GLUING p-PERMUTATION MODULES

RAPHAËL ROUQUIER

1. Introduction

We give a “local” construction of the stable category of p-permutation modules : a ppermutation kG-module gives rise, via the Brauer functor, to a family of p-permutation modules for kN_{G}(Q)/Q, where Q runs over the non-trivial p-subgroups of G, together with certain isomorphisms. Conversely, the data of a compatible family of kN_{G}(Q)/Q-modules comes from a p-permutation kG-module, unique up to a unique isomorphism in the stable category.

This should be the first half of a paper with a second part devoted to complexes of ppermutation modules.

2. The Brauer functor

Let G be a finite group and k a field of characteristic p > 0.

Let Q be a p-subgroup of G. We denote by Br_{Q} the Brauer functor Br_{Q} : kG − mod → kN_{G}(Q) − mod.

For V be a kG-module,

Br_{Q}(V) = V^{Q}/ (∑_{P <Q} Tr_{P}^{Q} V^{P} ) .

We write also V (Q) for Br_{Q}(V). For basic results about p-permutation modules and the Brauer functor, see [Br] and [Th, §27].

Restriction induces a fully faithful functor kN_{G}(Q)/Q − mod → kN_{G}(Q) − mod and we will identify kN_{G}(Q)/Q − mod with the full subcategory of kN_{G}(Q) − mod of the modules with a trivial action of Q.

We denote by kG−perm the full subcategory of kG−mod of p-permutation modules. This is the smallest full additive subcategory of kG−mod closed under direct summands and containing the permutation modules.

From now on, we will always consider the restriction of the Brauer functor Br_{Q} : kG−perm → kN_{G}(Q) − perm.

Let Ω be a G-set. The composition γ_{Ω} = γ_{Ω}^{Q} : kΩ^{Q} ↪ (kΩ)^{Q} ↠ (kΩ)(Q) is an isomorphism. It induces an isomorphism of functors γ : k(−)^{Q} →^{∼} Br_{Q} k(−), i.e., there is a diagram of functors, commutative up to isomorphism :

[figure: G − sets; k(−); (−)^{Q}; kG − perm; _{Br_{Q}}; N_{G}(Q) − sets; k(−); kN_{G}(Q) − perm]

The following easy result describes the effect of the Brauer construction on a permutation module. Let H and L be two subgroups of G. We put T_{G}(L, H) = {g ∈ G|L ≤ H^{g}}.

Date: August 1998.

Lemma 2.1. We have

(G/H)^{L} = ⋃_{g∈H\T_{G}(L,H)/N_{G}(L)} N_{G}(L)/ (N_{G}(L) ∩ H^{g})

as N_{G}(L)-sets.

2.1. Tensor product and duality. Let us describe the effect of the functor Br_{Q} on tensor products and duality.

Let V and W be two p-permutation kG-modules. The inclusion V^{Q} ⊗ W^{Q} → (V ⊗ W)^{Q} induces a map α_{V,W} : V (Q) ⊗ W (Q) → (V ⊗ W)(Q).

Restriction (V^{∗})^{Q} = Hom_{kQ}(V, k) → (V^{Q})^{∗} = Hom_{k}(V^{Q}, k) induces a map β_{V} : V^{∗}(Q) → V (Q)^{∗}.

We put ΔG = {(x, x)|x ∈ G} ⊆ G × G.

Lemma 2.2. The maps α_{V,W} and β_{V} are isomorphisms and induce isomorphisms of functors from kG − perm × kG − perm to kN_{G}(Q) − perm

α : Res_{ΔN_{G}(Q)}^{N_{G}(Q)×N_{G}(Q)} Br_{Q×Q} →^{∼} Br_{Q} Res_{ΔG}^{G×G}

and from kG − perm to kN_{G}(Q) − perm

β : Br_{Q}(−)^{∗} →^{∼} (−)^{∗} Br_{Q} .

Proof. In order to prove that α_{V,W} and β_{V} are isomorphisms, it is enough to consider permutation modules. So, let V = kΩ and W = kΨ, where Ω and Ψ are G-sets.

We have an isomorphism t_{Ω,Ψ} : kΩ ⊗ kΨ →^{∼} k(Ω × Ψ) given by ω ⊗ ψ ↦ (ω × ψ) for ω ∈ Ω and ψ ∈ Ψ.

There is a commutative diagram :

[figure: (kΩ^{Q}) ⊗ (kΨ^{Q}); ≃; t_{Ω^{Q},Ψ^{Q}}; ≃; (kΩ)^{Q} ⊗ (kΨ)^{Q}; t_{Ω,Ψ}; (kΩ)(Q) ⊗ (kΨ)(Q); α_{V,W}; k(Ω × Ψ)^{Q}; ≃; (k(Ω × Ψ))^{Q}; k(Ω × Ψ)(Q)]

Hence, α_{V,W} is an isomorphism.

