Projective Resolutions and Poincaré Duality Complexes.

D. J. Benson Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain

Jon F. Carlson^{∗} Dept. of Mathematics University of Georgia Athens, GA30602 U.S.A.

1 Introduction

For a finite dimensional algebra A, any finitely generated module is built from simple modules by a process of extensions. The process is regulated by a set of rules which are peculiar to the algebra. In general the rules can be very subtle and difficult to express in simple terms. Yet, they are recorded completely in any of several structures associated to the algebra and its module category. Three such structures are the form and constitution of the projective modules, the minimal projective resolutions of the simple modules, and the collection of extension groups Ext_{A}^{n} between simple modules, together with the (Yoneda and Massey) product information. Of course, there may be difficulties translating from one structure to another. It may not be easy to construct projective resolutions even with full knowledge of the projective modules. Each of the structures has both advantages and disadvantages with respect to problems such as computability and interpretation of information. However, it is certainly true that each of these structures, if understood thoroughly, would provide a complete picture of the module theory for the algebra.

Each of the structures mentioned above has been studied extensively for many individual algebras and classes of algebras. We are particularly interested in group algebras of finite groups. In this case, the amount of information needed to generate the structures is finite in the following sense. If k is a field of characteristic p and G is a finite group then a theorem of Evens [13] says that the cohomology ring H^{∗}(G, k) is a finitely generated k-algebra. Also if M is a finitely generated kG-module then H^{∗}(G, M) is finitely generated as a module over H^{∗}(G, k). From this it can be deduced that the minimal projective resolution of M has a polynomial growth rate. But what do the minimal resolutions look like? How much information is really needed to generate them?

∗Partially supported by a grant from the NSF.

Although the subject of this paper is the cohomology rings of finite groups, the roots and motivation lie firmly within the area of representation theory, and in particular the attempt to understand minimal resolutions. The study grew out of the authors’ work in [3] and [4], where the idea of using a homogeneous set of parameters to define filtrations on projective resolutions was first developed. In [4] we showed any module has a projective resolution which has the minimal polynomial growth rate, and which is formed by repeatedly splicing together certain finite complexes which are defined by the parameters. However, in general the resulting resolutions are not minimal, and we are left with the problem of understanding to what extent projectives may be stripped away to yield a smaller resolution. In Section 9 of [3], we answered this question for groups of p-rank two, provided certain conditions on the chain maps induced by the parameters are satisfied.

The primary purpose is to set up a framework for dealing with this problem in general. At present, we should honestly state that we open up more questions than we answer, but the theory shows some promise in that several extensions of the results have been obtained since the original manuscript for this paper was written. The study revolves around the investigation of finite complexes associated to systems of parameters in the cohomology ring. Each complex consists only of finitely generated projective modules, and is zero in all but a finite number of degrees. The complexes satisfy a Poincaré duality, and seem to encode an enormous amount of information about minimal resolutions, and in particular about the behaviour of the cohomology ring H^{∗}(G, k). We see, for example, that most finitely generated k-algebras cannot possibly be cohomology rings of finite groups. Although we work largely with coefficients in a field of characteristic p, the reader should notice that, with some modification, many of the techniques and results are applicable to the more general situation of cohomology with other coefficients.

After some generalities on chain complexes in Section 2, we introduce the finite complex associated to a system of parameters in Section 3. Using hypercohomology, we give a new proof in Section 4 of the basic result of [4], which states that the complexes consist of projective modules. Each complex looks homologically like a product of spheres with a free G-action. This analogy is taken further in Section 5, where we prove the Poincaré duality with respect to the top homology class. We pass from information about the finite complex to information about minimal resolutions and cohomology by using the hypercohomology spectral sequence. This is explained in Section 4, where we also determine enough about the differentials in the sequence to begin to understand what is happening (Theorem 5.5).

There is one case in which we do obtain very striking and easily statable results. This is for groups whose cohomology rings are Cohen–Macaulay. Section 6 is devoted to this case, and among other things we prove the following.

Theorem 1.1 Suppose that H^{∗}(G, k) is a Cohen–Macaulay ring. Let r be the p-rank of G, which by a theorem of Quillen [21] is equal to the Krull dimension of H^{∗}(G, k). So we may choose a homogeneous set of parameters ζ_{1}, …, ζ_{r} (i.e., H^{∗}(G, k) is a finitely generated algebra over the polynomial subring k[ζ_{1}, …, ζ_{r}]) with deg(ζ_{i}) ≥ 2. Then the quotient

H^{∗}(G, k)/(ζ_{1}, …, ζ_{r})

satisfies Poincaré duality in formal dimension s = ∑_{i=1}^{r}(deg(ζ_{i}) − 1). Moreover, the Poincaré series P_{k}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(G, k), regarded as a rational function of t, satisfies the functional equation

P_{k}(1/t) = (−t)^{r_{p}(G)}P_{k}(t).

One interpretation of this theorem is that the minimal projective resolution for the trivial module can be constructed by splicing together copies of the finite complex mentioned above.

Groups with abelian Sylow p-subgroups have cohomology rings which are Cohen– Macaulay. It also happens for some other groups. However it is important to realise that this is far from the generic case. Consider, for example, the following.

Proposition 1.2 If a finite group G has maximal elementary abelian p-groups of different ranks, then H^{∗}(G, k) is not Cohen–Macaulay.

Hence, in the defining characteristic, the cohomology ring of a finite group of Lie type is usually not Cohen–Macaulay. The proof of the proposition follows easily from Quillen’s Dimension Theorem [21], which asserts that the irreducible components of the maximal ideal spectrum of H^{∗}(G, k) are in one–one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups of G, and that the dimension of each component is equal to the rank of the corresponding subgroup. The proof is completed by recalling that for a ring to be Cohen–Macaulay it is necessary for the components of the maximal ideal spectrum all to have the same dimension (see for example Matsumura [17], Theorem 17.3). We should notice further that the converse of the proposition is false. Counterexamples include the semidihedral 2-groups (p = 2), and the extraspecial groups of order p^{3} and exponent p^{2} (for p odd).

Another interpretation of Theorem 1.1 is that if H^{∗}(G, k) is a Cohen–Macaulay ring, then it is a Gorenstein ring (see Stanley [27], Theorem 5.5). This is true not just in the usual (ungraded) sense, but in the sense that with the appropriate degree conventions, the canonical module is isomorphic to the ring as a graded module. Thus, for example, a polynomial ring is only Gorenstein in the graded sense if the generators are in degree one (cf. Corollary 6.6).

In the general case, where H^{∗}(G, k) is not necessarily Cohen–Macaulay, we can use the fact that the cohomology of an elementary abelian subgroup is Cohen– Macaulay to show that at least some part of the finite complex is visible in the cohomology of the group; namely the element of largest dimension. We call this element the last survivor. In Section 7, we prove the following theorem.

Theorem 1.3 If ζ_{1}, …, ζ_{r} is an irredundent homogeneous set of parameters for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2, then the quotient H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}) by the ideal generated by ζ_{1}, …, ζ_{r} has a non-zero element in degree ∑_{i=1}^{r}(n_{i} − 1). This element is a transfer from an elementary abelian subgroup.

To return to the representation theory, in Section 8 we show how to pass from a complex of projectives to a filtration on a projective module. In the case of the complexes introduced in Section 4, the E_{2} term of the spectral sequence associated to this filtration is the Koszul complex. A distillation of the philosophy of the paper is expressed in this section.

Section 9 discusses some structural questions suggested by our constructions. We discuss to what extent projective modules may be eliminated from one version of the complexes described in Section 4. If the questions have a positive answer, then the Poincaré series of the cohomology ring is given by a formula described at the end of the section. There is one situation in which these questions have a positive answer, which is formulated in the notion of a quasi-regular sequence in Section 10.

Finally, in Section 11 we show how certain secondary operations expressed in terms of matric Massey products are related to the differentials in the spectral sequence of Section 4.

2 Chain complexes and hypercohomology

In this section we develop some generalities concerning chain complexes over finite groups. Most of the results in the section are known in some sense, and those that aren’t can be easily derived. However the notation and statements of the results are very important for the sections which follow.

Suppose R is a commutative ring of coefficients and C and D are chain complexes of left R-modules, with differentials of degree −1. We define a new chain complex Hom_{R}(C, D) with

Hom_{R}(C, D)_{n} = ⨁_{i+n=j} Hom_{R}(C_{i}, D_{j})

with differential

∂_{n} : Hom_{R}(C, D)_{n} → Hom_{R}(C, D)_{n−1}

defined so that

∂_{j}(f(x)) = (∂_{n}f)(x) + (−1)^{n}f(∂_{i}(x))

for f ∈ Hom_{R}(C_{i}, D_{j}). In other words, ∂_{n} is defined by

(∂_{n}f)(x) = ∂_{j}(f(x)) − (−1)^{n}f(∂_{i}(x)).

This may be expressed by writing ∂f = [∂, f]. With this definition, an element f ∈ Hom_{R}(C, D)_{0} is a map of chain complexes if and only if f is a cycle (∂f = 0). Moreover, f and g are homotopic if and only if f − g is a boundary.

If M is a left R-module, we write M[n] for the chain complex consisting of M in degree n and zero elsewhere. We also write M for the complex M[0] consisting of M in degree 0 and zero elsewhere. If C is a chain complex of left R-modules, we write C[n] for R[n] ⊗_{R} C. Namely we have (C[n])_{i+n} = (C)_{i}. Note that the differential in C[n] is (−1)^{n} times the differential in C.

The dual of C is the chain complex Hom_{R}(C, R). Note that the differential on the dual is given by (∂_{n}f)(x) = (−1)^{n−1}f(∂_{−n+1}(x)). With these sign conventions, evaluation is a map of chain complexes

Hom_{R}(C, R) ⊗_{R} C → R
f ⊗ x ↦ f(x).

Finally, we regard chain complexes and cochain complexes as being the same thing. Namely, if (C_{n}, ∂_{n}) is a chain complex, then setting C^{n} = C_{−n} and δ^{n} = ∂_{−n}, we have a cochain complex (C^{n}, δ^{n}). In the end, whether we regard a particular complex as a chain complex or a cochain complex often depends on where it came from.

We now discuss Ext for chain complexes. This is sometimes also called hypercohomology, see for example Cartan and Eilenberg [9].

Definition 2.1 Suppose that C is a chain complex of left RG-modules, bounded below. Then a projective resolution of C is a chain complex P of projective left RG-modules, bounded below, together with a map of chain complexes P → C which is an isomorphism on homology.

Note that in the case C = M is a module concentrated in degree zero, this agrees with the usual definition of a projective resolution of a module. Existence of projective resolutions is easy to prove inductively using the definition of projective modules. Alternatively, one may tensor C with a projective resolution of the trivial RG-module R. The usual comparison theorem holds for projective resolutions of chain complexes. If R is a field (or more generally a local ring) then minimal resolutions exist and are unique, in the following sense. The resolution, as a complex of RG-modules, is uniquely determined, but the augmentation map P → C is only unique up to homotopy. Minimal resolutions are characterised by the property that the image of each term is contained in the radical of the next.

If D is another chain complex of left RG-modules, bounded above, we may define

Ext_{RG}^{n}(C, D) = H^{n}(Hom_{RG}(P, D)).

(Note that we are regarding this chain complex as a cochain complex by negating the degrees.) If I is an injective resolution of D (defined dually to the above definition) then

Hom_{R}(P, D) → Hom_{R}(P, I) ← Hom_{R}(C, I)

are homotopy equivalences, and so

Ext_{RG}^{n}(C, D) ≅ H^{n}(Hom_{RG}(C, I)).

We write H^{n}(G, C) for Ext_{RG}^{n}(R, C). Note that

Ext_{RG}^{n}(C, D) ≅ H^{n}(G, Hom_{R}(C, D)).

If D is bounded (i.e., bounded both above and below), then maps from P to D may be lifted to maps to a projective resolution of D. So if E is bounded above, then composition of maps in Hom gives rise to Yoneda composition

Ext_{RG}^{∗}(D, E) ⊗ Ext_{RG}^{∗}(C, D) → Ext_{RG}^{∗}(C, E),

which agrees with the usual Yoneda composition in case C, D and E are modules concentrated in degree zero. Indeed, this was part of the original motivation for the definition of the derived category, which is really what we are using in disguise (see [11], Appendix 1).

The tensor product map

Hom_{RG}(C, D) ⊗ Hom_{RG}(C′, D′) → Hom_{RG}(C ⊗ C′, D ⊗ D′)

gives rise to a cup product map

Ext_{RG}^{∗}(C, D) ⊗ Ext_{RG}^{∗}(C′, D′) → Ext_{RG}^{∗}(C ⊗ C′, D ⊗ D′)
ζ ⊗ η ↦ ζ ∪ η

For convenience of notation, we sometimes also denote the cup product by juxtaposition.

Lemma 2.2 If ζ ∈ Ext_{RG}^{m}(C, D), η ∈ Ext_{RG}^{n}(C′, D′) then the cup product

ζ ∪ η ∈ Ext_{RG}^{m+n}(C ⊗ C′, D ⊗ D′)

is equal to the Yoneda composite of

ζ ⊗ id_{D′} ∈ Ext_{RG}^{m}(C ⊗ D′, D ⊗ D′)

and

id_{C} ⊗ η ∈ Ext_{RG}^{n}(C ⊗ C′, C ⊗ D′).

Proof This follows easily from the corresponding (obvious) statement at the level of Hom.

Since tensor products are graded commutative, so are cup products, in the sense that the following diagram commutes.

Ext_{RG}^{m}(C, D) ⊗ Ext_{RG}^{n}(C′, D′) → Ext_{RG}^{m+n}(C ⊗ C′, D ⊗ D′)
↓≅  ↓≅
Ext_{RG}^{n}(C′, D′) ⊗ Ext_{RG}^{m}(C, D) → Ext_{RG}^{m+n}(C′ ⊗ C, D′ ⊗ D)

But in general Yoneda products are not graded commutative. So Ext_{RG}^{∗}(C, C) is in general a non-commutative graded ring and a module over the graded commutative ring H^{∗}(G, R) = Ext_{RG}^{∗}(R, R).

Regarding Hom_{RG}(C, I) as a double complex, we obtain a spectral sequence, called the hypercohomology spectral sequence

E_{2}^{pq} = Ext_{RG}^{p}(H_{q}(C), D) ⇒ Ext_{RG}^{p+q}(C, D). (2.3)

Similarly, regarding Hom_{RG}(P, D) as a double complex, we have a spectral sequence

E_{2}^{pq} = Ext_{RG}^{p}(C, H_{−q}(D)) ⇒ Ext_{RG}^{p+q}(C, D).

Theorem 2.4 Suppose that C and D are chain complexes of left RG-modules. If Hom_{R}(C, D) is a Noetherian R-module (and hence non-zero in only finitely many degrees) then Ext_{RG}^{∗}(C, D) is Noetherian as a module over H^{∗}(G, R).

Proof The proof given in Evens [13] for modules generalises directly to chain complexes.

Corollary 2.5 If R is a commutative Noetherian ring, and C and D are complexes of left RG-modules, each finitely generated as an R-module, then the ring Ext_{RG}^{∗}(C, C) is a finitely generated R-algebra, and Ext_{RG}^{∗}(C, D) is a finitely generated Ext_{RG}^{∗}(C, C)-module.

Proof It is a finitely generated module over the finitely generated R-algebra H^{∗}(G, R), and the products are compatible by Lemma 2.2.

Proposition 2.6 Suppose that k is a field and C has finite total dimension over k. If Ext_{kG}^{n}(C, C) = 0 for all n sufficiently large, then C has a finite projective resolution.

Proof If D is another complex of finite total dimension over k, then by Lemma 2.2 the action of H^{∗}(G, k) on Ext_{kG}^{∗}(C, D) factors as the map

H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) → Ext_{kG}^{∗}(C, C)

followed by Yoneda composition. Since Ext_{kG}^{∗}(C, D) is finitely generated as a module over H^{∗}(G, k) by Theorem 2.4, it is finitely generated as a module over Ext_{kG}^{∗}(C, C). The latter has finite total dimension by the hypothesis, and so Ext_{kG}^{n}(C, D) = 0 for all n sufficiently large.

