BOUNDED COMPLEXES OF PERMUTATION MODULES

DAVID J. BENSON AND JON F. CARLSON

Abstract. Let k be a field of characteristic p > 0. For G an elementary abelian p-group, there exist collections of permutation module such that if C^{∗} is any exact bounded complex whose terms are sums of copies of modules from the collection, then C^{∗} is contractible. A consequence is that if G is any finite group whose Sylow p-subgroups are not cyclic or quaternion, and if C^{∗} is a bounded exact complex such that each C^{i} is direct sum of one dimensional modules and projective modules, then C^{∗} is contractible.

1. Introduction

The study of complexes of permutation modules for finite groups has a long history. They occur naturally in the study of group actions on CW-complexes and manifolds, see for example Bredon [2], Illman [3]. Of particular interest are the complexes arising from collections of subgroups in the work of Quillen [4], Webb [6, 7] and others. They also arise in the theory of splendid equivalences between derived categories of blocks, in the work of Rickard [5].

In this paper we examine bounded exact complexes of permutation modules, and find conditions which force them to be contractible. Of course for a cyclic group G ≅ ℤ/p in characteristic p there exist bounded exact complexes that are not contractible, such as the periodicity complex 0 → k → kG → kG → k → 0, so the game is to find conditions prohibiting examples constructed from these.

Suppose that S is a collection of modules over the group algebra kG of a finite group G with coefficients in a field k of characteristic p > 0. We investigate the question of what collections S exist with the propery that any bounded exact complex of modules in the additive subcategory Add(S) is contractible. We show that if G is an elementary abelian pgroup having rank at least two, then there are collections of permutation modules that satisfy this property. In addition, for any finite group having p-rank at least two, the collection of all modules that are either projective or have dimension one has the property. There are many more such collections. Our main theorem is Theorem 3.1, which gives a sufficient condition on a collection of subgroups of an elementary abelian group for every bounded exact complex with these stabilisers to be contractible. It is easy to see, by inflating and inducing up the periodicity complex for a cyclic subquotient, that this condition is also necessary. The proof of the main theorem involves a spectral sequence argument and some commutative algebra. As an application of the main theorem, we show that for a finite group of p-rank at least two, every bounded exact complex of sums of projective modules and one dimensional modules is contractible, in contrast with the cyclic case discussed above.

2020 Mathematics Subject Classification. 20J06, 20C20.

Key words and phrases. Finite groups, permutation modules, bounded exact complex.

The second author was partially supported by Simons Foundation Grant 054813-01.

A recent paper by Paul Balmer and the first author is something of a complement to the results presented here. In [1], it is proved that every module over the modular group algebra of an elementary abelian has a finite resolution by permutation modules. By splicing together left and right resolutions obtained in this way, we obtain large supplies of exact complexes of permutation modules that do not split.

Acknowledgements. Both authors wish to thank Henning Krause and the University of Bielefeld for their hospitality and support during a visit when the initial ideas for this paper were developed. The authors would also like to thank the Newton Institute for Mathematical Sciences for support and hospitality during the programme “Groups, representations and applications: new perspectives,” when part of the work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1.

2. preliminaries

Let k be a field of characteristic p. Let G be an elementary abelian p-group of rank r. That is, G ≅ C_{p}^{r} is a direct product of r copies of the cyclic group C_{p} of order p.

Recall that if p = 2, then the cohomology ring of G, H^{∗}(G, k) ≅ Ext_{kG}^{∗}(k, k) is a polynomial ring

H^{∗}(G, k) = k[x_{1}, …, x_{r}]

with every x_{i} having degree one. If p is odd, then

H^{∗}(G, k) = k[x_{1}, …, x_{r}] ⊗ Λ(u_{1}, …, u_{r})

where Λ is the exterior algebra generated by elements u_{i} is degree one, and every x_{i} has degree 2.