We have an isomorphism d_{Ω} : kΩ →^{∼} (kΩ)^{∗} given by ω ↦ (∑_{ω′∈Ω} a_{ω′}ω′ ↦ a_{ω}) for ω ∈ Ω. The composition

kΩ^{Q} ↪ (kΩ)^{Q} ⟶^{d_{Ω}} ((kΩ)^{∗})^{Q} ↠ (kΩ)^{∗}(Q)

is an isomorphism. Since the following diagram is commutative

[figure: (kΩ)^{Q}; d_{Ω}; ((kΩ)^{∗})^{Q}; (kΩ)^{∗}(Q); β_{V}; kΩ^{Q}; d_{Ω^{Q}}; ≃; (kΩ^{Q})^{∗}; (kΩ)(Q)^{∗}; γ_{Ω}^{∗}; ≃]

we deduce that β_{V} is an isomorphism.

Let us now check that α_{V,W} and β_{V} induce natural transformations of functors.

Let V_{1}, V_{2}, W_{1} and W_{2} be p-permutation kG-modules and f_{i} ∈ Hom(V_{i}, W_{i}). The following diagram is commutative :

[figure: V_{1}(Q) ⊗ V_{2}(Q); f_{1}(Q)⊗f_{2}(Q); α_{V_{1},V_{2}}; W_{1}(Q) ⊗ W_{2}(Q); α_{W_{1},W_{2}}; V_{1}^{Q} ⊗ V_{2}^{Q}; f_{1}⊗f_{2}; W_{1}^{Q} ⊗ W_{2}^{Q}; (V_{1} ⊗ V_{2})^{Q}; f_{1}⊗f_{2}; (W_{1} ⊗ W_{2})^{Q}; (V_{1} ⊗ V_{2})(Q); (f_{1}⊗f_{2})(Q); (W_{1} ⊗ W_{2})(Q)]

Hence, α_{V,W} induces a morphism of functors : Res_{ΔN_{G}(Q)}^{N_{G}(Q)×N_{G}(Q)} Br_{Q×Q} → Br_{Q} Res_{ΔG}^{G×G}.

The commutativity of the diagram :

[figure: W_{1}^{∗}(Q); f_{1}^{∗}(Q); β_{W_{1}}; V_{1}^{∗}(Q); β_{V_{1}}; (W_{1}^{∗})^{Q}; f_{1}^{∗}; (V_{1}^{∗})^{Q}; (W_{1}^{Q})^{∗}; f_{1}^{∗}; (V_{1}^{Q})^{∗}; W_{1}(Q)^{∗}; f_{1}(Q)^{∗}; V_{1}(Q)^{∗}]

shows finally that β_{V} induces a morphism of functors : Br_{Q}(−)^{∗} →^{∼} (−)^{∗} Br_{Q}.

Proposition 2.3. There is a commutative diagram

[figure: Hom(V, W); _{Br_{Q}}; ≃; Hom(V ⊗ W^{∗}, k); _{Br_{Q}}; Hom((V ⊗ W^{∗})(Q), k); ≃; βα^{−1}; Hom(V (Q), W (Q)); ≃; Hom(V (Q) ⊗ W (Q)^{∗}, k)]

where the horizontal maps are isomorphisms provided by the adjoint pairs (− ⊗ W^{∗}, − ⊗ W) and (− ⊗ W (Q)^{∗}, − ⊗ W (Q)).

Proof. More explicitely, the first horizontal map is the composition

Hom(V, W) ⟶^{−⊗W^{∗}} Hom(V ⊗ W^{∗}, W ⊗ W^{∗}) ⟶^{tr(W)_{∗}} Hom(V ⊗ W^{∗}, k)

where tr(W) : W ⊗ W^{∗} → k is the trace map.

Thanks to Lemma 2.2, we have a commutative diagram

[figure: Hom(V, W); −⊗W^{∗}; _{Br_{Q}}; Hom(V ⊗ W^{∗}, W ⊗ W^{∗}); _{Br_{Q}}; Hom((V ⊗ W^{∗})(Q), (W ⊗ W^{∗})(Q)); α; −1; Hom(V (Q) ⊗ W^{∗}(Q), W (Q) ⊗ W^{∗}(Q)); β; Hom(V (Q), W (Q)); −⊗W^{∗}(Q); −⊗W (Q)^{∗}; Hom(V (Q) ⊗ W (Q)^{∗}, W (Q) ⊗ W (Q)^{∗})]

We will be done if we prove that the image of tr(W) : W ⊗ W^{∗} → k under the morphism

Hom(W ⊗ W^{∗}, k) ⟶^{βα^{−1} Br_{Q}} Hom(W (Q) ⊗ W (Q)^{∗}, k)

is tr(W (Q)). It is enough to prove this for W a permutation module.