Now let P be the minimal resolution of C. Then for each simple kG-module S,

Ext_{kG}^{n}(C, S) = Hom_{kG}(P_{n}, S)

is non-zero for only finitely many n. It follows that P_{n} = 0 for all but finitely many n.

3 From parameters to complexes

In [4], we introduced a construction for projective resolutions using as the basic building block a certain finite complex of projective modules. These complexes and others, which can be similarly derived, are the main focus of the paper. We present here a summary of the construction and introduce some ideas and notations which are needed later. The proof we gave that the modules in the complexes are projective used the machinery of varieties for modules, and in particular depended on theorems of Quillen [21]. In Section 4 we give an alternative proof avoiding this machinery.

The construction goes as follows. Suppose ζ ∈ H^{n}(G, R) ≅ Ext_{RG}^{n}(R, R) with n ≥ 2. We choose a cocycle ζ̂ : Ω̃^{n}R → R representing ζ, where Ω̃^{n}R is the nth kernel in a projective resolution P of R as an RG-module. By making P large enough, we may assume ζ̂ is surjective. We denote its kernel by L_{ζ}, and form the pushout diagram

0  0
↓  ↓
L_{ζ} =  L_{ζ}
↓  ↓
0 → Ω̃^{n}R → P_{n−1} → P_{n−2} → ⋯ → P_{0} → R → 0
↓ζ̂  ↓  ↓  ↓  ∥
0 → R → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{0} → R → 0.
↓  ↓
0  0

The bottom row of this diagram is an n-fold extension representing the element ζ ∈ Ext_{RG}^{n}(R, R). We denote by C_{ζ} the chain complex

0 → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{0} → 0

formed by truncating the bottom row of this diagram. Thus we have

H_{i}(C_{ζ}) ≅ \{ \begin{matrix}R & if i = 0, n − 1 \\ 0 & otherwise. \end{matrix}

We write ζ̃ for the generator of degree n − 1, and 1 for the generator of degree zero.

The reason why we demand that deg(ζ) ≥ 2 is that if deg(ζ) = 1 then C_{ζ} is a complex consisting of a single module in degree zero, which is the extension of R by R corresponding to ζ.

The complex C_{ζ} should be thought of as a sort of algebraic analogue of a sphere with G-action, with ζ being the transgression of the fundamental class of the sphere.

We also write C_{ζ}^{(∞)} for the chain complex

⋯ → P_{1} → P_{0} → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{1} → P_{0} → 0

obtained by splicing together infinitely many copies of C_{ζ} in positive degree. It is an exact complex except in degree zero, where the homology is R. This is the complex used in [4].

Lemma 3.1 Suppose that D is a bounded chain complex of RG-modules. In the two row spectral sequence

E_{2}^{pq}(ζ) = Ext_{RG}^{p}(H_{q}(C_{ζ}), D) ⇒ Ext_{RG}^{p+q}(C_{ζ}, D)

the differential d_{n} is given by

d_{n}(α.ζ̃) = α.ζ ∈ E_{n}^{p+n,0}(ζ) = H^{p+n}(G, D).

Proof We first remark that since H_{q}(C_{ζ}) = R if q = 0 or n−1 and is zero elsewhere, we have

E_{2}^{pq}(ζ) = \{ \begin{matrix}H^{p}(G, D) & if q = 0 or n − 1 \\ 0 & otherwise. \end{matrix}

So the only possible non-zero differential is d_{n}.

The differential d_{n} may be obtained as follows. We have a short exact sequence of chain complexes

0 → C_{ζ} → C_{ζ}^{(∞)} → C_{ζ}^{(∞)}[n] → 0,

where [n] indicates degree shift. If P is a projective resolution of R as an RG-module then we have a short exact sequence

0 → P ⊗ C_{ζ} → P ⊗ C_{ζ}^{(∞)} → P ⊗ C_{ζ}^{(∞)}[n] → 0

and hence a short exact sequence of cochain complexes

0→Hom_{RG}(P⊗C_{ζ}^{(∞)}, D)[−n] → Hom_{RG}(P⊗C_{ζ}^{(∞)}, D) → Hom_{RG}(P⊗C_{ζ}, D)→0.

Now P ⊗ C_{ζ}^{(∞)} is again a projective resolution of R as an RG-module, so the long exact sequence in cohomology of this short exact sequence of cochain complexes is

⋯ → Ext_{RG}^{r−1}(C_{ζ}, D) → H^{r−n}(G, D) →^{d_{n}} H^{r}(G, D) → Ext_{RG}^{r}(C_{ζ}, D) → ⋯

This is the long exact sequence associated to the above two-row spectral sequence, and so the marked homomorphism is d_{n}.

To identify this map as multiplication by ζ, we argue as follows. In general, multiplication by ζ may be thought of in the following way. A cocycle representing the element ζ lifts to a map of projective resolutions P → P[n]. If α ∈ H^{r−n}(G, D) is represented by a chain map P → D[r − n] then the product α.ζ is represented by shifting and composing

P → P[n] → D[r].

So all we need check is that the map of resolutions

P ⊗ C_{ζ}^{(∞)} → P ⊗ C_{ζ}^{(∞)}[n]

given above represents the element ζ in cohomology. But this follows from the diagram

⋯ → (P ⊗ C_{ζ}^{(∞)})_{n} → (P ⊗ C_{ζ}^{(∞)})_{n−1} → ⋯ → (P ⊗ C_{ζ}^{(∞)})_{0} → R → 0
↓
⋯ → (P ⊗ C_{ζ}^{(∞)})_{0} ↓ ↓ ↓
↓
0 →  R  → P_{n−1}/L_{ζ} → ⋯ →  P_{0}  → R → 0

since the bottom row is an exact sequence representing ζ.

Remark 3.2 There is a way of rewriting the above two-row spectral sequence in such a way that the two rows are adjacent and the differentials are unchanged. Namely, we replace the complex C_{ζ} by the truncated complex

B_{ζ} : 0 → P_{n−1}/L_{ζ} → P_{n−2} → 0

whose homology is k in degree one and Ω^{n−2}k in degree zero.

If ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, R) are homogeneous elements of degree ≥ 2, then we form the complex C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}}. The Künneth formula shows that H_{∗}(C) has a basis consisting of elements of the form x_{1} ⊗ ⋯ ⊗ x_{r} where each x_{i} is equal to either 1 or ζ̃_{i}. We also write ζ̃_{i} for the element 1 ⊗ ⋯ ⊗ ζ̃_{i} ⊗ ⋯ ⊗ 1 of H_{∗}(C). The complex C and the related complex B = B_{ζ_{1}} ⊗ ⋯ ⊗ B_{ζ_{r}} (see Section 8) are among the principal objects of study of this paper.

We are particularly interested in this construction in the following situation. If R is Noetherian and D is a bounded chain complex of finitely generated RG-modules, then by Theorem 2.4, H^{∗}(G, D) is a Noetherian module for H^{∗}(G, R). We say that elements ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, R) form a homogeneous set of parameters (h.s.o.p.) for H^{∗}(G, D) if H^{∗}(G, D) is Noetherian as a module for the subring of H^{∗}(G, R) generated by ζ_{1}, …, ζ_{r}. In case D = R, we simply speak of an h.s.o.p. for H^{∗}(G, R).

In case R = k is a field, the number r of parameters is at least the Krull dimension of the H^{∗}(G, k)-module H^{∗}(G, D). Equality holds if and only if ζ_{1}, …, ζ_{r} generate a polynomial subring k[ζ_{1}, …, ζ_{r}] of the quotient of H^{∗}(G, k) by the annihilator of H^{∗}(G, D). In this case, we say that ζ_{1}, …, ζ_{r} form an irredundent h.s.o.p. The Noether normalisation lemma for Noetherian graded modules over a finitely generated commutative k-algebra says that an irredundent h.s.o.p. always exists. Geometrically, it is the same as saying that the homogeneous hypersurfaces V_{G}(ζ_{i}) defined by the ζ_{i} in the maximal ideal spectrum V_{G} = maxH^{even}(G, k) intersect in the origin, and the number of hypersurfaces is exactly equal to the dimension of V_{G}.

The reader should beware at this stage that most texts on commutative algebra assume strict commutativity xy = yx. In our situation we have the sign conventions xy = (−1)^{deg(x) deg(y)}yx. One can check that the usual theorems of commutative algebra are also true in this situation after trivial alterations to the proofs.

4 The hypercohomology spectral sequence

Let ζ_{1}, …, ζ_{r} be an h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2. We assume for convenience that the n_{i} are even if p is odd, since in this case elements of odd degree square to zero. We set C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} as in the last section.

Suppose that D is a bounded chain complex of finitely generated kG-modules. Then from (2.3) we have a hypercohomology spectral sequence

E_{2}^{pq} = Ext_{kG}^{p}(H_{q}(C), D) ⇒ Ext_{kG}^{p+q}(C, D).

In this spectral sequence, the E_{2} page is a tensor product

E_{2}^{pq} ≅ H^{p}(G, D) ⊗ H^{q}(Hom_{k}(C, k))

and H^{∗}(Hom_{k}(C, k)) is an exterior algebra Λ^{∗}(ζ̃_{1}, …, ζ̃_{r}) on classes ζ̃_{i} of degree n_{i} − 1.

It should be remarked that in many cases the spectral sequence admits a product structure. Suppose that the complex C admits a diagonal approximation C → C ⊗ C. There is always a diagonal approximation on a projective resolution of k. Hence the spectral sequence with D = k (concentrated in degree zero)

H^{p}(G, k) ⊗ H^{q}(Hom_{k}(C, k)) ⇒ H^{p+q}(Hom_{kG}(C, k))

has a ring structure over which the spectral sequence

H^{p}(G, D) ⊗ H^{q}(Hom_{k}(C, k)) ⇒ H^{p+q}(Hom_{kG}(C, D))

is a module.

In Section 5 it is shown that C satisfies Poincaré duality of formal dimension s = ∑_{i=1}^{r}(n_{i} −1), so that instead of using C_{ζ_{i}} we may use the dual Hom_{k}(C_{ζ_{i}}, k)[n_{i} −1]. This turns out to be more convenient from the point of view of the diagonal approximation. This is because while C_{ζ_{i}} usually does not admit a diagonal approximation, it was shown in [7] that Hom_{k}(C_{ζ_{i}}, k)[n_{i} − 1] admits a diagonal approximation for which the augmentation is a counit if and only if ζ_{i} annihilates Ext_{kG}^{∗}(L_{ζ_{i}}, L_{ζ_{i}}). By tensoring and using Poincaré duality, we obtain a diagonal approximation C → C ⊗ C. It is also shown in [7] that as long as p is odd, ζ_{i} always annihilates Ext_{kG}^{∗}(L_{ζ_{i}}, L_{ζ_{i}}), so that the above spectral sequences have a multiplicative structure. For p = 2, the situation is less clear.

Now we can use the maps of spectral sequences

E_{2}^{pq}(ζ_{i}) = Ext_{kG}^{p}(H_{q}(C_{ζ_{i}}), k) ⇒ Ext_{kG}^{p+q}(C_{ζ_{i}}, k)
↓  ↓
E_{2}^{pq} = Ext_{kG}^{p}(H_{q}(C), k) ⇒ Ext_{kG}^{p+q}(C, k)

to see, using Lemma 3.1, that d_{n_{i}}(ζ̃_{i}) = ζ_{i} in E_{n_{i}}^{∗∗}. In the case where the spectral sequence has a multiplicative structure, this determines the differentials on all elements of E_{∗}^{0∗}, and shows that the spectral sequence is a sort of staggered version of the Koszul complex, together with possibly some further differentials. See Serre [25] Chapter IV, Matsumura [17] §16, or Carlson [8] Section 2 for discussions of the Koszul complex. We may make this into an actual Koszul complex without any staggering by using Remark 3.2. Namely, instead of C we use the complex B = B_{ζ_{1}} ⊗ ⋯ ⊗ B_{ζ_{r}}. In this case, we have ζ̃_{i} ∈ E_{2}^{∗,1} and d_{2}(ζ̃_{i}) = ζ_{i} ∈ E_{2}^{∗,0}, so that in this case

E_{3}^{∗,j} = Tor_{j}^{R[ζ_{1},...,ζ_{r}]}(R, H^{∗}(G, R)).

In particular, if H^{∗}(G, R)/(ζ_{1}, …, ζ_{r}) is a finitely generated R-module then E_{3}^{∗∗} is also a finitely generated R-module and hence so also is E_{∞}^{∗∗}. See Section 8 for more details.

Even when a spectral sequence such as given above does not have a multiplicative structure, it still has the structure of an H^{∗}(G, k)-module, and much of the same information can be obtained about the differentials by considering maps of spectral sequences as follows. Suppose as above that D is a bounded chain complex of RG-modules. Then we have maps of spectral sequences

E_{2}^{pq} =  Ext_{kG}^{p}(H_{q}(C), D)  ⇒  Ext_{kG}^{p+q}(C, D)
↑  ↑
E_{2}^{pq}(ζ_{i_{1}}, …, ζ_{i_{t}}) =Ext_{kG}^{p}(H_{q}(C_{ζ_{i_{1}}} ⊗ ⋯ ⊗ C_{ζ_{i_{t}}}), D)⇒Ext_{kG}^{p+q}(C_{ζ_{i_{1}}} ⊗ ⋯ ⊗ C_{ζ_{i_{t}}}, D)
↓  ↓
E_{2}^{p,q−ℓ}(ζ_{i_{t}}) =  Ext_{kG}^{p}(H_{q−ℓ}(C_{ζ_{i_{t}}}), D)  ⇒  Ext_{kG}^{p+q}(C_{ζ_{i_{t}}}, D)

(where ℓ = ∑_{j=1}^{t−1}(n_{i_{j}} − 1)). They are derived from the maps of complexes

C → C_{ζ_{i_{1}}} ⊗ ⋯ ⊗ C_{ζ_{i_{t}}} ← C_{ζ_{i_{t}}}[ℓ].

We deduce from this that if α.ζ̃_{i_{1}} … ζ̃_{i_{t}} survives to E_{n_{t}} then the coefficient of ζ̃_{i_{1}} … ζ̃_{i_{t−1}} in d_{n_{t}}(α.ζ̃_{i_{1}} … ζ̃_{i_{t}}) is αζ_{i_{t}}, with the appropriate sign, as required.

Theorem 4.1 Suppose that R is a field, or the ring of integers in an algebraic number field, or one of its localisations or completions. If ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, R) with deg(ζ_{i}) = n_{i} ≥ 2, then the tensor product

C = ⨂_{i=1}^{r} C_{ζ_{i}}

of the above complexes is a bounded complex of projective modules.

Proof Since the tensor product of any module with a projective module is projective, all the modules in C except possibly ⨂_{i=1}^{r} P_{n_{i}−1}/L_{ζ_{i}} = C_{s}, s = ∑_{i=1}^{r}(n_{i} − 1), are projective. The latter is at least projective as an R-module. By [10] Theorem 78.1, it suffices to prove the theorem in case R = k is a field. Since kG is self-injective, it is enough to demonstrate that C has a finite projective resolution.

So by Proposition 2.6 it suffices to prove that Ext_{kG}^{n}(C, C) = 0 for n sufficiently large.

We start by showing that Ext_{kG}^{n}(C, k) = 0 for n sufficiently large. Consider the spectral sequence

E_{2}^{pq} = Ext_{kG}^{p}(H_{q}(C), k) ⇒ Ext_{kG}^{p+q}(C, k).

The map C → C_{ζ_{i}} given by using the augmentation C_{ζ_{j}} → k for j ≠ i gives rise to a map of spectral sequences to the above spectral sequence from

E_{2}^{pq}(ζ_{i}) = Ext_{kG}^{p}(H_{q}(C_{ζ_{i}}), k) ⇒ Ext_{kG}^{p+q}(C_{ζ_{i}}, k).

The latter spectral sequence was examined in Lemma 3.1, where it was shown that the differential d_{n_{i}} is given by

d_{n_{i}}(α.ζ̃_{i}) = α.ζ_{i}.

