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 elementary 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 the 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 classes whose restriction to every proper subgroup is zero. The essential ideal is involved in the cohomological characterization of p-groups whose exponent p elements are all central [1], and it has featured prominently in investigations concerning the depth of the graded commutative ring H^{∗}(G, k) (see [6]).

The essential ideal also plays a key role in Carlson’s method [8] for computing 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 techniques. 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 p^{2} as a direct factor. If the essential ideal Ess(G) in H^{∗}(G, k) is nonzero, then it is a Cohen–Macaulay 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 ζ_{1}, … ζ_{d} are homogeneous elements of H^{∗}(G, k) such that the restrictions Res_{C}(ζ_{1}), …, Res_{C}(ζ_{d}) form a homogeneous 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[ζ_{1}, …, ζ_{d}].

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 theorem, programs based on Carlson’s method should run faster. Note that the same question 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 ζ_{1}, …, ζ_{d} satisfying the assumptions 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 consideration in §2, 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).

Date: 26 February 2004.

2000 Mathematics Subject Classification. Primary 20J06; Secondary 13C14.

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 abelian subgroup of G. Let A, I and J be the following ideals in H^{∗}(G):

• A is the annihilator ideal Ann_{H^{∗}(G)}(Ess(G));

• I is the kernel of restriction H^{∗}(G) → H^{∗}(C); and

• J is generated by all transfer classes from all proper subgroups.

Then \sqrt{A} = \sqrt{I} = \sqrt{J}, and this radical ideal is a prime ideal.

Proof. The cohomolgy ring of an elementary abelian p-group is a polynomial algebra (p = 2) or the tensor product over k of a polynomial algebra with an exterior algebra (p odd). So H^{∗}(C)/\sqrt{0} is a polynomial algebra, and \sqrt{I} a prime ideal. Quillen proved that x ∈ H^{∗}(G) is nilpotent if its restriction to every elementary abelian subgroup is nilpotent (Coroll. 8.3.4 in [9]). So \sqrt{I} consists of those classes with nilpotent restriction to the centre Z(G). It follows from Carlson’s theorem on varieties and transfer (§10.2 in [9]) that \sqrt{I} = \sqrt{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 will use an argument of Broto and Henn [5] to show that every element of H^{∗}(G) \ \sqrt{I} is regular, from which it follows that A ⊆ \sqrt{I}. Group multiplication μ: G × C → G is a group homomorphism, and so induces a map of k-algebras μ^{∗}: H^{∗}(G) → H^{∗}(G) ⊗_{k} H^{∗}(C). Moreover, (Id_{G} ×ε_{C}) ∘ μ^{∗} = Id_{G} and (ε_{G} ⊗ Id_{C}) ∘ μ^{∗} = Res_{C}^{G}, where Id_{G}: H^{∗}(G) → H^{∗}(G) is the identity map and ε_{G}: H^{∗}(G) → k the augmentation. So μ^{∗} is a split monomorphism; and μ^{∗}(x) ∈ 1 ⊗ Res_{C}(x) + H^{>0}(G) ⊗ H^{∗}(C) for all x ∈ H^{∗}(G). So if Res_{C}(x) is non-nilpotent, then it is a regular element of H^{∗}(C), and it follows that μ^{∗}(x) is a regular element of H^{∗}(G) ⊗_{k} H^{∗}(C). As μ^{∗} is injective, it follows that x ∈ 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 ζ_{1}, … ζ_{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[ζ_{1}, …, ζ_{d}].

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 the 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 that H^{∗}(G)/J is a finitely generated R-module.

If S ⊆ T are finitely generated (graded-)commutative k-algebras and L ⊆ T an ideal such that T/\sqrt{L} is a finitely generated S-module, then T/L is a finitely generated S-module too. So it suffices to show that H^{∗}(G)/\sqrt{J} is a finitely generated R-module. But we have seen that \sqrt{J} = \sqrt{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 ζ_{1}, … ζ_{d} of homogeneous elements of H^{∗}(G) such that (a) the restrictions Res_{C}(ζ_{1}), …, Res_{C}(ζ_{d}) 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[ζ_{1}, …, ζ_{d}].

