IDEMPOTENT kG-MODULES WITH INJECTIVE COHOMOLOGY

DAVE BENSON

Abstract. Let G be a finite group and k a field of characteristic p. This paper gives a short proof of a recent theorem of Benson and Greenlees, stating that the cohomology of the kappa module κ_{𝔭} of Benson, Carlson and Rickard is equal to the injective hull of H^{∗}(G, k)/𝔭 as a graded H^{∗}(G, k)-module. This shorter proof does not naturally extend to compact Lie groups, whereas the version of Benson and Greenlees is proved in this broader context.

Let G be a finite group and k a field of characteristic p. Benson, Carlson and Rickard [2] developed a theory of varieties for infinitely generated kG-modules where certain idempotent modules κ_{𝔭} play a central role (there, they are called κ_{V}, where V is the closed homogeneous irreducible subvariety of the cohomology variety corresponding to a homogeneous prime ideal 𝔭 in H^{∗}(G, k)). The purpose of κ_{𝔭} is to pick out the layer of the stable module category corresponding to 𝔭. From the definition of κ_{𝔭}, its most elementary properties such as its cohomology are not obvious. In [1], it was conjectured that Ĥ^{∗}(G, κ_{𝔭}) is the injective hull of H^{∗}(G, k)/𝔭 (for a precise statement, see Theorem 2 below) and this was proved in special cases, including the case where H^{∗}(G, k) is Cohen–Macaulay.

In a recent preprint of Benson and Greenlees [3], the general case was proved by interpreting it as a statement about localisation of Gorenstein duality on the cochains of BG in the sense of Dwyer, Greenlees and Iyengar [5]. The machinery used in that paper is sophisticated, not least because it uses the language of strictly commutative ring spectra in the sense of [6]. But the proof is rather natural and stands a good chance of generalising to other Gorenstein situations. The consequences of the theorem are spelled out in the papers [1, 3], so they will not be repeated here.

This paper gives a shorter, but in many ways cruder and less natural proof of the theorem. The point is to reduce to the Cohen–Macaulay case, which was dealt with in [1]. To do this, we use Quillen’s computation [8] of the cohomology of the general linear group over a finite field. Although the proof as presented here looks extremely short, it does rely on the proof given in [1] for the Cohen–Macaulay case, which is already somewhat technical.

This research was partly supported by a senior scientist prize from the Humboldt foundation.

It should also be remarked that the Gorenstein localisation theorem is proved in [3] in the greater generality of compact Lie groups, whereas the version proved here is strictly for finite groups.

Proposition 1. Given a finite group G and a field k, there exists an embedding of G into a finite group H with the property that H^{∗}(H, k) is Cohen–Macaulay.

Proof. Choose a prime ℓ different from the characteristic of k, and a faithful representation G → GL(n, 𝔽_{ℓ}). For example, we could use the regular representation with n = |G|. Quillen’s computation [8] of H^{∗}(GL(n, 𝔽_{ℓ}), k) shows that it is always a Cohen–Macaulay ring. For example, if ℓ is odd, it is a tensor product of a polynomial and an exterior algebra, and hence Cohen–Macaulay. □

Theorem 2. The Tate cohomology Ĥ^{∗}(G, κ_{𝔭}) is isomorphic to I_{𝔭}[d], the injective hull of H^{∗}(G, k)/𝔭 in the category of graded H^{∗}(G, k)-modules, shifted in degree by the Krull dimension d of H^{∗}(G, k)/𝔭.

Proof. It is shown in [1] that this statement is equivalent to the statement that κ_{𝔭} is isomorphic in the stable module category StMod(kG) to the module T (I_{𝔭}[d]) described in Benson and Krause [4].

Embed G into a finite group H such that H^{∗}(H, k) is Cohen–Macaulay, using Proposition 1. By Theorem 1.4 of [1], the theorem is true for H. So let 𝔮 = res_{H,G}^{−1} 𝔭 be the inverse image of 𝔭 under the restriction map

res_{H,G}: H^{∗}(H, k) → H^{∗}(G, k).

Then we have κ_{𝔮} ≅ T (I_{𝔮}[d]). There may also be other primes whose inverse image is 𝔮, but since the restriction map is a finite map by a theorem of Evens [7], there are only a finite number of such primes. Call them 𝔭_{1}, …, 𝔭_{s} with 𝔭_{1} = 𝔭.

By Theorem 7.1 of [4], we have

T (I_{𝔮})↓_{G}≅ ⨁_{i=1}^{s} T (I_{𝔭_{i}}).

By Lemma 8.2 of [2], we have

κ_{𝔮}↓_{G}≅ ⨁_{i=1}^{s} κ_{𝔭_{i}}.

Tensoring with κ_{𝔭} = κ_{𝔭_{1}}, we have

κ_{𝔭} ≅ κ_{𝔭} ⊗_{k} ⨁_{i=1}^{s} κ_{𝔭_{i}} ≅ κ_{𝔭} ⊗_{k} κ_{𝔮}↓_{G}

≅ κ_{𝔭} ⊗_{k} T (I_{𝔮}[d])↓_{G}≅ κ_{𝔭} ⊗_{k} ⨁_{i=1}^{s} T (I_{𝔭_{i}}[d]) ≅ T (I_{𝔭}[d]).

In this computation, we have used the fact that the varieties of both κ_{𝔭_{i}} and T (I_{𝔭_{i}}[d]) are equal to the singleton consisting of the variety of 𝔭_{i}. For κ_{𝔭_{i}} this is proved in Lemma 10.4 of [2]. For T (I_{𝔭_{i}}[d]), it is proved in Theorem 7.3 of [4]. □

References

1. D. J. Benson, Modules with injective cohomology, and local duality for a finite group, New York Journal of Mathematics 7 (2001), 201–215.

2. D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.

3. D. J. Benson and J. P. C. Greenlees, Localization and duality in topology and modular representation theory, Preprint, 2004.

4. D. J. Benson and H. Krause, Pure injectives and the spectrum of the cohomology ring of a finite group, J. Reine & Angew. Math. 542 (2002), 23–51.

5. W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar, Duality in algebra and topology, Preprint, 2002.

6. A. D. Elmendorf, I. Kříž, M. A. Mandell, and J. P. May, Rings, modules and algebras in stable homotopy theory, Surveys and Monographs, vol. 47, American Math. Society, 1996.

7. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224– 239.

8. D. G. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. 96 (1972), 552–586.

Department of Mathematics, University of Aberdeen, Aberdeen AB24 3EA, Scotland, UK

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