BECKER-GOTTLIEB TRANSFER FOR HOCHSCHILD COHOMOLOGY

FEI XU

Abstract. Let G be a finite group. Over any finite G-poset P we may define a transporter category G ∝ P as the corresponding Grothendieck construction. There exists a Becker-Gottlieb transfer from the ordinary cohomology of G ∝ P to that of G. We shall construct it using module-theoretic methods and then extend it to a transfer from the Hochschild cohomology of k(G ∝ P) to that of kG, where k is a base field.

1. Introduction

Let G be a finite group, P a finite G-poset and k a field. Here we always consider G as a category with one object •. Then having a G-poset P means there exists a functor F from G to the category of small categories such that F (•) = P. The Grothendieck construction on F is a finite category and is written as G ∝ P, equipped with a canonical functor π : G ∝ P → G. We may call such a category a transporter category as the usual transporter categories can be introduced in this way. Our concept is more general because moreover each subgroup H of G can be recovered as a transporter category G ∝ (G/H), for the G-set G/H, up to a category equivalence. It makes sense if we deem transporter categories as generalized subgroups for a fixed finite group. Topologically it is well known that B(G ∝ P) ≃ EG ×_{G} BP, and in particular B(G ∝ (G/H)) ≃ BH. Forming the transporter category over a G-poset eliminates the G-action, and thus is the algebraic analogy of introducing a Borel construction over a G-space. In this article we shall study the connections between G and G ∝ P through their category algebras kG and k(G ∝ P), see Section 2.3.

For any subgroup H and two kG-modules M and N, there is a transfer map

tr_{H}^{G} : Ext_{kH}^{∗}(M ↓_{H}^{G}, N ↓_{H}^{G}) → Ext_{kG}^{∗}(M, N).

It can be generalized to a Becker-Gottlieb transfer map

tr_{P} : Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}) → Ext_{kG}^{∗}(M, N),

where κ_{M} and κ_{N} are the restrictions of M and N, respectively. We shall construct the Becker-Gottlieb transfer in a way parallel to the previous one using module-theoretic methods, borrowing ideas from Dwyer-Wilkerson [6]. In this greater generality, tr_{G/H} = tr_{H}^{G}.

Using similar methods, we will construct another transfer map between Hochschild cohomology.

Theorem 1.1. Let π : G ∝ P → G be the canonical functor. Suppose Res_{π} : kG-mod → k(G ∝ P)-mod is the restriction along π and write κ_{M} = Res_{π}M for any M ∈ kG-mod. Then

(1) we have the following two maps, restriction and transfer,

Ext_{kG}^{∗}(M, N) →^{res_{P}}Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N})→^{tr_{P}}Ext_{kG}^{∗}(M, N),

which compose to χ(P; k)·1, multiplication by the Euler characteristic of (the order complex of) P; and

(2) there exists a map htr_{P} : Ext_{k(G∝P)^{e}}^{∗}(k(G ∝ P), k(G ∝ P)) → Ext_{kG^{e}}^{∗}(kG, kG) which induces tr_{P} : Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{kG}^{∗}(k, k).

2000 Mathematics Subject Classification. Primary 20C05; Secondary 20J99.

The author (徐斐) was supported in part by a Beatriu de Pinós research fellowship from the government of Catalonia of Spain.

Here \underline{k} is κ_{k}, the trivial k(G ∝ P)-module, and is of great importance. We comment that (2) makes sense because Ext_{kC}^{∗}(\underline{k}, \underline{k}) is a direct summand of Ext_{kC^{e}}^{∗}(kC, kC) [14]. Particularly it generalizes a transfer map of Linckelmann for P = G/H [8].

For other information about Ext_{k(G∝P)}^{∗}(𝔐, 𝔑), where 𝔐, 𝔑 ∈ k(G ∝ P)-mod are not necessarily restrictions of kG-modules, the reader is referred to [16].

2. Preliminaries

In this section, we recall the definition of a transporter category and some background in category algebras. Throughout this article we will only consider finite categories, in the sense that they have finitely many morphisms. Thus a group G, or a G-poset P, is always finite.