If M is a kG-module, then H^{∗}(G, M) is a finitely generated module over H^{∗}(G, k). For E a subgroup of G, let kG/E denote the permutation module on the cosets of E in G. By Frobenius Reciprocity, or the Eckmann–Shapiro Lemma, we have an isomorphism of H^{∗}(G, k)-modules H^{∗}(G, kG/E) ≅ H^{∗}(E, k), with the action of H^{∗}(G, k) on H^{∗}(E, k) given by the restriction map H^{∗}(G, k) → H^{∗}(E, k).

For the proof of the main theorm of the paper we require some technical facts. The first is easily verified by the reader.

Lemma 2.1. Let R be a ring and u ∈ R.

(i) If N ⊆ M are R-modules, and uN = 0 = u(M/N), then u^{2}M = 0.

(ii) If M and N are R-modules, uM = 0, and u is regular on N, then Hom_{R}(M, N) = 0.

Theorem 2.2. Let G be an elementary abelian p-group of rank r. Let s ⩽ r, and let M, N be finite direct sums of copies of H^{∗}(G, k)-modules of the form H^{∗}(E, k) where the E’s are subgroups of rank s. Let M_{0} ⊆ M, N_{0} ⊆ N be graded H^{∗}(G, k)-submodules and let X = M/M_{0}. Suppose that there are elements u and v in H^{∗}(G, k) such that u and v annihilate X but u, v is a regular sequence on N. Then

(i) Hom_{H^{∗}(G,k)}(X, N) = 0, (ii) Ext_{H^{∗}(G,k)}^{1}(X, N) = 0, (iii) Hom_{H^{∗}(G,k)}(M, N) → Hom_{H^{∗}(G,k)}(M_{0}, N) is an isomorphism, (iv) Hom_{H^{∗}(G,k)}(M_{0}, N_{0}) → Hom_{H^{∗}(G,k)}(M_{0}, N) is injective, and

(v) Hom_{H^{∗}(G,k)}(M_{0}, N_{0}) is zero in negative degrees, that is, there are no homomorphisms that lower degree.

Proof. To prove (i) and (ii), we use a standard depth argument from commutative algebra. Choose elements u, v ∈ H^{∗}(G, k) such that u and v annihilate X, but form a regular sequence on N. Consider the exact sequence

\xymatrix{0 \ar[r] & N \ar[r]^{u} & N \ar[r] & N/uN \ar[r] & 0.}

From this we get a long exact sequence

\xymatrix{0 \ar[r] & Hom_{H^{∗}(G,k)}(X, N) \ar[r]^{u} & Hom_{H^{∗}(G,k)}(X, N) \ar[r] & Hom_{H^{∗}(G,k)}(X, N/uN)}
[figure: Ext_{H^{∗}(G,k)}^{1}(X, N); u; Ext_{H^{∗}(G,k)}^{1}(X, N); …]

Since u annihilates X, it annihilates both Hom_{H^{∗}(G,k)}(X, N) and Ext_{H^{∗}(G,k)}^{1}(X, N). Thus Hom_{H^{∗}(G,k)}(X, N) = 0. By a similar agument, because v is regular on N/uN, we have that Hom_{H^{∗}(G,k)}(X, N/uN) = 0. From this it follows that Ext_{H^{∗}(G,k)}^{1}(X, N) = 0 as asserted.

From the exact sequence

\xymatrix{0 \ar[r] & M_{0} \ar[r] & M \ar[r] & X \ar[r] & 0}

we obtain an exact sequence

\xymatrix{0 \ar[r] & Hom_{H^{∗}(G,k)}(X, N) \ar[r] & Hom_{H^{∗}(G,k)}(M, N) \ar[r] & Hom_{H^{∗}(G,k)}(M_{0}, N) \ar[r] & Ext_{H^{∗}(G,k)}^{1}(X, N) \ar[r] & …}

Thus (iii) follows from (i) and (ii).