It follows that the same formula holds in the original spectral sequence E_{j}^{pq}.

Now E_{2}^{∗∗} is finitely generated as a module over E_{2}^{∗0} ≅ H^{∗}(G, k) by basis elements in E_{2}^{0∗} (which is dual to H_{∗}(C)). Since the differentials are E_{2}^{∗0}-module homomorphisms, it follows that E_{∞}^{∗∗} is also finitely generated as a module over E_{2}^{∗0}. The discussion preceding the theorem shows that ζ_{1}, …, ζ_{r} act as zero on E_{∞}^{∗∗}, and so E_{∞}^{∗∗} is finitely generated as a module over H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}). Since H^{∗}(G, k) is finitely generated as a module over the subring generated by ζ_{1}, …, ζ_{r}, the quotient by the ideal generated by ζ_{1}, …, ζ_{r} is finite dimensional. It follows that E_{∞}^{∗∗} is finite dimensional, and therefore so is Ext_{kG}^{∗}(C, k).

Finally, the spectral sequence

Ext_{kG}^{p}(C, H_{−q}(C)) ⇒ Ext_{kG}^{p+q}(C, C)

has as its E_{2} term a finite number of non-zero rows each isomorphic to Ext_{kG}^{∗}(C, k). Thus the E_{2} term is finite dimensional, and so is Ext_{kG}^{∗}(C, C). This completes the proof.

More generally, we have the following.

Theorem 4.2 Suppose that R is a field, or the ring of integers in an algebraic number field, or one of its localisations or completions. Suppose that D is a bounded complex of finitely generated RG-modules, and ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, R) form an h.s.o.p. for Ext_{RG}^{∗}(D, D) with deg(ζ_{i}) = n_{i} ≥ 2. Then letting C = ⨂_{i=1}^{r} C_{ζ_{i}}, the complex C ⊗ D is homotopy equivalent to a bounded complex of finitely generated projective modules. In particular, if D = M[n] is a module concentrated in a single degree, then C ⊗ D is a bounded complex of finitely generated projective modules.

Proof This theorem is proved in the same way as before. That is, we get it by reducing to the case where R = k is a field, and using the spectral sequences

Ext_{kG}^{p}(H_{q}(C), D^{∗} ⊗ D) ⇒ Ext_{kG}^{p+q}(C, D^{∗} ⊗ D)
∥≀  ∥≀
Ext_{kG}^{p}(H_{q}(C) ⊗ D, D) ⇒ Ext_{kG}^{p+q}(C ⊗ D, D)

and

Ext_{kG}^{p}(C ⊗ D ⊗ D^{∗}, H_{−q}(C)) ⇒ Ext_{kG}^{p+q}(C ⊗ D ⊗ D^{∗}, C)
∥≀  ∥≀
Ext_{kG}^{p}(C ⊗ D, H_{−q}(C) ⊗ D) ⇒ Ext_{kG}^{p+q}(C ⊗ D, C ⊗ D).

The following is in a similar vein, and will be used in Section 10.

Theorem 4.3 Suppose that k is a field, D is a bounded complex of finitely generated kG-modules, and ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, D) with deg(ζ_{i}) = n_{i} ≥ 2. Then letting C = ⨂_{i=1}^{r} C_{ζ_{i}}, we have

H^{∗}(G, L_{1} ⊗ ⋯ ⊗ L_{r} ⊗ D) = Ext_{kG}^{∗}(C, D) = 0,

where L_{i} is the cokernel of the injective map Ω^{−n_{i}}(ζ̂_{i}) : k → Ω^{−n_{i}}(k).

Proof It follows as in the last two theorems from the spectral sequence

Ext_{kG}^{p}(H_{q}(C), D) ⇒ Ext_{kG}^{p+q}(C, D)

that Ext_{kG}^{∗}(C, D) is non-zero in only finitely many degrees. Therefore, as in [5] Theorem 1.1, Ext_{kG}^{∗}(C, D) = 0. Since Ω(L_{i}) is the only non-projective module in the complex C_{ζ_{i}}^{∗}, and H^{∗}(G, C^{∗} ⊗D) = Ext_{kG}^{∗}(C, D) = 0, it follows that H^{∗}(G, L_{1} ⊗ ⋯ ⊗ L_{r} ⊗ D) = 0.

5 Poincaré duality

In this section, it is shown that if ζ_{1}, …, ζ_{r} is a homogeneous set of parameters for H^{∗}(G, k), then C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} is homotopy equivalent to its dual, suitably shifted in degree. We use the notation of Section 3. However, throughout this section the coefficient ring R = k is a field of characterestic p, and hence we can assume that the projective resolution of k as a kG-module is minimal. This implies that the kernels Ω^{n}(k) have no projective submodules.

We begin with a general lemma about homotopy equivalences.

Lemma 5.1 Suppose C and D are bounded chain complexes of finitely generated projective kG-modules, and f : C → D is a chain map. Then the following are equivalent.

(i) f : C → D induces an isomorphism in homology f_{∗} : H_{∗}(C) →^{≅} H_{∗}(D).

(ii) f is a homotopy equivalence.

(iii) There exist decompositions C = C′ ⊕ P′, D = D′ ⊕ Q′, where P′ and Q′ are exact sequences of projective modules, and the restriction of f to C′ is an isomorphism f : C′ →^{≅} D′.

Proof It is clear that (iii) ⇒ (ii) ⇒ (i), so we shall prove that (i) ⇒ (iii). Suppose f : C → D induces an isomorphism in homology. By adding an exact sequence of projective modules Q to C we may make f surjective, and still a homology isomorphism. Denote by P the kernel of f : C ⊕ Q → → D. The long exact sequence in homology shows that P is an exact sequence of projectives. Since projective kGmodules are also injective, P is a bounded exact sequence of injectives, and hence injective as a complex. So the sequence

0 → P → C ⊕ Q → D → 0

splits, and C⊕Q ≅ D⊕P. The result now follows from the Krull–Schmidt theorem for finite chain complexes of finitely generated kG-modules.

Now if ζ ∈ H^{n}(G, k) is represented by a cocycle ζ̂ : Ω^{n}k → k, we may dualise to obtain a map ζ̂^{∗} : k → Ω^{−n}k. Applying Ω^{n}, we get a map Ω^{n}(ζ̂^{∗}) : Ω^{n}k → k. The relationship between this homomorphism and the original map ζ̂ is given in the following proposition.

Proposition 5.2 If ζ ∈ H^{n}(G, k) then

Ω^{n}(ζ̂^{∗}) = (−1)^{n(n+1)/2}ζ̂.

Moreover there is a map of chain complexes

0 → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{1} →  P_{0}  → 0
↓  ↓  ↓  ↓
0 →  P_{0}^{∗}  → P_{1}^{∗} → ⋯ → P_{n−2}^{∗} → (P_{n−1}/L_{ζ})^{∗} → 0

inducing an isomorphism on homology.

Proof We build a commutative diagram

0 →  Ω^{n}k  →  P_{n−1}  → ⋯ →  P_{0}  → k → 0
↓ζ̂^{∗}⊗1  ↓ζ̂^{∗}⊗1  ↓ζ̂^{∗}⊗1  ↓ζ̂^{∗}
0 → Ω^{−n}k ⊗ Ω^{n}k → Ω^{−n}k ⊗ P_{n−1} → ⋯ → Ω^{−n}k ⊗ P_{0} → Ω^{−n}k → 0
↓γ  ↓  ↓  ∥
0 →  k  →  P_{0}^{∗}  → ⋯ →  P_{n−1}^{∗}  → Ω^{−n}k → 0.

In this diagram, the bottom set of vertical arrows has been filled in using the fact that both rows are exact sequences of projective modules, except at the ends, and γ is obtained by restricting the previous map. Now modulo maps which factor through a projective module, there is only one dimension of maps from Ω^{−n}k ⊗ Ω^{n}k to k. It follows that γ is some multiple, λ_{n}.ev, of the evaluation map (regarding Ω^{−n}k as the dual of Ω^{n}k). So the composite of the left-hand vertical maps in the above diagram is Ω^{n}(ζ̂^{∗}), and we have

Ω^{n}(ζ̂^{∗}) = λ_{n}.ev ∘ (ζ̂^{∗} ⊗ 1) = λ_{n}.ζ̂.

(There are no non-zero maps from Ω^{n}k to k which factor through a projective module).

It remains to determine the constants λ_{n}. Since the restriction of Ω^{n}k to a subgroup H of G is Ω^{n}k⊕ (projective) and the entire construction commutes with restriction, it suffices to restrict to a cyclic subgroup of G of order p = char k and evaluate λ_{n} there. By restricting from Z/p × Z/p to Z/p, it suffices to determine the λ_{n} for Z/p × Z/p. The cohomology of this group has the property that there are non-trivial products H^{m} × H^{n} → H^{m+n} for every pair of positive integers m and n.

Since the map ζ ↦ λ_{n}.ζ is an anti-automorphism of H^{∗}(G, k) (duality reverses Yoneda composition) we have

λ_{m}ξ.λ_{n}ζ = λ_{m+n}ζ.ξ,

so that

λ_{m}λ_{n} = (−1)^{mn}λ_{m+n}.

It remains only to determine λ_{1}. We leave to the reader the easy exercise of showing that λ_{1} = −1. Then by induction λ_{n} = (−1)^{n(n+1)/2}, as desired.

To prove the second statement of the lemma, we observe that the left-hand map, γ ∘(ζ̂^{∗}⊗1) = (−1)^{n(n+1)/2}ζ̂, in the above diagram has kernel L_{ζ}, while the right-hand map ζ̂^{∗} has cokernel L_{ζ}^{∗}, and so by passing to the appropriate quotient on the top row and subcomplex on the bottom row, we obtain a map of complexes

0 → k → P_{n−1}/L_{ζ} → P_{n−2} → ⋯ → P_{1} →  P_{0}  → k → 0
(−1)^{n(n+1)/2} ↓  ↓  ↓  ↓  ↓  ∥
0 → k → P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{n−2}^{∗} → (P_{n−1}/L_{ζ})^{∗} → k → 0

and hence the required homology equivalence.

Theorem 5.3 Suppose that ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2. Then the complex C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} is a direct sum of a complex C′ satisfying Poincaré duality in formal dimension s = ∑_{i=1}^{r}(n_{i} − 1),

Hom_{k}(C′, k)[s] ≅ C′

and an exact complex P′ of projective modules.

Proof By the proposition, we have homology equivalences

Hom_{k}(C_{ζ_{i}}, k)[n_{i} − 1] → C_{ζ_{i}}.

Putting these together, we obtain a homology equivalence

Hom_{k}(C, k)[s] ≅ C

By Theorem 4.1, C is a finite complex of projective modules, and so the theorem now follows by applying Lemma 5.1.

Corollary 5.4 If S is a simple kG-module and C and s are as in the theorem, then the Poincaré series p_{S}(t) = ∑_{i=0}^{s} t^{i} dim_{k} Hom_{kG}(C_{i}, S) satisfies the equation t^{s}p_{S}(1/t) = p_{S^{∗}}(t).

We now summarise what we have established so far about the hypercohomology spectral sequence of the complex C.

Theorem 5.5 Suppose that ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2. Then the complex C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} = C′ ⊕ P′ is a finite complex of projective modules with cohomology an exterior algebra

H^{∗}(Hom_{k}(C, k)) = H^{∗}(Hom_{k}(C′, k)) = Λ^{∗}(ζ̃_{1}, …, ζ̃_{r})

with deg(ζ̃_{i}) = n_{i}−1. The complex C′ satisfies Poincaré duality in formal dimension s = ∑_{i=1}^{r}(n_{i} − 1), and P′ is a split exact sequence of projectives.

There is a spectral sequence with

E_{2}^{∗∗} = H^{∗}(G, k) ⊗ Λ^{∗}(ζ̃_{1}, …, ζ̃_{r})

and converging to H^{∗}(Hom_{kG}(C, k)), which also satisfies Poincaré duality in formal dimension s. In this spectral sequence we have d_{n_{i}}(ζ̃_{i}) = ζ_{i}.

If char k is odd, then the above spectral sequence has a multiplicative structure (but see also Section 3 for a discussion of the remaining case).

Further information about the differentials in this spectral sequence is given in Section 11.

6 The Cohen–Macaulay case

We begin with a brief review of Cohen–Macaulay rings.

Definition 6.1 Suppose that A = ⨁_{n≥0} A_{n} is a finitely generated graded commutative k-algebra, and M = ⨁_{n≥0} M_{n} is a (non-zero) finitely generated graded A-module. A sequence ζ_{1}, …, ζ_{r} of homogeneous elements of degree n_{i} > 0 in A is said to be a regular sequence for M if for each i = 1, …, r the map

M_{n}/M_{n} ∩ (ζ_{1}, …, ζ_{i−1})M → M_{n+n_{i}}/M_{n+n_{i}} ∩ (ζ_{1}, …, ζ_{i−1})M

induced by multiplication by ζ_{i} is injective, for all n. Note that we are not asking that M_{n}/M_{n} ∩ (ζ_{1}, …, ζ_{i−1})M be non-zero.

The depth of M is the length of the longest regular sequence. M is said to be Cohen–Macaulay if its depth is equal to its Krull dimension. The ring A is said to be Cohen–Macaulay if it is Cohen–Macaulay as a module over itself. This is equivalent to the condition that there is a polynomial subring k[ζ_{1}, …, ζ_{r}] ⊆ A generated by homogeneous elements ζ_{i}, such that A is a finitely generated free module over k[ζ_{1}, …, ζ_{r}].

The following theorem is proved in Serre [25], p. IV–20, Theorem 2; see also Stanley [26], Proposition 3.1. Actually the proof given in Serre [25] is for strictly commutative local rings, but the proof carries over verbatim to the graded commutative case.

Theorem 6.2 Suppose that A is a finitely generated graded commutative k-algebra. Then the following are equivalent.

(i) There exists a polynomial subring k[ζ_{1}, …, ζ_{r}] ⊆ A generated by homogeneous elements ζ_{i}, such that A is a finitely generated free module over k[ζ_{1}, …, ζ_{r}].

(ii) For every polynomial subring k[ζ_{1}, …, ζ_{r}] ⊆ A generated by homogeneous elements ζ_{i}, such that A is finitely generated as a module over k[ζ_{1}, …, ζ_{r}] (i.e., ζ_{1}, …, ζ_{r} is an h.s.o.p. for A), A is a free module.

If E is an elementary abelian p-group then H^{∗}(E, k) is Cohen–Macaulay. Quillen’s calculations [23] show that the cohomology rings of general linear groups at primes other than the natural one are Cohen–Macaulay, and his work on extraspecial 2-groups [22] shows that these also have Cohen–Macaulay cohomology rings. If G is a semidihedral 2-group then H^{∗}(G, k) is not Cohen–Macaulay, (see Evens and Priddy [14]). Also, if G is a split metacyclic p-group with p odd, then usually H^{∗}(G, k) is not Cohen–Macaulay, (see Diethelm [12]).

It is interesting to note that, until recently (see [1]), in every example which had been computed, H^{∗}(G, k) was Cohen–Macaulay whenever G was simple. This is not a general fact because, as noted in the introduction, the Cohen–Macaulay condition requires that all irreducible components of the maximal ideal spectrum of H^{∗}(G, k) have the same dimension. Hence if G is any group having maximal elementary abelian p-subgroups of unequal ranks, then H^{∗}(G, k) is not Cohen–Macaulay.

Theorem 6.3 Suppose that H^{∗}(G, k) is Cohen–Macaulay, and that ζ_{1}, …, ζ_{r} is an irredundant h.s.o.p. with deg(ζ_{i}) = n_{i} ≥ 2. Then the quotient

H^{∗}(G, k)/(ζ_{1}, …, ζ_{r})

satisfies Poincaré duality with formal dimension s = ∑_{i=1}^{r}(n_{i} − 1).

Proof Let C = C_{ζ_{1}} ⊗⋯⊗C_{ζ_{r}} = C′ ⊕P′ as in Theorem 5.3. Then by Theorem 5.5 we have a spectral sequence with

E_{2}^{∗∗} = H^{∗}(G, k) ⊗ Λ^{∗}(ζ̃_{1}, …, ζ̃_{r}).

