Resolutions and Poincaré Duality for Finite Groups

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

April 7, 1995

Abstract

This talk is a survey of some recent joint work with Jon Carlson on cohomology of finite groups. I shall describe how, for an arbitrary finite group G, one can produce an algebraic analogue of a free G-action on a product of spheres. If k is the field of coefficients, one can use this to build a resolution of k as a kG-module, which consists of a finite Poincaré duality piece and a polynomial piece. This resolution has the same rate of growth as the minimal resolution, but in general is not quite minimal. The deviation from minimality is measured by secondary operations in group cohomology expressible in terms of matric Massey products.

1 Introduction

This talk is intended as a survey of some recent joint work with Jon Carlson [1, 2, 3] on cohomology of finite groups. Let me begin with an example.

Let G be the quaternion group Q_{8} of order eight, and k be the field of two elements. Then it is well known that

H^{∗}(G, k) = k[x, y, z]/(x^{2} + xy + y^{2}, x^{3}, y^{3})

with deg(x) = deg(y) = 1 and deg(z) = 4, so that we have

\begin{matrix}n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & ⋯ \\ dim_{k} H^{n}(G, k) & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & ⋯ \end{matrix}

Thus the cohomology consists of a finite symmetric piece in degrees zero to three, and a polynomial generator in degree four.

There is an easy topological interpretation of this structure, as follows. We can regard the quaternion group as a subgroup of the multiplicative group of unit quaternions. These unit quaternions form a 3-sphere S^{3}, and so G has a free action on S^{3} by left multiplication. If we choose an equivariant CW-decomposition of S^{3} and take cellular chains with coefficients in k, we obtain a complex of free kG-modules of length four, with homology one copy of k at the beginning and the end. Thus we may take the Yoneda splice of this complex with itself infinitely often to obtain a projective resolution of k as a kG-module. Taking cohomology, we obtain

H^{∗}(G, k) = H^{∗}(S^{3}/G, k) ⊗ k[z],

where z is a polynomial generator in degree four. Since S^{3}/G is a compact manifold, its cohomology satisfies Poincaré duality, which explains the symmetry in the first four degrees.

What I wish to outline here is how this picture can be generalised to all finite groups. Given a finite group G and a field k of characteristic p dividing the order of G (otherwise there is no cohomology), there is a construction of a finite Poincaré duality complex which is a sort of algebraic analogue of a free action of G on a product of spheres. This gives rise to a projective resolution of k as a kG-module which consists of a finite Poincaré duality part and a “polynomial” part. The number of polynomial generators required is equal to the number of spheres; this is the p-rank r_{p}(G) (the maximal rank of an elementary abelian p-subgroup of G), in accordance with a theorem of Quillen [9, 10] on the Krull dimension of the cohomology ring. The resolution we construct has the same polynomial rate of growth as the minimal resolution, but in general it is not minimal. The deviation from minimality is measured by certain secondary operations in group cohomology which are like Massey products. These secondary operations only appear in case the cohomology is not Cohen-Macaulay, so that in case the ring is Cohen-Macaulay we can read off from the Poincaré duality statement a functional equation for the Poincaré series. Thus we have the following theorem from Benson and Carlson [3].

Theorem 1.1 Suppose that the cohomology ring H^{∗}(G, k) is Cohen-Macaulay. Then the Poincaré series P_{k}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(G, k) satisfies the functional equation

P_{k}(1/t) = (−t)^{r_{p}(G)}P_{k}(t).

If H^{∗}(G, k) is not Cohen-Macaulay, we still obtain information in terms of a spectral sequence converging to a finite Poincaré duality ring, and in which the differentials are determined by the secondary operations mentioned above.

2 Resolutions and complexity

We begin with a gentle introduction to the cohomology of finite groups, to set the scene. Let G be a finite group, and k be a field. We think of finitely generated kG-modules as being the same thing as representations of G as matrices with entries in k. Recall that Maschke’s theorem says that if the characteristic of k does not divide the order of G then every short exact sequence

0 → M′ → M′′ → M → 0

of kG-modules splits. If the characteristic of k is a prime p dividing the order of G, then there are always counterexamples to Maschke’s theorem, and this is one of the fundamental differences between ordinary and modular representation theory. A module M is projective if Maschke’s theorem holds for short exact sequences ending in M, and M′ is injective if Maschke’s theorem holds for short exact sequences beginning with M′. In fact in this context of finite group algebras over a field, a module is projective if and only if it is injective.