Part (iv) follows from the fact that the map N_{0} → N is injective. To prove part (v), we note that Hom_{H^{∗}(G,k)}(M, N) is zero in negative degrees, so by (iii) the same is true of Hom_{H^{∗}(G,k)}(M_{0}, N), and then by (iv) the same is true of Hom_{H^{∗}(G,k)}(M_{0}, N_{0}). □

3. The main theorem

Our objective in this section is to prove the main theorem of the paper. If S is a collection of finitely generated kG-modules, then Add(S) is the full additive subcategory of the module category consisting of all kG-modules that are isomorphic to finite direct sums of copies of objects in S.

Theorem 3.1. Let G be an elementary abelian p-group and k a field of characteristic p. Suppose that H is a collection of subgroups of G with the property that there is no pair E, F of elements in H such that E ⊆ F with index p. If

\xymatrix{0 \ar[r] & C^{0} \ar[r] & C^{1} \ar[r] & ⋯ \ar[r] & C^{ℓ} \ar[r] & 0}

is a bounded exact sequence of kG-modules in Add({kG/E | E ∈ H}) then C^{∗} is contractible.

Before beginning the proof we note the following.

Lemma 3.2. Suppose that H is a collection of subgroups of G as in Theorem 3.1. Let s be the maximum rank of an element of H. Let H′ be the subset of subgroups in H of rank s and let H′′ = H \ H′. If s > 1 then there exist homogeneous elements u and v of H^{∗}(G, k) such that for every E′ ∈ H′, u and v form a regular sequence on H^{∗}(E′, k), and u and v restrict to zero on every H^{∗}(E′′, k) for all E′′ ∈ H′′.

Proof. This is an easy exercise in prime avoidance, using the hypothesis on H in the theorem.

□

We are now prepared to prove our theorem.

Proof of Theoerm 3.1. Suppose that C^{∗} is an exact complex of permutation modules as in the statement of Theorem 3.1. Without loss of generality, we may assume that C^{∗} does not have a nontrivial direct summand that is contractible. Let

\xymatrix{… \ar[r] & P_{2} \ar[r] & P_{1} \ar[r] & P_{0} \ar[r] & k \ar[r] & 0}

be a projective resolution of k, the trivial kG-module. We consider the two spectral sequences whose E_{0}-term is the double complex

E_{0}^{i,j} = Hom_{kG}(P_{j}, C^{i}).

If we first take the differential that comes from the complex C^{∗}, then we get that E_{1}^{∗,∗} is identically zero. This is because P_{j} is projective and the complex C^{∗} is exact.

It follows that the spectral sequence obtained by first starting with the differential on P_{∗} converges to zero. Indeed, it converges to zero after a finite number of steps, since there are only a finite number of columns. The E_{1} term has the form

[figure: E_{1}^{i,j} = H^{j}(G, C^{i}); 0,]

and all differentials on this and subsequent pages are homomorphisms of H^{∗}(G, k)-modules. Each C^{i} is a direct sum of permutation modules on subgroups E ∈ H. The corresponding summand of H^{∗}(G, C^{i}) is H^{∗}(G, kG/E) ≅ H^{∗}(E, k). Note that the larger the subroup the smaller is the dimension of the permutation module, and the larger is the Krull dimension of its cohomology.

Let s be the maximum p-rank of any element of H. Let H′ be the set of subgroups in H that have rank s, and let H′′ = H \ H′. We decompose each C^{i} as D^{i} ⊕ U^{i}, where D^{i} is a direct sum of permutation modules k(G/E) with E ∈ H′ having rank s, and where U^{i} is a direct sum of permutation modules on cosets of subgroups in H′′. By Lemma 3.2, there are homogeneous elements u, v ∈ H^{∗}(G, k) which form a regular sequence on every H^{∗}(G, D^{i}) but annihilate all H^{∗}(G, U^{i}). In particular, since u is regular on H^{∗}(G, D^{i}) but annihilates H^{∗}(G, U^{i}), there are no nonzero H^{∗}(G, k)-module homomorphisms from H^{∗}(G, U^{i}) to H^{∗}(G, D^{i+1}). Hence, d_{1}(H^{∗}(G, U^{∗})) ⊆ H^{∗}(G, U^{∗}) and we obtain a short exact sequence of complexes

\xymatrix{0 \ar[r] & H^{∗}(G, U^{∗}) \ar[r] & E_{1}^{∗,∗} \ar[r]^{ϑ_{1}} & H^{∗}(G, D^{∗}) \ar[r] & 0.}

Our aim is to prove that if C^{∗} is not a contractible complex, then the above spectral sequence cannot converge to zero, thus giving us a contradiction. Again, we are assuming that C^{∗} has no contractible direct summands. We proceed by induction on the pages of the spectral sequence. The induction statement is the following.

