Associated Primes for Cohomology Modules

Jonathan Elmer

1. Introduction

Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V. Then the dual vector space V^{∗} is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V^{∗}), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V] of polynomial functions on V. The G-fixed points of R under this action form a ring, which we denote by R^{G} and call the ring of invariants. If k is algebraically closed, R^{G} can be regarded as the set of G-invariant polynomial functions on V, or the ring of coordinate functions on the quotient space V/G. The ring of invariants R^{G} is the central object of study in invariant theory. The situation becomes modular when we assume p divides the order of G. Let P be a (fixed) Sylow-p-subgroup of G.

Since the ring of invariants R^{G} coincides with the zeroth cohomology H^{0}(G, R), we can regard R^{G} as the zeroth degree part of the cohomology ring H^{∗}(G, R), and as such, the higher cohomology modules H^{i}(G, R) become R^{G}-modules via the cup product. One can often learn more about the structure of modular rings of invariants by studying these higher cohomology modules; for example, in [3] Ellingsrud and Skjelbred showed that H^{i}(G, R) is Cohen-Macaulay for G cyclic of order p. They then used this result to find a formula for the depth the ring of invariants R^{G} in this case. This approach was also used in [7], [9] and [10] to answer questions about the depth or Cohen-Macaulay property of modular invariant rings.

If X < G, we may define a mapping Tr_{X}^{G} : R^{X} → R^{G} as follows: let S be a set of right coset representatives of X in G. Then we define

Tr_{X}^{G}(x) := ∑_{g∈S} xg.  (1)

This mapping is often called the relative transfer, and induces mappings Tr_{X}^{G} : H^{i}(X, R) → H^{i}(G, R) also called the relative transfer. Both are surjective when the index of X in G is coprime to p. The image of the transfer map Tr_{X}^{G}(R^{X}) is an ideal in R^{G} called the relative transfer ideal which we denote by I_{X}^{G}. We may generalise this definition and define

I_{χ}^{G} := ∑_{X∈χ} I_{X}^{G}

for any set χ of subgroups of G. Relative transfer ideals and their radicals have been studied widely in connection with modular invariant theory. For example it is known that the quotient ring R^{G}/\sqrt{I_{<P}^{G}}^{1} is always Cohen-Macaulay (see [6]).

Let H^{+}(G, R) denote the set of positive degree elements of the cohomology ring H^{∗}(G, R), that is, we define H^{+}(G, R) := ⊕_{i>0}H^{i}(G, R). The main purpose of this paper is to prove the following:

Theorem 1.1. Let 𝔭 be an associated prime ideal of the R^{G}-module H^{+}(G, R). Then 𝔭 = \sqrt{I_{χ}^{G}} for some set χ of subgroups of G.

Remark: The relative transfer ideals I_{χ}^{G} defined above were first studied by Fleischmann ([5]), who proved the following formulae:

\sqrt{I_{χ}^{G}} = ( ⋂_{X∈χ′}((g − 1)V^{∗}| g ∈ X)R) ∩ R^{G} = ( ⋂_{X∈χ′} I(V^{X})) ∩ R^{G}  (2)

where χ′ := {Q ≤ P | Q ≰ X^{g} for any g ∈ G and X ∈ χ}, and for a subspace W of V, I(W) denotes {f ∈ k[V]| f(W) = 0}. So we should be able to use these formulae along with Theorem 1.1 to construct some associated primes of cohomology modules.

Brief digression: Consider for a moment the cohomology ring H := H^{∗}(G, k) of a finite group G with coefficients in a field k whose characteristic divides the order of G. It is known (see, for example [2], Theorem 12.7.1) that the associated

primes of the ring H take the form \sqrt{ker(res_{E}^{G})} for certain elementary abelian subgroups E of G. Using a result of Benson ([1], Theorem 1.1), one can show

this is equal to \sqrt{∑_{X∈χ} Tr_{X}^{G}(H^{∗}(X, k))} where χ is the set of subgroups of G not contained in any Sylow-p-subgroup of C_{G}(E). So the associated primes of H are also radicals of relative transfer ideals. Whether this result and Theorem 1.1 are two examples of a more general phenomenon remains to be seen.

2. Annihilators in Cohomology

The following lemma is an observation of Lorenz and Pathak ([11], Lemma 1.3). It is a simple consequence of the transfer-restriction formula for cup products ([2], Theorem 4.4.2) and the starting point for our investigations. Throughout this section, let m be a strictly positive integer.

Lemma 2.1. Suppose α ∈ H^{m}(G, R) satisfies res_{N}^{G}(α) = 0. Then Ann_{R^{G}}(α) ≥ I_{N}^{G}.

1Here, “< P ” means the set of all proper subgroups of P.

Proof. Let x ∈ R^{N}. Then we have Tr_{N}^{G}(x) · α = Tr_{N}^{G}(x · res_{N}^{G}(α)) = 0. □

Corollary 2.2. Suppose α ∈ H^{m}(G, R) and define

χ(α) := {X ≤ P | res_{X}^{G}(α) = 0}.  (3)

Then Ann_{R^{G}}(α) ≥ I_{χ(α)}^{G}.

Remark: Since the Sylow-p-subgroups of G are conjugate and Tr_{X^{g}}^{G}(x) = Tr_{X}^{G}(xg), we gain nothing by considering the set of all subgroups X ≤ G on which res_{X}^{G}(α) = 0.

The following result on annihilators is the key to proving our main theorem.

Lemma 2.3. Let 0 ≠ α ∈ H^{m}(G, R). Then we have

Ann_{R^{G}}(α) ≤ \sqrt{I_{<P}^{G}} = I(V^{P}) ∩ R^{G}.

