A split exact sequence of Mackey functors

P.J. Webb

1. The main theorems

We prove a theorem which allows a Mackey functor on a finite group to be computed in terms of its values on certain subgroups in a quite explicit way. The ingredients which make up the statement are an action of the group on a simplicial complex, and projectivity of the Mackey functor relative to certain subgroups. The result may be regarded as a refinement of the result of Dress [7] which shows that relative projectivity of a Mackey functor with respect to a set of subgroups implies computability in terms of those subgroups. Viewed differently, one can also see the theorem as an extension of work of K.S. Brown [3] which gives the cohomology of a group in terms of its action on the simplicial complex of psubgroups.

THEOREM A. Let G be a finite group, M a Mackey functor, X and Y classes of subgroups of G which are closed under taking subgoups and conjugation, and Δ a Gsimplicial complex of dimension d. Suppose that

(i) For every simplex σ ∈ Δ the vertices of σ lie in distinct G-orbits. (ii) For every subgroup H ∈ X − Y, Δ^{H} is contractible. (iii) M is projective relative to X. (iv) For every Y ∈ Y, M(Y) = 0. Then there are split exact sequences

\begin{matrix}0 ⟶ M(G) ⟶ ⨁_{σ∈G\Γ_{0}(Δ)} M(G_{σ}) ⟶^{(ϕ_{τσ})} ⨁_{τ∈G\Γ_{1}(Δ)} & M(G_{τ}) ⟶ \\ ⋯ ⟶ ⨁_{σ∈G\Γ_{d}(Δ)} M(G_{σ}) ⟶ 0 \end{matrix}

and

\begin{matrix}0 ⟵ M(G) ⟵ ⨁_{σ∈G\Γ_{0}(Δ)} M(G_{σ}) ⟵^{(ψ_{στ})} ⨁_{τ∈G\Γ_{1}(Δ)} & M(G_{τ}) ⟵ \\ ⋯ ⟵ ⨁_{σ∈G\Γ_{d}(Δ)} M(G_{σ}) ⟵ 0 \end{matrix}

where if σ is a face of gτ then

\begin{aligned}ϕ_{τσ} & = (σ|gτ) · c_{g^{−1}} · res_{G_{gτ}}^{G_{σ}} \\ ψ_{στ} & = (σ|gτ) · ind_{G_{gτ}}^{G_{σ}} · c_{g} \end{aligned}

and otherwise ϕ_{τσ} and ψ_{στ} are zero. Here (σ|gτ) is +1 or −1 according to the orientation of the embedding of σ in gτ, and Γ_{i}(Δ) is the set of i-simplices in Δ.

In this theorem, all of the mappings are given in the way we have indicated between the 0 and 1 terms. The other mappings were left unlabelled to save on notation. By a Gsimplicial complex we mean a simplicial complex on which G acts simplicially, that is G permutes the simplices amongst themselves preserving the face relationships and sending simplices of a given dimension to simplices of the same dimension. We use the term orientation of the embedding to mean the following. It is possible to put a G-invariant partial order on the vertices of Δ in such a fashion that the vertices of every simplex are totally ordered (see Lemma 4.1.1). If τ is an r-simplex of Δ with vertices (v_{0}, …, v_{r}) taken in order, and if ρ = (v_{0}, …, v_{j−1}, v_{j+1}, …, v_{r}) is a face of τ of dimension r − 1 then we put (ρ|τ) = (−1)^{j}. In the statement of the theorem we are composing maps on the left and this will generally be our convention.

The sequences in the theorem are in fact natural with respect to G and this gives rise to a stronger but more abstract statement, which we now give. In the notation of Dress [7], we may regard Mackey functors as being defined on G-sets rather than on subgroups of G. Thus we may write M(G/H) instead of M(H) to denote the value of the Mackey functor M at the subgroup H. If Ω is a G-set we let M_{Ω} be the Mackey functor with M_{Ω}(Ψ) = M(Ψ × Ω).

THEOREM B. In the situation of Theorem A we have split exact sequences of Mackey functors

0 → M → M_{Γ_{0}(Δ)} → M_{Γ_{1}(Δ)} → ⋯ → M_{Γ_{d}(Δ)} → 0

and

0 ← M ← M_{Γ_{0}(Δ)} ← M_{Γ_{1}(Δ)} ← ⋯ ← M_{Γ_{d}(Δ)} ← 0.

The mappings in this theorem are described as follows. We give the component maps with respect to the decomposition

M_{Γ_{i}(Δ)} = ⨁_{σ∈G\Γ_{i}(Δ)} M_{G/G_{σ}}

obtained by choosing representatives of the orbits of G on Δ. The component mapping M_{G/G_{σ}} → M_{G/G_{τ}} is zero unless σ is a face of gτ for some g ∈ G, in which case its evaluation at a G-set Ω is

(σ|gτ)M^{∗}(1 × π_{G_{gτ}}^{G_{σ}} · c_{g}) : M(Ω × G/G_{σ}) → M(Ω × G/G_{τ})

in the case of the first sequence, or

(σ|gτ)M_{∗}(1 × π_{G_{gτ}}^{G_{σ}} · c_{g}) : M(Ω × G/G_{τ}) → M(Ω × G/G_{σ})

in the case of the second sequence, where c_{g} : G/G_{τ} → G/G_{gτ} is conjugation and π_{G_{gτ}}^{G_{σ}} : G/G_{gτ} → G/G_{σ} is the canonical quotient map. On evaluation at G (or, equivalently, at the G-set consisting of a single point) these sequences become the sequences in Theorem A. The sequences in Theorem B are resolutions of M by induced functors, and they take a particularly nice form. They invite comparison with the resolutions by functors M_{Ω} constructed by Dress [7]. The sequences here have the advantage that they are smaller and easier to compute with in specific examples.

In Section 2 we describe many situations in which Theorem A can be applied, to do with computing the values of specific Mackey functors, and also deducing a structure theorem for the chain complex of K.S. Brown’s simplicial complex of p-subgroups of G. I have found Theorem A most useful for computing cohomology groups, and indeed this was the origin of this paper. Theorem A implies the conclusion of the Theorem A in [20] which expresses the cohomology as an alternating sum. I noticed that this alternating sum could be given a more satisfactory conceptual basis by arranging the terms in an exact sequence. I then found that the same thing could be done with Mackey functors. More recently Theorem A has found application in other areas, and we can refer the reader to [19] for an application in connection with Alperin’s conjecture.

In Section 4 we give the proofs of Theorems A and B, and for this Section 3 on Mackey functors is a prerequisite. Finally in Section 5 we make deductions about the chain complex of Δ, and in particular about Brown’s complex. The main result here is stated as Theorem 2.7.1, which describes the chain complex of Brown’s complex. There is also a result, stated as 2.6.1, which says that the quotient by G of Brown’s simplicial complex is mod p acyclic.

2. Applications

2.1 Suitable simplicial complexes

We start by describing some specific examples of simplicial complexes which can be used in Theorems A and B. In our applications Δ will always be obtained as the simplicial complex of chains in a partially ordered set. If P is a poset the corresponding simplicial complex Δ = Δ(P) has as its n-simplices the set

Γ_{n}(Δ) = {x_{0} < … < x_{n} | x_{i} ∈ P}.

The subsimplices of an n-simplex are the shorter subchains. If G acts on P as a group of poset automorphisms there is induced a simplicial action of G on Δ(P). A G-simplicial complex constructed in this way will always satisfy condition (i) of the theorem, provided the poset is finite, because if x, y are two vertices of a simplex σ then they are comparable as elements of the poset, and they must therefore lie in distinct G-orbits.

There are three posets which we use all the time, consisting of p-subgroups of G. We fix a prime p and put

S_{p}(G) = {all non-identity p-subgroups of G}
A_{p}(G) = {all non-identity elementary abelian p-subgroups of G}
B_{p}(G) = {H ≤ G | H = O_{p}N_{G}(H), H ≠ 1}.

G acts on each of these posets by conjugating the subgroups. It turns out that from a certain point of view we get effectively the same answer in our applications no matter which of these three posets we use. This is a consequence of the following result.

2.1.1 THEOREM (Quillen [14], Bouc [1], Thévenaz, Webb [18]). The three simplicial complexes Δ(S_{p}(G)), Δ(A_{p}(G)) and Δ(B_{p}(G)) are all G-homotopy equivalent.

For a discussion of what this means see [22]. A consequence is that if any one of these simplicial complexes satisfies (ii) then so do the others. The significance of B_{p}(G) is that when G is a finite Chevalley group in defining characteristic p, Δ(B_{p}(G)) is the barycentric subdivision of the building of G. In particular Δ(B_{p}(G)) and the building are G-homotopic.

As well as the simplicial complexes associated to the above posets we will be using posets of the form P = S_{p}(G) − Y, where Y is some set of subgroups of G closed under conjugation and taking subgroups. For these simplicial complexes the fixed point condition (ii) in the theorem will be verified using:

2.1.2 LEMMA. (i) Let Δ be the simplicial complex of one of S_{p}(G), A_{p}(G) or B_{p}(G), and let H be a subgroup of G which has a non-trivial normal p-subgroup. Then Δ^{H} is contractible. (ii) Let Y be a set of subgroups of G closed under taking subgroups and conjugation, and let Δ = Δ(S_{p}(G) − Y). Let H be a subgroup of G which has a non-trivial normal p-subgroup H_{p} with H_{p} ∈ S_{p}(G) − Y. Then Δ^{H} is contractible.

Proof. Part (i) in the case of Δ(S_{p}(G)) follows from (ii) by taking Y to be empty. Now the result for Δ(A_{p}(G)) and Δ(B_{p}(G)) follows in view of 2.1.1.

We prove (ii) by contracting Δ^{H} onto {H_{p}}. Now (Δ(S_{p}(G) − Y))^{H} = Δ((S_{p}(G) − Y)^{H}). Take any K ∈ (S_{p}(G) − Y)^{H}. The chain of inequalities K ≤ K.H_{p} ≥ H_{p} in (S_{p}(G) − Y)^{H} contracts (S_{p}(G) − Y)^{H} onto {H_{p}}, as in [4, p.268].

Of course, one may try to apply the theorem with whatever simplicial complex and Mackey functor one may have in mind. Other candidate simplicial complexes come from the sporadic geometries which people construct for certain simple groups [16].