Induction Statement for page n. There exist homogeneous elements u_{n}, v_{n} in H^{∗}(G, k) and, for every i, j, there is a k-subspace U_{n}^{i,j} ⊆ E_{n}^{i,j} such that the following hold. IS(n, 1) : U_{n}^{i,∗} is an H^{∗}(G, k)-submodule of E_{n}^{i,∗}. Let D_{n}^{i,j} = E_{n}^{i,j}/U_{n}^{i,j}. IS(n, 2) : u_{n}, v_{n} form a regular sequence on H^{∗}(G, D^{∗}). IS(n, 3) : u_{n}, v_{n} annihilate U_{n}^{∗,∗}. IS(n, 4) : There are natural inclusions D_{n}^{i,j} ⊆ D_{n−1}^{i,j} ⊆ ⋯ ⊆ D_{1}^{i,j} = H^{∗}(G, D^{∗}) as modules

over H^{∗}(G, k). Moreover, H^{∗}(G, D^{∗})/D_{n}^{∗,∗} is annihilated by both u_{n} and v_{n}.

IS(n, 5) : U_{n}^{∗,∗} is a subcomplex of (E_{n}^{∗,∗}, d_{n}) so that

\xymatrix{0 \ar[r] & U_{n}^{∗,∗} \ar[r] & E_{n}^{∗,∗} \ar[r]^{ϑ_{n}} & D_{n}^{∗,∗} \ar[r] & 0}

is an exact sequence of complexes of H^{∗}(G, k)-modules. IS(n, 6) : d_{n}(E_{n}^{i,j}) ⊆ U_{n}^{i+n,i−n+1} for all i and j.

We begin the induction with n = 1. Let U_{1}^{i,j} = H^{j}(G, U_{i}), D_{1}^{i,j} = H^{j}(G, D_{i}), u_{1} = u, v_{1} = v. Then conditions IS(1, 1) and IS(1, 5) have already been proved. Conditions IS(1, 2) and IS(1, 3) are true by the choice of u and v. Condition IS(1, 4) is obvious. The only remaining task is to verify IS(1, 6) which is easily seen to be equivalent to the condition that the induced map

\xymatrix{d̂_{1} : D_{1}^{i,∗} ≅ H^{∗}(G, D^{i}) \ar[r] & D_{1}^{i+1,∗} ≅ H^{∗}(G, D^{i+1})}

is the zero map for all i = 0, …, ℓ − 1. This is a degree zero map of H^{∗}(G, k)-modules induced by the coboundary map on C^{∗}. But note that if E, F ∈ H′ and E ≠ F, then there are no nonzero homomorphisms from H^{∗}(G, k(G/E)) to H^{∗}(G, k(G/F)). Consequently, the only way that d̂_{1} could be nonzero is if, for some i, C^{i} and C^{i+1} both had direct summands isomorphic to k(G/E) for some E ∈ H′, and the composition of the injection of one followed by the coboundary followed by the projection onto the other is an isomorpism. However, such a situation does not occur because it violates our assumption that the complex C^{∗} has no contractible direct summands. Thus, the induction statement is true in the case that n = 1.