If M is not projective, we can measure the extent to which it “glues” to other modules in the above sense by forming a projective resolution; namely an exact sequence

⋯ → P_{2} → P_{1} → P_{0} → M → 0

in which all the P_{i} are projective. There is a unique minimal resolution for a given module M, and all other resolutions may be obtained from this one by adding on a split exact sequence of projective modules. The nth kernel in the minimal resolution is denoted Ω^{n}(M).

If M′ is another module we can take homomorphisms

Hom_{kG}(P_{0}, M′) → Hom_{kG}(P_{1}, M′) → Hom_{kG}(P_{2}, M′) → ⋯

and define Ext_{kG}^{i}(M, M′) to be the kernel of the (i+1)st map in this sequence modulo the image of the ith map. This is independent of choice of resolution. We define H^{i}(G, M) to be Ext_{kG}^{i}(k, M). This is a finitely generated graded commutative k-algebra in which the multiplication may be described in terms either of cup products or Yoneda compositions.

Let us illustrate with an example. Let G = GL_{3}(2) be the simple group of order 168 and k be the field of two elements. Then there are four simple kG-modules k, M, N, St of dimensions 1, 3, 3, 8 respectively. The Steinberg module St is projective, while the projective covers P_{k}, P_{M}, P_{N} have the following structures.

\begin{matrix}k \\ / \ \\ M N \\ \ / \\ k \end{matrix}  \begin{matrix}M \\ / \ \\ k \begin{matrix}N \\ | \\ M \\ | \\ N \end{matrix} \\ \ / \\ M \end{matrix}  \begin{matrix}N \\ / \ \\ k \begin{matrix}M \\ | \\ N \\ | \\ M \end{matrix} \\ \ / \\ N \end{matrix}

This means P_{k} has a unique top and bottom composition factor each isomorphic to k, and Rad(P_{k})/Soc(P_{k}) ≅ M ⊕ N. Similarly, P_{M} has a unique top and bottom composition factor each isomorphic to M, and Rad(P_{M})/Soc(P_{M}) is a direct sum of k and a uniserial module (i.e., a module with a unique composition series) with composition factors N, M, N.

To obtain a projective resolution of k, we begin with P_{k}, so that the first kernel is

Ω(k) = \begin{matrix}M N \\ \ / \\ k \end{matrix} . Thus Ω(k)/RadΩ(k) ≅ M ⊕ N, so that the projective cover of Ω(k) is P_{M} ⊕ P_{N}. The next kernel is obtained by cutting and pasting the diagrams for P_{M} and P_{N} as follows.

0 → \begin{matrix}N  M \\ |  | \\ M  N \\ |  | \\ N  k  M \\ \ / \ / \\ M  N \end{matrix} → \begin{matrix}M \\ / \ \\ \begin{matrix}N \\ | \\ M \\ | \\ N \end{matrix} k \\ \ / \\ M \end{matrix} ⊕ \begin{matrix}N \\ / \ \\ k \begin{matrix}M \\ | \\ N \\ | \\ M \end{matrix} \\ \ / \\ N \end{matrix} → \begin{matrix}M N \\ \ / \\ k \end{matrix} → 0.

Thus the left-hand module in this sequence is Ω^{2}(k). This module modulo its radical is N ⊕ k ⊕ M, and so the projective cover is P_{N} ⊕ P_{k} ⊕ P_{M} with kernel

Ω^{3}(k) = \begin{matrix}k  M  N  k \\ \ / \ / \ / \\ N  k  M \end{matrix}

and so on. At this stage, the reader must presumably have started to wonder about the exact meanings of the diagrams we are drawing. I do not wish to explain that in detail here, but the method is explained in detail in Benson and Carlson [1]. The reader may also get the impression that to calculate an entire projective or injective resolution this way would take infinitely much work. However, after a while the patterns start “repeating.” This is formalised, at least for groups of p-rank at most two, in Section 9 of Benson and Carlson [1].