2.2 Overview of the Mackey functors to which the theorem applies

The notion of a Mackey functor will be described in Section 3. We first describe in general terms a proceedure we can follow which produces a Mackey functor M and simplicial complex Δ satisfying the conditions of Theorem A. In order to apply Theorem A to some candidate Mackey functor M we may have in mind, the first step is to find a set X of subgroups of G with respect to which M is projective, and preferably a minimal such set. Now choose a simplicial complex Δ satisfying (i) of Theorem A, and such that Δ^{H} is contractible for as many of the subgroups H ∈ X as possible. To avoid triviality, G should not stabilize any simplex of Δ. Take Y to be a set of subgroups of G closed under conjugation and taking subgroups so that Y ⊇ {H ∈ X | Δ^{H}is not contractible}. We define two more Mackey functors

M(H, Y) = ∑_{Y ∈Y,Y ≤H} ind_{Y}^{H}M(Y), M_{/Y}(H) = M(H)/M(H, Y).

One sees that if the original Mackey functor M were to satisfy M(Y) = 0 for all Y ∈ Y, then we would have M = M_{/Y}.

2.2.1 PROPOSITION. Conditions (i) - (iv) of Theorem A are satisfied for the quadruple M_{/Y}, X, Y and Δ.

This result is proved in Section 3.6, where it is also shown that M_{/Y} is characterised as the largest quotient Mackey functor of M which vanishes on the subgroups in Y.

We now give a list of examples of how the above construction turns out in particular cases. These will be examined in greater detail in the ensuing sections. We fix a prime p. We use the following symbols to denote some familiar Mackey functors.

r(G) – the ring of virtual complex characters
Br(G) – the ring of Brauer characters in characteristic p
a(G) – the Green ring of F G-modules, where F is a field of characteristic p
A(G) – the Green ring Q ⊗_{Z Z} a(G) with rational coefficients
Ĥ^{n}(G, V) – Tate cohomology with coefficient module V
Ĥ^{n}(G, V)_{p} – the p-torsion subgroup of Tate cohomology
H_{Y}^{n}(G, V) – cohomology relative to Y

In the case of a(G) we will need to suppose F is a splitting field for G, but not in the case of A(G).

In the next statement we list X and Y so that the reader can see what is going on, but as far as practical applications are concerned we might as well forget them because they do not appear in the conclusions of Theorems A or B. For the sake of generality we have put Δ(S_{p}(G) − Y) all the time, but the simplest situation is when Y consists just of the p′-subgroups of G, and in this case S_{p}(G) − Y = S_{p}(G).

We will use the following terminology:

Definition. Let Z be a class of groups. We say a group H is Z mod p if H has a normal p-subgroup H_{p} ◁ H with H/H_{p} ∈ Z. Recall that a subgroup is said to be Brauer elementary if it is the direct product of a cyclic group with a q-group, for some prime q.

2.2.2 THEOREM. Conditions (i) – (iv) of Theorem A are satisfied with the following choices of M, X, Y and Δ.

M(G)  type of subgroups in X  Y  Δ
r_{/Y}(G)  Brauer elementary  ⊇ p′-subgroups Δ(S_{p}(G) − Y)
Br_{/Y}(G)  Brauer elementary  ⊇ p′-subgroups Δ(S_{p}(G) − Y)
a_{/Y}(G)  Brauer elementary mod p ⊇ p′-subgroups Δ(S_{p}(G) − Y)
A_{/Y}(G)  cyclic mod p  ⊇ p′-subgroups Δ(S_{p}(G) − Y)
Ĥ^{n}(G, V)_{p}  p-subgroups  {1}  Δ(S_{p}(G))
H_{/Y}^{n}(G, V)_{p}  p-subgroups  ⊇ {1}  Δ(S_{p}(G) − Y)
H_{Y}^{n}(G, V)_{p}, n ≥ 1  p-subgroups  ⊇ {1}  Δ(S_{p}(G) − Y)

The meaning of ⊇ in the third column is that we may take Y to be any set of subgroups closed under taking subgroups and conjugation which contains the named set of subgroups.

Proof. We deduce this from 2.2.1. We must show that Δ satisfies (i) and (ii), which it does from 2.1.2 and the remarks in Section 2.1. Note in this connection that if a Brauer elementary subgroup is not a p′-group then it has a non-trivial normal p-subgroup, so 2.1.2 may be applied. We must also show that M is X -projective in the various cases. This is discussed in Section 3.4.

2.2.3 Remark. It will be apparent from Proposition 3.4.2 that if we start with an X - projective Mackey functor N, let Z be some class of subgroups closed under taking subgroups and conjugation, and put M(H) = N(H, Z) then M_{/Y} is also X -projective. By this means quadruples such as

M(G) = a(G, Z)/a(G, Y),
X = subgroups ‘Brauer elementary mod p’,
Y = any subconjugacy closed set such that {p′-subgroups} ⊆ Y ⊆ Z,
Δ = Δ(S_{p}(G))

may be included in the list in Theorem 2.2.2. There is also some freedom in the choice of Δ in that we may take Δ = Δ(P) where P is any G-invariant poset satisfying S_{p}(G)−Y ⊆ P ⊆ S_{p}(G) (see Lemma 2.1.2(ii)).

2.3 Character rings

In using Theorem A to compute either a character ring or a Green ring in the manner of Theorem 2.2.2 it is necessary to factor out the span of the projective modules (or the characters of the projective modules, in the case of a character ring). To be explicit about this, let R be a complete discrete valuation ring in characteristic 0 containing a primitive |G|th root of unity with residue field of characteristic p. Then

r(G, p′-subgroups) = the ZZ-span of characters of projective RG-modules
Br(G, p′-subgroups) = the ZZ-span of Brauer characters of projective F G-modules
a(G, p′-subgroups) = the ZZ-span of projective F G-modules
A(G, p′-subgroups) = the Q-span of projective F G-modules.

For Br and a we require F to be a splitting field. These statements follow from [8] and [5]. In view of this, Br_{/p′-subgroups}(G) = ZZ^{ℓ(G)}/C(G)ZZ^{ℓ(G)} where C(G) is the Cartan matrix of G and ℓ(G) is the number of p-regular conjugacy classes of G, and this is a finite abelian p-group whose torsion coefficients are the elementary divisors of C(G). Theorem A now states that this group is computable as the end term in an exact sequence where all the remaining information is p-locally determined.

2.3.1 COROLLARY. With the above notation, there is a short exact sequence

0 → ZZ^{ℓ(G)}/C(G)ZZ^{ℓ(G)} → ⨁_{σ∈G\Γ_{0}(Δ)}ZZ^{ℓ(G_{σ})}/C(G_{σ})ZZ^{ℓ(G_{σ})} →

⋯ → ⨁_{σ∈G\Γ_{d}(Δ)} ZZ^{ℓ(G_{σ})}/C(G_{σ})ZZ^{ℓ(G_{σ})} → 0

where Δ is any one of Δ(S_{p}(G)), Δ(A_{p}(G)) or Δ(B_{p}(G)), and where the maps correspond to restrictions of Brauer characters, some with minus signs. There is a similar short exact sequence with maps in the opposite direction corresponding to induction of characters.

By equating the alternating sum of the terms in either of the above sequences to zero one obtains a formula for the number of elementary divisors of C(G) of a given size as an alternating sum of these numbers for the groups G_{σ}. This formula has been observed by Robinson [15].

In a similar way to the above, r_{/p′-subgroups}(G) is the quotient of r(G) by the span of the columns of the characteristic p decomposition matrix D. Since the rank of D equals the number of p-regular conjugacy classes in G, r_{/p′-subgroups}(G) has torsion free rank equal to the number of classes of elements of order divisible by p. We obtain a result which was first observed in a different form by Knörr.

2.3.2 COROLLARY. Let p be a prime and for each finite group G let m(G) be the number of conjugacy classes of elements of G of order divisible by p. Then

m(G) = ∑_{σ∈G\Δ}(−1)^{dim σ}m(G_{σ})

where Δ is any one of Δ(S_{p}(G)), Δ(A_{p}(G)) or Δ(B_{p}(G)).

2.4 Cohomology

In [20] and [21] I obtained the cohomology Ĥ^{n}(G, V)_{p} additively as an alternating sum ∑_{σ∈G\Δ}(−1)^{dim σ}Ĥ^{n}(G_{σ}, V)_{p} valid in K_{0}(Ab, 0), the Grothendieck group of finite abelian groups with relations given only by direct sum decompositions. Such an expression as an alternating sum is evidently a consequence of the result here, but I demonstrated the existence of the alternating sum in [21] under a weaker hypothesis on Δ, which in fact is a necessary and sufficient condition for its existence. The split exact sequence formulation has certain advantages. In [21] I required the coefficient module V to be finitely generated in order that the cohomology groups be finite, and hence that the alternating sum would make sense. With the exact sequence here we may perfectly well have non-finitely generated modules. With this sequence it is also possible to determine the cup product structure of cohomology. To do this, note that in the start of the sequence

0 → Ĥ^{∗}(G, ZZ)_{p} ⟶^{(res_{Gσ}^{G})} ⨁_{σ∈G\Γ_{0}(Δ)} Ĥ^{∗}(G_{σ}, ZZ)_{p} → ⋯

the first morphism embeds Ĥ^{∗}(G, ZZ)_{p} by a sum of restriction maps, and this is a ring homomorphism. We know the cohomology ring of G once we know the cohomology rings of the G_{σ} for σ ∈ Γ_{0}(Δ), and once we have computed the kernel of the map ⨁_{σ∈G\Γ_{0}(Δ)} Ĥ^{∗}(G_{σ}, ZZ)_{p} → ⨁_{σ∈G\Γ_{1}(Δ)} Ĥ^{∗}(G_{σ}, ZZ)_{p}.

2.5 Equivariant cohomology

In the situation where G acts on a simplicial complex Δ there is defined the equivariant cohomology Ĥ_{G}^{n}(Δ, V) and in case Δ = Δ(S_{p}(G)) it is a result of Brown [4,X.7.2] that this has the same p-torsion subgroup as Ĥ^{n}(G, V)_{p}. The restriction sequences of Theorem A applied to cohomology groups M(G) = Ĥ^{n}(G, V), n ∈ ZZ, but with the left hand terms M(G) removed, make up the E_{1} page of the equivariant cohomology spectral sequence associated to this situation, the restriction maps giving the differential [4, VII.8.1]. Because of exactness of these sequences (except where M(G) has been removed), the p-torsion part of the homology of the E_{1} page consists just of groups Ĥ^{n}(G, V)_{p} concentrated on the fibre. This is then the E_{2} page, and its differential is the zero map. We thus obtain the following result.

2.5.1 COROLLARY. Let G be a finite group and Δ a G-simplicial complex such that for every simplex σ ∈ Δ the vertices of σ lie in distinct G-orbits. Suppose that for every non-identity p-subgroup H ≤ G, Δ^{H} is contractible. Then the equivariant cohomology spectral sequence

E_{1}^{r,s} = ⨁_{σ∈G\Γ_{r}(Δ)} Ĥ^{s}(G_{σ}, V)_{p} ⇒ Ĥ_{G}^{r+s}(Δ, V)_{p}

stops at the E_{2} page, and the only non-zero terms E_{2}^{r,s} lie on the fibre.