Let W = kΩ, Ω a G-set. The claim follows from the commutativity of the diagram

[figure: (kΩ ⊗ kΩ)^{Q}; 1⊗d_{Ω}; (kΩ ⊗ (kΩ)^{∗})^{Q}; kΩ^{Q} ⊗ kΩ^{Q}; 1⊗d_{Ω^{Q}}; (kΩ ⊗ (kΩ)^{∗})(Q); βα^{−1}; kΩ^{Q} ⊗ (kΩ^{Q})^{∗}; γ_{Ω}⊗(γ_{Ω}^{−1})^{∗}; kΩ(Q) ⊗ kΩ(Q)^{∗}]

2.2. Compatibilities. Let us define a category T_{G}. Its objects are the non-trivial p-subgroups of G. Let P and Q be two non-trivial p-subgroups of G. Then, the set of maps between P and Q in T_{G} is {_{P} g_{Q}|g ∈ T_{G}(P, Q)}. The composition of maps is the product in G : (_{Q}h_{R}) · (_{P} g_{Q}) = _{P} (hg)_{R}. We put ϕ̄ = g for ϕ = _{P} g_{Q} and ϕ(P) = ^{g}P.

We call a map ϕ = _{P} g_{Q} in Hom_{T_{G}}(P, Q) normal if ^{g}P is normal in Q. Every normal map can be expressed uniquely as the composition of an isomorphism with a normal inclusion ϕ = ϕ_{⊲}ϕ_{∼} where ϕ_{∼} = _{P} g_{^{g}P} and ϕ_{⊲} = _{^{g}P} 1_{Q}.

An important property of the normal maps is that they generate the category T_{G}, i.e., every map in T_{G} is a composition of normal maps.

For g ∈ G and H a subgroup of G, we denote by

g_{∗} : kH − mod →^{∼} k^{g}H − mod

the isomorphism of categories induced by the group isomorphism H →^{∼} ^{g}H, x ↦ ^{g}x. We also denote by

g_{∗} : H − sets →^{∼} ^{g}H − sets

the isomorphism of categories induced by this group isomorphism. We have the obvious compatiblity with the previous isomorphism of categories.

Let V be a p-permutation kH-module and ϕ ∈ Hom_{T_{G}}(P, Q) invertible. Then, the isomorphism of N_{^{g}H}(Q)-modules ϕ̄_{∗}(V^{P}) →^{∼} (ϕ̄_{∗}V)^{Q}, v ↦ v, induces an isomorphism

⟨ϕ⟩_{V}^{0} : ϕ̄_{∗}(V (P)) →^{∼} (ϕ̄_{∗}V)(Q).

If H = G, then we have an isomorphism of kG-modules

ι_{ϕ̄,V} : V ↦ ϕ̄_{∗}V, v ↦ ϕ̄v

and an isomorphism of kN_{G}(Q)-modules

⟨ϕ⟩_{V} = Br_{Q}(ι_{ϕ̄,V}^{−1} ) · ⟨ϕ⟩_{V}^{0} : ϕ̄_{∗}(V (P)) →^{∼} V (Q).

Let now P ⊴ Q and ϕ = _{P} 1_{Q}. The canonical map V^{Q} ↪ V^{P} ↠ V (P) factors through the inclusion V (P)^{Q} ↪ V (P) to give a map V^{Q} → V (P)^{Q}. Composing with the canonical map V (P)^{Q} ↠ V (P)(Q), we get a map V^{Q} → V (P)(Q) which factors through the canonical map V^{Q} ↠ V (Q). The induced map V (Q) → V (P)(Q) is an isomorphism and we denote the inverse isomorphism by ⟨ϕ⟩_{V}

⟨ϕ⟩_{V} : V (P)(Q) →^{∼} V (Q)

Let us summarize the construction by the following commutative diagram :

[figure: V^{P}; V (P); V^{Q}; V (P)^{Q}; V (Q); ≃; V (P)(Q)]

For ϕ ∈ Hom_{T_{G}}(P, Q) a normal map with P ≤ H, Q ≤ ϕ̄H and V a p-permutation kHmodule, we put

⟨ϕ⟩_{V}^{0} = ⟨ϕ_{⊲}⟩_{ϕ̄_{∗}V} · Br_{Q}(⟨ϕ_{∼}⟩_{V}^{0} ) : (ϕ̄_{∗}(V (P))) (Q) →^{∼} (ϕ̄_{∗}V)(Q).

If V is a p-permutation kG-module, we put

⟨ϕ⟩_{V} = ⟨ϕ_{⊲}⟩_{V} · Br_{Q}(⟨ϕ_{∼}⟩_{V}) : (ϕ̄_{∗}(V (P))) (Q) →^{∼} V (Q).