Although many constructions can be made over a commutative ring with identity, we work throughout this article over a field k for convenience.

2.1. Transporter categories as Grothendieck constructions. We usually deem a group as a category with one object, usually denoted by •. The identity of a group is named e. We say a poset P is a G-poset if there exists a functor F from G to sCats, the category of small categories, such that F (•) = P. The Grothendieck construction on F is called a transporter category. In the following explicit definition, the morphisms in a poset is customarily denoted by ≤.

Definition 2.1. Let G be a group and P a G-poset. The transporter category G ∝ P has the same objects as P, that is, Ob(G ∝ P) = Ob P. For x, y ∈ Ob(G ∝ P), a morphism from x to y is a pair (g, gx ≤ y) for some g ∈ G.

The symbol G ∝ P is used because this particular Grothendieck construction resembles a semidirect product, yet is different. From the definition one can easily see that there is a natural embedding ι_{P} : P ↪ G ∝ P via (x ≤ y) ↦ (e, x ≤ y). On the other hand, the transporter category admits a natural functor π_{P} : G ∝ P → G, given by x ↦ • and (g, gx ≤ y) ↦ g. The functor π_{P} is obtained by functoriality of the Grothendieck construction via the unique functor P → • and the isomorphism G ∝ • ≅ G. We always have a sequence of functors

P↪^{ι_{P}}G ∝ P→^{π_{P}}G

such that π_{P} ∘ ι_{P}(P) is the trivial subgroup or subcategory of G. For convenience, in the rest of this article we often neglect the subscript P and write ι = ι_{P}, π = π_{P}. The classifying spaces of the above sequence form a fibration

BP → B(G ∝ P) ≅ EG ×_{G} P → BG.

Example 2.2. If G acts trivially on P, then G ∝ P = G × P.

Example 2.3. Let G be a finite group and H a subgroup. We consider the set of left cosets G/H which can be regarded as a G-poset: G acts via left multiplication. The transporter category G ∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H. In this way one can recover all subgroups of G, up to category equivalences.

A category equivalence D → C induces a Morita equivalence between the category algebras, to be recalled in 2.3, kD ≃ kC (see [13]), as well as a homotopy equivalence BD ≃ BC. It means there is no essential difference between kH and k(G ∝ (G/H)) (and their module categories), or between BH and B(G ∝ (G/H)).

2.2. Nerves and homology representations of kG. Suppose C is a small category. Its nerve NC is a simplicial set, from which one can construct a (normalized) chain complex ℂ_{∗}(C, k) such that, for each i > 0, ℂ_{i}(C, k) is a k-vector space with a basis the set of i-chains of (non-identity) morphisms in C, while ℂ_{0}(C, k) = k Ob C. The topological realization of NC is called the classifying space of C, and is written as BC. The homology of the complex ℂ_{∗}(C, k) gives rise to the homology of BC, i.e. H_{∗}(BC, k). The k-dual, ℂ_{∗}(C, k)^{∧}, calculates H^{∗}(BC, k), the singular cohomology of BC. The Euler characteristic χ(C) = χ(C, k) is defined to be ∑_{i≥0}(−1)^{i} dim_{k} ℂ_{i}(C, k), assuming ℂ_{∗}(C, k) is a finite complex.

When C admits a G-action, ℂ_{∗}(C, k) becomes a complex of permutation kG-modules. The homology and cohomology groups of BC are also kG-modules. This way of constructing kGmodules has been studied intensively when the category is a G-poset. In the present article we shall mainly focus on constructing kG-modules from G ∝ P, which is closely related to certain constructions over ℂ_{∗}(P, k).

Example 2.4. In Example 2.3, ℂ_{∗}(G/H) is a stalk complex ℂ_{∗}(G/H) = ℂ_{0}(G/H) ≅ k ↑_{H}^{G} with χ(G/H) = [G : H].