2.6 Relative cohomology and the groups H_{/Y}^{n}(G, V)

Let Y be a class of subgroups of G closed under taking subgroups and conjugation. The cohomology of G relative to Y is defined as follows. Let

P. = … → P_{2} → P_{1} → P_{0} → ZZ → 0

be a Y-split Y-projective resolution of ZZ. This means that the sequence splits on restriction to every Y ∈ Y, and every indecomposable summand of every module which appears is projective relative to some Y ∈ Y. The relative cohomology is defined as

H_{Y}^{n}(G, V) = H_{n}(Hom_{Z ZG}(P., V)).

These are not the same as the groups H_{/Y}^{n}(G, V) defined previously as H_{/Y}^{n}(G, V) = H^{n}(G, V)/ ∑_{Y ∈Y} cores_{Y}^{G}H^{n}(Y, V). As an example of the difference, take G = A × B where A and B are both cyclic of order 2 and let Y = {A, {1}}. One readily sees that a minimal Y-split Y-projective resolution of ZZ is periodic, being the splice of short exact sequences 0 → ZZ → ZZ ↑_{A}^{G}→ ZZ → 0, and hence H_{Y}^{n}(G, V) is periodic. On the other hand putting k = ZZ/2ZZ we have dim_{k} H^{n}(G, k) = n + 1 and dim_{k} H^{n}(A, k) = 1 so dim_{k} H_{/Y}^{n}(G, k) is unbounded as a function of n. This demonstrates that H_{Y}^{n}(G, V) and H_{/Y}^{n}(G, V) behave quite differently. They do, however, share the important property that they vanish if G ∈ Y.

The following is a corollary of Theorem A applied to H_{/Y}^{0}(G, ZZ)_{p}. Recall that a space is said to be mod p acyclic if it has the same homology with ZZ/pZZ coefficients as a point.

2.6.1 COROLLARY. Let Y be a set of p-subgroups of G closed under conjugation and taking subgroups. If Y does not contain a Sylow p-subgroup of G then G\Δ(S_{p}(G) − Y) is mod p acyclic.

Proof. We apply Theorem A with M(G) = H_{/Y}^{0}(G, ZZ)_{p}. In this context the mapping ‘ind’ of the Mackey functor is cores, and H_{/Y}^{0}(G, ZZ) is the quotient of H^{0}(G, ZZ) = ZZ by the sum of the images Im(cores_{Y}^{G}) = |G : Y |ZZ, Y ∈ Y. Thus H_{/Y}^{0}(G, ZZ)_{p} = ZZ/p^{α}ZZ where p^{α} is the highest power of p dividing all |G : Y |, Y ∈ Y. I claim that p | |G_{σ} : Y | for every simplex σ ∈ Δ(S_{p}(G) − Y) and every subgroup Y ≤ G_{σ}, Y ∈ Y. This is equivalent to saying that no Sylow p-subgroup of G_{σ} is in Y. To see this, suppose that σ is the chain P_{0} < P_{1} < ⋯ < P_{r} where the P_{i} are all p-subgroups not in Y. Then P_{0} ≤ G_{σ} and it follows that no Sylow p-subgroup of G_{σ} is in Y, otherwise one of them would contain P_{0}, and P_{0} would be in Y, a contradiction.

Write Δ for Δ(S_{p}(G) − Y). Theorem A now states that the sequence

0 → H_{/Y}^{0}(G, ZZ)_{p} → ⨁_{σ∈G\Γ_{0}(Δ)} H_{/Y}^{0}(G_{σ}, ZZ)_{p} → ⋯ → ⨁_{σ∈G\Γ_{d}(Δ)} H_{/Y}^{0}(G_{σ}, ZZ)_{p} → 0

is split exact. For each orbit of simplices there corresponds in this sequence a non-trivial cyclic p-group. We now reduce the sequence modulo p (to reduce all cyclic groups to ZZ/pZZ). Because of the splitting the sequence remains exact:

0 → ZZ/pZZ → ⨁_{σ∈G\Γ_{0}(Δ)} ZZ/pZZ → ⋯ → ⨁_{σ∈G\Γ_{d}(Δ)} ZZ/pZZ → 0.

The component morphisms are the reductions mod p of the (σ | gτ) · c_{g^{−1}} · res_{G_{gτ}}^{G_{σ}}, which acting on ZZ are just (σ | gτ), so we have the augmented cellular cochain complex of G\Δ with ZZ/pZZ coefficients. Its dual is the chain complex, and is also acyclic.

By Theorem 2.1.1, G\Δ(S_{p}(G)), G\Δ(A_{p}(G)) and G\Δ(B_{p}(G)) are homotopy equivalent, and hence they are all mod p acyclic, provided p | |G|. Corollary 2.6.1 thus improves on 8.2(i) in [20] where it was shown that these spaces had Euler characteristic one. It is conceivable that they are always contractible.

2.7 A structure theorem for the chain complex of S_{p}(G)

If C. = ⋯ ⟶^{d_{3}} C_{2} ⟶^{d_{2}} C_{1} ⟶^{d_{1}} C_{0} is a chain complex of modules over some ring R, we say C. is acyclic split if it has zero homology and for every r the sequence 0 → ker d_{r} → C_{r} → Im d_{r} → 0 is split. If C. is augmented by an epimorphism C_{0} → R, by saying C. is acyclic split augmented we mean the same condition applied to the augmented complex (i.e. zero reduced homology, all the sequences split, and the augmentation splits). Let ZZ_{p} be the ring of p-adic integers. The following theorem applies when Δ is the simplicial complex of S_{p}(G) , A_{p}(G) or B_{p}(G) .

2.7.1 THEOREM. Let Δ be a G-simplicial complex satisfying

(i) For every simplex σ ∈ Δ the vertices of σ lie in distinct G-orbits. (ii) For every non-identity p-subgroup H ≤ G, Δ^{H} is contractible. Let C.(Δ) be the chain complex of Δ over ZZ_{p}, as a complex of ZZ_{p}G-modules. Then C.(Δ) = D.⊕P. where D. is an acyclic split augmented subcomplex, and P. is a subcomplex of projective ZZ_{p}G-modules.

We may immediately deduce that C.(Δ) and P. have the same reduced homology. Another implication of the theorem is obtained by considering the Lefschetz module ∑_{i=0}^{d}(−1)^{i}C_{i}(Δ).

2.7.2 COROLLARY. In the circumstances of the last theorem,

ZZ_{p} ≡ ∑_{σ∈G\Δ}(−1)^{dim σ}ZZ_{p} ↑_{G_{σ}}^{G} (mod projectives).

This congruence holds in the Green ring of ZZ_{p}G-lattices.

This was the conclusion of Theorem A in [20] and one implication of the theorem in [21].

We derive Theorem 2.7.1 from our split exact sequences of cohomology groups. The proof is immediate from the next two results, proved in Section 5.

2.7.3 PROPOSITION. Let M(G) = H^{n}(G, V)_{p} for some integer n ≥ 1 and ZZGmodule V. In this situation the sequences in Theorem A are Ext_{Z Z_{p}G}^{n}(C̃.(Δ), V ⊗_{Z Z} ZZ_{p}) and Ext_{Z Z_{p}G}^{n}(C̃.(Δ)^{∗}, V ⊗_{Z Z}ZZ_{p}) where ∗ denotes the contragredient and C̃.(Δ) is the augmented chain complex of Δ.

2.7.4 THEOREM. Let R denote either ZZ_{p} or ZZ/pZZ and let C. be a finite dimensional chain complex of RG-modules which are permutation modules. The following conditions are equivalent:

(i) For some integer n ≥ 1, Ext_{RG}^{n}(C., V) is acyclic for all RG-modules V. (ii) For some integer n ≥ 1, Ext_{RG}^{n}(C., V) is acyclic split for all RG-modules V.

(iii) C. has a subcomplex D. which is acyclic split over RG so that the quotient C./D. is a complex of projective modules. (iv) C. = D. ⊕ P. as complexes of RG-modules where D. is acyclic split over RG, and P. is a complex of projective modules.

2.8 An Example

We take an uncomplicated example to illustrate the principles involved in the previous sections, where the statements we make were more fully explained. Let G = GL(3, 2) and let Δ be the building of G. This is the graph whose vertices are the subgroups Σ_{4}, the edges are the subgroups D_{8}, and the two end vertices of an edge are the two Σ_{4} subgroups containing the D_{8}. These are the proper parabolic subgroups of G.

The quotient G\Δ is the graph

\begin{matrix}Σ_{4} & D_{8} & Σ_{4} \\ • & & • \end{matrix}

with representative stabilizers as shown. We choose the representative vertices and edge so that the two vertices are actually the end vertices in Δ of the edge, and equivalently so that the particular copies of the subgroups Σ_{4} contain the D_{8} subgroup. These Σ_{4} subgroups are representatives of the two conjugacy classes in G of such subgroups. For a Mackey functor M as prescribed in Section 2.2 we obtain split exact sequences as follows:

0 → M(G) \underrightarrow{\binom{res}{res}} M(Σ_{4}) ⊕ M(Σ_{4}) ⟶^{(−res,res)} M(D_{8}) → 0
0 ← M(G) ⟵^{(ind,ind)} M(Σ_{4}) ⊕ M(Σ_{4}) \underleftarrow{\binom{−ind}{ind}} M(D_{8}) ← 0.

Taking M(G) to be Tate cohomology Ĥ^{n}(G, V)_{2} and specializing to the case Ĥ^{−2}(G, ZZ)_{2} = H_{1}(G, ZZ)_{2} = (G/G′)_{2} we obtain the sequence

0 → (G/G′)_{2} → Σ_{4}/Σ′_{4} ⊕ Σ_{4}/Σ′_{4} → D_{8}/D′_{8} → 0

with ± the classical transfer homomorphism as the component morphisms, and also the sequence

0 ← (G/G′)_{2} ← Σ_{4}/Σ′_{4} ⊕ Σ_{4}/Σ′_{4} ← D_{8}/D′_{8} ← 0

with the component homomorphisms given by ± inclusion of subgroups. We obtain similar sequences with the 2-torsion subgroup of the Schur multiplier Ĥ^{−3}(G, ZZ)_{2} = H_{2}(G, ZZ)_{2}, one of which identifies as

0 → C_{2} → C_{2} ⊕ C_{2} → C_{2} → 0.

The Mackey functor M(G) = Br/p_{′}-subgroups(G) takes as its value the abelian group which is a direct sum of cyclic groups whose orders are the elementary divisors of the Cartan matrix. For this Mackey functor the first sequence is

0 → C_{8} → C_{8} ⊕ C_{8} → C_{8} → 0.