This gives an isomorphism of functors from kH − perm to kN_{ϕ̄H}(ϕ(P), Q) − perm

⟨ϕ⟩^{0} : Br_{Q} ϕ̄_{∗} Br_{P} →^{∼} Res_{N_{ϕ̄H}(ϕ(P),Q)}^{N_{ϕ̄H}(Q)} Br_{Q} ϕ̄_{∗}.

and an isomorphism of functors from kG − perm to kN_{G}(ϕ(P), Q) − perm

⟨ϕ⟩ : Br_{Q} ϕ̄_{∗} Br_{P} →^{∼} Res_{N_{G}(ϕ(P),Q)}^{N_{G}(Q)} Br_{Q} .

When V = kΩ, Ω a G-set, we have a commutative diagram

[figure: k(ϕ̄_{∗}Ω^{P})(Q); Br_{Q} ϕ̄_{∗}γ_{Ω}^{P}; (ϕ̄_{∗}(kΩ)(P))(Q); ⟨ϕ⟩_{V}; k(ϕ̄_{∗}Ω^{P})^{Q}; γ_{ϕ̄∗Ω^{P}}^{Q}; k⟨ϕ⟩_{Ω}; kΩ^{Q}; γ_{Ω}^{Q}; (kΩ)(Q)]

where

⟨ϕ⟩_{Ω} : (ϕ̄_{∗}Ω^{P})^{Q} →^{∼} Ω^{Q}, ω ↦ ϕ̄^{−1}ω.

Note that this justifies the claim that ⟨ϕ⟩_{V} is an isomorphism for V a permutation module and consequently for V an arbitrary p-permutation module.

We will now check a transitivity property of the isomorphisms constructed above.

Lemma 2.4. Let ϕ ∈ Hom_{T_{G}}(P, Q) and ψ ∈ Hom_{T_{G}}(Q, R) such that ϕ, ψ and ψϕ are normal maps and let V be a p-permutation kG-module. Then, the following diagram is commutative

[figure: ψ̄_{∗} ((ϕ̄_{∗}(V (P)))(Q)) (R); Br_{R} ψ̄_{∗}⟨ϕ⟩_{V}; ⟨ψ⟩_{ϕ̄∗(V (P))}^{0}; (ψ̄_{∗}(V (Q)))(R); ⟨ψ⟩_{V}; ((ψ̄ϕ̄)_{∗}(V (P)))(R); ⟨ψϕ⟩_{V}; V (R)]

Proof. Indeed, it is enough to check commutativity for V = kΩ a permutation module. It follows from the commutativity of the following diagram

[figure: ψ̄_{∗} ((ϕ̄_{∗}(Ω^{P}))^{Q})^{R}; ω↦ψ̄ϕ̄^{−1}ψ̄^{−1}ω; ω↦ω; (ψ̄_{∗}(Ω^{Q}))^{R}; ω↦ψ̄^{−1}ω; ((ψ̄ϕ̄)_{∗}(Ω^{P}))^{R}; ω↦ψ̄ϕ̄^{−1}ω; Ω^{R}]

The difference between ⟨ϕ⟩ and ⟨ϕ⟩^{0} is is given by the following lemma :

Lemma 2.5. Let ψ ∈ Hom_{T_{G}}(Q, R) be a normal map with ψ̄ = 1, g ∈ G and V a p-permutation kG-module. Then, the following diagram is commutative

[figure: (V (Q))(R); ⟨ψ⟩_{V}; V (R); ((g_{∗}V)(Q))(R); ⟨ψ⟩_{g∗V}; Br_{R} Br_{Q}(ι_{g,V}^{−1} ); (g_{∗}V)(R); Br_{R}(ι_{g,V}^{−1} )]

In particular, if ϕ ∈ Hom_{T_{G}}(P, Q) is any normal map, then ⟨ϕ⟩_{V} = Br_{Q}(ι_{ϕ̄,V}^{−1} ) · ⟨ϕ⟩_{V}^{0} .

Proof. Again, it is enough to deal with V = kΩ a permutation module. Then, the lemma reduces to the commutativity of

[figure: (Ω^{Q})^{R}; ω↦ω; Ω^{R}; ((g_{∗}Ω)^{Q})^{R}; ω↦ω; ω↦g^{−1}ω; (g_{∗}Ω)^{R}; ω↦g^{−1}ω]

For the second part of the lemma, we take ψ = ϕ_{⊲} and g = ϕ̄ in the commutative diagram.

3. A category of sheaves on p-subgroups complexes

3.1. Definition. Let F be a subcategory of T_{G}. We define a category S_{F} of “sheaves” on F.

Its objects are families {V_{Q}, [ϕ]}_{Q,ϕ} where Q runs over the objects of F and ϕ over the normal maps of F. Here, V_{Q} is a p-permutation kN_{G}(Q)/Q-module and for ϕ ∈ Hom_{F}(P, Q) normal, [ϕ] is an isomorphism of kN_{G}(ϕ(P), Q)-modules

[ϕ] : (ϕ̄_{∗}V_{P})(Q) →^{∼} Res_{N_{G}(ϕ(P),Q)}^{N_{G}(Q)} V_{Q}.