In Section 3, we described enough differentials in this spectral sequence to be able to identify it with a staggered version of the Koszul complex for H^{∗}(G, k) as a module over k[ζ_{1}, …, ζ_{r}]. Since H^{∗}(G, k) is Cohen–Macaulay, it is a free module over k[ζ_{1}, …, ζ_{r}], and so the E_{∞} page of this spectral sequence is the quotient H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}) concentrated along the bottom row. It follows that we have an isomorphism

H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}) ≅ H^{∗}(Hom_{kG}(C, k)),

so that this quotient of the cohomology ring satisfies Poincaré duality with formal dimension s = ∑_{i=1}^{r}(n_{i} − 1).

Remark 6.4 The proof of this theorem also shows that if D is a bounded chain complex of kG-modules with the property that H^{∗}(G, D) is Cohen–Macaulay as a module over H^{∗}(G, k), then

H^{∗}(G, D)/(ζ_{1}, …, ζ_{r}) ≅ H^{∗}(Hom_{kG}(C, D)).

Corollary 6.5 Suppose that H^{∗}(G, k) is Cohen–Macaulay, and that ζ_{1}, …, ζ_{r} is an irredundant h.s.o.p. with deg(ζ_{i}) = n_{i} ≥ 2. Then the Poincaré series

P_{k}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(G, k)

is of the form p(t)/ ∏_{i=1}^{r}(1−t^{n_{i}}), where p(t) is a polynomial satisfying p(t) = t^{s}p(1/t) where s = ∑_{i=1}^{r}(n_{i} − 1). As rational functions of t, we have

P_{k}(1/t) = (−t)^{r}P_{k}(t).

Proof Basis elements of the k-vector space H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}) may be lifted to give free generators of H^{∗}(G, k) as a module over k[ζ_{1}, …, ζ_{r}]. So the Poincaré series P_{k}(t) is of the form p(t)/ ∏_{i=1}^{r}(1−t^{n_{i}}), where p(t) is the Poincaré series (polynomial) of H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}).

By the above theorem, H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}) satisfies Poincaré duality with formal dimension s = ∑_{i=1}^{r}(n_{i} − 1), and so we have

t^{s}p(1/t) = p(t).

This means that

P_{k}(1/t) = p(1/t). ∏_{i=1}^{r} 1/(1 − t^{−n_{i}})
= t^{−s}p(t). ∏_{i=1}^{r} t^{n_{i}}/(t^{n_{i}} − 1)
= (−t)^{r}P_{k}(t).

Remark By a theorem of Quillen [21], the integer in the above theorem, namely the Krull dimension, r, of H^{∗}(G, k), is equal to the p-rank r_{p}(G).

Corollary 6.6 If H^{∗}(G, k) is a polynomial ring, then char k = 2, the generators are in degree one and G/O(G) is an elementary abelian 2-group (here, O(G) denotes the largest odd order normal subgroup of G). Hence we have a complete proof to Theorem 1.1.

Proof If H^{∗}(G, k) ≅ k[ζ_{1}, …, ζ_{r}] with deg(ζ_{i}) = n_{i} then the Poincaré series is

P_{k}(t) = 1/ ∏_{i=1}^{r}(1 − t^{n_{i}})

and so

P_{k}(1/t) = (−t)^{n_{1}+···+n_{r}}P_{k}(t).

It thus follows from the above theorem that n_{1} +⋯+n_{r} = r, so that each n_{i} = 1. In odd characteristic, elements of degree one square to zero, so char k = 2. Moreover, examining H^{1}(G, k), we see that G has an elementary abelian quotient E with the property that inflation H^{∗}(E, k) → H^{∗}(G, k) is an isomorphism, so the kernel has odd order.

We finish this section by showing that for H^{∗}(G, k) to be Cohen–Macaulay it is sufficient (but not necessary) for the cohomology of the Sylow p-subgroup to be Cohen–Macaulay. The following lemma is well known.

Lemma 6.7 Let P be a Sylow p-subgroup of G. Then

res_{G,P} : H^{∗}(G, k) → H^{∗}(P, k)

is injective, and makes H^{∗}(P, k) into a finitely generated H^{∗}(G, k)-module, which has H^{∗}(G, k) as a direct summand.

Proof This follows from the fact that the transfer map

tr_{P,G} : H^{∗}(P, k) → H^{∗}(G, k)

satisfies (i) res_{G,P} (tr_{P,G}(x)) = |G : P |x (ii) x.tr_{P,G}(y) = tr_{P,G}(res_{G,P} (x).y)

for x ∈ H^{∗}(G, k) and y ∈ H^{∗}(P, k). Finite generation follows from Evens’ theorem (cf. Theorem 2.4) and the fact that the Shapiro isomorphism

H^{∗}(P, k) ≅ H^{∗}(G, k_{P} ↑^{G})

is an isomorphism of H^{∗}(G, k)-modules, where k_{P} ↑^{G} is the permutation module for G on the cosets of P.

Proposition 6.8 Let P be a Sylow p-subgroup of G. If H^{∗}(P, k) is Cohen–Macaulay then so is H^{∗}(G, k).

Proof We may choose the polynomial subring k[ζ_{1}, …, ζ_{r}] ⊆ H^{∗}(P, k) of part (ii) of Theorem 6.2 to lie inside H^{∗}(G, k), by the lemma. Thus H^{∗}(G, k) is a direct summand of a finitely generated free module over k[ζ_{1}, …, ζ_{r}] and is hence free.

Remark The converse of this proposition is not true. For example, the Sylow 2subgroup of M_{11} is a semidihedral group of order 16, whose cohomology in characteristic two is not Cohen–Macaulay (Evens and Priddy [14]), whereas the cohomology of M_{11} in characteristic two is Cohen–Macaulay, see [3].

Corollary 6.9 If a Sylow p-subgroup P of G is abelian then H^{∗}(G, k) is Cohen– Macaulay.

7 The last survivor

In this section we prove Theorem 1.3. The idea is that the top degree element in the cohomology of Hom_{kG}(⨂_{i=1}^{r} C_{ζ_{i}}, k) lifts to a non-zero element of H^{∗}(G, k), which we think of as the “last survivor” of the cohomology of this finite complex.

So let ζ_{1}, …, ζ_{r} be an irredundent h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2, and let C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} be the finite complex of projective modules discussed in Theorem 5.5. Let P be a projective resolution of k as a kG-module. Since P is exact and C consists of projective modules, there is a chain map C → P inducing an isomorphism in degree zero homology (the degree zero homology of both complexes is k), and any two such maps are chain homotopic. So if D is a bounded complex of kG-modules, we obtain a well defined map

e : H^{∗}(Hom_{kG}(P, D)) = H^{∗}(G, D) → H^{∗}(Hom_{kG}(C, D))

which is the horizontal edge homomorphism of the spectral sequence

E_{2}^{pq} = Ext_{kG}^{p}(H_{q}(C), D) ⇒ H^{∗}(Hom_{kG}(C, D))

that was discussed in Section 4. The map e commutes with restriction to subgroups.

We saw in Section 6 (see Remark 6.4) that if H^{∗}(G, D) is Cohen–Macaulay as a module over H^{∗}(G, k) then the E_{∞} page of this spectral sequence consists of the quotient H^{∗}(G, D)/(ζ_{1}, …, ζ_{r}) concentrated along the bottom row, and so the face map e is surjective in this case.

We may interpret this as follows. Write C as C′ ⊕ P′, where P′ is an exact sequence of projective modules and C′ has no summands which are exact sequences of projective modules (so that according to Section 5, C′ is self-dual). Then for S a simple kG-module we have

H^{∗}(Hom_{kG}(C, S)) ≅ Hom_{kG}(C′, S).

If P is a minimal resolution of k then

H^{∗}(G, S) = Hom_{kG}(P, S).

Thus if H^{∗}(G, S) is a Cohen–Macaulay module of Krull dimension r over H^{∗}(G, k) for each simple kG-module S in the principal block, the edge homomorphisms

Hom_{kG}(P, S) → Hom_{kG}(C′, S)

are all surjective, and so the original map of complexes C → P is injective. We have thus proved the following embeddability theorem.

Theorem 7.1 Suppose that for each simple kG-module S in the principal block of kG, H^{∗}(G, S) is a Cohen–Macaulay module of Krull dimension r = r_{p}(G) over H^{∗}(G, k). For an irredundent h.s.o.p. ζ_{1}, …, ζ_{r} in H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2, we express C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}} as a direct sum C′ ⊕ P′, where P′ is an exact sequence of projective modules and C′ has no summands which are exact sequences of projective modules. Then there is an embedding of C′ in the minimal resolution of k as a kG-module, which induces an isomorphism on H_{0}.

Remark The hypothesis of this theorem fails to hold if H^{∗}(G, S) has Krull dimension smaller than that of H^{∗}(G, k). For example, if G = SL(2, 4) ≅ A_{5} and char(k) = 2, then the simple modules in the principal block are k, M and N, with dim_{k}(M) = dim_{k}(N) = 2 and with projective covers P_{k}, P_{M} and P_{N}. If we take ζ_{1} and ζ_{2} to be non-zero elements of degrees two and three, then C′ is as follows

0 → P_{k} → P_{M} ⊕ P_{N} → P_{M} ⊕ P_{N} → P_{k} → 0,

whereas the minimal resolution is of the form

⋯ → P_{k} → P_{M} ⊕ P_{N} → P_{k} → 0.

All of this can be verified using diagrams ([3]).

Now suppose that ζ_{1}, …, ζ_{r} is an irredundent h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2, and let C = C_{ζ_{1}} ⊗⋯⊗C_{ζ_{r}} as above. By Quillen’s theorem [21] on nilpotent elements in H^{∗}(G, k), if E is an elementary abelian p-subgroup of G of maximal rank, restriction to E is injective on the subring k[ζ_{1}, …, ζ_{r}] of H^{∗}(G, k). Since H^{∗}(E, k) is Cohen–Macaulay, it follows that if we set C↓_{E}= C′ ⊕ P′ as above, then C′ embeds in the minimal resolution of k as a kE-module.

Now in degree s = ∑_{i=1}^{r}(n_{i} − 1), H^{s}(Hom_{kG}(C, k)) is one dimensional, and so we have a canonical map

γ : H^{s}(G, k) →^{e} H^{s}(Hom_{kG}(C, k)) →^{≅} k.

By Tate duality, this corresponds to an element

γ(ζ_{1}, …, ζ_{r}) ∈ Ĥ^{−s−1}(G, k) ≅ Hom_{k}(H^{s}(G, k), k).

The above discussion shows that

res_{G,E}γ(ζ_{1}, …, ζ_{r}) = γ(res_{G,E}(ζ_{1}), …, res_{G,E}(ζ_{r})) ≠ 0

in Ĥ^{−s−1}(E, k), and hence γ(ζ_{1}, …, ζ_{r}) ≠ 0 in Ĥ^{−s−1}(G, k). Since the edge homomorphism e, and hence also the map γ, factors through H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}), it follows that there is a non-zero element of degree s in H^{∗}(G, k) not in the ideal generated by ζ_{1}, …, ζ_{r}. Moreover, since the Tate dual of restriction is transfer, this non-zero element may be taken to be a transfer from H^{∗}(E, k). This completes the proof of Theorem 1.3.

In fact in the Cohen–Macaulay case, the element γ(ζ_{1}, …, ζ_{r}) ∈ Ĥ^{−s−1}(G, k) described above determines the ideal (ζ_{1}, …, ζ_{r}) of H^{∗}(G, k), as we now prove.

Theorem 7.2 Suppose that H^{∗}(G, k) is Cohen–Macaulay, and that ζ_{1}, …, ζ_{r} is an irredundent h.s.o.p. Let C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}}. Then

H_{∗}(Hom_{kG}(k, C)) ≅ γ(ζ_{1}, …, ζ_{r}) ∩ H^{∗}(G, k)

(cap product). If ζ′_{1}, …, ζ′_{r} is another irredundent h.s.o.p. with

γ(ζ_{1}, …, ζ_{r}) = γ(ζ′_{1}, …, ζ′_{r})

then setting D = C_{ζ′_{1}} ⊗ ⋯ ⊗ C_{ζ′_{r}}, we have isomorphisms

H_{∗}(Hom_{kG}(k, C)) ≅ H_{∗}(Hom_{kG}(k, D)),
H^{∗}(Hom_{kG}(C, k)) ≅ H^{∗}(Hom_{kG}(D, k)),

and the ideals (ζ_{1}, …, ζ_{r}) and (ζ′_{1}, …, ζ′_{r}) are equal.

Proof First note that γ(C) = γ(ζ_{1}, …, ζ_{r}) is a class in Ĥ^{−s−1}(G, k) ≅ H_{s}(G, k), and that cap product is the map H_{s}(G, k) × H^{t}(G, k) → H_{s−t}(G, k) for s ≤ t induced by cup product in Tate cohomology. Let P be the minimal projective resolution of k as a kG-module. Set C = C′ ⊕ P′ with P′ an exact sequence of projective modules and C′ having no summands of this form. Then there is a map C′ →^{ϕ} P which induces an isomorphism in H_{0}. Moreover, by the same argument as used in the proof of Theorem 7.1, ϕ^{∗} : Hom_{kG}(P, k) → Hom_{kG}(C′, k) is surjective. The dual Hom_{k}(P, k) is the minimal injective resolution of k and we have a map ϕ^{∗} : Hom_{k}(P, k) → Hom_{k}(C′, k) which is a surjection on G-fixed points. Let ψ be the composite map

Hom_{k}(P, k)[s] →^{ϕ^{∗}} Hom_{k}(C′, k)[s] →^{θ} C′ ↪^{ϕ} P

where s = ∑_{i=1}^{r}(deg(ζ_{i}) − 1) and θ is the isomorphism given in Theorem 5.3. Now the homology of Hom_{k}(P, k) is given by the dual ε^{∗} of the augmentation in degree zero. Now ψ ∘ ε^{∗} is a non-zero multiple a.γ(C), so the map

H^{t}(G, k)  ⟶  H_{s−t}(G, k)
∥≀  ∥≀
H_{−t}(Hom_{kG}(P, k)) ⟶^{ψ_{∗}} H_{s−t}(Hom_{kG}(k, P))

is cap product with a.γ(C). By Theorem 7.1, the map

ϕ^{∗} : H_{∗}(Hom_{kG}(P, k)) → H_{∗}(Hom_{kG}(C′, k))

is surjective,

θ_{∗} : H_{∗}(Hom_{kG}(C′, k))[s] → H_{∗}(Hom_{kG}(k, C′)) = H_{∗}(Hom_{kG}(k, C))

is an isomorphism, and

ϕ_{∗} : H_{∗}(Hom_{kG}(k, C′)) → H_{∗}(Hom_{kG}(k, P))

is injective. So we have proved that

H_{∗}(Hom_{kG}(k, C)) ≅ γ(ζ_{1}, …, ζ_{r}) ∩ H^{∗}(G, k).

It follows that if γ(ζ_{1}, …, ζ_{r}) is equal to γ(ζ′_{1}, …, ζ′_{r}) as an element of Ĥ^{−s−1}(G, k), then we have an isomorphism H_{∗}(Hom_{kG}(k, C)) ≅ H_{∗}(Hom_{kG}(k, D)), and dually we have an isomorphism H^{∗}(Hom_{kG}(C, k)) ≅ H^{∗}(Hom_{kG}(D, k)) as modules over H^{∗}(G, k). The annihilator ideals of these modules are therefore equal. We saw in Section 6 that H^{∗}(Hom_{kG}(C, k)) ≅ H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}), so that the annihilator of this module is the ideal (ζ_{1}, …, ζ_{r}).