The group r/p_{′}-subgroups(G) has torsion-free rank the number of conjugacy classes of elements in G of order divisible by p. With p = 2 the first sequence for this Mackey functor is

0 → ZZ^{2} → ZZ^{3} ⊕ ZZ^{3} → ZZ^{4} → 0.

Turning to the chain complex C.(Δ) of Δ over ZZ_{2}, one knows from the Solomon-Tits theorem that the reduced homology is zero except in dimension 2, where it is the Steinberg module St(G), a projective ZZ_{2}G-module. In the notation of Theorem 2.7.1 this is the homology of the subcomplex P. so we have an exact sequence

0 → St(G) → P_{2} → P_{1} → P_{0} → 0

and since all modules are projective, it splits. It follows from 2.7.1 that the augmented chain complex C̃.(Δ) is the direct sum of an acyclic split complex and the homology group St(G). This splitting has also been observed for Chevalley groups by Kuhn and Mitchell [11].

3. Mackey functors

3.1 First notions

The notion of a Mackey functor is designed to capture the common features of such constructions as character rings, cohomology groups, certain K-groups, various representation rings, etc., and in the form in which we will take them they were first defined by Dress [7]. Dress’s work followed soon after work of Green [9], who made a similar definition. The Mackey functors here may be subsumed into Green’s theory of G-functors by considering them as G-functors with zero multiplication. We choose not to do this because Green’s definition of relative projectivity is not the appropriate one to use here. In the present context the definition of relative projectivity given by Dress is more fundamental and convenient, and it is for this reason that we follow Dress. Our concepts agree with those of Yoshida [23,24] and Sasaki [17] but we differ in terminology. They use Green’s term ‘G-functor’ to mean a Mackey functor regarded as a being defined on the subgroups of G, but this differs from Green’s usage. They reserve the term ‘Mackey functor’ for a Mackey functor regarded as being defined on G-sets, which is the same as the terminology of Dress. Here we prefer to regard these two ways of viewing Dress’s Mackey functors as being little more than a change of notation, and so we will call them both Mackey functors.

We will summarise the work of Dress [7], often taking a different point of view.

Definition. A Mackey functor for G over a ring R is a function

M : {subgroups of G} → R-mod

with morphisms

\begin{aligned}ind_{K}^{H} & : M(K) → M(H) \\ res_{K}^{H} & : M(H) → M(K) \\ c_{g} & : M(H) → M(^{g}H) \end{aligned}

whenever K ≤ H are subgroups of G and g ∈ G, such that

(0) ind_{H}^{H}, res_{H}^{H}, c_{h} : M(H) → M(H) are the identity morphisms for all subgroups H and h ∈ H

(1) res_{J}^{K}res_{K}^{H} = res_{J}^{H} (2) ind_{K}^{H}ind_{J}^{K} = ind_{J}^{H}} for all subgroups J ≤ K ≤ H

(3) c_{g}c_{h} = c_{gh} for all g, h ∈ G

(4) res_{^{g}K}^{^{g}H}c_{g} = c_{g}res_{K}^{H} (5) ind_{^{g}K}^{^{g}H}c_{g} = c_{g}ind_{K}^{H}} for all subgroups K ≤ H and g ∈ G

(6) res_{J}^{H}ind_{K}^{H} = ∑_{x∈J\H/K} ind_{J∩^{x}K}^{J}c_{x}res_{J^{x}∩K}^{K} for all subgroups J, K ≤ H.

The most important of these axioms is (6), which is the Mackey decomposition formula. As is well-known, this definition is equivalent to the definition given by Dress, except that Dress works with a more general domain of definition. We now remind the reader of Dress’s version.

Let C and D be categories. A bifunctor M = (M_{∗}, M^{∗}) : C → D is a pair consisting of a covariant functor M_{∗} : C → D and a contravariant functor M^{∗} : C → D such that on objects, M_{∗}(C) = M^{∗}(C) for all C ∈ Ob(C). We write M(C) for the common value of these functors.

Let G-set denote the category of finite left G-sets. The morphisms in this category are the G-equivariant mappings. A Mackey functor is a bifunctor M : G-set → R-mod such that the following two conditions hold:

(1) for every pullback diagram

Ω_{1} ⟶^{α} Ω_{2}
β↓  ↓^{γ}
Ω_{3} ⟶_{δ} Ω_{4}

in G-set we have M^{∗}(δ)M_{∗}(γ) = M_{∗}(β)M^{∗}(α).

(2) The two mappings Ω → Ω ∪ Ψ ← Ψ into the disjoint union define an isomorphism M(Ω ∪ Ψ) ≅ M(Ω) ⊕ M(Ψ) via M_{∗}.

The purpose of giving these definitions is that we will sometimes switch notation and write either M(H) or M(G/H) for the value of the Mackey functor M, using either the subgroup H as the argument as in the first definition, or the G-set G/H as in the second. On morphisms the connection between these two notations is that if K ≤ H and π : G/K → G/H is the natural quotient map then

M_{∗}(π) = ind_{K}^{H}  and  M^{∗}(π) = res_{K}^{H}

while if c_{g} : G/H → G/^{g}H is a conjugation map of G-sets then

M_{∗}(c_{g}) = c_{g}  and  M^{∗}(c_{g}) = c_{g^{−1}}.

In technical arguments it is usually easier to work with G-set notation. On the other hand it is generally the case that whenever one has a naturally occuring example of a Mackey functor, one most easily recognizes that it is a Mackey functor by verifying the first definition. Thus both definitions have their uses.

3.2 Relative projectivity

For the sake of establishing notation we recall the definition of relative projectivity from [7]. Using the G-set notation, let S be a fixed G-set and M a Mackey functor. Then

\begin{matrix}M_{S} : T → M(T × S) \\ M_{S}^{∗}(f) = M^{∗}(f × id_{S}), M_{S∗}(f) = M_{∗}(f × id_{S}) \end{matrix}

defines a Mackey functor M_{S}, as one easily checks. Projection pr : T × S → T onto the first coordinate defines a natural transformation of bifunctors

θ_{S} : M_{S} → M, θ_{S}(T) = M_{∗}(pr).

Definition. The Mackey functor M is called S-projective if θ_{S} is split-surjective as a natural transformation of Mackey functors. If X is a set of subgroups of G we put X = ⋃_{H∈X} G/H and say M is X -projective if it is X-projective.

The most powerful general method we have of showing that a particular Mackey functor M is projective relative to a set of subgroups X depends on the theorem of Dress [7], which states that in case M happens to be a Green functor (in the sense of Dress) then it is sufficient that

∑_{H∈X} ind_{H}^{G} : ⨁_{H∈X} M(H) → M(G)

be surjective. Furthermore, a Green module over a Green functor which is X -projective is itself X -projective. Using this result we are able to determine the relative projectivity of all the Mackey functors we consider here, and we summarize this information in the following table.

Mackey functor projective relative to
r(G)  Brauer elementary subgroups
Br(G)  Brauer elementary subgroups
a(G)  subgroups Brauer elementary mod p
A(G)  subgroups cyclic mod p
Ĥ^{n}(G, V)_{p}  p-subgroups
H_{Y}^{n}(G, V)_{p}, n ≥ 1 p-subgroups

These assertions of relative projectivity follow, respectively, from Brauer’s induction theorem, Brauer’s induction theorem together with surjectivity of the decomposition map, Dress’s induction theorem [8], Conlon’s induction theorem [5], and for the cohomology groups the fact that corestriction from a Sylow p-subgroup is surjective on the p-torsion subgroup.

We will combine this with the work of Section 3.4 to deduce the X -projectivity of the Mackey functors in the list in Theorem 2.2.2.

3.3 Induction and restriction of Mackey functors

It will be important for us later on to interpret the notion of relative projectivity in terms of induction and restriction of Mackey functors. These are introduced in [17], where they are attributed to Yoshida. We present them in a different way using G-set notation. First we go back to G-sets, where we have a notion of induction and restriction

↑_{H}^{G}: H-set → G-set

and

↓_{H}^{G}: G-set → H-set

whenever H is a subgroup of G. Induction is defined by the formula

Ω ↑_{H}^{G}= G ×_{H} Ω

which, by definition, is the set of equivalence classes of G×Ω under the equivalence relation (gh, ω) ∼ (g, hω), g ∈ G, h ∈ H, ω ∈ Ω. It has the property that (H/K) ↑_{H}^{G}= G/K. The restriction Ω ↓_{H}^{G} is defined simply to be Ω regarded as an H-set by restriction of the action.

We define induction and restriction of Mackey functors. We denote the category of Mackey functors on G by Mack(G). Now induction and restriction are functors

↑_{H}^{G}: Mack(H) → Mack(G)

and

↓_{H}^{G}: Mack(G) → Mack(H).

They are defined in a formal fashion by

\begin{aligned}M ↑_{H}^{G} (Ω) & = M(Ω ↓_{H}^{G}) \\ M ↓_{H}^{G} (Ω) & = M(Ω ↑_{H}^{G}). \end{aligned}

In terms of subgroup notation, M ↓_{H}^{G} simply requires that one forget those subgroups of G which are not subgroups of H. Thus M ↓_{H}^{G} (K) = M(K) if K ≤ H. To make the connection with the notion of relative projectivity we will need the following lemma.

3.3.1 LEMMA. (i) Ω ↓_{H}^{G}↑_{H}^{G} ≅ Ω × G/H (ii) M_{G/H} ≅ M ↓_{H}^{G}↑_{H}^{G}

Proof. (i) The isomorphism is given by

\begin{matrix}G ×_{H} Ω & → Ω × G/H \\ (g, ω) & ↦ (ω, gH). \end{matrix}

(ii) Evaluating at a G-set Ω we have

M_{G/H}(Ω) = M(Ω × G/H)
≅ M(Ω ↓_{H}^{G}↑_{H}^{G})
= M ↓_{H}^{G}↑_{H}^{G} (Ω).

Because the isomorphism in part (i) is natural with respect to Ω we deduce that we have a natural isomorphism between M_{G/H} and M ↓_{H}^{G}↑_{H}^{G}.

It is because of the identification of M_{G/H} just given that we refer to the sequences in Theorem B as resolutions of M by induced functors. We point out a further property of induction and restriction, but we will not use it in this paper. It turns out that ↑_{H}^{G} is both the left and the right adjoint of ↓_{H}^{G}. With the identification of M_{G/H} just given the natural transformation θ_{G/H} : M_{G/H} → M, which arises in the definition of projectivity relative to G/H, corresponds to the counit M ↓_{H}^{G}↑_{H}^{G}→ M of one of these adjunctions.

In Section 4.2 we will need to know the following result, which has an easy direct proof.