We require the following two conditions to be satisfied :

For ϕ ∈ Hom_{F}(Q, Q), we have

[ϕ] = ι_{ϕ̄,V_{Q}}^{−1}.  (1)

Let ϕ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, R) such that ϕ, ψ and ψϕ are normal maps. Then, the following diagram should be commutative

[figure: ψ̄_{∗} ((ϕ̄_{∗}V_{P})(Q)) (R); Br_{R} ψ̄_{∗}[ϕ]; ⟨ψ⟩_{ϕ̄∗V_{P}}^{0}; (ψ̄_{∗}V_{Q})(R); [ψ]; ((ψ̄ϕ̄)_{∗}V_{P})(R); [ψϕ]; V_{R}]  (2)

For V = {V_{Q}, [ϕ]} and V′ = {V′_{Q}, [ϕ]′} two objects of S_{F}, Hom_{S_{F}} (V, V′) is the set of families Λ = {λ_{Q}}_{Q}, where Q runs over the objects of F. Here, λ_{Q} ∈ Hom_{kN_{G}(Q)}(V_{Q}, V′_{Q}). Furthermore, Λ should have the following property : for every normal map ϕ ∈ Hom_{F}(P, Q), the following diagram is commutative

[figure: (ϕ̄_{∗}V_{P})(Q); [ϕ]; Br_{Q} ϕ̄_{∗}λ_{P}; (ϕ̄_{∗}V′_{P})(Q); [ϕ]′; V_{Q}; λ_{Q}; V′_{Q}]  (3)

Thanks to the results of §2.2, we have a functor

Br : kG − perm → S, V ↦ {V (Q), ⟨ϕ⟩_{V} }

where S = S_{T_{G}}.

We can now state our main result :

Theorem 3.1. The functor Br induces an equivalence of categories

kG − perm /kG − proj →^{∼} S.

3.2. Some properties of S_{F}. Let us give a special case of the commutative diagram (2).

Lemma 3.2. Let ϕ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, R) be two normal maps with ψ̄ ∈ N_{G}(ϕ(P)). Then, the following diagram is commutative

[figure: ψ̄_{∗} ((ϕ̄_{∗}V_{P})(Q)) (R); Br_{R} ψ̄_{∗}[ϕ]; ⟨ψ⟩_{ϕ̄∗V_{P}}; (ψ̄_{∗}V_{Q})(R); [ψ]; (ϕ̄_{∗}V_{P})(R); [ψ_{⊲} ϕ]; V_{R}]

Proof. Let ψ′ = ψϕ and ϕ′ = _{P} (ϕ̄^{−1}ψ̄^{−1}ϕ̄)_{P}. We have ψ_{⊲}ϕ = ψ′ϕ′. By (1), we have [ϕ′] = ι_{ϕ̄′,V_{P}}^{−1} . We have also ψ̄_{∗}ϕ̄_{∗}ι_{ϕ̄′,V_{P}}^{−1} = ι_{ψ̄,ϕ̄_{∗}V_{P}}. The commutativity of the diagram (2) applied to ψ′ and ϕ′ gives the commutative diagram and we obtain the commutativity of the diagram of the lemma since ⟨ψ⟩_{ϕ̄_{∗}V_{P}} = Br_{R}(ι_{ψ̄,ϕ̄_{∗}V_{P}}^{−1} ) · ⟨ψ⟩_{ϕ̄_{∗}V_{P}}^{0} by Lemma 2.5.

[figure: (ϕ̄_{∗}V_{P})(R); [ψ_{⊲} ϕ]; Br_{R} ι_{ψ̄,ϕ̄∗V_{P}}; ((ψ̄ϕ̄)_{∗}V_{P})(R); [ψϕ]; V_{R}]

The diagram (3) need be checked only on a generating set :

Lemma 3.3. Let E be a set of normal maps in F such that every normal map of F is a product of elements of E and of inverses of invertible elements of E.

