x Cp, and that H has no direct factor which is cyclic of order p. Denote by C0 the largest central elementary abelian subgroup of H. Then C = C0x Cp, and Ess(H) is a H*(C0)-comodule by the above argument. Applying Lemma 2.2 to H, C0 one obtains a sequence i1, . .,.id-1 2 H*(H) which satisfies condition (2a) of Lemma 1.3 for H, C0. Let t be a homogeneous system * *of parameters for the one-dimensional k-algebra H*(Cp). Then i1, . .,.id-1, t sati* *sfies condition (2a) for G, C. Set S = k[i1, . .,.id-1] H*(H) and R = S[t] H*(G). Observe that Ess(H) is a free S-module by the case we have already proved. Let b1, . .,.bn be a finite free generating set of the k[t]-module H*(Cp). We take * *each b` to be homogeneous.LLet B H*(Cp) be the k-vector space with basis b1, . .,.bn. Then H*(Cp) = j 0tjB. L i Set Fi := H*(H) k j=0tjB. Define Mi and Ni by Mi := Ess(G) \ Fi and Ni := Mi-1+ Mi-1t. Both Mi and Ni are submodules of the free and finitely generated H*(H)-module Fi. Moreover they are subcomodules of the H*(C0)- comodule Fi, as C0 lies in every maximal subgroup of G. So Mi=Ni is a free S- module by Lemma 2.1. Observe that the R-modules k[t]Ni and k[t]Mi-1 coincide. Moreover, the noetherian R-module Ess(G) is the union of the R-modules k[t]Mi, and so k[t]Mi= Ess(G) for large enough i. It would suffice to show that each k[t]Mi is a free R-module, and this follows by induction if we can show that each (k[t]Mi)=(k[t]Ni) is a free R-module. (No* *te that N0 = {0}.) We shall show this by proving that (k[t]Mi)=(k[t]Ni) is the free S[t]-module k[t] k (Mi=Ni). Note that Ess(G) Ess(H) k H*(Cp), since K x Cp is a maximal subgroup of G for each maximal subgroup K of H. (For H = 1 one has Ess(1) = k = H*(1).) Let y be an element of Mi in degree m. Then thereParePhomogeneous classes aj`2PEss(H) for 0 j i and 1 ` n with y = ij=0 n`=1aj` b`tj. Call `ai` b` the leading coefficient of y, and let Ii Ess(H) k B denote the set* * of all such leading coefficients. Then Ii is an H*(H)-module, and y lies in Ni if * *and only if its leading coefficient lies in Ii-1. So we may pick a basis for a comp* *lement of the subspace Ni of Mi consisting of classes whose leading terms constitute a basis for a complement of the subspace Ii-1of Ii. So the natural surjective map* * of S[t]-modules k[t] k (Mi=Ni) ! (k[t]Mi)=(k[t]Ni) is injective too. But the doma* *in is a free S[t]-module. 3. Concluding remarks Application to computer calculations. In [8], Carlson describes a series of tes* *ts on a partial presentation for H*(G) and proves that the presentation is complete if it passes the tests. This is of crucial importance for the computer calcula* *tion of group cohomology via minimal resolutions, for otherwise one would never know when to stop. However Carlson's tests depend on two conjectures about the structure of the cohomology ring, which have to be checked for the group in question as part of * *the calculation. This means that there could conceivably be some groups where there is no complete presentation that the tests can detect as being complete. One of these conjectures concerns the Koszul complex associated to a homogeneous system of parameters. The other is now proven, as our Theorem B: note that Carlson's 6 DAVID J. GREEN tests assume that the cohomology rings of all subgroups are known, and so the cohomology ring of a product group is known by the Künneth theorem. References [1]Alejandro Adem and Dikran Karagueuzian, Essential cohomology of finite grou* *ps, Comment. Math. Helv. 72 (1997), no. 1, 101-109. [2]Alejandro Adem and R. 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[7]____, Problems in the calculation of group cohomology, Computational method* *s for repre- sentations of groups and algebras (Essen, 1997) (P. Dräxler, G. O. Michler, * *and C. M. Ringel, eds.), Birkhäuser, Basel, 1999, pp. 107-120. [8]____, Calculating group cohomology: Tests for completion, J. Symbolic Compu* *t. 31 (2001), no. 1-2, 229-242. [9]Leonard Evens, The cohomology of groups, Oxford Univ. Press, Oxford, 1991. [10]David J. Green, On Carlson's depth conjecture in group cohomology, Math. Z.* * 244 (2003), no. 4, 711-723. [11]Hu`ynh Mui, The mod p cohomology algebra of the extra-special group E(p3), * *Unpublished essay, 1982. [12]Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Ben* *jamin, Inc., New York, 1969. Department of Mathematics, University of Wuppertal, D-42097 Wuppertal, Germany E-mail address: green@math.uni-wuppertal.de