3.3.2 LEMMA. Induction of Mackey functors is exact.

Proof. Given an exact sequence of Mackey functors for H

0 → L → M → N → 0

the induced sequence

0 → L ↑_{H}^{G}→ M ↑_{H}^{G}→ N ↑_{H}^{G}→ 0

is exact if and only if it is exact on evaluation at each G-set Ω. But this is

0 → L(Ω ↓_{H}^{G}) → M(Ω ↓_{H}^{G}) → N(Ω ↓_{H}^{G}) → 0

which is exact because it is an evaluation of the original sequence.

3.4 The construction of M_{/Y} and the proof of 2.2.1

In Section 2.2, starting with a Mackey functor M and a set Y of subgroups of G closed under taking subgroups and conjugation, we defined Mackey functors

M(H, Y) = ∑_{Y ∈Y,Y ≤H} ind_{Y}^{H}M(Y), M_{/Y}(H) = M(H)/M(H, Y).

We should observe first that these are Mackey functors, and for this it suffices to note that M(H, Y) is stable under ind and res and c_{g}. This is straightforward from the axioms. We now prove 2.2.1. Conditions (i) and (ii) of Theorem A hold immediately by construction. It remains to verify conditions (iii) and (iv). These will follow from 3.4.1 and 3.4.2.

3.4.1 PROPOSITION. M_{/Y} is the largest quotient of M which vanishes on the subgroups in Y.

Proof. First, M_{/Y} vanishes on Y because if H ∈ Y then M(H, Y) contains M(H) = ind_{H}^{H}M(H) so M_{/Y}(H) = M(H)/M(H) = 0. To say it is the largest means that whenever we have an epimorphism of Mackey functors ν : M → N (i.e. a natural transformation) and N(Y) = 0 for all Y ∈ Y then there is a factorisation

M  ⟶^{ν}  N
↘  ↗
M_{/Y}

Consider the short exact sequence of Mackey functors

0 → M( , Y) → M → M_{/Y} → 0.

Now for any Y ∈ Y, ν(ind_{Y}^{H}(M(Y))) = ind_{Y}^{H}ν(M(Y)) = 0 since ν is a natural transformation, and νM(Y) ⊆ N(Y) = 0. Thus ν vanishes on M( , Y), and hence ν lifts to a mapping M_{/Y} → N as required.

3.4.2 PROPOSITION. Suppose M is X -projective. Then M( , Y) and M_{/Y} are X -projective.

We first need to prove:

3.4.3 LEMMA. Let T = ⋃_{Y ∈Y} G/Y. Then M( , Y) is the image of the natural transformation θ_{T} : M_{T} → M.

Proof. Recall from 3.2 that M_{T} (Ω) = M(Ω × T) for any G-set Ω, and θ_{T} = M_{∗}(pr) : M(Ω × T) → M(Ω). To say that M( , Y) is the image of θ_{T} means that M(Ω, Y) is the image of θ_{T} for every G-set Ω. Using a mix of subgroup and G-set notation,

\begin{aligned}θ_{T} M_{T} (H) & = θ_{T} M((G/H) × T) \\ & = ∑_{Y ∈Y} ∑_{x∈H\G/Y} ind_{H∩^{x}Y}^{H} M(H ∩ ^{x}Y) \end{aligned}

and this is M(H, Y) since all terms in the above sum are contained in M(H, Y); and conversely taking the terms with x = 1 we obtain ind_{H∩Y}^{H} (M(H ∩ Y)) for all Y ∈ Y in the above sum, and this includes every term ind_{Y}^{H}M(Y) with Y ∈ Y, Y ≤ H.

Proof of 3.4.2. Let S = ⋃_{X∈X} G/X and T = ⋃_{Y ∈Y} G/Y. Consider the diagram

0 → M( , Y)_{S} → M_{S} → M_{/YS} → 0
↓^{θ_{S}}  ↓^{θ_{S}}  ↓^{θ_{S}}
0 → M( , Y) → M → M_{/Y} → 0.

The bottom row is exact, by definition of M_{/Y}. The top row is obtained by evaluating the bottom on Ω × S, and it follows that the top row is exact, because the bottom row is exact on every evaluation. The diagram commutes, because the horizontal arrows are natural transformations and the vertical arrows are obtained from evaluation on Ω × S → Ω. We are given that M is X -projective, so there is a natural transformation α : M → M_{S} which splits θ_{S}, i.e. θ_{S}α = 1_{M}. We show that α sends M( , Y) to M( , Y)_{S}. This is because for any G-set Ω,

α(M(Ω, Y)) = α Im(M(Ω × T) → M(Ω))
⊆ Im(M_{S}(Ω × T) → M_{S}(Ω))
= Im(M(Ω × T × S) → M(Ω × S))
= Im(M(Ω × S × T) → M(Ω × S))
= M(Ω, Y)_{S}.

by 3.4.3

from the diagram below

by swapping S and T

The containment above follows by considering the commutative diagram

M_{S}(Ω × T) → M_{S}(Ω)
↑^{α_{Ω×T}}  ↑^{α_{Ω}}
M(Ω × T) → M(Ω).

It follows that α restricts to a splitting of θ_{S} on M( , Y)_{S} and induces a splitting of θ_{S} on M_{/YS}. Thus M( , Y) and M_{/Y} are both X -projective.

4. Proof of the main theorems

4.1 Semisimplicial constructions

Throughout the constructions we consider we will be working with a G-simplicial complex Δ which satisfies

(*) for every simplex σ ∈ Δ, the vertices of σ lie in distinct G-orbits.

If Δ does not have this property then its barycentric subdivision does.

We will be interested in two consequences of (*), the first of which is the fact that in the presence of (*) every fixed point set Δ^{H} is also a simplicial complex. The other consequence is the following lemma.

4.1.1 LEMMA. Let Δ be a G-simplicial complex satisfying (*). Then we may put a G-invariant partial order on the vertices of Δ in such a way that in every simplex the vertices are totally ordered.

Proof. Totally order the set of orbits of vertices. The partial order we require has x < y if and only if the orbit of x is less than the orbit of y.

A G-simplicial complex satisfying the conclusion of this lemma is sometimes called an ordered G-simplicial complex, and we might as well be working in the category of ordered G-simplicial complexes.

We will need to use the language of semisimplicial objects, and for this we refer the reader to [13]. Starting with an ordered G-simplicial complex Δ we may form the associated simplicial set

S(Δ) = ⋯ Γ_{2}(Δ) → ; ← ; → ; ← ; → Γ_{1}(Δ) → ; ← ; → Γ_{0}(Δ)

where the face maps ∂_{j} : Γ_{n}(Δ) → Γ_{n−1}(Δ) are ∂_{j}(v_{0}, …, v_{n}) = (v_{0}, …, v_{j−1}, v_{j+1}, …, v_{n}) and the degeneracy maps are s_{j}(v_{0}, …, v_{n}) = (v_{0}, …, v_{j}, v_{j}, …, v_{n}). It is immediate from the fact that G preserves the order of vertices in every simplex that all the ∂_{j} and s_{j} are G-equivariant mappings. We thus see that in fact S(Δ) is a simplicial G-set. We really wish to consider the augmented simplicial object associated to Δ, which is

S̃(Δ) = ⋯ Γ_{2}(Δ) → ; ← ; → ; ← ; → Γ_{1}(Δ) → ; ← ; → Γ_{0}(Δ) → Γ_{−1}(Δ)

where Γ_{−1}(Δ) is a single point, and this is also a simplicial G-set.

We now perform some further operations. These rely on the observation that whenever we are given a functor Θ : G-set → C where C is some category then the diagram ΘS̃(Δ) is a simplicial object in C. Suppose we are given a Mackey functor M for G (over a commutative ring R). Let Mack(G) denote the category of Mackey functors for G over the ring R. We define two functors Θ_{M}, Θ^{M} : G-set → Mack(G) in the following fashion, such that Θ_{M} is covariant and Θ^{M} is contravariant. For a G-set Ω we put Θ_{M}(Ω) = Θ^{M}(Ω) = M_{Ω}. Given a morphism α : Ω → Ψ of G-sets we define Θ_{M}(α) : M_{Ω} → M_{Ψ} and Θ^{M}(α) : M_{Ψ} → M_{Ω} as follows. The effect of Θ_{M}(α) on the Mackey functor M_{Ω} evaluated at a G-set X is M_{∗}(1 × α) : M(X × Ω) → M(X × Ψ). The effect of Θ^{M}(α) is similarly M^{∗}(1 × α) : M(X × Ψ) → M(X × Ω). In Mack(G) we may now form the chain complexes of Mackey functors associated to Θ_{M} S̃(Δ) and Θ^{M} S̃(Δ). These will be denoted CC(Θ_{M} S̃(Δ)) and CC(Θ^{M} S̃(Δ)). Explicitly, we mean the objects in the first chain complex to be the Θ_{M}(Γ_{r}(Δ)) with differential ∑_{j=0}^{r}(−1)^{j}Θ_{M}(∂_{j}), and similarly for the second one. We summarise the operations we have just performed, writing Θ instead of Θ_{M} or Θ^{M}:

G-simplicial complex⟶^{S̃}simplicial G-set
⟶^{Θ}simplicial Mackey functor
⟶^{CC}chain complex of Mackey functors.

We will show that the chain complexes we finish with are the ones specified in Theorem B.

The chain complexes in Theorem A are the evaluations at the trivial G-set (or, equivalently, at G) of the chain complexes in Theorem B. We can therefore construct them by repeating what we have just done, and evaluating always at the trivial G-set. A more direct proceedure is indicated by the following scheme:

G-simplicial complex⟶^{S̃}simplicial G-set
⟶^{M}simplicial R-module
⟶^{CC}chain complex of R-modules.

Here M denotes either M_{∗} or M^{∗}. We apply M_{∗} and M^{∗} to all the mappings and objects which make up S̃(Δ). We thus obtain two simplicial R-modules denoted M_{∗}S̃(Δ) and M^{∗}S̃(Δ). In the category of R-modules, we may now form the chain complexes of Rmodules associated to M_{∗}S̃(Δ) and M^{∗}S̃(Δ) and these are denoted CC(M_{∗}S̃(Δ)) and CC(M^{∗}S̃(Δ)).

4.1.2 LEMMA. The sequences in Theorem A are CC(M^{∗}S̃(Δ)) and CC(M_{∗}S̃(Δ)). The sequences in Theorem B are CC(Θ^{M} S̃(Δ)) and CC(Θ_{M} S̃(Δ)).

Proof. We give the proof for the sequences in Theorem B and then indicate the modifications we must make for the sequences in Theorem A.

