VARIETIES AND COHOMOLOGY OF INFINITELY GENERATED MODULES

DAVID J. BENSON AND JON F. CARLSON

1. Introduction

In a series of papers [2, 3, 4] with Jeremy Rickard, the authors developed a theory of support varieties for infinitely generated modules over the group algebra of a finite group. It was conjectured in [1] that the variety V_{G}(M) could be computed as the small support (see below for the definition) of the cohomology H^{∗}(G, M) as a graded module over H^{∗}(G, k), at least if G is a p-group. The first counterexample to this was given in [5] with G = (ℤ/2)^{3}, by a rather complicated construction. The purpose of this paper is to provide a simpler and more general method for producing counterexamples. Our examples are Rickard idempotent modules [6].

Theorem 1.1. Let G be a finite group of p-rank at least three, and k be a field of characteristic p. Suppose that there is a regular sequence of length two, say (x, y), of homogeneous elements of H^{∗}(G, k). Then the Rickard idempotent module F (x, y) has the property that V_{G}(F (x, y)) consists of the prime ideals in H^{∗}(G, k) not contained in (x, y), while supp_{H^{∗}(G,k)}H^{∗}(G, F (x, y)) consists of all homogeneous primes in H^{∗}(G, k).

The definition of small support of a module N over a graded commutative Noetherian ring R is the set of homogeneous primes occuring in the minimal injective resolution of N over R. More precisely,

supp_{R}N = {𝔭 | Ext_{R}^{∗}(k(𝔭), N_{𝔭}) ≠ 0}

where k(𝔭) is the field of fractions of R/𝔭. If N happens to be finitely generated, this coincides with the set of primes containing in the annihilator of N, but generally supp_{R}N does not have to be closed under specialisation.

The research of both authors was supported by senior research prizes from the Humboldt Foundation in Germany. The research of the second author was partly supported by a grant from the NSF.

2. Proof of the Theorem

The proof of the theorem is a simple calculation with a Mayer– Vietoris triangle. In Section 8 of Rickard [6], it is shown that there is a triangle in the stable module category

(2.1)  F (x, y) → F (x) ⊕ F (y) → F (x) ⊗ F (y).

It follows from the description of F (x) in Section 6 of [6] that

Ĥ^{∗}(G, F (x)) = H^{∗}(G, k)[x^{−1}],

namely the localisation of H^{∗}(G, k) given by inverting x. The long exact sequence in Tate cohomology of the triangle (2.1) is therefore as follows:

⋯ → Ĥ^{∗}(G, F (x, y)) → H^{∗}(G, k)[x^{−1}] ⊕ H^{∗}(G, k)[y^{−1}]

→ H^{∗}(G, k)[x^{−1}y^{−1}] → …

Since the sequence x, y is regular, the kernel of

(2.2)  H^{∗}(G, k)[x^{−1}] ⊕ H^{∗}(G, k)[y^{−1}] → H^{∗}(G, k)[x^{−1}y^{−1}]

is equal to H^{∗}(G, k). So we obtain a short exact sequence

(2.3)  0 → U → Ĥ^{∗}(G, F (x, y)) → H^{∗}(G, k) → 0

where U is the cokernel of (2.2), shifted in degree by one. As a sequence of H^{∗}(G, k)-modules, the sequence (2.3) splits, since the right hand side is free. So we obtain

Ĥ^{∗}(G, F (x, y)) ≅ H^{∗}(G, k) ⊕ U.

It follows that H^{∗}(G, F (x, y)) also has H^{∗}(G, k) as a direct summand. So supp_{H^{∗}(G,k)}H^{∗}(G, F (x, y)) = supp_{H^{∗}(G,k)}Ĥ^{∗}(G, F (x, y)) consists of all homogeneous primes, as required. The computation of V_{G}(F (x, y)) is given in [3].

References

1. D. J. Benson, Commutative algebra in the cohomology of groups, Trends in Commutative Algebra, MSRI Publications, vol. 51, Cambridge University Press, 2004, pp. 1–50.

2. D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, I, Math. Proc. Camb. Phil. Soc. 118 (1995), 223–243.

3. , Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.

4. , Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

5. D. J. Benson, S. Iyengar, and H. Krause, Local cohomology and support for triangulated categories, Annales Scientifiques de l’Ecole Normale Supérieur, to appear.

6. J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149–170.