(3) Every sequence ζ_{1}, … ζ_{d} which satisfies condition (2a) also satisfies condi-

tion (2b). If these equivalent conditions hold, then d and the Krull dimension of the module 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:

(2a′) The images of ζ_{1}, …, ζ_{d} in H^{∗}(G)/A are algebraically independent over k,

and Ess(G) is a finitely generated module over R = k[ζ_{1}, …, ζ_{r}]. Since \sqrt{I} = \sqrt{A}, the images of ζ_{1}, …, ζ_{d} 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. Conversely, (2a′) implies that Ess(G) and therefore H^{∗}(G)/A are finitely generated R-modules. Since \sqrt{I} = \sqrt{A}, it follows that H^{∗}(G)/\sqrt{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) and (2a′) 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 (2a′) 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), (2a′) 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 deduces that H^{∗}(C) is a free and finitely generated R-module.

2. Freeness

As in [5], observe that group multiplication μ: G × C → G turns H^{∗}(G) into a comodule over the coalgebra H^{∗}(C). As was noted above, the comodule structure map μ^{∗}: H^{∗}(G) → H^{∗}(G) ⊗_{k} H^{∗}(C) is simultaneously a map of k-algebras.

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. Lemma 2.1. Let G, C, R be as in Lemma 1.2, and let F = ⨁_{i≥i_{0}} F_{i} be the free graded H^{∗}(G)-module on generators e_{1}, …, e_{s} (not necessarily in degree zero). Hence F is a comodule over H^{∗}(C), with structure map ψ given by ψ(ae_{j}) = μ^{∗}(a) · (e_{j} ⊗ 1) ∈ 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 ψ̄: M/N → M/N ⊗_{k} H^{∗}(C) by ψ̄(a + N) = ψ(a) + N ⊗_{k} H^{∗}(C). By the assumptions, (M/N, ψ̄) is an H^{∗}(C)-comodule. Moreover ψ̄ is a map of H^{∗}(G)- modules if one gives M/N ⊗_{k} H^{∗}(C) the H^{∗}(G)-module structure induced from the obvious H^{∗}(G) ⊗_{k} H^{∗}(C)-module structure by μ^{∗}. In fact, ψ̄ is a split monomorphism of H^{∗}(G)-modules, the splitting map being Id_{M/N} ⊗ε, where ε: H^{∗}(C) → k is the augmentation map.

Inclusion of R in H^{∗}(G) induces an R-module structure on M/N ⊗_{k} H^{∗}(C). We shall show that this R-module is free. It then follows that ζ_{1}, …, ζ_{z} is a regular sequence for this R-module, and therefore for the R-module M/N too. Hence M/N is a free R-module.

For i ≥ i_{0} set S_{i} := ⨁_{j≥i}(M/N)_{i}, and let T_{i} be the R-submodule S_{i} ⊗_{k} H^{∗}(C) of M/N ⊗ H^{∗}(C). Then M/N ⊗_{k} H^{∗}(C) = T_{i_{0}} ⊇ T_{i_{0}+1} ⊇ T_{i_{0}+2} ⊇ ⋯ and ⋂_{i≥i_{0}} T_{i} = {0}. Now, T_{i}/T_{i+1} is H^{∗}(C)-module, and as such it is free of rank dim_{k}(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 all ρ ∈ R, x ∈ S_{i} and y ∈ H^{∗}(C) one has ρ·(x⊗y +T_{i+1}) = x⊗Res_{C}(ρ)·y +T_{i+1}. We observed above that H^{∗}(C) is a free R-module, and so T_{i}/T_{i+1} is a free R-module. By induction, T_{i_{0}}/T_{i} is a free R-module for all i. Since T_{i} is confined to degree ≥ i, this means that T_{i_{0}} = M/N ⊗_{k} H^{∗}(C) is itself a free R-module. □

Lemma 2.2. In Lemma 1.3 there is always a sequence ζ_{1}, …, ζ_{d} satisfying condition (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 restriction map.