8 Filtrations on projective modules

The aim of this section is to show how a complex C of kG-modules which has at most one non-projective term M gives rise to a filtration on M ⊕ P, for some projective module P, where the composition factors are Heller translates Ω^{±n} of the homology of C. In the event that C is a complex of projective modules, this is a filtration of a projective module. The construction given here is a rudimentary form of the equivariant cohomology spectral sequence—without the cohomology. It would be possible to make a construction by embedding the complex in a projective resolution or something similar. This would be equivalent to resolving the homology of C piece by piece as in [4]. However, the method given here is cleaner and gives a more satisfactory result.

Of course, any filtration on a module leads naturally to a spectral sequence on the cohomology of the module. If the complex C is one of those constructed in Section 4 then the E_{2}-term of the spectral sequence corresponding to the filtration is a very familiar object, namely a Koszul complex.

Suppose that

C : 0 → C_{s} → ⋯ → C_{1} → C_{0} → 0

is a bounded complex of kG-modules. Let

P̂ : ⋯ → P_{1} →^{∂} P_{0} →^{∂} P_{−1} →^{∂} P_{−2} → ⋯

be a doubly infinite projective resolution of the trivial kG-module k. This means that the complex is exact, and that ∂(P_{0}) ≅ k. Now consider the double complex C ⊗ P̂. The total complex of C ⊗ P̂ has no homology and hence is a doubly infinite exact sequence. Also since for each i and j, C_{i} and P_{j} are finitely generated, so is (C ⊗ P̂)_{m} = ⨁_{i+j=m} C_{i} ⊗ P_{j}.

For any index t let (C ⊗ P̂)^{(t)} denote the complex

(C ⊗ P̂)^{(t)} : ⋯ → (C ⊗ P̂)_{t+2} → (C ⊗ P̂)_{t+1} → (C ⊗ P̂)_{t} → 0.

It is the truncation of the total complex C ⊗ P̂ at the tth term.

Lemma 8.1 (i) For any bounded complex C of kG-modules, the homology

H_{i}((C ⊗ P̂)^{(t)}) = 0

for i ≠ t.

(ii) If 0 → A → B → C → 0 is a short exact sequence of complexes of kGmodules, then

0 → (A ⊗ P̂)^{(t)} → (B ⊗ P̂)^{(t)} → (C ⊗ P̂)^{(t)} → 0

is an exact sequence, and we have an exact sequence

0 → H_{t}(A ⊗ P̂)^{(t)} → H_{t}(B ⊗ P̂)^{(t)} → H_{t}(C ⊗ P̂)^{(t)} → 0.

(iii) If H_{∗}(C) = 0 then H_{t}((C ⊗ P̂)^{(t)}) is projective.

(iv) Suppose that C is concentrated in degree i (that is, C_{j} = 0 if j ≠ i). Then

H_{t}((C ⊗ P̂)^{(t)}) = Ω^{t−i}(C_{i}) ⊕ (projective).

Proof The first statement follows from the fact that C ⊗ P̂ is exact. The second is easily verified because, for any i and j,

0 → A_{i} ⊗ P_{j} → B_{i} ⊗ P_{j} → C_{i} ⊗ P_{j} → 0

is exact and hence so is

0 → ⨁_{i+j=m} A_{i} ⊗ P_{j} → ⨁_{i+j=m} B_{i} ⊗ P_{j} → ⨁_{i+j=m} C_{i} ⊗ P_{j} → 0.

The last statement of (ii) is a consequence of the long exact sequence on homology. To demonstrate (iii), let

P : ⋯ → P_{2} → P_{1} → P_{0} → 0

be the non-negative part of P̂, and note that C ⊗ P has no homology. So C ⊗ P and hence also C ⊗ P̂ are totally split exact sequences of projective modules. Hence ∂((C ⊗ P̂)_{t}) = H_{t}((C ⊗ P̂)^{(t)}) is a projective module. To prove (iv) we need only notice that in this case (C ⊗ P̂)^{(t)} is the complex

⋯ → C_{i} ⊗ P_{t−i+2} → C_{i} ⊗ P_{t−i+1} → C_{i} ⊗ P_{t−i} → 0,

which is a projective resolution of

C_{i} ⊗ ∂(P_{t−i}) ≅ C_{i} ⊗ (Ω^{t−i}(k) ⊕ (projective)) ≅ Ω^{t−i}(C_{i}) ⊕ (projective).

Theorem 8.2 Suppose that

C : 0 → C_{s} → C_{s−1} → ⋯ → C_{1} → C_{0} → 0

is a bounded complex of kG-modules. Let M = C_{t} and suppose that C_{i} is projective whenever i ≠ t. Then there exists a projective module P and a projective resolution Q of M ⊕ P (i.e., H_{∗}(Q) = H_{0}(Q) = M ⊕ P) such that Q has a filtration by complexes

Q^{(s)} ⊆ Q^{(s−1)} ⊆ ⋯ ⊆ Q^{(0)} = Q

with the following properties.

(i) For every i ≥ 0 and 1 ≤ j ≤ s, Q_{i}^{(j)} (and hence also Q_{i}^{(j)}/Q_{i}^{(j+1)}) is projective. (ii) H_{∗}(Q^{(j)}/Q^{(j+1)}) = H_{0}(Q^{(j)}/Q^{(j+1)}) ≅ Ω^{t−j}(H_{j}(C)) ⊕ (projective). (iii) For every j the induced map

H_{0}(Q^{(j)}) → H_{0}(Q) ≅ M ⊕ P

is injective. Denoting its image by M_{j}, this gives a filtration

0 ⊆ M_{s} ⊆ ⋯ ⊆ M_{1} ⊆ M_{0} ≅ M ⊕ P

where M_{j}/M_{j+1} ≅ Ω^{t−j}(H_{j}(C)) ⊕ (projective).

Proof As before, let

P̂ : ⋯ → P_{1} →^{∂} P_{0} →^{∂} P_{−1} →^{∂} P_{−2} → ⋯

be a doubly infinite projective resolution of k. We claim that the complex (C ⊗ P̂)^{(t)}[t] = Q satisfies the conclusion of the theorem. This is the complex (C ⊗ P̂)^{(t)} as in Lemma 8.1, shifted down in degree by t. Clearly (C ⊗ P̂)^{(t)} is a complex of projectives with homology only in degree zero. So we must first show that ∂((C ⊗ P̂)_{t}) = M ⊕ (projective).

Let A^{(j)} ↪ C be the subcomplex such that A_{i}^{(j)} = C_{i} if i ≤ j and A_{i}^{(j)} = 0 if i > j. Then we have an exact sequence for j ≤ k

0 → A^{(j)} → A^{(k)} → A^{(k)}/A^{(j)} → 0

and by part (ii) of the last lemma we have an exact sequence

0 → (A^{(j)} ⊗ P̂)^{(t)} → (A^{(k)} ⊗ P̂)^{(t)} → (A^{(k)}/A^{(j)} ⊗ P̂)^{(t)} → 0.

If j = k − 1 then A^{(k)}/A^{(k−1)} is concentrated in degree k, and so

H_{t}((A^{(k)}/A^{(k−1)} ⊗ P̂)^{(t)}) = Ω^{t−k}(C_{k}) ⊕ (projective).

If k = t, then Ω^{t−k}(C_{k}) ≅ M, while if k ≠ t then this homology is projective. Likewise, for i < j < k we have an exact sequence

0 → H_{t}((A^{(j)}/A^{(i)} ⊗ P̂)^{(t)}) → H_{t}((A^{(k)}/A^{(i)} ⊗ P̂)^{(t)}) → H_{t}((A^{(k)}/A^{(j)} ⊗ P̂)^{(t)}) → 0.

This last sequence must split because one of the two end terms must be projective (hence also injective). The right hand end is projective if j ≥ t, while the left hand end is projective if j < t. We conclude that

H_{t}((C ⊗ P̂)^{(t)}) ≅ ⨁_{j=1}^{s} H_{t}((A^{(j)}/A^{(j−1)} ⊗ P̂)^{(t)}) ≅ M ⊕ (projective).

For each i let B^{(i)} ↪ C be the subcomplex

B^{(i)} : 0 → C_{s} → ⋯ → C_{i+1} → Ker∂_{i} → 0

which is non-zero only in degrees i to s. Of course B^{(s)} is the module Ker∂_{s} concentrated in degree s. Also B^{(0)} = C. Now the complex

B^{(i)}/B^{(i+1)} : 0 → C_{i+1}/Ker∂_{i+1} → Ker∂_{i} → 0

has a subcomplex

D^{(i)} : 0 → C_{i+1}/Ker∂_{i+1} →^{≅} ∂(C_{i+1}) → 0

and the quotient (B^{(i)}/B^{(i+1)})/D^{(i)} is the complex consisting of the one module H_{i}(C) in degree i. By Lemma 8.1 (iii), H_{t}((D^{(i)} ⊗ P̂)^{(t)}) is projective. So from Lemma 8.1 (ii) we get that

H_{t}((B^{(i)}/B^{(i+1)} ⊗ P̂)^{(t)}) ≅ Ω^{t−i}(H_{i}(C)) ⊕ (projective).

Again by Lemma 8.1 we see that the rest of the homology groups of the complex

(B^{(i)}/B^{(i+1)} ⊗ P̂)^{(t)} ≅ (B^{(i)} ⊗ P̂)^{(t)}/(B^{(i+1)} ⊗ P̂)^{(t)}

are zero.

So let Q^{(j)} = (B^{(j)} ⊗ P̂)^{(t)}[t] ⊆ Q. It is clear that we have a filtration

Q^{(s)} ⊆ Q^{(s−1)} ⊆ ⋯ ⊆ Q^{(0)} = Q

as in statement (i). Statement (ii) is a consequence of the above paragraph. The third condition follows directly from Lemma 8.1 (ii). This completes the proof.

Now suppose that ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i} ≥ 2. Let C_{ζ_{i}} be the complex described in Section 3. Then Theorem 8.2 (iii) tells us that there exists a projective module P (i) and a filtration of P_{n_{i}−1}/L_{ζ_{i}} ⊕ P (i) of length two (i.e., a short exact sequence)

0 → k → P_{n_{i}−1}/L_{ζ_{i}} ⊕ P (i) → Ω^{n_{i}−1}(k) → 0. (8.3)

Here, we have removed the projective summands from the end two terms, at the same time removing summands from P (i), as we may since projective kG-modules are also injective.

In this case the sequence (8.3) is obvious because the map k → P_{n_{i}−1}/L_{ζ_{i}} is the injection of H_{n_{i}−1}(C_{ζ_{i}}). Moreover the sequence represents ζ_{i} as an extension in

Ext_{kG}^{1}(Ω^{n_{i}−1}(k), k) ≅ Ext_{kG}^{n_{i}}(k, k) ≅ H^{i}(G, k).

The filtration on the projective module

P = P_{n_{1}−1}/L_{ζ_{1}} ⊗ ⋯ ⊗ P_{n_{r}−1}/L_{ζ_{r}} ⊕ (projective)

(see Theorem 4.1) is the product filtration. It can be written as

{0} ⊆ M_{r} ⊆ ⋯ ⊆ M_{1} ⊆ M_{0} = P (8.4)

where M_{r} ≅ k and

M_{r−i}/M_{r−i+1} ≅ ⨁_{\begin{matrix}S⊆\{ζ_{1},...,ζ_{r}\} \\ |S|=i \end{matrix}} Ω^{deg(S)−i}(k) ⊕ (projective)

where deg(S) = ∑_{ζ_{i}∈S} deg(ζ_{i}).

As we noted earlier, any filtration on a module gives rise to a spectral sequence on the cohomology of that module. This can be done in either ordinary or Tate cohomology. Let P̂ be the doubly infinite projective resolution of k. For any kG-module N, we have a two-row spectral sequence coming from the filtration (8.3). If we write it as a cohomology spectral sequence and if we shift the degrees appropriately, then the rows on the E_{0} page are given by the formulas

Ê_{0}^{t,1} = Hom_{kG}(P̂_{t−n_{i}+2} ⊗ k, N)
Ê_{0}^{t,0} = Hom_{kG}(P̂_{t−n_{i}+1} ⊗ Ω^{n_{i}−1}(k), N).

The E_{2} page has rows

Ê_{2}^{t,1} = Êxt_{kG}^{t−n_{i}+2}(k, N) = Ĥ^{t−n_{i}+2}(G, N)
Ê_{2}^{t,0} = Êxt_{kG}^{t−n_{i}+1}(Ω^{n_{i}−1}(k), N) = Ĥ^{t}(G, N).

Then

d_{2} :  Ê_{2}^{t,1}  ⟶  Ê_{2}^{t+2,0}
∥  ∥
Ĥ^{t−n_{i}+2}(G, N) ⟶ Ĥ^{t+2}(G, N)

is the connecting homomorphism in the long exact sequence associated to (8.3). Specifically it is given by cup product with ζ_{i}. Of course, the spectral sequence converges to Êxt_{kG}^{∗}(P_{n_{i}−1}/L_{ζ_{i}}, N) suitably shifted.

On the other hand, suppose we consider the filtration given in (8.4). Here the corresponding spectral sequence converges to zero. It has r + 1 rows. Again if we shift the degrees appropriately then the terms on the E_{2} page are given by

Ê_{2}^{t,i} = Êxt_{kG}^{t−s′+3i}(M_{i}/M_{i−1}, N)
= ∑ Êxt_{kG}^{t−s′+3i}(Ω^{deg(S)−i}(k), N)
= ∑ Ĥ^{t+deg(S)−s′+2i}(G, N),

where the sum is over all subsets S of {ζ_{1}, …, ζ_{r}} having exactly r − i elements. Here s′ = ∑_{i=1}^{r} deg(ζ_{i}). The d_{2} differential is obvious from the product structure, or from the extension classes which connect the terms in the filtration. An easy analysis then shows the following.

Proposition 8.5 In the above spectral sequence the Ê_{2} page is the Koszul complex of Ĥ^{∗}(G, N) as a k[ζ_{1}, …, ζ_{r}]-module. More specifically it is isomorphic to Ĥ^{∗}(G, N)⊗_{A}K, where K is the Koszul complex of free modules over A = k[ζ_{1}, …, ζ_{r}] which is the usual A-projective resolution of k. Thus the Ê_{3} page of the spectral sequence has terms

Tor_{A}^{∗}(k, Ĥ^{∗}(G, N)).

Remark The Koszul complexes are well documented in the literature. See for example Mac Lane [16]. For another treatment which is both leisurely and considers spectral sequences as above, see [8]. It should be remarked that the spectral sequence in the proposition is identical to the hypercohomology spectral of the complex B considered in Sections 3 and 4.

9 Eliminating projectives from complexes

If 0 ≠ ζ ∈ H^{n}(G, k), we write ζ̂ for the corresponding map Ω^{n}k → k, where Ω^{n}k is the nth kernel in the minimal projective resolution of the trivial kG-module k. We write L_{ζ} for the kernel of ζ̂, so that there is a short exact sequence

0 → L_{ζ} → Ω^{n}k →^{ζ̂} k → 0.

We proved in Section 4 that if ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k) then L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}} is projective (and hence also injective). It follows from this fact, that if we

tensor together the complexes 0 → Ω^{n_{i}}k →^{ζ̂_{i}} k → 0 and the module Ω^{n}k, and insert the module L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}} ⊗ Ω^{n}k to make it exact, we obtain an exact complex each term of which is a sum of modules of the form Ω^{s}k and projective modules (note that Ω^{s}k ⊗ Ω^{t}k ≅ Ω^{s+t}k ⊕ (projective)). The question is, to what extent may we strip off the projective modules to leave a complex of modules of the form Ω^{s}k?

The following questions are an attempt to formalise this, and the main result of this section is that these questions are equivalent. In the next section, we give some conditions under which a positive answer is guaranteed. In particular, a negative answer implies that r_{p}(G) ≥ 3. For p = 2 we know of no example where these questions fail to have a positive answer, but for p odd we shall show in a later paper that for the extraspecial groups of order p^{1+2n}, n ≥ 2, and exponent p, the answer is negative.

Question 9.1 Suppose ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k). Given n ≥ 0, does there exist an exact sequence of the following form?

X^{(n)} : 0 → X_{r} →^{ν_{r}} X_{r−1} →^{ν_{r−1}} ⋯ → X_{1} →^{ν_{1}} X_{0} → 0

where

(i) X_{i}^{(n)} = X_{i} = ⨁_{\begin{matrix}S⊆\{ζ_{1},...,ζ_{r}\} \\ |S|=i \end{matrix}} X̃_{S}.