2.3. Category algebras, representations and cohomology. We recall some facts about category algebras. The reader is referred to [13, 15] for further details. Let C be a finite category and k a field. One can define category algebra kC, which has a basis the set of all morphisms in C. The multiplication is given by compositions of base elements. When C is a group, kC is exactly the group algebra. A k-representation of C is a covariant functor from C to V ect_{k}, the category of finite dimensional k-vector spaces. All representations of C form the functor category V ect_{k}^{C}. By a theorem of B. Mitchell, the finitely generated left kC-modules are the same as the k-representations of C, in the sense that there exists a natural equivalence

V ect_{k}^{C} ≃ kC-mod.

In the module category, there is a distinguished module \underline{k}, sometimes called the trivial module, which can be defined as a constant functor taking k as its value at every object of C. Since V ect_{k} is a symmetric monoidal category, V ect_{k}^{C} inherits this structure. There exists an (internal) tensor product, or the pointwise tensor product, written as ˆ⊗, such that for any two kC-modules 𝔐, 𝔑, (𝔐 ˆ⊗𝔑)(x) = 𝔐(x) ⊗_{k} 𝔑(x). If α ∈ Mor C is a base element in kC, then α acts on 𝔐 ˆ⊗𝔑 via α ⊗ α. Obviously \underline{k} is the identity with respect to ˆ⊗ and 𝔐 ˆ⊗𝔑 ≅ 𝔑 ˆ⊗𝔐. We can also construct a function object, the internal hom, Hom(𝔐, 𝔑) ∈ kC-mod such that

Hom_{kC}(𝔏 ˆ⊗𝔐, 𝔑) ≅ Hom_{kC}(𝔏, Hom(𝔐, 𝔑)),

for any 𝔏 ∈ kC-mod, and the isomorphism is natural in 𝔐 and 𝔑. The monoidal structure allows us to define a cup product on Ext_{kC}^{∗}(\underline{k}, \underline{k}). The resulting ring is isomorphic to H^{∗}(BC, k) in a natural way.

The following functors for comparing category algebras and their modules play a key role in our constructions of transfer maps. When τ : D → C is a functor between two finite categories. There are functors for comparing there representations. The functor τ induces a restriction

Res_{τ} : kC-mod → kD-mod.

If we regard a kC-module as a functor, then its restriction is the precomposition of it with τ. The functor Res_{τ} is equipped with two adjoints: the left and right Kan extensions

LK_{τ} , RK_{τ} : kD-mod → kC-mod.

The definition of the left and right Kan extensions depend on the so-called over-categories and under-categories, respectively. For each x ∈ Ob C, one can define an overcategory τ/x whose objects are pairs {(d, α) | α ∈ Hom_{C}(τ(d), x)}. A morphism from (d, α) to some (d′, β) is a morphism f ∈ Hom_{D}(d, d′) such that the following diagram commutes

\xymatrix{τ(d) \ar[dr]^{α} \ar[dd]_{τ(f)} \\  & x \\ τ(d′) \ar[ur]_{β} & .}

One can always build a canonical projective resolution B_{∗}^{C}, the bar resolution, of \underline{k} ∈ kC-mod. For each n ≥ 0, B_{n}^{C} := ℂ_{n}(Id_{C}/−, k). It means, for each x ∈ Ob C, B_{n}^{C}(x) = ℂ_{n}(Id_{C}/x, k) is the k-vector space spanned over the n-chains of morphisms in Id_{C}/x. A well known fact is that, given a functor τ : D → C,

LK_{τ} B_{∗}^{D} ≅ ℂ_{∗}(τ/−, k)

is a (usually not exact) complex of projective kC-modules. To deal with category cohomology, one often has to understand the structure of such LK_{τ} B_{∗}^{D} for various τ : D → C. The reader will see that calculating the structure of various chain complexes like ℂ_{∗}(τ/−, k) is our major methods in this article. In fact, we do so by analysing the structure of those overcategories τ/x, x ∈ Ob C, which determine the chain complexes. For convenience we often omit the base field in these chain complexes.