Proof. The second statement is just (2) applied to the set {< P } of all proper subgroups of P. The first is [7], Corollary 2.2, which is itself a consequence of a much more general result of Kemper ([10], Proposition 1.2). □

Lemma 2.4. Let 0 ≠ α ∈ H^{m}(G, R). Then we have

\sqrt{I_{χ(α)}^{G}} = ⋂_{X∈χ′(α)}(I(V^{X})) ∩ R^{G}

where χ′(α) := {X ≤ P | res_{X}^{G}(α) ≠ 0}.

Proof. Using (2), we must show that χ′(α) as defined above is equal to

{X ≤ P |X ≰ Y^{g} for any g ∈ G and Y ∈ χ(α)}.

This is tantamount to proving that res_{X}^{G}(α) = 0 implies res_{X^{g}}^{G} = 0 for all g ∈ G, which is well known and follows from the fact that conjugation map (−)^{g^{−1}} : X^{g} → X induces an isomorphism i : H^{∗}(X, R) → H^{∗}(X^{g}, R) satisfying res_{X}^{G} = i ∘ res_{X^{g}}^{G}. □

Our main theorem now follows from the following result:

Proposition 2.5. Suppose α ∈ H^{m}(G, R) and χ(α) is defined as in Corollary 2.2. Then

\sqrt{Ann_{R^{G}}(α)} = \sqrt{I_{χ(α)}^{G}}

Remark: Suppose α ∈ H^{1}(G, k). Then α can be viewed as a homomorphism from G to k, which has a well-defined kernel N. Kemper ([9], Proposition 3.4)^{2}

proved that \sqrt{Ann_{R^{G}}(α)} = \sqrt{I_{N}^{G}}. For any subgroup X ≤ G, we have res_{X}^{G}(α) = 0 if and only if X ≤ N^{g} for some g ∈ G. So Proposition 2.5 may be viewed as a generalisation of this result.

2Kemper actually proved this result under the assumption that k is algebraically closed, although since Fleischmann’s formulae (2) hold for an arbitrary field of characteristic p, the generalisation of his result to any field of characteristic p is easily obtained.

4 Jonathan Elmer

Proof. That \sqrt{Ann_{R^{G}}(α)} ≥ \sqrt{I_{χ(α)}^{G}} is an immediate consequence of Corollary 2.2. To prove the reverse, let y^{n} ∈ Ann_{R^{G}}(α) for some n ≥ 0. Let Q ∈ χ′(α) and define β := res_{Q}^{G}(α) ≠ 0. Then we have

0 = y^{n} · α = res_{Q}^{G}(y^{n} · α) = y^{n} · β

since res_{Q}^{G} : H^{∗}(G, R) → H^{∗}(Q, R) is a ring homomorphism which specialises to the inclusion R^{G} → R^{Q} on the degree zero part. This means that y^{n} ∈ Ann_{R^{Q}}(β), so y^{n} ∈ I(V^{Q}) ∩ R^{Q} by Lemma 2.3, and since this holds for every Q ∈ χ′(α) we have

y^{n} ∈ R^{G} ∩ ⋂_{X∈χ′(α)}((I(V^{X})) ∩ R^{X}) = R^{G} ∩ ⋂_{X∈χ′(α)}(I(V^{X})) = \sqrt{I_{χ(α)}^{G}}

where the final equality follows from Lemma 2.4. Therefore y ∈ \sqrt{I_{χ(α)}^{G}} as required. This completes the proof of Proposition 2.5, and since the associated primes of H^{+}(G, R) are those annihilators of homogeneous α ∈ H^{+}(G, R) which are prime ideals, this completes the proof of Theorem 1.1 too.

□

References

[1] D. J. Benson, The Image of the Transfer Archiv der Mathematik 61 (1993), pp 7 – 11.

[2] Jon F. Carlson et al, Cohomology Rings of Finite Groups, Kluwer Academic Publications (2003).

[3] G. Ellingsrud and T.Skjelbred, Profondeur d’Anneaux d’Invariants en Caractéristique p, Compos. Math.. 41, (1980), pp 233–244.

[4] J. Elmer and P. Fleischmann, On the Depth of Modular Invariant Rings for the Groups C_{p} × C_{p}, in Proc. Symmetry and Space, Fields Institute (2006), preprint (2007).

[5] P. Fleischmann, Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings in Computational Methods for Representations of Groups and Algebras, Euroconference in Essen 1997, Progress in Mathematics, 173 Birkhäuser, Basel (1999).

[6] P. Fleischmann, On Invariant Theory Of Finite Groups, CRM Proceedings and Lecture Notes, 35, (2004), pp 43–69.

[7] P. Fleischmann, G. Kemper, R. J. Shank, Depth and Cohomological Connectivity in Modular Invariant Theory, Transactions of the American Mathematical Society, 357, (2005), pp 3605–3621.

[8] P. Fleischmann, G. Kemper, R. J. Shank, On the Depth of Cohomology Modules, Quaterly Journal of Mathematics, 55(2), (2004), pp 167 – 184.

[9] G. Kemper On the Cohen-Macaulay Property of Invariant Rings, Journal of Algebra 215, (1999), pp 330 – 351.

Associated Primes for Cohomology Modules 5

[10] G. Kemper, with an appendix by K. Magaard, The Depth of Invariant Rings and Cohomology, Journal of Algebra 245, (2001), pp 463 – 531.

[11] M. Lorenz, J. Pathak, On Cohen-Macaulay Rings of Invariants, Journal of Algebra, 245, (2001), pp 247–264.

Jonathan Elmer Department of Mathematics, Meston Hall University of Aberdeen Aberdeen AB24 3UE

e-mail: j.elmer@maths.abdn.ac.uk