(ii) X̃_{S} ≅ Ω^{n+deg(S)}(k) where deg(S) = ∑_{ζ_{i}∈S} deg(ζ_{i}).

(iii) If S′ = S ∪ {ζ_{j}} with ζ_{j} ∉ S, then the part of the boundary map ν_{|S′|} going from X̃_{S′} to X̃_{S} is (−1)^{t} times a map representing ζ_{j}, where t = |{i | ζ_{i} ∈ S, i < j}|.

(iv) If |S′| = |S| + 1 but S ⊈ S′ then the part of the boundary map ν_{|S′|} going from X̃_{S′} to X̃_{S} is zero.

Question 9.2 Suppose ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k). Let M be a kGmodule, n ≥ 0, and consider the cochain complex Y_{(n)} = Y(M; n; ζ_{1}, …, ζ_{r}) defined as follows.

Y_{(n)} : 0 → Y^{0} → Y^{1} → ⋯ → Y^{r−1} → Y^{r} → 0

where

(i) Y_{(n)}^{i} = Y^{i} = ⨁_{\begin{matrix}S⊆\{ζ_{1},...,ζ_{r}\} \\ |S|=i \end{matrix}} Ỹ^{S}.

(ii) Ỹ^{S} ≅ H^{n+deg(S)}(G, M) where deg(S) = ∑_{ζ_{i}∈S} deg(ζ_{i}).

(iii) If S′ = S ∪ {ζ_{j}} with ζ_{j} ∉ S, then the part of the coboundary map going from Ỹ^{S} to Ỹ^{S′} is (−1)^{t} times cup product with ζ_{j}, where t = |{i | ζ_{i} ∈ S, i < j}|.

(iv) If |S′| = |S| + 1 but S ⊈ S′ then the part of the coboundary map going from Ỹ^{S} to Ỹ^{S′} is zero.

Is Y_{(n)} an exact sequence? (It is easy to check that Y_{(n)} is a cochain complex.)

The complex Y_{(n)} is actually a singly graded piece of the Ê_{2} page of the spectral sequence considered at the end of the last section. That is, it is the complex

0 → Ê_{2}^{n+s′−2r,r} → Ê_{2}^{n+s′−2r+2,r−1} → ⋯ → Ê_{2}^{n+s′,0} → 0

where the boundary map is d_{2} and M is replaced by N. Interestingly, Proposition 8.5 implies that Y_{(n)} is an exact sequence for all n sufficiently large. This is a consequence of the fact that the Tor_{A}^{∗}(k, N) has finite k-dimension. Moreover the question of whether Question 9.2 has an affirmative answer for all n > 0 is independent of the choice of the set of parameters. See [8] for more details.

Proposition 9.3 Suppose that ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k). Then the following are equivalent.

(i) Question 9.1 has an affirmative answer for the elements ζ_{1}, …, ζ_{r} and for all n ≥ 0.

(ii) Question 9.2 has an affirmative answer for the elements ζ_{1}, …, ζ_{r}, all n ≥ 0 and all simple kG-modules M.

Proof We first prove that (i) implies (ii). Assume that for each n ≥ 0 there is a complex X^{(n)} having the prescribed properties. Recall that if M is a simple kG-module then

Hom_{kG}(Ω^{m}k, M) ≅ H^{m}(G, M)

and

Ext_{kG}^{l}(Ω^{m}k, M) ≅ Hom_{kG}(Ω^{m+l}k, M) ≅ H^{m+l}(G, M).

Moreover if μ : Ω^{m+l}k → Ω^{m}k represents an element ζ ∈ H^{l}(G, k) then

μ^{∗} : Hom_{kG}(Ω^{m}k, M) → Hom_{kG}(Ω^{m+l}k, M)

is given by cup product with ζ. It follows that the complex Hom_{kG}(X^{(n)}, M) is isomorphic to Y_{(n)} = Y(M, n; ζ_{1}, …, ζ_{r}) by making the obvious identifications. Moreover

Ext_{kG}^{l}(X_{i}^{(n)}, M) ≅ Y_{(n+l)}^{i}.

So the complex Hom_{kG}(X^{(n)}, M) can be written as

0 →Hom_{kG}(X_{0}^{(n)}, M)⟶^{ν_{1}^{(n)∗}}Hom_{kG}(X_{1}^{(n)}, M)→ ⋯ →Hom_{kG}(X_{r}^{(n)}, M) → 0
∥≀  ∥≀  ∥≀
0 →  Y_{(n)}^{0}  ⟶  Y_{(n)}^{1}  → ⋯ →  Y_{(n)}^{r}  → 0.

Since Hom is left exact, ν_{1}^{(n)∗} is injective. Hence

H^{0}(Y_{(n)}) ≅ H^{0}(Hom_{kG}(X^{(n)}, M)) = 0.

The short exact sequence

0 → ν_{2}^{(n)}(X_{2}^{(n)}) ⟶^{i_{1,n}} X_{1}^{(n)} ⟶^{ν_{1}^{(n)}} X_{0}^{(n)} → 0.

gives rise to the long exact sequence

0 → Hom_{kG}(X_{0}^{(n)}, M) ⟶^{ν_{1}^{(n)∗}} Hom_{kG}(X_{1}^{(n)}, M) ⟶^{i_{1,n}^{∗}} Hom_{kG}(ν_{2}^{(n)}(X_{2}^{(n)}), M)

→ Ext_{kG}^{1}(X_{0}^{(n)}, M) → Ext_{kG}^{1}(X_{1}^{(n)}, M) → · · · .

Now for every j > 0 the map

Ext_{kG}^{j}(X_{0}^{(n)}, M) ⟶^{ν_{1}^{(n)∗}} Ext_{kG}^{j}(X_{1}^{(n)}, M)
∥≀  ∥≀
Y_{(n+j)}^{0}  ⟶^{ν_{1}^{(n+j)∗}}  Y_{(n+j)}^{1}

is injective. So in the long exact sequence, all connecting homomorphisms are zero and in particular i_{1,n}^{∗} is surjective. We also have a sequence

0 → ν_{3}^{(n)}(X_{3}^{(n)}) ⟶^{i_{2,n}} X_{2}^{(n)} ⟶^{ε_{2,n}} ν_{2}^{(n)}(X_{2}^{(n)}) → 0,

and the corresponding long exact sequence is

0 → Hom_{kG}(ν_{2}^{(n)}(X_{2}^{(n)}), M) ⟶^{ε_{2,n}^{∗}} Hom_{kG}(X_{2}^{(n)}, M) ⟶^{i_{2,n}^{∗}} Hom_{kG}(ν_{3}^{(n)}(X_{3}^{(n)}), M)

→ Ext_{kG}^{1}(ν_{2}^{(n)}(X_{2}^{(n)}), M) ⟶^{ε_{2,n}^{∗}} Ext_{kG}^{1}(X_{2}^{(n)}, M) → · · · . (9.4)

So the map ε_{2,n}^{∗} is injective, and hence the kernel of

ν_{2}^{(n)∗} = ε_{2,n}^{∗} ∘ i_{1,n}^{∗} : Hom_{kG}(X_{1}^{(n)}, M) → Hom_{kG}(X_{2}^{(n)}, M)

is precisely

Ker(i_{1,n}^{∗}) = ν_{1}^{(n)∗}(Hom_{kG}(X_{0}^{(n)}, M)) = ν_{1}^{(n)∗}(Y_{(n)}^{0}).

Therefore H^{1}(Y_{(n)}) = 0. Notice further that

Ext_{kG}^{j}(ν_{2}^{(n)}(X_{2}^{(n)}), M) ≅ Y_{(n+j)}^{1}/ν_{1}^{(n+j)∗}(Y_{(n+j)}^{0}).

So because H^{1}(Y_{(n+j)}) = 0, the map

Ext_{kG}^{j}(ν_{2}^{(n)}(X_{2}^{(n)}), M) ⟶^{ε_{2,n}^{∗}} Ext_{kG}^{j}(X_{2}^{(n)}, M)
∥≀  ∥≀
Y_{(n+j)}^{1}/ν_{1}^{(n+j)∗}(Y_{(n+j)}^{0}) ⟶  Y_{(n+j)}^{2}

is injective for all n, j ≥ 0. Consequently in the long exact sequence (9.4), the connecting homomorphisms are all zero, and so for every n, j ≥ 0 the map

Ext_{kG}^{j}(X_{2}^{(n)}, M) ⟶^{i_{2,n}^{∗}} Ext_{kG}^{j}(ν_{3}^{(n)}(X_{3}^{(n)}), M)

is surjective, so that Ext_{kG}^{j}(ν_{3}^{(n)}(X_{3}^{(n)}), M) ≅ Y_{(n+j)}^{2}/ν_{2}^{(n+j)}(Y_{(n+j)}^{1}). Continuing in this way, we see that ε_{3,n}^{∗} is injective and so H^{2}(Y_{(n)}) = 0, and so on. It follows that the homology of Y_{(n)} is zero and hence Y_{(n)} is an exact sequence. This completes the proof that (i) implies (ii).

We now give an alternative proof that (i) implies (ii). Let (P, ∂) be a projective resolution of the trivial kG-module k. Consider the double complexes X^{(n)} ⊗_{k} P. Notice that in each such complex the rows

0 → X_{r}^{(n)} ⊗ P_{i} → ⋯ ⟶^{ν⊗1} X_{1}^{(n)} ⊗ P_{i} ⟶^{ν⊗1} X_{0}^{(n)} ⊗ P_{i} → 0

are totally split exact sequences of projective modules. So the total complex of the double complex Hom_{kG}(X_{∗}^{(n)} ⊗ P_{∗}, M) is an exact sequence and hence has zero homology.

So consider the spectral sequence of the double complex

E_{0}^{pq} = Hom_{kG}(X_{p}^{(n)} ⊗ P_{q}, M).

We obtain one such spectral sequence for each value of n ≥ 0. Taking the boundary with respect to the differential d_{0} = (1 ⊗ ∂)^{∗}, we obtain the E_{1} term

E_{1}^{pq} = Ext_{kG}^{q}(X_{p}^{(n)}, M) ≅ Y_{(n+q)}^{p}.

This is because (X_{p}^{(n)} ⊗P, 1⊗∂) is a projective resolution of X_{p}^{(n)}. The isomorphism with Y_{(n+q)}^{p} was discussed earlier. But notice that the map induced from (ν ⊗ 1)^{∗} on E_{1}^{pq} is simply the differential on Y_{(n+q)}. So the homology with respect to this differential is

E_{2}^{pq} = H^{p}(Y_{(n+q)}).

Since this is a first quadrant cohomological spectral sequence, the terms E_{2}^{0,0} and E_{2}^{1,0} live until the E_{∞} term. By the previous paragraph the spectral sequence converges to zero. So E_{∞}^{pq} = 0 for all p and q. Therefore H^{0}(Y_{(n)}) = E_{2}^{0,0} = 0 and H^{1}(Y_{(n)}) = E_{2}^{1,0} = 0. Since this happens for every value of n ≥ 0, we must have E_{2}^{0,q} = 0 and E_{2}^{1,q} = 0 for all q. But now E_{2}^{2,0} and E_{2}^{3,0} must live until the E_{∞} term. So again H^{2}(Y_{(n)}) = E_{2}^{2,0} = 0 and H^{3}(Y_{(n)}) = E_{2}^{3,0} = 0 for every n ≥ 0. Continuing this argument we see that H^{∗}(Y_{(n)}) = 0 for all n ≥ 0.

We now turn our attention to proving that (ii) implies (i). We are given an h.s.o.p. ζ_{1}, …, ζ_{r}. For each i = 1, …, r, let U^{(i)} be the chain complex

0 → Ω^{deg(ζ_{i})}(k) →^{ζ̂_{i}} k → 0
∥  ∥
0 →  U_{1}^{(i)}  → U_{0}^{(i)} → 0

having homology H_{∗}(U^{(i)}) = H_{1}(U^{(i)}) = L_{ζ_{i}}. Let U = U^{(1)} ⊗ ⋯ ⊗ U^{(r)}, which has homology

H_{∗}(U) = H_{n}(U) = L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}},

which is a projective kG-module (see the first paragraph of this section). The implication (ii) ⇒ (i) is a consequence of the following proposition.

Proposition 9.5 Suppose that Question 9.2 has an affirmative answer for the elements ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, k) and all simple kG-modules M. Then for all n ≥ 0 there exist complexes X^{(n)} and Z^{(n)} such that the following hold.

(i) U ⊗ Ω^{n}k ≅ X^{(n)} ⊕ Z^{(n)}.

(ii) X^{(n)} is exact (H_{∗}(X^{(n)}) = 0) and no X_{i}^{(n)} has a projective summand.

(iii) Z^{(n)} is a complex of projective kG-modules with H_{∗}(Z^{(n)}) = H_{r}(Z^{(n)}) = L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}} ⊗ Ω^{n}k.

Proof First note that we have an exact sequence

0 → H_{r}(U ⊗ Ω^{n}k) →^{i} U_{r} ⊗ Ω^{n}k → ⋯ → U_{0} ⊗ Ω^{n}k → 0

where the map i is the inclusion of the homology, and

H_{r}(U ⊗ Ω^{n}k) = L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}} ⊗ Ω^{n}k,

which is a projective module. If we let Z′ be the complex with Z′_{r} = H_{r}(U ⊗ Ω^{n}k), and Z′_{i} = 0 for i ≠ r, then we have a split injective map of complexes Z′ → U⊗Ω^{n}k. Let V^{(n)} be the cokernel, so that U ⊗ Ω^{n}k ≅ V^{(n)} ⊕ Z′ and V^{(n)} is exact. The main step in the proof is to show that the complex Hom_{kG}(V^{(n)}, M) is exact for all simple kG-modules M.

First we need some notation. If A and B are kG-modules, we denote by the expression P Hom_{kG}(A, B) the subspace of Hom_{kG}(A, B) consisting of maps which factor through some projective module. Let

\underline{Hom}_{kG}(A, B) = Hom_{kG}(A, B)/P Hom_{kG}(A, B).

Of course if A or B is projective then \underline{Hom}_{kG}(A, B) = 0. In the event that A = Ω^{s}k ⊕ (projective), we have \underline{Hom}_{kG}(A, B) = H^{s}(G, B). In particular, for M simple, \underline{Hom}_{kG}(U_{i}^{(n)}, M) ≅ Y^{i}(M; n; ζ_{1}, …, ζ_{r}). So we have an exact sequence

0 → \underline{Hom}_{kG}(V_{0}^{(n)}, M) → ⋯ → \underline{Hom}_{kG}(V_{r}^{(n)}, M) → 0

for any n ≥ 0. Similarly

Ext_{kG}^{t}(V_{i}^{(n)}, M) ≅ Y^{i}(M; n + t; ζ_{1}, …, ζ_{r}).

So for M simple, the sequence

0 → Ext_{kG}^{t}(V_{0}^{(n)}, M) → ⋯ → Ext_{kG}^{t}(V_{r}^{(n)}, M) → 0

is exact, for all t > 0 and n ≥ 0.

Now let (P, ∂) be a projective resolution of k and let M be a simple kG-module. Consider the spectral sequence of the double complex

E_{0}^{pq} = Hom_{kG}(V_{p}^{(n)} ⊗ P_{q}, M).

As V^{(n)} is an exact sequence, the homology of the total complex is zero and so the spectral sequence converges to zero. Taking the boundary (1 ⊗ ∂)^{∗} we get the E_{1} term

E_{1}^{pq} = Ext_{kG}^{q}(V_{p}^{(n)}, M).

Now take the boundary induced by (ν ⊗ 1)^{∗} to get

E_{2}^{pq} = H^{p}(Ext_{kG}^{q}(V_{∗}^{(n)}, M), (ν ⊗ 1)^{∗}).