2.4. Factorization category and Hochschild cohomology. Let C be a finite category. We can consider its Hochschild cohomology HH^{∗}(kC) = Ext_{kC^{e}}^{∗}(kC, kC), where C^{e} = C × C^{op} and kC^{e} is the enveloping algebra of kC. The kC^{e}-module kC as a functor C^{e} → V ect_{k} is given by

kC(x, y) = kHom_{C}(y, x), ∀(x, y) ∈ Ob C^{e}.

In [14] we proved that the Hochschild cohomology of kC is closely related with the ordinary cohomology. In order to state the result we need the category of factorizations in C, written as F (C). The objects are the morphisms in C, and in order to avoid confusions if a morphism α ∈ Mor C is considered as an object in F (C) we shall denote it by [α] ∈ Ob F (C). Then a morphism [α] → [β] is a pair (u, v) of morphisms in C such that β = uαv. The category F (C) plays the role of the diagonal subgroup ΔG ⊂ G × G ≅ G^{e} in group cohomology. There exists a commutative diagram of functors

\xymatrix{F (C) \ar[rr]^{∇} \ar[dr]_{t} &  & C^{e} \ar[dl]^{p} \\  & C & .}

On objects the functor t is given by t([α]) = y, if α ∈ Hom_{C}(x, y), the functor ∇ is given by ∇([α]) = (y, x) and p is the projection onto the first component. One can easily figure out their actions on morphisms. By using relevant restriction and Kan extensions along these functors we obtained the following isomorphisms [14]

Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k})

(particularly t induces a homotopy equivalence BF (C) ≃ BC) and

Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{∇}kC).

Moreover Res_{∇}kC has \underline{k} as a direct summand and thus the ordinary cohomology is a direct summand of the Hochschild cohomology.

For future reference, we record the following result.

Lemma 2.5. If τ : D → C is a functor. Then it induces a new functor F (τ) : F (D) → F (C).

Proof. For each [α] ∈ Ob F (D) we assign to it [τ(α)] ∈ Ob F (C), while for each morphism (u, v) : [α] → [β] we have (τ(u), τ(v)) : [τ(α)] → [τ(β)], a morphism in F (C). One can verify that these assignments define a functor, written as F (τ). □

3. Becker-Gottlieb transfer

In this article we will focus on the functor π : G ∝ P → G. The accompanying restriction Res_{π} and its left adjoint LK_{π} will be our main tools. For instance, if M is a kG-module, then Res_{π}M becomes a k(G ∝ P)-module. Since this kind of modules are of particular importance in the present article, we shall denote them as κ_{M} = Res_{π}M ∈ k(G ∝ P)-mod, following [11]. As a functor κ_{M} takes constant value M at each object of G ∝ P, but the k(G ∝ P)-action on it is not trivial in general. However the kP-action on κ_{M} is trivial because each morphism in P is given by the identity element e ∈ G. In the end, if k is the trivial kG-module κ_{k} is identified with the trivial k(G ∝ P)-module \underline{k}.

As a convention, throughout this article, the kG-modules are usually written as M, N etc, while the modules of a (non-group) category algebra kC are denoted by 𝔐, 𝔑 etc., except those special modules, namely \underline{k} and κ_{M}, which we have just mentioned.

3.1. Modules constructed over categories. Suppose P is a G-poset and π is the natural functor from the transporter category G ∝ P to G, regarded as a category with one object •. The fibre of the functor π [10], i.e. π^{−1}(•), is exactly P. Also by [10], the overcategory π/• provides the fibre of the map Bπ in the sense that the sequence of categories

π/• → G ∝ P → G

corresponds to the fibration

B(π/•) → B(G ∝ P) → BG.

It leads to an action of G = π_{1}(BG) on B(π/•), which is realized by a G-action on the category π/•, defined shortly after Proposition 3.1. We shall see that there exists an inclusion P → π/•, although not a G-functor, inducing a category equivalence.