We work first with CC(Θ^{M} S̃(Δ)). The typical Mackey functor in CC(Θ^{M} S̃(Δ)) is Θ^{M}(Γ_{r}(Δ)) = M_{Γ_{r}(Δ)} and since

Γ_{r}(Δ) ≅ ⋃_{σ∈G\Γ_{r}(Δ)} G/G_{σ}

broken up according to the given set of representatives σ for the G-orbits on Γ_{r}(Δ), we have

M_{Γ_{r}(Δ)} ≅ ⨁_{σ∈G\Γ_{r}(Δ)} M_{G_{σ}}.

The differentials in the chain complexes are ∑_{j=0}^{r}(−1)^{j}Θ^{M}(∂_{j}). If τ = (v_{0} … v_{r}) is an r-simplex with the vertices taken in order then ∂_{j}(τ) = (v_{0} … v_{j−1}, v_{j+1} … v_{r}). Putting ∂_{j}(τ) = ρ, say, we use the notation (ρ|τ) = (−1)^{j}, and this is what we mean by the ‘orientation of the embedding of ρ in τ’. Thus the component of the differential between the orbit containing τ in Γ_{r}(Δ) and the orbit containing ρ in Γ_{r−1}(Δ) is (ρ|τ)Θ^{M}(∂_{j}) where on the orbit containing τ, ∂_{j} is the map

\begin{matrix}∂_{j} : G/G_{τ} & → G/G_{ρ} \\ gG_{τ} & ↦ gG_{ρ}. \end{matrix}

Suppose that in fact ρ was not one of our originally chosen orbit representatives, but that σ was, and that σ = gρ for some g ∈ G. Then σ is a face of gτ. Identifying the orbit containing ρ and σ with G/G_{σ} we now write ∂_{j} as

\begin{matrix}∂_{j} : G/G_{τ} & → G/G_{σ} \\ xG_{τ} & ↦ xg^{−1}G_{σ} \end{matrix} .

This factorizes as

G/G_{τ} ⟶^{c_{g}} G/^{g}G_{τ} = G/G_{gτ} ⟶^{π_{Ggτ}^{Gσ}} G/G_{σ},

where c_{g} is the map xG_{τ} ↦ xg^{−1}^{g}G_{τ} and π_{G_{gτ}}^{G_{σ}} is xG_{gτ} ↦ xG_{σ}. Applying Θ^{M} to this we conclude that the component morphism M_{G_{σ}} → M_{G_{τ}} is

(σ|gτ)Θ^{M}(π_{G_{gτ}}^{G_{σ}} c_{g}) = (σ|gτ)M^{∗}(1 × π_{G_{gτ}}^{G_{σ}} · c_{g}) : M(? × G/G_{σ}) → M(? × G/G_{τ}).

The proof for the second chain complex CC(Θ_{M} S̃(Δ)) is similar to this, but we apply Θ_{M} instead of Θ^{M}. Thus the component morphism M_{G_{τ}} → M_{G_{σ}} is

(σ|gτ)Θ_{M}(π_{G_{gτ}}^{G_{σ}} c_{g}) = (σ|gτ)M_{∗}(1 × π_{G_{gτ}}^{G_{σ}} · c_{g}) : M(? × G/G_{τ}) → M(? × G/G_{σ}).

These are the mappings described in Theorem B.

For the sequences in Theorem A we proceed in the same way but we only have to apply the functors M^{∗} and M_{∗}. Thus the component morphisms are

ϕ_{τσ} = (σ|gτ)M^{∗}(π_{G_{gτ}}^{G_{σ}} c_{g}) = (σ|gτ) · c_{g^{−1}} · res_{G_{gτ}}^{G_{σ}}

and

ψ_{στ} = (σ|gτ)M_{∗}(π_{G_{gτ}}^{G_{σ}} c_{g}) = (σ|gτ) · ind_{G_{gτ}}^{G_{σ}} · c_{g}.

Remark. It is clear from the above discussion that the sequences in Theorem A are obtained by evaluating the sequences in Theorem B at the trivial G-set. Because of this, Theorem A is a consequence of Theorem B, since a sequence of Mackey functors which is exact and split must be exact and split on each evaluation. In view of this we will only prove Theorem B in the next sections. A direct proof of Theorem A may be obtained by following the arguments we give and replacing Θ_{M} with M_{∗} and Θ^{M} with M^{∗}.

4.2 The reduction step

We suppose we are given a G-simplicial complex Δ, a Mackey functor M and sets of subgroups X and Y satisfying the conditions of Theorem A. We will show

4.2.1 PROPOSITION. To prove Theorems A and B it suffices to assume the structure of G is such that G ∈ X.

The proof of this is obtained by combining the next two results. If H is a subgroup of G we let M ↓_{H}^{G} denote M with its domain of definition restricted to the subgroups of H. Similarly Δ ↓_{H}^{G} denotes Δ regarded as an H-simplicial complex by restriction of the action. We put

X_{H} = {K ∈ X | K ≤ H} andY_{H} = {K ∈ Y | K ≤ H}.

It is an elementary observation that the hypotheses of Theorem A are inherited by the quadruple M ↓_{H}^{G}, Δ ↓_{H}^{G}, X_{H} and Y_{H}. The immediate implication of the next results is that it suffices to show that CC(Θ_{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G} and CC(Θ^{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G} are split acyclic for all H ∈ X. Since induction is exact (and hence preserves direct sums), it suffices to show that CC(Θ_{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) and CC(Θ^{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) are split acyclic for all H ∈ X, and in view of the observation made just previously this is the same as assuming that G ∈ X.

4.2.2 LEMMA. CC(Θ_{M} S̃(Δ)) is a direct summand of

⨁_{H∈X} CC(Θ_{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G} .

Similarly CC(Θ^{M} S̃(Δ)) is a direct summand of

⨁_{H∈X} CC(Θ^{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G} .

4.2.3 LEMMA. A direct sum of chain complexes is split acyclic if and only if its summands are split acyclic.

Proof of 4.2.2. This is an exercise in using the relative projectivity of M. We work first in an abstract setting where we have a functor Θ : G-set → C for some abelian category C, thus obtaining a simplicial object Θ(S̃(Δ)) in C. In our application we will take Θ = Θ_{M} or Θ^{M}. Suppose we have another functor Ξ : G-set → C and natural transformations α : Ξ → Θ, β : Θ → Ξ such that αβ = 1. Then α, β give rise to simplicial maps

Θ(S̃(Δ))⟶^{β}Ξ(S̃(Δ))⟶^{α}Θ(S̃(Δ))

and hence to maps of chain complexes

CC(Θ(S̃(Δ)))⟶^{β}CC(Ξ(S̃(Δ)))⟶^{α}CC(Θ(S̃(Δ)))

such that the composite of the two maps is the identity. By this means we deduce that CC(Θ(S̃(Δ))) is a direct summand of CC(Ξ(S̃(Δ))) as chain complexes.

In our situation we put X = ⋃_{H∈X} G/H, and take Θ = Θ_{M} and Ξ = Θ_{M_{X}}. The projectivity of M relative to X means that the natural transformation θ_{X} : M_{X} → M is a split epimorphism. We will show that θ_{X} gives rise to a natural transformation α : Θ_{M_{X}} → Θ_{M}. At a G-set Ω we define the effect of α to be another natural transformation α_{Ω} : Θ_{M_{X}} (Ω) = (M_{X})_{Ω} → M_{Ω} = Θ_{M}(Ω), whose effect at a G-set Ψ is

(M_{X})_{Ω}(Ψ) = M_{X}(Ψ × Ω)⟶^{θ_{X}}M(Ψ × Ω) = M_{Ω}(Ψ).

It is apparent that α is natural with respect to Ω because when we expand the G-sets as direct products the G-set morphisms which take place in the Ω factor commute with morphisms which take place in the other factors. In a similar fashion the natural transformation M → M_{X} which splits θ_{X} gives rise to a natural transformation β : Θ_{M} → Θ_{M_{X}}, and we have αβ = 1. We have now shown that the abstract situation just described in the previous paragraph is in force here, and we deduce that CC(Θ_{M} S̃(Δ)) is a direct summand of CC(Θ_{M_{X}} S̃(Δ)).

Evidently CC(Θ_{M_{X}} S̃(Δ)) ≅ ⨁_{H∈X} CC(Θ_{M_{G/H}} S̃(Δ)) and it remains to show that Θ_{M_{G/H}} S̃(Δ) ≅ (Θ_{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G}. This is immediate from the fact that Θ_{M_{G/H}} factorizes as a composite

G-set⟶^{↓_{H}^{G}}H-set ⟶^{Θ_{M↓{G \atop H}}} Mack(H)⟶^{↑_{H}^{G}} Mack(G).

We see this by considering the chain of isomorphisms

Θ_{M_{G/H}} (Ω)(Ψ) = (M_{G/H})_{Ω}(Ψ)
= M_{G/H}(Ψ × Ω)
= M(Ψ × Ω × G/H)
≅ M((Ψ × Ω) ↓_{H}^{G}↑_{H}^{G})
= M ↓_{H}^{G} (Ψ ↓_{H}^{G} ×Ω ↓_{H}^{G})
= (M ↓_{H}^{G})_{Ω↓_{H}^{G}} (Ψ ↓_{H}^{G})
= ((M ↓_{H}^{G})_{Ω↓_{H}^{G}}) ↑_{H}^{G} (Ψ).

The argument which shows that CC(Θ^{M} S̃(Δ)) is a direct summand of

⨁_{H∈X} CC(Θ^{M↓_{H}^{G}} S̃(Δ ↓_{H}^{G})) ↑_{H}^{G}

is similar.

Proof of 4.2.3. Suppose C_{1} ⊕ C_{2} is a direct sum of chain complexes. Then H_{∗}(C_{1} ⊕ C_{2}) ≅ H_{∗}(C_{1}) ⊕ H_{∗}(C_{2}) so if C_{1} ⊕ C_{2} is acyclic, so are its summands.

As for the splitting, we recall from [22, 7.1] (and it is easy to prove) that a chain complex with differential d is split if and only if there is a chain map α of degree +1 with dαd = d. The differential on C_{1} ⊕ C_{2} is d = (d_{1}, d_{2}), where d_{1} and d_{2} are the differentials on C_{1} and C_{2}. Suppose that C_{1} ⊕ C_{2} is split by a map α of degree +1 with dαd = d. Then i_{1}απ_{1} splits C_{1}, where i_{1} : C_{1} → C_{1} ⊕ C_{2} and π_{1} : C_{1} ⊕ C_{2} → C_{1} are inclusion and projection, since d_{1}i_{1}απ_{1}d_{1} = i_{1}dαdπ_{1} = i_{1}dπ_{1} = d_{1}.