One explicit sequence is ζ_{i} = mth Chern class of the regular representation of G, for m = 2(p^{n} − p^{n−i}), where |G| = p^{n}. This construction features in Venkov’s topological proof of the Evens–Venkov theorem (§3.10 in [4]). □

Proof of Theorems A and B. Lemma 1.3 shows that the two theorems are equivalent, and that to prove them it suffices to prove the existence of a sequence ζ_{1}, …, ζ_{d} 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. Then every maximal subgroup of G contains C, and so Ess(G) is a subcomodule of the 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 × C_{p}, and that H has no direct factor which is cyclic of order p. Denote by C′ the largest central elementary abelian subgroup of H. Then C = C′ × C_{p}, and Ess(H) is a H^{∗}(C′)-comodule by the above argument. Applying Lemma 2.2 to H, C′ one obtains a sequence ζ_{1}, …, ζ_{d−1} ∈ H^{∗}(H) which satisfies condition (2a) of Lemma 1.3 for H, C′. Let t be a homogeneous system of parameters for the one-dimensional k-algebra H^{∗}(C_{p}). Then ζ_{1}, …, ζ_{d−1}, t satisfies condition (2a) for G, C. Set S = k[ζ_{1}, …, ζ_{d−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 b_{1}, …, b_{n} be a finite free generating set of the k[t]-module H^{∗}(C_{p}). We take each b_{ℓ} to be homogeneous. Let B ⊆ H^{∗}(C_{p}) be the k-vector space with basis b_{1}, …, b_{n}. Then H^{∗}(C_{p}) = ⨁_{j≥0} t^{j}B.

Set F_{i} := H^{∗}(H) ⊗_{k} ⨁_{j=0}^{i} t^{j}B. Define M_{i} and N_{i} by M_{i} := Ess(G) ∩ F_{i} and N_{i} := M_{i−1} + M_{i−1}t. Both M_{i} and N_{i} are submodules of the free and finitely generated H^{∗}(H)-module F_{i}. Moreover they are subcomodules of the H^{∗}(C′)- comodule F_{i}, as C′ lies in every maximal subgroup of G. So M_{i}/N_{i} is a free Smodule by Lemma 2.1. Observe that the R-modules k[t]N_{i} and k[t]M_{i−1} coincide. Moreover, the noetherian R-module Ess(G) is the union of the R-modules k[t]M_{i}, and so k[t]M_{i} = Ess(G) for large enough i.

It would suffice to show that each k[t]M_{i} is a free R-module, and this follows by induction if we can show that each (k[t]M_{i})/(k[t]N_{i}) is a free R-module. (Note that N_{0} = {0}.) We shall show this by proving that (k[t]M_{i})/(k[t]N_{i}) is the free S[t]-module k[t] ⊗_{k} (M_{i}/N_{i}).

Note that Ess(G) ⊆ Ess(H) ⊗_{k} H^{∗}(C_{p}), since K × C_{p} 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 M_{i} in degree m. Then there are homogeneous classes a_{jℓ} ∈ Ess(H) for 0 ≤ j ≤ i and 1 ≤ ℓ ≤ n with y = ∑_{j=0}^{i} ∑_{ℓ=1}^{n} a_{jℓ} ⊗ b_{ℓ}t^{j}. Call ∑_{ℓ} a_{iℓ} ⊗ b_{ℓ} the leading coefficient of y, and let I_{i} ⊆ Ess(H) ⊗_{k} B denote the set of all such leading coefficients. Then I_{i} is an H^{∗}(H)-module, and y lies in N_{i} if and only if its leading coefficient lies in I_{i−1}. So we may pick a basis for a complement of the subspace N_{i} of M_{i} consisting of classes whose leading terms constitute a basis for a complement of the subspace I_{i−1} of I_{i}. So the natural surjective map of S[t]-modules k[t] ⊗_{k} (M_{i}/N_{i}) → (k[t]M_{i})/(k[t]N_{i}) is injective too. But the domain is a free S[t]-module. □

3. Concluding remarks

Application to computer calculations. In [8], Carlson describes a series of tests 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 calculation 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 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.

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Department of Mathematics, University of Wuppertal, D-42097 Wuppertal, Germany

E-mail address: green@math.uni-wuppertal.de