The objects of π/• are of the form (x, h), in which x ∈ Ob(G ∝ P) = Ob P and h ∈ G. A morphism from (x, h) to (x′, h′) is a morphism (g, gx ≤ x′) ∈ Mor(G ∝ P) such that h′g = h. It implies that each object (x, h) ≅ (x′, h′) if and only if x′ = gx and h′ = hg^{−1} for some g ∈ G. Indeed the objects isomorphic to (x, h) are {(gx, hg^{−1}) | g ∈ G}. Particularly (x, h) ≅ (hx, e) for the identity e ∈ G.

Proposition 3.1. The category π/• is isomorphic to (Id_{G}/•) × P.

Proof. We establish an isomorphism ϕ : (Id_{G}/•)×P → π/• as follows. The objects of (Id_{G}/•)×P are {(g, x) | g ∈ G, x ∈ Ob P}. We define ϕ((g, x)) = (gx, g^{−1}) ∈ Ob(π/•). For a morphism (g_{2}g_{1}^{−1}, x_{1} ≤ x_{2}) : (g_{1}, x_{1}) → (g_{2}, x_{2}) we put

ϕ((g_{2}g_{1}^{−1}, x_{1} ≤ x_{2})) := (g_{2}g_{1}^{−1}, (g_{2}g_{1}^{−1})(g_{1}x_{1}) ≤ g_{2}x_{2}),

a morphism from (g_{1}x_{1}, g_{1}^{−1}) to (g_{2}x_{2}, g_{2}^{−1}). The inverse ϕ^{−1} : π/• → (Id_{G}/•) × P is given by ϕ^{−1}((y, h)) := (h^{−1}, hy) and ϕ^{−1}((h_{2}^{−1}h_{1}, h_{2}^{−1}h_{1}x_{1} ≤ x_{2})) := (h_{2}^{−1}h_{1}, h_{1}x_{1} ≤ h_{2}x_{2}). □

The G-action on π/• can be explicitly described. For any object (x, h) ∈ Ob π/• and u ∈ G, we have u ∘ (x, h) = (ux, hu^{−1}), and for any morphism (g, gx ≤ x′) : (x, h) → (x′, h′) we have u ∘ (g, gx ≤ x′) = (ugu^{−1}, (ugu^{−1})ux ≤ ux′) : (ux, hu^{−1}) → (ux′, h′u^{−1}).

The proposition actually implies that P and π/• are equivalent. We write out the functors that induce the equivalence below. Part (i) of the following corollary was discussed and used in [15]. Since it is a direct consequence from the preceding proposition, for completion we record it here.

Corollary 3.2. (1) There exists a natural embedding P ↪ π/• making P a skeleton of π/•.

Consequently LK_{π}𝔐 ≅ \underrightarrow{lim}_{P} 𝔐 and RK_{π}𝔐 ≅ \underleftarrow{lim}_{P} 𝔐 for any 𝔐 ∈ k(G ∝ P)-mod.

(2) The natural functor π/• ↠ P is a G-functor.

Proof. The natural functor P → π/•, given by x ↦ (x, e) and x ≤ y ↦ (e, x ≤ y), is an embedding, sending P to a skeleton of π/•. It is straightforward to check that there is a natural surjective functor π/• → P, induced by (x, h) ↦ hx, which is a quasi inverse to the previous embedding. In fact these two functors provide an equivalence between π/• and P. The existence of an equivalence between these two categories forces \underrightarrow{lim}_{π/•} 𝔐 ≅ \underrightarrow{lim}_{P} 𝔐 and \underleftarrow{lim}_{π/•} 𝔐 ≅ \underleftarrow{lim}_{P} 𝔐.

It follows from our description of the G-action on π/• that the natural functor π/• ↠ P is a G-functor. □

The embedding P ↪ π/• is not a G-functor, and one reason is that an object x ∈ Ob P is mapped isomorphically to ux ∈ Ob P for any u ∈ G, while its image (x, e) ∈ Ob(π/•) must be sent to (ux, u^{−1}) ∈ Ob(π/•), not (ux, e).