4.3 The direct argument with subgroups in X

Our argument uses notions from topology, and in particular the notion of G-homotopy. Given topological spaces X and Y on which G acts as a group of homeomorphisms, we say that G-equivariant maps f, g : X → Y are G-homotopic if they are homotopic by a homotopy H : X × I → Y which is itself a G-equivariant map, where the unit interval I has the trivial G-action. The spaces X and Y are G-homotopy equivalent if there are Gequivariant maps between them in each direction so that the composites in both directions are G-homotopic to the identity.

The final lines in our argument will rely on the following observation.

4.3.1 LEMMA. Let Δ_{1} and Δ_{2} be G-simplicial complexes satisfying (*) which are G-homotopy equivalent, and let Θ denote one of the functors Θ_{M}, Θ^{M}, M_{∗} or M^{∗} as in Section 4.1. Then CC(ΘS̃(Δ_{1})) and CC(ΘS̃(Δ_{2})) are chain homotopy equivalent.

Proof. For each G-simplicial complex Δ we will need to consider the simplicial Gset of singular simplices of |Δ|, which we will denote Σ(Δ), and also the corresponding augmented simplicial G-set Σ̃(Δ). We will show two things, firstly that Δ_{1} ≃_{G} Δ_{2} implies CC(ΘΣ̃(Δ_{1})) ≃ CC(ΘΣ̃(Δ_{2})); and secondly that CC(ΘS̃(Δ)) ≃ CC(ΘΣ̃(Δ)) for any Gsimplicial complex Δ. Putting these pieces together we obtain the conclusion of Lemma 4.3.1, since CC(ΘS̃(Δ_{1})) ≃ CC(ΘΣ̃(Δ_{1})) ≃ CC(ΘΣ̃(Δ_{2})) ≃ CC(ΘS̃(Δ_{2})).

To start, it is immediate that Δ_{1} ≃_{G} Δ_{2} implies Σ̃(Δ_{1}) ≃ Σ̃(Δ_{1}) as in [25, p.12], and since all mapings are G-equivariant we have a homotopy equivalence of simplicial G-sets. We now wish to apply the functor Θ to both sides. It is first neccessary to say what this would mean, since Θ has so far only been defined on finite G-sets and we need to extend the domain of definition to infinite ones. We define Θ on an infinite G-set Ω by first expressing it as a union of its orbits Ω = ⋃ Ω_{i} and then putting Θ(Ω) = ⨁ Θ(Ω_{i}). On morphisms we define Θ in terms of its components with respect to this direct sum decomposition, each component being the effect of Θ on the restriction of the morphism to the corresponding orbit. Now to show that CC(ΘΣ̃(Δ_{1})) ≃ CC(ΘΣ̃(Δ_{2})) we proceed as in [25, p.100]. The essential matter is to show that if we have two semisimplicial mappings

f_{0}, f_{1} : Σ̃(Δ_{1}) → Σ̃(Δ_{2})

which are homotopic by a homotopy H then the chain mappings CC(Θ(f_{0})) and CC(Θ(f_{1})) are chain homotopic. Using notation borrowed from [25], each of the mappings

\begin{matrix}κ_{i} : Σ_{n}(Δ_{1}) & → Σ_{n+1}(Δ_{2}) \\ x & ↦ H(s_{i}x, (0 … ^{i}01 … 1)) \end{matrix}

is G-equivariant, so we may form Θ(κ_{i}). Now we put

k = ∑_{i=0}^{n}(−1)^{i}Θ(κ_{i})

and calculate in the usual way that dk + kd = CC(Θ(f_{1})) − CC(Θ(f_{0})). This is the same as the usual calculation except that all the terms have Θ applied to them. To be explicit about this, we have to show that

∑_{j=0}^{n+1}(−1)^{j}Θ(∂_{j}) ∑_{i=0}^{n}(−1)^{i}Θ(κ_{i}) + ∑_{i=0}^{n−1}(−1)^{i}Θ(κ_{i}) ∑_{j=0}^{n}(−1)^{j}Θ(∂_{j}) = Θ(f_{1}) − Θ(f_{0}).

We rely on the identities

\begin{aligned}∂_{j}κ_{i} & = \{ \begin{matrix}κ_{i−1}∂_{j} if j < i \\ κ_{i}∂_{j−1} if j > i + 1 \end{matrix} \\ ∂_{i+1}κ_{i} & = ∂_{i+1}κ_{i+1} \\ ∂_{0}κ_{0} & = f_{1} \\ ∂_{n+1}κ_{n} & = f_{0} \end{aligned}

which immediately show that terms cancel in pairs except for two of them, and hence the above equation holds. Now the desired assertion about chain homotopy equivalence of the complexes follows in the usual way from what we have just shown about mappings.

Finally we need to show that CC(ΘS̃(Δ)) ≃ CC(ΘΣ̃(Δ)) for any G-simplicial complex Δ. To do this we follow a standard treatment which show that the singular and simplicial chain complexes of Δ are chain homotopy equivalent, such as [26]. One has the natural inclusion CC(S̃(Δ)) ↪ CC(Σ̃(Δ)) and in [26, p.115] there is constructed a map in the reverse direction which is a homotopy inverse to the inclusion. We have to observe that in this construction the homotopy inverse may be obtained on each orbit of simplices as an integer linear combination of G-equivariant maps. This is because the constructions at every stage are natural, and we may always proceed by making a definition on one element in an orbit and extending to the rest of the orbit so as to be equivariant for the action of G. We may now take instead the same linear combinations of these mappings but with Θ applied to them. The result is a chain map CC(ΘΣ̃(Δ)) → CC(ΘS̃(Δ)), and it is a homotopy inverse to the canonical map CC(ΘS̃(Δ)) → CC(ΘΣ̃(Δ)). One shows these facts by exactly the same arguments as in [26] except that Θ is applied to all the terms in the equations. For the validity of these equations after Θ has been applied one relies on the fact that in every case the equations simplify because terms cancel in pairs. Thus it makes no difference if we apply Θ everywhere. We omit the precise technicalities of this argument because they are complicated, and well-known.

We now give the proof of Theorem B. By virtue of 4.2.1 we may assume that G ∈ X, and so we are reduced to proving the following special case.

4.3.2 THEOREM. Let G be a finite group, M a Mackey functor for G, Y a set of subgroups of G closed under taking subgroups and conjugation, and Δ a G-simplicial complex of dimension d. Suppose that

(i) For every simplex σ ∈ Δ the vertices of σ lie in distinct G-orbits. (ii) For every subgroup H ∉ Y, Δ^{H} is contractible. (iv) For every Y ∈ Y, M(Y) = 0. Then the complexes of Mackey functors

0 → M → M_{Γ_{0}(Δ)} → M_{Γ_{1}(Δ)} → ⋯ → M_{Γ_{d}(Δ)} → 0

and

0 ← M ← M_{Γ_{0}(Δ)} ← M_{Γ_{1}(Δ)} ← ⋯ ← M_{Γ_{d}(Δ)} ← 0

are split acyclic.

The assumptions of this theorem will now remain in force throughout this section. The key to the proof of the theorem is the following result.

4.3.3 LEMMA. Suppose G ∉ Y. Then Δ has a G-subcomplex E which is Gcontractible and such that every simplex σ not in E has its stabilizer G_{σ} in Y.

Proof. We define E = ⋃_{H≤G,H∉Y} Δ^{H}. This is a subcomplex, since if σ is fixed by H then so is every face of σ; and it is preserved under the action of G. Evidently all simplices outside E have their stabilizers in Y. We show that E is G-contractible.

Because Δ^{G} is contractible it is non-empty, so E is non-empty. Choose a vertex x ∈ Δ^{G} and consider the inclusion mapping i : {x} → E. We will show that this is a G-homotopy equivalence. For each subgroup H ≤ G there is a mapping of fixed points i^{H} : {x} → E^{H}. By a theorem of Bredon ([2,sec.II] see also [22]) it suffices to show that each map i^{H} is a homotopy equivalence. For this we only need show that E^{H} is contractible for all subgroups H ≤ G. There are two cases. When H ∉ Y we have E^{H} = Δ^{H}, and this is contractible by hypothesis. When H ∈ Y we have E^{H} = ⋃_{H≤K,K∉Y} Δ^{K} since the right hand side is clearly contained in the left, and if σ ∈ E^{H} then H ≤ G_{σ} ∉ Y by construction of E, so σ ∈ Δ^{G_{σ}} which is in the right hand side. Thus E^{H} is a union of contractible spaces, every non-empty intersection of which is also contractible since Δ^{K} ∩Δ^{J} = Δ^{<K,J>}. Hence E^{H} is homotopic to the nerve of the covering [3,p.50]. Since the intersection of all the sets in the covering is non-empty, the nerve consists of a single simplex, which is contractible.

4.3.4 LEMMA. Suppose G ∉ Y. The complexes of Mackey functors obtained from Δ and E are the same.

Proof. In the complex

0 → M → M_{Γ_{0}(Δ)} → M_{Γ_{1}(Δ)} → ⋯ → M_{Γ_{d}(Δ)} → 0

the typical term has the form

M_{Γ_{i}(Δ)} = ⨁_{σ∈G\Γ_{i}(Δ)} M_{G/G_{σ}}.

If σ ∉ E then G_{σ} ∈ Y and hence M_{G/G_{σ}} = M ↓_{G_{σ}}^{G}↑_{G_{σ}}^{G}= 0 since M is zero on subgroups of G_{σ}. Thus M_{Γ_{i}(Δ)} = M_{Γ_{i}(E)}. Furthermore the restriction to M_{Γ_{i}(E)} of the differential coming from Δ is the same as the differential coming from E, so the two complexes are the same.

Proof of Theorem 4.3.2. If G ∈ Y then M is the zero Mackey functor, so we may assume G ∉ Y. We work with Θ_{M} the argument for Θ^{M} being similar. Let E be the subcomplex of Δ in Lemma 4.3.3. We have that CC(Θ_{M} S̃(Δ)) = CC(Θ_{M} S̃(E)). But E is G-contractible, so by 4.3.1 CC(Θ_{M} S̃(E)) is chain homotopic to the zero complex. Therefore it has zero homology, and is split since splitting is preserved under chain homotopy equivalence [22, 7.2].

5. Proof of theorems on the chain complex of Δ

5.1 Proof of Proposition 2.7.3

Let ZZ_{p} denote the p-adic integers. For any ZZG-module V we will write V_{p} = V ⊗_{Z Z}ZZ_{p}. The statement we are due to prove is:

2.7.3 PROPOSITION. Take M(G) = H^{n}(G, V)_{p} for some integer n ≥ 1 and ZZGmodule V. The sequences in Theorem A are Ext_{Z Z_{p}G}^{n}(C̃.(Δ), V_{p}) and Ext_{Z Z_{p}G}^{n}(C̃.(Δ)^{∗}, V_{p}) where ∗ denotes the contragredient and C̃.(Δ) is the augmented chain complex of Δ.