Then, the commutativity of the diagram (3) for ϕ ∈ E implies the commutativity for every normal map ϕ of F

Proof. Let ϕ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, R) such that ψ and ψϕ are normal maps. Then, ϕ is a normal map and the following diagram is commutative

[figure: ((ψ̄ϕ̄)_{∗}V_{P})(R); Br_{R}(ψ̄ϕ̄)_{∗}λ_{P}; [ψϕ]; V_{R}; λ_{R}; ψ̄_{∗} ((ϕ̄_{∗}V_{P})(Q)) (R); Br_{R} ψ̄_{∗}[ϕ]; Br_{R} ψ̄_{∗} Br_{Q} ϕ̄_{∗}λ_{P}; ⟨ψ⟩_{ϕ̄∗V_{P}}^{0}; (ψ̄_{∗}V_{Q})(R); [ψ]; Br_{R} ψ̄_{∗}λ_{Q}; ψ̄_{∗} ((ϕ̄_{∗}V′_{P})(Q)) (R); Br_{R} ψ̄_{∗}[ϕ]′; ⟨ψ⟩_{ϕ̄∗V′_{P}}^{0}; (ψ̄_{∗}V′_{Q})(R); [ψ]′; ((ψ̄ϕ̄)_{∗}V′_{P})(R); [ψϕ]′; V′_{R}]

The lemma follows.

We now define a restriction functor from G to N_{G}(P)/P.

Let P be an object in F. We assume _{P} 1_{Q} ∈ Hom_{F}(P, Q) for every Q in F with P ⊲ Q. Let F(P) be the subcategory of T_{N_{G}(P)/P} whose objects are the Q/P where Q is in F and P ⊲ Q, P ≠ Q and where Hom_{F(P)}(Q/P, R/P) is given by the image in T_{N_{G}(P)/P} (Q/P, R/P) of Hom_{F}(Q, R).

Let V = {V_{Q}, [ϕ]} be an object of S_{F}. Let V′_{Q/P} = V_{Q}. For ϕ′ ∈ Hom_{F(P)}(Q/P, R/P), we put [ϕ′] = [ϕ] where ϕ ∈ Hom_{F}(Q, R) has image ϕ′ in Hom_{F(P)}(Q/P, R/P). It follows from (1) (and from the diagram (2)) that this is independent of the choice of ϕ.

The restriction functor is

Res_{F(P)}^{F} : S_{F} → S_{F(P)}, {V_{Q}, [ϕ]} ↦ {V′_{Q/P} , [ϕ′]}.

We denote by E_{P} : T_{F} → kN_{G}(P)/P − perm the functor sending V on V_{P}.

The commutative diagram in Lemma 3.2 says that objects can be “glued locally” :

Lemma 3.4. For V in S_{F}, we have an isomorphism {[_{P} 1_{Q}]}_{Q} : Br E_{P} (V) →^{∼} Res_{F(P)}^{F} V. This induces an isomorphism of functors Br E_{P} →^{∼} Res_{F(P)}^{F}. So, we have a diagram, commutative up to isomorphism :

[figure: S_{F}; ^{Res{F \atop F(P)}}; E_{P}; S_{F(P)}; kN_{G}(P)/P − perm; Br]

Proof. Let V = {V_{Q}, [ϕ]} in S_{F} and V′ = Res_{F(P)}^{F} V. We have Br(V_{P}) = {W_{Q/P} , ⟨ψ⟩_{V_{P}} } with W_{Q/P} = V_{P} (Q).

Let λ_{Q/P} = [_{P} 1_{Q}] : W_{Q/P} →^{∼} V′_{Q/P} and Λ = {λ_{Q/P} }.

Let ϕ = _{P} 1_{Q} in F and ψ ∈ Hom_{F}(Q, R) be two normal maps with ψ̄ ∈ N_{G}(P).

We have a commutative diagram (Lemma 3.2)

[figure: (ψ̄_{∗}V_{P} (Q))(R); ⟨ψ⟩_{V_{P}}; Br_{R} ψ̄_{∗}[ϕ]; (ψ̄_{∗}V_{Q})(R); [ψ]; V_{P} (R); [ψ_{⊲} ϕ]; V_{R}]

This shows that Λ defines a map Br(V_{P}) → V′. This induces an isomorphism between Br ·E_{P} and Res_{F(P)}^{F}.

3.3. Proof of Theorem 3.1. Let us first note that Br(V) = 0 if V is projective, hence Br induces indeed a functor B̄r : kG − perm /kG − proj → S.

Lemma 3.5. The functor B̄r is fully faithful.

Proof. We have to prove that B̄r induces an isomorphism

\overline{Hom}(V, W) →^{∼} Hom(Br V, Br W)

for V and W any p-permutation kG-modules.

Thanks to Proposition 2.3, we have a commutative diagram :

[figure: \overline{Hom}(V, W); ≃; Br; \overline{Hom}(V ⊗ W^{∗}, k); Br; Hom(Br V, Br W); ≃; Hom(Br(V ⊗ W^{∗}), Br k)]

So, it is enough to consider the case W = k.

Since the modules k(G/Q), Q a p-subgroup of G, generate kG−perm as an additive category closed under taking direct summands, we may assume V = k(G/Q). We may take Q ≠ 1 since otherwise V is projective.

Now, Hom(kG/Q, k) ≃ \overline{Hom}(kG/Q, k) is a one-dimensional vector space, generated by the unique map f between the G-sets G/Q and G/G.