In the above example, the number of summands of the projective modules in the minimal resolution

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

is growing linearly. Because cohomology of a finite group with coefficients in a finitely generated module is finitely generated (Evens [5]), minimal resolutions always have “polynomial growth” in the following sense. If P_{n} is the nth module in a minimal resolution of M, we form the Poincaré series

f(t) = ∑_{n=0}^{∞} t^{n} dim_{k} P_{n}.

Then f(t) is the power series expansion of a rational function of t of the form

(polynomial with integer coefficients)/ ∏_{i=1}^{r}(1 − t^{k_{i}}).

If f(t) has a pole of order c at t = 1 then for some positive real constant μ we have

dim_{k} P_{n} ≤ μn^{c−1},

but the exponent c − 1 may not be decreased to c − 1 − ε for any positive value of ε. We say that the integer c is the complexity of the module M.

If we look at the above example more closely, we see that the given resolution is the total complex of a double complex of the form

⋮  ⋮  ⋮  ⋮
↓  ↓  ↓  ↓
P_{k} ← P_{N} ← P_{M} ← P_{k} ← ⋯
↓  ↓  ↓  ↓
P_{N} ← P_{M} ← P_{k} ← P_{N} ← ⋯
↓  ↓  ↓  ↓
P_{M} ← P_{k} ← P_{N} ← P_{M} ← ⋯
↓  ↓  ↓  ↓
P_{k} ← P_{N} ← P_{M} ← P_{k} ← ⋯

Thus there are vertical and horizontal differentials d_{0} and d_{1} squaring to zero and anticommuting, so that the total differential d_{0} + d_{1} squares to zero.

More generally, a c-fold multiple complex is like a double complex, except that it has c different directions (vertical, horizontal, …) and differentials d_{0}, …, d_{c−1} which anticommute and square to zero, so that the total differential, namely the sum of the d_{i}, squares to zero. The following theorem is proved in Benson and Carlson [2].

Theorem 2.1 Let M be a finitely generated kG-module of complexity c. Then there is a resolution of M which is the total complex of a c-fold multiple complex in which each row, column, etc. is periodic.

Note that the growth rate of the total complex of such a c-fold multiple complex is c, so that this is the same as the growth rate of the minimal resolution. In fact, the resolution produced in the proof of the above theorem is a tensor product of periodic complexes, so that the minimal resolution is usually not of the form produced here. In all the examples we have looked at, one can find a c-fold multiple complex whose total complex is the minimal resolution, but one must allow the rows, columns, etc. to be eventually periodic rather than strictly periodic. It is unclear whether one should expect to be able to prove a theorem of this sort.

3 The construction

It is worth sketching the construction used in the proof of the theorem at the end of the last section, because this provides us with the algebraic product of spheres with free G-action mentioned in the introduction.

If M is a finitely generated kG-module, we have a map

Ext_{kG}^{∗}(k, k) ⟶^{⊗M} Ext_{kG}^{∗}(M, M).

given by tensoring with M. Note that while Ext_{kG}^{∗}(M, M), as a ring under Yoneda composition, is not in general graded commutative (even is M is simple it may have complete matrix rings as quotients), the image of this map is in the graded centre, and by the theorem of Evens [5], Ext_{kG}^{∗}(M, M) is finitely generated as a module over its image. Thus by Noether’s normalisation lemma, one can find a polynomial subring k[ζ_{1}, …, ζ_{c}] of Ext_{kG}^{∗}(k, k) with the following properties.

(i) The above map is injective on the subring k[ζ_{1}, …, ζ_{c}].

(ii) Ext_{kG}^{∗}(M, M) is finitely generated as a module over the image of k[ζ_{1}, …, ζ_{c}]. (iii) The elements ζ_{1}, …, ζ_{c} are homogeneous (i.e., live in a single degree, say ζ_{i} lives

in degree m_{i}).

The number c is the Krull dimension of Ext_{kG}^{∗}(M, M) as a module over Ext_{kG}^{∗}(k, k), and in fact this is equal to the complexity of M as a kG-module (Carlson [4]).