We verify first that the groups in the complexes are what they should be, then that the maps are correct. A typical chain group C_{r}(Δ) can be written ⨁_{σ∈G\Γ_{r}(Δ)} ZZ_{p} ↑_{G_{σ}}^{G}, because it is the free ZZ_{p}-module on the simplices in dimension r and these divide up into orbits, each giving a submodule ZZ_{p} ↑_{G_{σ}}^{G}. Applying Ext we obtain

Ext_{Z Z_{p}G}^{n}(C_{r}(Δ), V_{p}) ≅ ⨁_{σ∈G\Γ_{r}(Δ)} Ext_{Z Z_{p}G}^{n}(ZZ_{p} ↑_{G_{σ}}^{G}, V_{p})

≅ ⨁_{σ∈G\Γ_{r}(Δ)} Ext_{Z Z_{p}G_{σ}}^{n}(ZZ_{p}, V_{p})

≅ ⨁_{σ∈G\Γ_{r}(Δ)} Ext_{Z ZG_{σ}}^{n}(ZZ, V)_{p}

= ⨁_{σ∈G\Γ_{r}(Δ)} H^{n}(G_{σ}, V)_{p},

by means of standard isomorphisms. The fact that we can take completion at p outside the Ext term follows from [10, p.233]. There is a similar chain of isomorphisms for the contragredient representations C_{r}(Δ)^{∗}, since they are permutation modules and C_{r}(Δ)^{∗} ≅ C_{r}(Δ).

We have to show that ϕ_{τσ} and ψ_{στ} are the maps induced on the Ext groups by the homomorphism ZZ_{p} ↑_{G_{τ}}^{G} → ZZ_{p} ↑_{G_{σ}}^{G} and its dual. We suppose here that σ is a face of gτ. Evidently the maps have the right sign from the definition of the boundary operator. Regarding the modules now as free ZZ_{p}-modules on the simplices, the map is hgτ → hσ, h ∈ G, and its dual is hσ̂ → hzgτ̂, where z is the sum of a set of representatives for the cosets of G_{gτ} in G. A hat indicates the element of the dual basis corresponding to the bare-headed symbol. Because these maps take place in the first Ext variable and this commutes with what happens in the second variable, they give natural transformations of the cohomological functors H^{n}(G_{σ}, V_{p}) and H^{n}(G_{τ} , V_{p}), so it suffices to check that they coincide with the original definitions of ϕ_{τσ} and ψ_{στ} on fixed points (see [10]). On fixed points the maps are

Hom_{Z Z_{p}G}(ZZ_{p} ↑_{G_{σ}}^{G}, V_{p}) → Hom_{Z Z_{p}G}(ZZ_{p} ↑_{G_{τ}}^{G} , V_{p})
(σ ↦ v) ↦ (τ ↦ g^{−1}v)

and

Hom_{Z Z_{p}G}(ZZ_{p} ↑_{G_{τ}}^{G} , V_{p}) → Hom_{Z Z_{p}G}(ZZ_{p} ↑_{G_{σ}}^{G}, V_{p})
(τ ↦ v) ↦ (σ ↦ zgv).

These are respectively c_{g^{−1}} · res_{G_{gτ}}^{G_{σ}} and ind_{G_{gτ}}^{G_{σ}} · c_{g}.

5.2 Proof of Theorem 2.7.4

We first need a technical lemma. We work with modules over a finite dimensional algebra Λ, and if A and B are Λ-modules we use the notation \underline{Hom}_{Λ}(A, B) for the group of homomorphisms from A to B modulo those homomorphisms which factor through a projective module.

5.2.1 LEMMA. Let Λ be a self-injective algebra finite dimensional over a field k, and let μ : A → B be a monomorphism between finite dimensional Λ-modules. If the induced map \underline{Hom}_{Λ}(B, V) → \underline{Hom}_{Λ}(A, V) is surjective for all Λ-modules V then μ is split.

With full homomorphism groups this result is immediate, and the point is that it works with homomorphisms modulo projectives.

Proof. Taking V = A, there exists ϕ : B → A whose image in \underline{Hom}_{Λ}(A, V) is the same as that of 1_{A}, i.e. ϕμ ≡ 1 (mod projective homomorphisms). For some n, (ϕμ)^{n} has the same image as (ϕμ)^{n+1}, so A = A_{0} ⊕A_{1} where (ϕμ)^{n} is an automorphism on A_{0} and is zero on A_{1}, and again (ϕμ)^{n} ≡ 1 (mod projective homomorphisms). When we restrict the domain of μ to A_{0} the monomorphism μ : A_{0} → B is split by [(ϕμ)^{n}]^{−1}(ϕμ)^{n−1}ϕ, where [(ϕμ)^{n}]^{−1} denotes the inverse of (ϕμ)^{n} on A_{0}. Because (ϕμ)^{n} is zero on A_{1}, 1_{A_{1}} factors through an injective (= projective) module, and so A_{1} is injective and μ : A_{1} → B is split. Hence μ is split as a morphism A → B.

We now prove Theorem 2.7.4. The implications (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) are clear. We first prove (i) ⇒ (iii). Because trivial source modules and all morphisms between them are liftable from ZZ/pZZ to ZZ_{p} [12, II, 12.4], it suffices to prove the decomposition of C. in (iii) when R = ZZ/pZZ = k, say.

We proceed by induction on the dimension d of the complex

C. = C_{d} → ⋯ → C_{0}.

When d = 0 then Ext_{kG}^{n}(C_{0}, V) = 0 for all V, so C_{0} is projective. Now suppose d > 0 and the result is true for smaller dimensions. There is an isomorphism \underline{Hom}_{kG}(A, B) ≅ Ext_{kG}^{1}(A, Ω^{−1}B) and we apply the condition in (i) to these groups. Let H_{d} be the top homology of C., so

0 → H_{d} → C_{d} → C_{d−1}

is exact. Then the composite

\underline{Hom}_{kG}(C_{d−1}, V) → \underline{Hom}_{kG}(C_{d}, V) → \underline{Hom}_{kG}(H_{d}, V)

is zero and the left hand map is epi by the hypothesis (i), so the right hand map is zero and the image of 1_{C_{d}} in Hom_{kG}(H_{d}, C_{d}) (i.e. the inclusion map H_{d} → C_{d}) factors through an injective module. Hence there is a factorisation H_{d} → I → C_{d} where I is the injective hull of H_{d}, and I → C_{d} is injective since I and H_{d} have the same socle and this embeds in C_{d}. This means that C_{d} ≅ I ⊕ Y for some submodule Y of C_{d}, and H_{d} ⊆ I. Also Y embeds in C_{d−1} by restriction of the map C_{d} → C_{d−1} and we have a commutative diagram

\underline{Hom}_{kG}(C_{d−1}, V) → \underline{Hom}_{kG}(C_{d}, V)
↘  ↓≅
\underline{Hom}_{kG}(Y, V)

since the summand I contributes zero to the homomorphisms modulo injectives. Hence the monomorphism Y → C_{d−1} splits by Lemma 5.2.1 and C_{d−1} ≅ Y ⊕ W for some submodule W. Now

\underline{Hom}_{kG}(C_{d−2}, V) → \underline{Hom}_{kG}(Y ⊕ W, V) → \underline{Hom}_{kG}(Y, V) → 0

is exact, and so

\underline{Hom}_{kG}(C_{d−2}, V) → \underline{Hom}_{kG}(W, V)

is epi. Thus W → C_{d−2} → ⋯ → C_{0} is a complex of dimension d − 1 satisfying the induction hypothesis, so there exists an acyclic split subcomplex X → D_{d−2} → ⋯ → D_{0} with C_{i} = D_{i} ⊕P_{i} for i ≤ d−2, W = X ⊕P_{d−1}. Put D_{d−1} = X ⊕Y, D_{d} = Y. Then D_{d} → ⋯ → D_{0} is our acyclic split complex D. and C_{d} ≅ Y ⊕ I, C_{d−1} ≅ Y ⊕ W = D_{d−1} ⊕ P_{d−1} as required. This completes the proof of the implication (i) ⇒ (iii).

We now prove (iii) ⇒ (iv), showing by induction on n that for all r with 0 ≤ r ≤ n we may choose a projective submodule P_{r} so that C_{r} = D_{r} ⊕ P_{r} and such that the differential d_{r} sends D_{r} into D_{r−1} and P_{r} into P_{r−1}. The induction starts at n = 0. Here we only need the additive decomposition C_{0} = D_{0} ⊕ C_{0}/D_{0} which results from the fact that C_{0}/D_{0} is projective. Suppose now that n > 0 and the result holds for smaller values. Because C_{n}/D_{n} is projective we may write C_{n} = D_{n} ⊕ Q_{n} for some projective module Q_{n} and represent d_{n}|_{Q_{n}} in component form (β_{n}, α_{n}) corresponding to the decomposition C_{n−1} = D_{n−1} ⊕ P_{n−1}. The picture of C. is

⋯ ⟶ Q_{n}  ⟶^{α_{n}} P_{n−1} ⟶^{α_{n−1}} P_{n−2} ⟶ ⋯
⊕ β_{n} ↘ ⊕  ⊕
⋯ ⟶ D_{n}  ⟶^{γ_{n}} D_{n−1} ⟶^{γ_{n−1}} D_{n−2} ⟶ ⋯

where for 0 ≤ r < n, d_{r} = γ_{r} ⊕ α_{r} with d_{r}|_{D_{r}} = γ_{r}. For any element x ∈ Q_{n}, d_{n−1}d_{n}(x) = (γ_{n−1}β_{n}(x), α_{n−1}α_{n}(x)) = 0, from which we see that γ_{n−1}β_{n} = 0 and Im β_{n} ⊆ ker γ_{n−1} = Im γ_{n}. Hence by projectivity of Q_{n} there exists a map ϕ_{n} : Q_{n} → D_{n} so that β_{n} = γ_{n}ϕ_{n}. Define P_{n} = {(−ϕ_{n}(x), x) | x ∈ Q_{n}}. This is an isomorphic image of Q_{n} under the map x ↦ (−ϕ_{n}(x), x), so P_{n} ≅ Q_{n} is a projective submodule, and plainly C_{n} = D_{n} ⊕ P_{n}. Furthermore for y = (−ϕ_{n}(x), x) ∈ P_{n},

d_{n}(y) = d_{n}((−ϕ_{n}(x), x))
= (−γ_{n}ϕ_{n}(x) + β_{n}(x), α_{n}(x))
= (0, α_{n}(x)) ∈ P_{n−1}

thus completing the induction step.

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School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455