The map Br_{Q}(f) : V (Q) = k(G/Q)^{Q} → k is induced by the unique map between the sets N_{G}(Q)/Q and N_{G}(Q)/N_{G}(Q). In particular, it is non-zero, hence Br(f) ≠ 0.

Let Λ ∈ Hom(Br kG/Q, Br k). Since Hom(V (Q), k) is one-dimensional, we have λ_{Q} = α Br_{Q}(f) for some α ∈ k. So, Λ − α Br(f) vanishes on V (Q).

We assume now λ_{Q} = 0. We will prove that λ_{P} = 0 for all P. This is clear if P is not conjugate to a subgroup of Q, since then V (P) = 0. We will now prove the result for P ≤ Q by induction on [Q : P].

Let P < Q and g ∈ T_{G}(P, Q). Let R be a p-subgroup of G such that P ⊲ R ≤ Q^{g}, P ≠ R. Then, (N_{G}(P)/N_{G}(P) ∩ Q^{g})^{R} ≠ ∅. Now, λ_{P} and λ_{R} have the same restriction to (N_{G}(P)/N_{G}(P) ∩ Q^{g})^{R}. Consequently, λ_{P} is zero on (N_{G}(P)/N_{G}(P) ∩ Q^{g})^{R}, by induction. By Lemma 2.1, we deduce that λ_{P} = 0.

The first part of following lemma is essentially due to Bouc [Bou1, Bou2] (cf also [Li, Lemma 5.4]).

Lemma 3.6. (i) Let f : V → W be a morphism between two p-permutation kG-modules such that Br_{Q}(f) is injective for all Q ≠ 1. Then, there is a morphism σ : W → V such that σf is a stable automorphism of V. In particular, if V has no projective direct summand, then f is a split injection.

(ii) Let Λ : V → W be a morphism between two objects of S. Assume for all Q, there is a projective direct summand L_{Q} of the kN_{G}(Q)/Q-module V_{Q} such that the restriction of λ_{Q} to L_{Q} is injective and V_{Q}/L_{Q} has no projective direct summand. Then, for all Q, λ_{Q} is a split injection.

Proof. Let us prove part (i) of the lemma. We can assume V has no projective direct summand. Assume G is a p-group. Let us first consider a morphism f : V = k(G/Q) → W = k(G/R) such that Br_{Q}(f) is

injective, where Q and R are non-trivial p-subgroups of G. The module V (Q) is a projective indecomposable kN_{G}(Q)/Q-module. Since W (Q) ≠ 0, Q is contained in R, up to conjugacy. Without changing W, we can assume Q ≤ R.

Assume Q ≠ R. Then, for any g ∈ T_{G}(Q, R), there is S such that P ⊲ S ≤ Q^{g}, P ≠ S. So, kN_{G}(Q)/(N_{G}(Q) ∩ R^{g}) is not a projective kN_{G}(Q)/Q-module. By Lemma 2.1, V (Q) is not isomorphic to a submodule of W (Q) and we have reached a contradiction.

If Q = R, then f becomes invertible in the quotient End_{kN_{G}(Q)/Q}(V (Q)) of the local ring End_{kG}(V), hence f is invertible.

Let now f : V → W be a morphism between two p-permutation modules such that Br_{Q}(f) is injective for Q non trivial. In order to prove that f is injective, we may assume V is indecomposable, i.e., V = k(G/Q) for some subgroup Q of G. We can assume Q ≠ 1, otherwise V is projective. Since V (Q) is indecomposable, there is an indecomposable direct summand W′ of W such that if f′ is the composition V ⟶^{f} W ↠ W′, then Br_{Q}(f′) is injective. Now, the considerations above show that f′ is an isomorphism and we are done.

Take now G an arbitrary finite group. Let U = ker f and let S be a Sylow p-subgroup of G. We know that the inclusion Res_{S}^{G} U → Res_{S}^{G} V is projective. So, the inclusion U → V is projective, hence U = 0 since we assume V has no projective direct summand. Finally, the short exact sequence 0 → V → W → W/V → 0 splits, since it splits by restriction to S.

Let us come to part (ii) of the lemma.

We prove the result by inverse induction on the order of Q. When Q is a Sylow p-subgroup of G, then V_{Q} = L_{Q}, so λ_{Q} is a split injection.

Assume now Q is not a Sylow p-subgroup of G. Consider the restriction f : M_{Q} → W_{Q} of λ_{Q}, where V_{Q} = L_{Q} ⊕ M_{Q}. By induction, Br_{R/Q}(f) is injective, for all p-subgroups R with Q ⊲ R, Q ≠ R. By part (i) of the Lemma applied to N_{G}(Q)/Q, we deduce that f is a split injection. Hence, λ_{Q} is a split injection.