Now we assume for some n ⩾ 1 that the Induction Statement for page n is true. We need to show that it holds also for page n + 1. Let u_{n+1} = u_{n}^{2} and v_{n+1} = v_{n}^{2}. Note that IS(n + 1, 2) is automatic from IS(n, 2). Recall that E_{n+1}^{i,j} is the quotient of K^{i,j} which is the kernel of d_{n} on E_{n}^{i,j} by the submodule d_{n}(E_{n}^{i−n,j+n−1}). Let D_{n+1}^{i,j} = ϑ_{n}(K^{i,j}), the image in D_{n}^{i,j} of the kernel of d_{n} on E_{n}^{i,j}. By IS(n, 6), we have that ϑ_{n}(d_{n}(E_{n}^{i−n,j+n−1})) = 0. Hence there is a well defined homomorphism ϑ_{n+1}, and an exact sequence

\xymatrix{0 \ar[r] & U_{n+1}^{∗,∗} \ar[r] & E_{n+1}^{∗,∗} \ar[r]^{ϑ_{n+1}} & D_{n+1}^{∗,∗} \ar[r] & 0.}

That is, we defined U_{n+1}^{∗,∗} to be the kernel ϑ_{n+1}, which is the map induced by ϑ_{n}.

Because ϑ_{n} is an H^{∗}(G, k)-homomorphism and both the kernel and image of d_{n} are H^{∗}(G, k)-submodules, we have that ϑ_{n+1} is also an H^{∗}(G, k)-homomorphism. This proves IS(n + 1, 1). Condition IS(n + 1, 3) is a consequence of the fact that U_{n+1}^{∗,∗} is a quotient of U_{n}^{∗,∗} by a submodule thereof. So it is annihilated by u_{n} and v_{n}. For condition IS(n + 1, 4), notice that if x ∈ D_{n}^{i,j}, then x = ϑ_{n}(y) for some y in E_{n}^{i,j}. By IS(n, 6) and IS(n, 2), d_{n}(y) ∈ U_{n}^{i+n,j−n+1} is annihilated by u_{n} and v_{n}. That is, u_{n}y and v_{n}y are in the kernel of d_{n}. So ϑ_{n}(u_{n}y) = u_{n}x and ϑ_{n}(v_{n}y) = v_{n}x are in D_{n+1}^{i,j}. Thus, D_{n}^{∗,∗}/D_{n+1}^{∗,∗} is annihilated by u_{n} and v_{n}. It follows from IS(n, 4) and Lemma 2.1, that H^{∗}(G, D^{∗}) is annihilated by u_{n+1} and v_{n+1}.

We can see that there are no nonzero homomorphisms from U_{n+1}^{∗,∗} to D_{n+1}^{∗,∗}. The reason is that by IS(n + 1, 2) and IS(n + 1, 4), u_{n+1} is a regular element on D_{n+1}^{∗,∗}, while it annihilates U_{n+1}^{∗,∗} by IS(n + 1, 3) and Lemma 2.1. This proves IS(n + 1, 5).

Finally, we observe that d_{n+1}(D_{n+1}^{∗,∗}) ⊆ U_{n+1}^{∗,∗}. The reason for this is that D_{n+1}^{∗,∗} is an H^{∗}(G, k)-submodule of H^{∗}(G, D^{∗}), and the homomorphism

\xymatrix{ϑ_{n+1} d_{n+1} : D_{n+1}^{∗,∗} \ar[r] & D_{n+1}^{∗,∗},}

which lowers the degrees, is the zero map by Theorem 2.2(5). This implies that d_{n+1}(E_{n+1}^{∗,∗}) ⊆ U_{n+1}^{∗,∗}. Consequently, Condition IS(n + 1, 6) holds.

Thus, we have shown that the Induction Statement for page n of the spectral sequence implies that of page n + 1. Hence, the statement holds for all pages. Because the complex C^{∗} has only ℓ + 1 nonzero terms, the spectral sequence has only ℓ + 1 nonzero columns, and it must stop after ℓ + 2 steps. That is, E_{ℓ+2}^{∗,∗} = E_{∞}^{∗,∗} = {0}. However, this is a contradiction. By IS(ℓ + 2, 4), D_{ℓ+2}^{∗,∗} is an H^{∗}(G, k)-submodule of H^{∗}(G, D^{∗}) such that H^{∗}(G, D^{∗})/D_{ℓ+2}^{∗,∗} is annihilated by u_{ℓ+2} which is a regular element on H^{∗}(G, D^{∗}). Thus D_{ℓ+2}^{∗,∗} and also E_{ℓ+2}^{∗,∗} cannot be zero. This proves the theorem. □

4. An application

We present one easy application of the main theorem. There are numerous similar variations. As before assume that k is a field of characteristic p > 0.