For each generator ζ ≠ 0 (say of degree m) of this polynomial ring, we perform the following construction. Let

⋯ → P_{2} → P_{1} → P_{0} → k → 0

be the minimal resolution of k as a kG-module. Then ζ is represented by a cocycle P_{m} → k which is surjective and vanishes on the image of P_{m+1} → P_{m}. It may therefore be regarded as a surjective map ζ̂ : Ω^{m}(k) → k (which is uniquely determined since we are working with the minimal resolution). We denote the kernel of this map by L_{ζ}. Thus L_{ζ} may be regarded as a submodule of P_{m−1} via the inclusion of Ω^{m}(k) in P_{m−1}, and we obtain a diagram of the following form, in which the rows are exact.

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

We denote by C_{ζ} the chain complex

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

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

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

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

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_{m−1}/L_{ζ} → P_{m−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 k.

Theorem 3.1 If ζ_{1}, …, ζ_{c} are chosen as above, then the complex

C_{ζ_{1}} ⊗ C_{ζ_{2}} ⊗ ⋯ ⊗ C_{ζ_{c}} ⊗ M

is a finite complex of projective modules, whose homology consists of 2^{c} copies of the module M.

Note that since the tensor product of a projective module with any module is again projective, there is only one module in the above complex which might conceivably not be projective, namely the module

P_{m_{1}−1}/L_{ζ_{1}} ⊗ ⋯ ⊗ P_{m_{c}−1}/L_{ζ_{c}} ⊗ M.

There are two available proofs for the projectivity of this module. One involves the machinery of varieties for modules, and involves showing that the choice of ζ_{1}, …, ζ_{c} means that the hypersurfaces defined by these elements intersected with the variety of M give the zero variety, and then observing that modules with the zero variety are projective. This proof may be found in Benson and Carlson [2]. The other available proof uses the hypercohomology spectral sequence and Evens’ finite generation theorem. This may be found in Benson and Carlson [3].

It follows from the above proposition that the complex

C_{ζ_{1}}^{(∞)} ⊗ ⋯ ⊗ C_{ζ_{c}}^{(∞)} ⊗ M

consists of projective modules, and by the Künneth theorem it is exact everywhere except in degree zero, where its homology is M. In other words, it is a projective resolution of M, which is c-fold multiply periodic in the sense of Theorem 2.1. Thus it has the same polynomial growth rate as the minimal resolution.

4 Poincaré duality

Now let us concentrate on the case of the trivial module k. By a theorem of Quillen [9, 10], the Krull dimension of H^{∗}(G, k) is r = r_{p}(G), the maximal rank of an elementary abelian p-subgroup of G. Thus the construction of the last section tells us that we should choose a polynomial subring k[ζ_{1}, …, ζ_{r}] over which H^{∗}(G, k) is finitely generated as a module. Let C denote the complex C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}}. Then C is a Poincaré duality complex in the sense that it is chain homotopy equivalent to its dual, suitably shifted in degree. More precisely, the following theorem is proved in Benson and Carlson [3].

Theorem 4.1 The complex C is a direct sum of an exact sequence of projective modules P′ and a complex C′ having no such summands, and

C′ ≅ Hom_{k}(C′, k)[s],

where s = ∑_{i=1}^{r}(deg ζ_{i} − 1) and [s] denotes a shift in degree by s.

Of course, before the removal of P′, the complex C is not isomorphic to its shifted dual in the above sense, because the modules at one end are much larger than those at the other. But the above theorem is equivalent to saying that C is chain homotopy equivalent to its shifted dual. In this form, the statement remains true for more general commutative rings of coefficients.

5 A spectral sequence

The chain complex C gives rise to a spectral sequence as follows. Let P be a resolution of k as a kG-module, and let C be the chain complex described in the last section. We examine the spectral sequence of the double complex Hom_{kG}(P⊗C, M), for a kG-module M. If we look at the spectral sequence in which the horizontal differential is performed first, we find that the E_{1} page consists of the complex Hom_{kG}(C, M) up the left-hand wall, and zero elsewhere. It follows that the spectral sequence converges to H^{∗}(Hom_{kG}(C, M)). On the other hand, if we perform the vertical differential first, then the E_{1} page is Hom_{kG}(H_{∗}(C)⊗ P, M). Thus the E_{2} page takes the form

Ext_{kG}^{∗}(H_{∗}(C), M) ≅ H^{∗}(G, M) ⊗ Λ^{∗}(ζ̃_{1}, …, ζ̃_{r}).