If q > 0 then we have already seen that Ext_{kG}^{q}(V_{∗}^{(n)}, M) ≅ Y_{(n+q)}^{∗}, which is an exact sequence. So E_{2}^{pq} = 0 for q > 0. Therefore the spectral sequence collapses onto the first row of the E_{2} page, and all further differentials are zero. Because the spectral sequence converges to zero, we have

H^{i}(Hom_{kG}(V^{(n)}, M), (ν ⊗ 1)^{∗}) = E_{2}^{i,0} = 0.

So the complex Hom_{kG}(V^{(n)}, M) is exact.

Now notice that P Hom_{kG}(V^{(n)}, M) is a subcomplex of Hom_{kG}(V^{(n)}, M). Moreover, the quotient complex is

\underline{Hom}_{kG}(V^{(n)}, M) ≅ Y_{(n)}.

So we have an exact sequence of complexes

0 → P Hom_{kG}(V^{(n)}, M) → Hom_{kG}(V^{(n)}, M) → Y_{(n)} → 0

in which the last two terms have no cohomology. So by the long exact sequence in cohomology we have H^{∗}(P Hom_{kG}(V^{(n)}, M)) = 0.

With this information we proceed to strip the projective modules from the sequence V^{(n)}. Recall that we have an exact sequence

0 → V_{r}^{(n)} →^{ν} V_{r−1}^{(n)} →^{ν} ⋯ →^{ν} V_{0}^{(n)} → 0.

Let P_{M} be the projective cover of the simple module M. Then

dim_{k} P Hom_{kG}(V_{i}^{(n)}, M)

is the number of summands isomorphic to P_{M} in any direct sum decomposition of V_{i}^{(n)} into indecomposable pieces. Let i be the least index such that we have P Hom_{kG}(V_{i}^{(n)}, M) ≠ 0, and let 0 ≠ θ ∈ P Hom_{kG}(V_{i}^{(n)}, M). Then θ extends to a homomorphism ψ : V_{i}^{(n)} → P_{M} (i.e., if ε : P_{M} → M then ε ∘ ψ = θ). Since P Hom_{kG}(V^{(n)}, M) is exact, we have θν ≠ 0. So we have a map of chain complexes

0 → V_{r}^{(n)} → V_{r−1}^{(n)} → ⋯ → V_{i+1}^{(n)} → V_{i}^{(n)} → ⋯ → V_{0}^{(n)} → 0.
ψν ↓  ↓ψ
0 → P_{M} = P_{M} → 0

Note that ψ and ψν are surjective because εψ = θ ≠ 0 and εψν = θν ≠ 0. So if Q_{M} is the complex (0 → P_{M} →^{id} P_{M} → 0) concentrated in degrees i and i + 1, then we have a split short exact sequence of complexes

0 → V^{(n)}′ → V^{(n)} → Q_{M} → 0.

So V^{(n)} ≅ V^{(n)}′ ⊕ Q_{M}. In this way we can eliminate (in pairs) all projective summands from V^{(n)} and obtain an exact sequence of the prescribed form.

We now show how the questions above have implications for the Poincaré series of the cohomology of G.

Proposition 9.6 Suppose that ζ_{1}, …, ζ_{r} is an h.s.o.p. for H^{∗}(G, k) with deg(ζ_{i}) = n_{i}, and let n ≥ 0. Suppose that Question 9.2 has an affirmative answer. Then

∑_{i=0}^{r}(−1)^{i} ∑_{\begin{matrix}S⊆\{ζ_{1},...,ζ_{r}\} \\ |S|=i \end{matrix}} dim_{k} H^{n+deg(S)}(G, M) = 0.  (9.7)

for all n ≥ 0 and all simple modules M, where deg(S) = ∑_{ζ_{i}∈S} deg(ζ_{i}).

Proof This is an easy consequence of the exactness of Y(M; n; ζ_{1}, …, ζ_{r}).

Now let P_{M}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(G, M) be the Poincaré series for M. Set T = {ζ_{1}, …, ζ_{r}}, multiply equation (9.7) by t^{n+deg(T)} and sum over n. We get

0 = ∑_{n=0}^{∞} ∑_{S⊆T}(−1)^{|S|}t^{n+deg(T)} dim_{k} H^{n+deg(S)}(G, M)
= ∑_{n+0}^{∞} ∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)}(t^{n+deg(S)} dim_{k} H^{n+deg(S)}(G, M))

= ∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)} ( ∑_{i=deg(S)}^{∞} t^{i} dim_{k} H^{i}(G, M))

so that

∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)}P_{M}(t) = ∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)}( ∑_{i=0}^{deg(S)−1} t^{i} dim_{k} H^{i}(G, M)).

Now we have

∏_{j=1}^{r}(1 − t^{n_{j}}) = (−1)^{r}t^{deg(T)} ∏_{j=1}^{r}(1 − t^{−n_{j}})
= (−1)^{r} ∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)}

and so substituting in the previous equation we find that

∏_{j=1}^{r}(1 − t^{n_{j}})P_{M}(t) = (−1)^{r} ∑_{S⊆T}(−1)^{|S|}t^{deg(T)−deg(S)} ( ∑_{i=0}^{deg(S)−1} t^{i} dim_{k} H^{i}(G, M)) .

Replacing S by T \ S and noting that (−1)^{r}(−1)^{|S|} = (−1)^{|T \S|}, this becomes

P_{M}(t) = \frac{∑_{S⊆T}(−1)^{|S|}t^{deg(S)} ( ∑_{i=0}^{deg(T)−deg(S)−1} t^{i} dim_{k} H^{i}(G, M))}{∏_{j=1}^{r}(1 − t^{n_{j}})} .

Note that the numerator in the above expression is a polynomial.

10 Quasi-regular sequences

There is a notion weaker that that of a regular sequence, which we now introduce, and which is good enough to give a positive answer to the questions posed in Section 9.

Definition 10.1 Suppose that A = ⨁_{n≥0} A_{n} is a finitely generated graded commutative k-algebra, and M = ⨁_{n≥0} M_{n} is a finitely generated graded A-module. A sequence ζ_{1}, …, ζ_{r} of homogeneous elements of degree n_{i} in A is said to be a quasiregular sequence for M if for each i = 1, …, r the map

M_{n}/M_{n} ∩ (ζ_{1}, …, ζ_{i−1})M → M_{n+n_{i}}/M_{n+n_{i}} ∩ (ζ_{1}, …, ζ_{i−1})M

induced by multiplication by ζ_{i} is injective, whenever n ≥ n_{1} + ⋯ + n_{i−1}. In particular, ζ_{1} : M_{n} → M_{n+n_{1}} is injective for all n ≥ 0.

Serre has investigated a similar definition in which the generators are assumed to be in the same degree. His conclusions are rather different from ours. For further information see the letter from Serre reproduced in the appendix to Guillemin and Sternberg [15]. Note that our terminology has nothing to do with the terminology of quasi-regular sequences used in Matsumura [17].

Proposition 10.2 Suppose that A = ⨁_{n≥0} A_{n} is a finitely generated graded commutative k-algebra, M = ⨁_{n≥0} M_{n} is a finitely generated graded A-module, and ζ_{1}, …, ζ_{r} is a quasi-regular sequence for M. For each n ≥ 0, consider the cochain complex Y_{(n)} = Y(M; n; ζ_{1}, …, ζ_{r}) defined as follows.

Y_{(n)} : 0 → Y^{0} → Y^{1} → ⋯ → Y^{r} → 0

where

(i) Y_{(n)}^{i} = Y^{i} = ⨁_{\begin{matrix}S⊆\{ζ_{1},...,ζ_{r}\} \\ |S|=i \end{matrix}} Ỹ^{S}.

(ii) Ỹ^{S} ≅ M_{n+deg(S)} where deg(S) = ∑_{ζ_{i}∈S} deg(ζ_{i}).

(iii) If S′ = S ∪ {ζ_{j}} with ζ_{j} ∉ S, then the part of the coboundary map going from Ỹ^{S} to Ỹ^{S′} is (−1)^{t} times multiplication by ζ_{j}, where t = |{i | ζ_{i} ∈ S, i < j}|.

(iv) If |S′| = |S| + 1 but S ⊈ S′ then the part of the coboundary map going from Ỹ^{S} to Ỹ^{S′} is zero.

Then H^{∗}(Y) is concentrated in degree r, and

H^{r}(Y) = M_{n+deg(T)}/M_{n+deg(T)} ∩ (ζ_{1}, …, ζ_{r})M,

where T = {ζ_{1}, …, ζ_{r}}.

Here the complexes Y_{(n)} are the singly graded pieces of the Koszul complex of M as a k[ζ_{1}, …, ζ_{r}]-module, as also in Question 9.2.

Proof We proceed by induction on r. Note that if r = 1 then the complex Y is simply

0 → M_{n} →^{ζ_{1}} M_{n+deg(ζ_{1})} → 0

and because multiplication by ζ_{1} is injective, the proposition is obvious in this case.

So assume that r > 1. It is easy to check that Y = Y(M; n; ζ_{1}, …, ζ_{r}) is a complex. The subcomplex consisting of all the Ỹ^{S} such that ζ_{r} ∈ S is

Y′[1] ≅ Y(M; n + deg(ζ_{r}); ζ_{1}, …, ζ_{r−1})[1],

and the quotient is

Y/Y′[1] ≅ Y(M; n; ζ_{1}, …, ζ_{r−1}).

By induction, Y′ and Y/Y′[1] satisfy the conclusion of the proposition. Thus the long exact sequence in cohomology of the short exact sequence 0 → Y′[1] → Y → Y/Y′[1] → 0 of cochain complexes takes the form

0 → H^{r−1}(Y) → H^{r−1}(Y/Y′[1]) →^{θ} H^{r−1}(Y′) → H^{r}(Y) → 0.

But θ is the map

M_{n+deg(T′)}/M_{n+deg(T′)} ∩ (ζ_{1}, …, ζ_{r−1})M ⟶^{(−1)^{r−1}ζ_{r}}

M_{n+deg(T)}/M_{n+deg(T)} ∩ (ζ_{1}, …, ζ_{r−1})M

where T′ = {ζ_{1}, …, ζ_{r−1}} and T = {ζ_{1}, …, ζ_{r}}. This map is injective, according to the definition of a quasi-regular sequence. This completes the proof of the proposition.

We now consider the case in which A = H^{∗}(G, k) for G a finite group and k a field of characteristic p, and M = H^{∗}(G, D) for D a bounded complex of finitely generated kG-modules.

Proposition 10.3 Let D be a bounded complex of finitely generated kG-modules, and suppose that ζ_{1}, …, ζ_{r} are non-zero elements of H^{∗}(G, k) forming an h.s.o.p. for H^{∗}(G, D), with the property that ζ_{1}, …, ζ_{r−1} is a quasi-regular sequence for H^{∗}(G, D). Then ζ_{1}, …, ζ_{r} is a quasi-regular sequence for the module H^{∗}(G, D) and the complex Y = Y(H^{∗}(G, D); n; ζ_{1}, …, ζ_{r}) is exact (i.e., H^{∗}(Y) = 0) for all n ≥ 0.

Proof Each ζ_{i} is represented by a unique non-zero homomorphism Ω^{−n_{i}}(ζ̂_{i}) : k → Ω^{−n_{i}}(k) (cf. Proposition 6.2). Let L_{i} be the cokernel of Ω^{−n_{i}}(ζ̂_{i}), so that we have a short exact sequence

0 → k ⟶^{Ω^{−n_{i}}(ζ̂_{i})} Ω^{−n_{i}}(k) → L_{i} → 0.

Let V_{(j,n)} = Y(H^{∗}(G, D); n; ζ_{1}, …, ζ_{j}). We claim that V_{(j,n)} only has cohomology in degree j, and that its cohomology there is

H^{j}(V_{(j,n)}) = H^{n+∑_{i=1}^{j} n_{i}}(G, D)/H^{n+∑_{i=1}^{j} n_{i}}(G, D) ∩ (ζ_{1}, …, ζ_{j})H^{∗}(G, D)
≅ H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j}).

The proof of this fact follows the scheme of the proof of the last proposition. For j = 1 we have an exact sequence of chain complexes

0 → D ⟶^{1⊗Ω^{−n_{1}}(ζ̂_{1})} D ⊗ Ω^{−n_{1}}(k) ⟶ D ⊗ L_{1} → 0

and hence a long exact sequence

⋯ → H^{n}(G, D) ⟶^{(1⊗Ω^{−n_{1}}(ζ̂_{1}))_{∗}} H^{n}(G, D ⊗ Ω^{−n_{1}}(k)) → H^{n}(G, D ⊗ L_{1}) → · · · .

But H^{n}(G, D⊗Ω^{−n_{1}}(k)) ≅ H^{n+n_{1}}(G, D) and the map (1⊗Ω^{−n_{1}}(ζ̂_{1}))_{∗} is cup product with ζ_{1}, which is injective. So all the connecting homomorphisms are zero, and the proposition is proved in this case.

Now suppose that 1 < j < r. Then we have an exact sequence of chain complexes

0 → D⊗L_{1}⊗⋯⊗L_{j−1} ⟶^{1⊗Ω^{−n_{j}} (ζ̂_{j})} D⊗L_{1}⊗⋯⊗L_{j−1}⊗Ω^{−n_{j}}(k) → D⊗L_{1}⊗⋯⊗L_{j} → 0

and the long exact sequence in cohomology takes the form

⋯ → H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j−1})⟶^{1⊗Ω^{−n_{j}} (ζ̂_{j})}H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j−1} ⊗ Ω^{−n_{j}}(k))
↘ζ_{j}  ∥≀
H^{n+n_{j}}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j−1})

and by induction multiplication by ζ_{j} is injective.

As in the proof of the last proposition, we have an exact sequence of complexes

0 → V_{(j−1,n+n_{j})}[1] → V_{(j,n)} → V_{(j−1,n)} → 0

and so taking cohomology we have

0 → H^{j−1}(V_{(j−1,n)}) →^{δ}  H^{j−1}(V_{(j−1,n+n_{j})})  → H^{j}(V_{(j,n)}) → 0
∥≀  ∥≀
H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j−1})→^{ζ_{j}}H^{n+n_{j}}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j−1}).

So the connecting homomorphism δ, which is equal to cup product with ζ_{j}, is injective with cokernel H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{j}), which proves the claim.

Now consider the case in which j = r. Note that by Theorem 4.3, we have H^{∗}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{r}) = 0. Thus multiplication by ζ_{r} is an isomorphism from H^{n}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{r−1}) onto H^{n+n_{r}}(G, D ⊗ L_{1} ⊗ ⋯ ⊗ L_{r−1}). From the exact sequence of complexes

0 → V_{(r−1,n+n_{r})}[1] → V_{(r,n)} → V_{(r−1,n)} → 0

we get an exact sequence in cohomology

0 → H^{r−1}(V_{(r,n)}) → H^{r−1}(V_{(r−1,n)}) → H^{r−1}(V_{(r−1,n+n_{r})}) → H^{r}(V_{(r,n)}) → 0.

The middle map is an isomorphism, and so H^{∗}(V_{(r,n)}) = 0. Hence

V_{(r,n)} = Y(H^{∗}(G, D); n; ζ_{1}, …, ζ_{r})

is an exact sequence.

Corollary 10.4 Let G be a finite group, k a field of characteristic p, and M any kGmodule. Suppose that ζ_{1}, …, ζ_{r} are non-zero elements of H^{∗}(G, k) forming a h.s.o.p. for H^{∗}(G, M), with the property that ζ_{1}, …, ζ_{r−1} is a quasi-regular sequence for H^{∗}(G, M). Then Question 9.2 has an affirmative answer for the elements ζ_{1}, …, ζ_{r} and all n ≥ 0 for the module M.

Proof This follows immediately from Proposition 10.3 by taking for D the module M concentrated in degree zero.

Corollary 10.5 Suppose that G has p-rank 2. Let ζ_{1}, ζ_{2} be a h.s.o.p. for H^{∗}(G, k). Then either ζ_{1}, ζ_{2} is a quasi-regular sequence for H^{∗}(G, k) or ζ_{2}, ζ_{1} is.

Proof By the proposition, it suffices to show that either ζ_{1} or ζ_{2} is not a zero divisor in H^{∗}(G, k). But this follows from an easy spectral sequence argument.