Theorem 4.1. Suppose that G is a finite group having an elementary abelian subgroup E of p-rank 2. Let S be the collection consisting of all indecomposable projective kG-modules and all one dimensional kG-modules. If

\xymatrix{0 \ar[r] & C^{0} \ar[r]^{∂} & C^{1} \ar[r] & ⋯ \ar[r] & C^{ℓ−1} \ar[r]^{∂_{ℓ−1}} & C^{ℓ} \ar[r] & 0}

is a bounded exact sequence of kE-modules in Add(S), then C^{∗} is contractible.

Proof. Without loss of generality we may assume that C^{∗} has no nonzero direct summands other than itself. In particular, this means that C^{ℓ} is a sum of one dimensional modules. That is, if C^{ℓ} has a submodule P that is projective, then P is a direct summand of C^{ℓ} and ∂_{ℓ−1} followed by the projection on to P splits. Thus, the complex 0 → P → P → 0 is a direct summand of C^{∗}, violating our assumption.

Next, we notice that the theorem is true if it holds in the case that G is a p-group. For suppose that Q is a Sylow p-subgroup of G. Assume that the restriction of C^{∗} to Q is contractible. Then there is a kQ-homomorphism θ: C^{ℓ} → C^{ℓ−1} such that ∂_{ℓ−1} θ = Id_{C^{ℓ}}. Let ψ be the map ψ = (1/|G : Q|) ∑_{x∈G/Q} xθx^{−1} : C^{ℓ} → C^{ℓ−1}. Then ψ is a kG-homomorphism that splits ∂_{ℓ−1}.

We assume now that G is a p-group and that each C^{i} is a direct sum of trivial modules and a projective module. By Theorem 3.1, the restriction of C^{∗} to the elementary abelian subgroup E is contractible. Hence, there is kE-homomorphism θ: C^{ℓ} → C^{ℓ−1} such that ∂_{ℓ−1} θ = Id_{C^{ℓ}}. Now write, C^{ℓ−1} = M ⊕ P where M ≅ k^{n} is a sum of trivial kG-modules and P is a projective module. Let ρ_{P} : C^{ℓ−1} → P be the projection, and ι_{P} be the inclusion of P into C^{ℓ−1}. Let ρ_{M} and ι_{M} be the same for M so that ι_{M} ρ_{M} + ι_{P} ρ_{P} = Id_{C^{ℓ−1}}. Notice that ∂_{ℓ−1} ι_{P} ρ_{P} θ is the zero map. The reason is that the image of θ is in the space of fixed points of E on P, and since P is a free module, this is a subset of the radical of P. Because C^{ℓ} is a sum of trivial modules, the radical of P is in the kernel of ∂_{ℓ−1}. It follows that ∂_{ℓ−1} ι_{M} ρ_{M} θ = Id_{C^{ℓ}}. That is, ψ = ι_{M} ρ_{M} θ splits ∂_{ℓ−1}. To finish the proof we only need to notice that ψ is a kG-homomorphism, since it is a linear map between sums of trivial modules. □

Remark 4.2. The above theorem is not true for groups of p-rank one. This means groups whose Sylow p-subgroups are cyclic or quaternion. Indeed, for such a group the trivial module is periodic, and so there is a non-contractible exact complex beginning and ending with the trivial module, part of a projective resolution of the trivial module, with all intermediate modules projective.

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Institute of Mathematics, Fraser Noble Building, Univeristy of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Department of Mathematics, University of Georgia, Athens GA 30602, USA