Lemma 3.7. Let G and H be two full subcategories of T_{G} closed under inverse inclusion with G ⊆ H. Let V, V′ ∈ S_{H}. Then, the restriction map

Hom_{S_{H}}(V, V′) → Hom_{S_{G}}(Res_{G}^{H} V, Res_{G}^{H} V′)

is surjective.

Proof. It is enough to prove the lemma when H has one more object, Q, than G. Let λ ∈ Hom_{S_{G}}(Res_{G}^{H} V, Res_{G}^{H} V′).

Assume first there is g ∈ G such that Q^{g} ∈ G and let ψ = _{Q}g_{Q^{g}} ∈ Hom_{H}(Q, Q^{g}). Let λ′_{R} = λ_{R} for R ≠ Q and λ′_{Q} = [ψ^{−1}]′λ_{Q^{g}}[ψ]. In order to prove that {λ′_{R}} gives a map between V and V′ (extending λ), it is enough to check commutativity of the diagram (3) for the map ψ, thanks to Lemma 3.3. This is immediate.

Assume now Q^{g} ∉ G for all g ∈ G. Let f : Br(V_{Q}) → Br(V′_{Q}) be the restriction of λ to S_{H(Q)} (cf Lemma 3.4). By the fullness of Br applied to N_{G}(Q)/Q (Lemma 3.5), there is a map λ′_{Q} : V_{Q} → V′_{Q} such that Br(λ′_{Q}) = f. Let λ′_{R} = λ_{R} for R ∈ G. Since every map in H starting from Q is the composition of a map from Q to R, with Q a strict normal subgroup of R and of a map in G, Lemma 3.3 shows that {λ′_{R}} defines a map between V and V′, extending λ.

We now complete the proof of Theorem 3.1 by showing the essential surjectivity of Br. Let V ∈ S. We will prove by induction on the cardinality of {Q|V_{Q} ≠ 0} that V is in the image of Br.

For Q ∈ T_{G}, let L_{Q} be a projective direct summand of V_{Q} such that V_{Q}/L_{Q} has no projective direct summand. We denote by α_{Q} : V_{Q} → L_{Q} the canonical surjection.

Let M = Ind_{N_{G}(Q)}^{G} L_{Q} and M = Br M. We have M(Q) ≃ L_{Q} by Green’s correspondence.

Let ζ : V_{Q} ⟶^{α_{Q}} L_{Q} →^{∼} M(Q). Let H = T_{G} and G be the full subcategory of T_{G} with objects the p-subgroups containing Q. Let ζ′_{R} = 0 for R ∈ G, R ≠ Q and ζ′_{Q} = ζ. Then, {ζ′_{R}} ∈ Hom_{S_{G}}(Res_{G}^{H} V, Res_{G}^{H} M). By Lemma 3.7, there is χ ∈ Hom_{S}(V, M) extending {ζ′_{R}}.

Let now V′ = ⨁_{Q∈T_{G}/G} Ind_{N_{G}(Q)}^{G} L_{Q}, V′ = Br V′ and λ : V → V′ be the sum of the morphisms constructed for each Q above.

Then, for all Q, the restriction of λ_{Q} to L_{Q} is injective. By Lemma 3.6, (ii), we deduce that λ_{Q} is a split injection for all Q. Let then W be the cokernel of λ.

Take R with V_{R} = 0. Then, V_{Q} = 0 whenever R is contained up to G-conjugation in Q. So, V′ has no direct summand with vertex R, hence W_{R} = 0. Let now Q be maximal such that V_{Q} ≠ 0. Then, λ_{Q} : V_{Q} = L_{Q} → (Ind_{N_{G}(Q)}^{G} L_{Q}) (Q) is an isomorphism. So, W_{Q} = 0. It follows that {Q|W_{Q} ≠ 0} is strictly contained in {Q|V_{Q} ≠ 0}. By induction, there is a p-permutation kG-module W (without projective direct summand) such that W = Br W. By fulness of Br (Lemma 3.5), the canonical morphism V′ → W comes from a morphism f : V′ → W. By Lemma 3.6, (i), dualized, f is a split surjection. Hence, ker f is a p-permutation kG-module and we have an isomorphism V →^{∼} Br(ker f). This finishes the proof of Theorem 3.1.

References

[Bou1] S. Bouc, Résolutions de foncteurs de Mackey, in “Groups, representations : cohomology, group actions and topology”, Proc. of Symp. in Pure Math., p. 31–83, vol. 63, AMS, 1998.

[Bou2] S. Bouc, Le complexe de chaînes d’un G-complexe simplicial acyclique, preprint (1998).

[Br] M. Broué, On Scott modules and p-permutation modules : an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401–408.

[Li] M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, preprint (1997).

[Th] J. Thévenaz, “G-algebras and modular representation theory”, Oxford University Press, 1995.