Proposition 10.6 Let P be a Sylow p-subgroup of G. If a sequence of homogeneous elements ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, k) ⊆ H^{∗}(P, k) is a quasi-regular sequence for H^{∗}(P, k), then it is also a quasi-regular sequence for H^{∗}(G, k).

Proof This follows directly from Lemma 6.7 and the definition of quasi-regular sequence.

Remark In [8] it is shown that Corollary 10.4 has a strong converse. That is, if the answer to Question 9.2 is affirmative for some set of parameters and a particular module M, then H^{∗}(G, M) has a quasi-regular sequence which is also a sytem of parameters for H^{∗}(G, M) in H^{∗}(G, k).

11 Secondary operations

In this section, we determine some more of the differentials in the spectral sequence described in Theorem 5.5. These differentials are given in terms of some secondary operations which we describe in terms of matric Massey products. Accounts of the theory of matric Massey products and their use in determining differentials in spectral sequences may be found in May [18], Ravenel [24] (Appendix A1.4) and McCleary [19] §8.3.3.

In case H^{∗}(G, D) is not Cohen–Macaulay as a module over H^{∗}(G, k), the first example of a possibility for a non-trivial differential not determined by the relations d_{n_{i}}(ζ̃_{i}) = ζ_{i} is the differential d_{n_{i}+n_{j}−1}(α.ζ̃_{i}ζ̃_{j}) in case α.ζ_{i} = α.ζ_{j} = 0.

We suppose for simplicity that the multiplication on cohomology is given by a strictly coassociative diagonal approximation Δ : P → P⊗P. This can be arranged by taking for P the bar resolution. Note, however, that the diagonal approximation Δ cannot in general be chosen to be strictly cocommutative, and it is this fact that is responsible for the existence of Steenrod operations. Similarly, the lack of strict coassociativity for the minimal resolution is responsible for the usual Massey triple products.

Suppose that D is a chain complex of kG-modules, bounded above, and α ∈ H^{a}(G, D) has the property that α.ζ_{i} = α.ζ_{j} = 0. Choose cocycles u representing α and η_{i} representing ζ_{i}. Since α.ζ_{i} = 0 we have u.η_{i} = du_{i}, and similarly u.η_{j} = du_{j}, for suitable cochains u_{i} and u_{j}. Since

ζ_{i}ζ_{j} − (−1)^{n_{i}n_{j}}ζ_{j}ζ_{i} = 0,

we have

η_{i}η_{j} − (−1)^{n_{i}n_{j}}η_{j}η_{i} = dη_{ij}.

To simplify some signs, we shall assume that if p is odd, the n_{i} are even. This will be true in the case we are interested in, namely if the ζ_{i} generate a polynomial ring, since elements of odd degree square to zero in case p is odd. So we have

d(u_{i}η_{j} − u_{j}η_{i} − (−1)^{a}uη_{ij}) = 0

and we define

⟨α|ζ_{i}, ζ_{j}⟩ = [u_{i}η_{j} − u_{j}η_{i} − (−1)^{a}uη_{ij}]
∈ H^{a+n_{i}+n_{j}−1}(G, D).

This operation may be expressed in terms of matric Massey products as follows:

⟨α|ζ_{i}, ζ_{j}⟩ = ⟨α, (ζ_{i}, ζ_{j}), ( \begin{matrix}ζ_{j} \\ −ζ_{i} \end{matrix} )⟩.

Lemma 11.1 In the spectral sequence

E_{2}^{pq}(ζ_{i}, ζ_{j}) = Ext_{kG}^{p}(H_{q}(C_{ζ_{i}} ⊗ C_{ζ_{j}}), D) ⇒ Ext_{kG}^{p+q}(C_{ζ_{i}} ⊗ C_{ζ_{j}}, D)

if α ∈ H^{∗}(G, D) with α.ζ_{i} = α.ζ_{j} = 0, then

d_{n_{i}+n_{j}−1}(α.ζ̃_{i}ζ̃_{j}) = ⟨α|ζ_{i}, ζ_{j}⟩.

Proof Consider the commutative diagram of chain complexes

0  0  0
↓  ↓  ↓
0→Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj}^{(∞)},D)[n_{i}+n_{j}]→^{ζ̂i}Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj}^{(∞)},D)[n_{j}]→^{πi}Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj} ,D)[n_{j}]→0
↓ζ̂j ↓ζ̂j ↓ζ̂j
0→ Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj}^{(∞)},D)[n_{i}] →^{ζ̂i} Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj}^{(∞)},D) →^{πi} Hom_{kG}(P⊗C_{ζi}^{(∞)}⊗C_{ζj} ,D) →0
↓π_{j} ↓π_{j} ↓π_{j}
0→ Hom_{kG}(P⊗C_{ζi}⊗C_{ζj}^{(∞)},D)[n_{i}] →^{ζ̂i} Hom_{kG}(P⊗C_{ζi}⊗C_{ζj}^{(∞)},D) →^{πi} Hom_{kG}(P⊗C_{ζi}⊗C_{ζj} ,D) →0
↓  ↓  ↓
0  0  0

Taking homology, we obtain a mesh of long exact sequences, part of which is as follows.

→ Ext_{kG}^{n_{i}+a−1}(k, D) ⟶^{(π_{i})_{∗}} Ext_{kG}^{n_{i}+a−1}(C_{ζ_{i}}, D) → Ext_{kG}^{a}(k, D) →^{ζ_{i}}Ext_{kG}^{n_{i}+a}(k, D)
↓ζ_{j}  ↓ζ_{j}  ↓ζ_{j}  ↓
→Ext_{kG}^{n_{i}+n_{j}+a−1}(k, D)⟶^{(π_{i})_{∗}}Ext_{kG}^{n_{i}+n_{j}+a−1}(C_{ζ_{i}}, D)→Ext_{kG}^{n_{j}+a}(k, D)→
↓

The differential

d_{n_{i}+n_{j}−1} : Ext_{kG}^{a}(k, D) → Ext_{kG}^{n_{i}+n_{j}+a−1}(k, D)

is the switchback map in this diagram. By Lemma 3.1, the maps marked ζ_{i}, ζ_{j} are given by multiplication by ζ_{i}, ζ_{j}. By the definition of the connecting homomorphism in cohomology, [π_{i}u_{i}] ∈ Ext_{kG}^{n_{i}+a−1}(C_{ζ_{i}}, D) has image α = [u] in Ext_{kG}^{a}(k, D). We have

(π_{i})_{∗}⟨α|ζ_{i}, ζ_{j}⟩ = [π_{i}(u_{i}η_{j} − u_{j}η_{i} − (−1)^{a}uη_{ij})]
= [π_{i}u_{i}η_{j} − (π_{i} ∘ ζ̂_{i})u_{j}]
= [π_{i}u_{i}].ζ_{j}

in Ext_{kG}^{n_{i}+n_{j}+a−1}(C_{ζ_{i}}, D), and so ⟨α|ζ_{i}, ζ_{j}⟩ is the image of u under this switchback map. Note that only one of the terms in the expression for ⟨α|ζ_{i}, ζ_{j}⟩ survives in this calculation; the rest of the terms are to ensure that it is a cocycle.

The next example of a non-trivial differential is given as follows. If α ∈ H^{a}(G, D) satisfies ⟨α|ζ_{i}, ζ_{j}⟩ = ⟨α|ζ_{i}, ζ_{k}⟩ = ⟨α|ζ_{j}, ζ_{k}⟩ = 0, then there is a possibility for a non-trivial differential d_{n_{i}+n_{j}+n_{k}−2}(α.ζ̃_{i}ζ̃_{j}ζ̃_{k}). Since ⟨α|ζ_{i}, ζ_{j}⟩ = 0, we may choose a cochain u_{ij} with

du_{ij} = u_{i}η_{j} − u_{j}η_{i} − (−1)^{a}uη_{ij}.

Now the element

η_{ij}η_{k} − η_{ik}η_{j} + η_{jk}η_{i} − η_{i}η_{jk} + η_{j}η_{ik} − η_{k}η_{ij}

is a coboundary dη_{ijk}. To see this, note that the complex Hom_{kG}(P ⊗ P ⊗ P, P) is exact in positive degrees (the negative homology in degree −n is Ext_{kG}^{n}(k, k) = H^{n}(G, k)), and that the map sending η_{i} ⊗ η_{j} ⊗ η_{k} to the above element is a map of degree +1 whose coboundary is zero.

The cochain

u_{ij}η_{k} − u_{ik}η_{j} + u_{jk}η_{i} − u_{i}η_{jk} + u_{j}η_{ik} − u_{k}η_{ij} − (−1)^{a}uη_{ijk}

is a cocycle, and we denote by ⟨α|ζ_{i}, ζ_{j}, ζ_{k}⟩ the cohomology class it represents. In terms of matric Massey products, this is given as follows:

⟨α|ζ_{i}, ζ_{j}, ζ_{k}⟩ = ⟨α, (ζ_{i}, ζ_{j}, ζ_{k}), ( \begin{matrix}0 & −ζ_{k} & ζ_{j} \\ ζ_{k} & 0 & −ζ_{i} \\ −ζ_{j} & ζ_{i} & 0 \end{matrix}) , ( \begin{matrix}ζ_{i} \\ ζ_{j} \\ ζ_{k} \end{matrix} )⟩.

The proof of the following is similar to the proof of Lemma 11.1.

Lemma 11.2 In the spectral sequence

E_{2}^{pq}(ζ_{i}, ζ_{j}, ζ_{k}) = Ext_{kG}^{p}(H_{q}(C_{ζ_{i}} ⊗ C_{ζ_{j}} ⊗ C_{ζ_{k}}), D) ⇒ Ext_{kG}^{p+q}(C_{ζ_{i}} ⊗ C_{ζ_{j}} ⊗ C_{ζ_{k}}, D)

if α ∈ H^{∗}(G, D) with ⟨α|ζ_{i}, ζ_{j}⟩ = ⟨α|ζ_{i}, ζ_{k}⟩ = ⟨α|ζ_{j}, ζ_{k}⟩ = 0, then

d_{n_{i}+n_{j}+n_{k}−2}(α.ζ̃_{i}ζ̃_{j}ζ̃_{k}) = ⟨α|ζ_{i}, ζ_{j}, ζ_{k}⟩.

The general case of the above construction goes as follows. We define operations

⟨α|ζ_{i_{1}}, …, ζ_{i_{t}}⟩

under the conditions that αζ_{i_{1}} = ⋯ = αζ_{i_{t}} = 0, and all operations ⟨α| ⋯⟩ are zero for ⋯ a proper subsequence of ζ_{i_{1}}, …, ζ_{i_{t}}. The operation is described as a matric Massey product

⟨α, M_{1}, …, M_{t}⟩

where M_{1}, …, M_{t} are matrices whose entries are either zero or plus or minus one of the ζ_{i_{j}}. The rows and columns of M_{s} (1 ≤ s ≤ t) are indexed by subsets I and J of {i_{1}, …, i_{t}} of sizes s − 1 and s respectively. If I ⊈ J then the corresponding entry is zero, while if J = I ∪ {i_{j}} then the corresponding entry is plus or minus ζ_{i_{j}}. The sign is (−1) to the power of the number of elements of J which come before i_{j}. It is easy to check that M_{s}M_{s+1} = 0. Since the chain complex Hom_{kG}(P^{⊗t}, P) is exact in positive degrees, the matric Massey products of consecutive elements of the list M_{1}, …, M_{t} vanish. So by the conditions on α, the matric Massey product ⟨α, M_{1}, …, M_{t}⟩ is defined, and we write ⟨α|ζ_{i_{1}}, …, ζ_{i_{t}}⟩ for this expression. An argument similar to the proof of Lemma 11.1 shows that if α.ζ̃_{i_{1}} … ζ̃_{i_{t}} lives to E_{n_{i_{1}}+···+n_{i_{t}}−t+1}^{∗∗} then

d_{n_{i_{1}}+···+n_{i_{t}}−t+1}(α.ζ̃_{i_{1}} … ζ̃_{i_{t}}) = ⟨α|ζ_{i_{1}}, …, ζ_{i_{t}}⟩.

An example involving the last survivor described in Section 7 is as follows. We look at the spectral sequence

Ê_{2}^{pq} = Êxt_{kG}^{p}(H_{q}(C), k) ⇒ 0

of the double complex Hom_{kG}(P̂ ⊗ C, k), where P̂ is a complete resolution of the trivial module. Let α be a non-zero element of the one dimensional space Ĥ^{−1}(G, k). Then α.ζ̃_{1} … ζ̃_{r} (r = r_{p}(G)) transgresses to a non-zero element in the base Ê_{s,0}^{s} (s = ∑_{i=1}^{r} n_{i}) of this spectral sequence, representing the last survivor:

⟨α|ζ_{1}, …, ζ_{r}⟩ ∈ E_{∞}^{s,0} ≅ H^{s}(G, k)/Ker(e).

A non-trivial example involving elements of positive degree is described in [6].

References

[1] A. Adem and J. Milgram. A_{5} invariants and the cohomology of L_{3}(4). Preprint.

[2] G. S. Avrunin and L. L. Scott. Quillen stratification for modules. Invent. Math. 66 (1982), 277–286.

[3] D. J. Benson and J. F. Carlson. Diagrammatic methods for modular representations and cohomology. Comm. in Algebra 15 (1987), 53–121.

[4] D. J. Benson and J. F. Carlson. Complexity and multiple complexes. Math. Zeit. 195 (1987), 221–238.

[5] D. J. Benson, J. F. Carlson and G. R. Robinson. On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73.

[6] D. J. Benson and J. F. Carlson. Products in negative cohomology. Preprint.

[7] J. F. Carlson. Products and projective resolutions. Proc. Symp. Pure Maths 47 (Part 1), A.M.S. 1987, 399–408.

[8] J. F. Carlson. Projective resolutions and degree shifting for cohomology and group rings. Proc. Tsukuba workshop on Representations of Algebras (1990) (to appear).

[9] H. Cartan and S. Eilenberg. Homological Algebra. Princeton University Press, 1956.

[10] C. W. Curtis and I. Reiner. Representation theory of finite groups and associative algebras. Wiley-Interscience 1962.

[11] P. Deligne. Séminaire de Géométrie Algébrique du Bois-Marie SGA 4\frac{1}{2}. Springer Lecture Notes in Mathematics 569, Springer-Verlag, Berlin/New York, 1977.

[12] T. Diethelm. The mod p cohomology rings of the nonabelian split metacyclic p-groups. Arch. Math. 44 (1985), 29–38.

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

[14] L. Evens and S. Priddy. The cohomology of the semi-dihedral group. Conf. on Algebraic Topology in honour of Peter Hilton, ed. R. Piccinini and D. Sjerve, Contemp. Math. 37, A.M.S. 1985.

[15] V. W. Guillemin and S. Sternberg. An algebraic model of transitive differential geometry. Bull. Amer. Math. Soc. 70 (1964), 16–47.

[16] S. Mac Lane. Homology. Springer-Verlag, 1963.

[17] H. Matsumura. Commutative ring theory. C.U.P., 1986.

[18] J. P. May. Matric Massey products. J. Algebra 12 (1969), 533–568.

[19] McCleary. User’s guide to spectral sequences. Publish or Perish, Inc., Delaware, 1985.

[20] M. Nakaoka. Homology of the infinite symmetric group. Ann. of Math. 73 (1961), 229–257.

[21] D. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–602.

[22] D. Quillen. The mod 2 cohomology rings of extra-special 2-groups and the spinor groups. Math. Ann. 194 (1971), 197–212.

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

[24] D. Ravenel. Complex cobordism and stable homotopy groups of spheres. Academic press, 1986.

[25] J.-P. Serre. Algèbre locale—multiplicités. Springer Lecture Notes in Mathematics 11, Springer-Verlag, Berlin/New York, 1965.

[26] R. P. Stanley. Invariants of finite groups and their applications to combinatorics. Bull. A.M.S. 1 (3) (1979), 475–511.

[27] R. P. Stanley. Hilbert functions of graded algebras. Advances in Math. 28 (1978), 57–83.