Thus the spectral sequence takes the form

H^{∗}(G, M) ⊗ Λ^{∗}(ζ̃_{1}, …, ζ̃_{r}) ⇒ H^{p+q}(Hom_{kG}(C, M)).

Some of the differentials in this spectral sequence are given as follows. If α ∈ H^{∗}(G, M), then we have

d_{n_{i}}(αζ̃_{i}) = αζ_{i}.

Now if the ζ_{i} form a regular sequence for H^{∗}(G, M) as a module over H^{∗}(G, k) (for example, this happens in case M = k and H^{∗}(G, k) is a Cohen-Macaulay ring), then these differentials determine all the differentials, and the E_{∞} page is H^{∗}(G, M)/(ζ_{1}, …, ζ_{r}) concentrated along the bottom line. So in this case, we have

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

So for example if H^{∗}(G, k) is a Cohen-Macaulay ring (see for example Matsumura [7] or Serre [11] for standard properties of Cohen-Macaulay rings), then the Poincaré series of the cohomology ring P_{k}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(G, k) takes the form P_{k}(t) = (polynomial of degree s with symmetric coefficients)/ ∏_{i=1}^{r}(1 − t^{n_{i}}). This is easily seen to imply the functional equation given in Theorem 1.1.

Many cohomology rings are Cohen-Macaulay, but for an example of a cohomology ring which is not Cohen-Macaulay, see for example Evens and Priddy [6].

6 Secondary operations

In general, in the spectral sequence of the last section, not all differentials are described by the formula

d_{n_{i}}(αζ̃_{i}) = αζ_{i}.

For example, if α ∈ H^{∗}(G, M) with αζ_{1} = αζ_{2} = 0, then we have not yet described d_{n_{1}+n_{2}−1}(αζ̃_{1}ζ̃_{2}). This is given by a certain secondary operation related to the Massey triple product, and defined as follows. Choose cocycles u, and η_{i} with α = [u] and ζ_{i} = [η_{i}]. Then uη_{1} is a coboundary, say uη_{1} = du_{1}, and similarly uη_{2} = du_{2}. Also, since H^{∗}(G, k) is graded commutative, we may write

η_{1}η_{2} − (−1)^{n_{1}n_{2}}η_{2}η_{1} = dη_{12}.

It is easily seen that

u_{1}η_{2} − (−1)^{n_{1}n_{2}}u_{2}η_{1} − (−1)^{deg α}uη_{12}

is a cocycle, and we denote its cohomology class by ⟨α|ζ_{1}, ζ_{2}⟩. This is well defined modulo the submodule of H^{∗}(G, M) generated by ζ_{1} and ζ_{2}. The differential asked for above is now given by

d_{n_{1}+n_{2}−1}(αζ̃_{1}ζ̃_{2}) = ⟨α|ζ_{1}, ζ_{2}⟩.

There are similar definitions of secondary operations ⟨α|ζ_{1}, …, ζ_{j}⟩, defined when the secondary operations involving proper subsets of the ζ’s vanish, and defined modulo the images of the previous differentials, with the property that

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

When I gave this talk in Sant Feliu, Doug Ravenel was in the audience, and pointed out that in fact the above secondary operations can be expressed in terms of the usual matric Massey products [8]. For example, the first two cases are given by

⟨α|ζ_{1}, ζ_{2}⟩ = (α, (ζ_{1}, ζ_{2}), ( \begin{matrix}ζ_{2} \\ −ζ_{1} \end{matrix} ))

⟨α|ζ_{1}, ζ_{2}, ζ_{3}⟩ = (α, (ζ_{1}, ζ_{2}, ζ_{3}), ( \begin{matrix}0 & −ζ_{3} & ζ_{2} \\ ζ_{3} & 0 & −ζ_{1} \\ −ζ_{2} & ζ_{1} & 0 \end{matrix} ) , ( \begin{matrix}ζ_{1} \\ ζ_{2} \\ ζ_{3} \end{matrix} )).

References

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[6] 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.

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

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

[9] D. Quillen. The spectrum of an equivariant cohomology ring, I. Ann. of Math. 94 (1971), 549–572.

[10] D. Quillen. The spectrum of an equivariant cohomology ring, II. Ann. of Math. 94 (1971), 573–602.

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