By Proposition 4.1, if P = G/G = •, then π_{G/G} = Id_{G} and π_{G/G}/• = Id_{G}/• is the Cayley graph, giving rise to the total space EG whose complex is the bar resolution B_{∗}^{G}. More generally for P = G/H, we have an isomorphism of complexes of kG-modules ℂ_{∗}(π/•) ≅ B_{∗}^{G} ⊗ k(G/H), which provide a projective resolution of k(G/H) = ℂ_{0}(G/H) = ℂ_{∗}(G/H) (a stalk complex).

Corollary 3.3. The complex LK_{π}B_{∗}^{G∝P} ≅ ℂ_{∗}(π/•) ≃ B_{∗}^{G} ⊗ ℂ_{∗}(P) is a projective resolution of the finite complex of kG-modules ℂ_{∗}(P).

Proof. Since LK_{π} preserves projectives, LK_{π}B_{∗}^{G∝P} ≅ ℂ_{∗}(π/•) is a complex of projective kGmodules. The existing G-functor π/• → P gives rise to a chain map of complexes of kG-modules LK_{π}B_{∗}^{G∝P} ≅ ℂ_{∗}(π/•) ≃ B_{∗}^{G} ⊗ ℂ_{∗}(P) → ℂ_{∗}(P). However since that G-functor is a category equivalence, it induces an isomorphism between the homology of complexes. □

Corollary 3.4. Suppose M ∈ kG-mod and 𝔑 ∈ k(G ∝ P)-mod. Then LK_{π}(κ_{M} ˆ⊗𝔑) ≅ M ⊗ LK_{π}𝔑 and RK_{π}(κ_{M} ˆ⊗𝔑) ≅ M ⊗ RK_{π}𝔑 as kG-modules. In particular LK_{π}(κ_{M}) ≅ M ⊗ LK_{π}\underline{k} and RK_{π}(κ_{M}) ≅ M ⊗ RK_{π}\underline{k}, where LK_{π}\underline{k} ≅ H_{0}(BP, k) ≅ H^{0}(BP, k) ≅ RK_{π}\underline{k} of dimension equal to the number of connected components of P.

Proof. For the left Kan extension we have

LK_{π}(κ_{M} ˆ⊗𝔑) ≅ \underrightarrow{lim}_{P}(κ_{M} ˆ⊗𝔑) ≅ M ⊗ \underrightarrow{lim}_{P}𝔑 ≅ M ⊗ LK_{π}𝔑.

The second isomorphism is true because κ_{M} as a kP-module admits trivial action. The statement for the right Kan extension is similar. □

The above corollary implies that LK_{π}κ_{M} ≅ RK_{π}κ_{M}, suggesting the existence of a transfer map, which we will construct later on.

The next result is needed to construct transfer maps. It is [16, Theorem 4.1]

Theorem 3.5. Let 𝔓 ∈ k(G ∝ P)-mod be a projective module and κ_{M} = Res_{π}M for some M ∈ kG-mod. Then 𝔓 ˆ⊗κ_{M} is a projective k(G ∝ P)-module. Consequently B_{∗}^{G∝P} ˆ⊗κ_{M} → \underline{k} ˆ⊗κ_{M} = κ_{M} → 0 is a projective resolution.

Assume ℂ_{∗} is a complex of kG-modules. Then we naturally obtain a complex of k(G ∝ P)- modules via restriction, written as κ_{ℂ_{∗}}. For any M ∈ kG-mod, there is naturally a chain map, unique up to homotopy, as follows

Π_{M} : {B_{∗}^{G∝P} ˆ⊗κ_{M} → κ_{M} → 0} → {κ_{B_{∗}^{G}} ˆ⊗κ_{M} = κ_{B_{∗}^{G}⊗M} → κ_{M} → 0}.

In the proof of Theorem 3.8, for simplicity, we shall also use Π_{M} to denote its brutal truncation

Π_{M} : {B_{∗}^{G∝P} ˆ⊗κ_{M}} → {κ_{B_{∗}^{G}} ˆ⊗κ_{M} = κ_{B_{∗}^{G}⊗M}}.

Remark 3.6. Using Dan Swenson’s definition [12] (see also [15]) of the internal hom one verifies that

κ_{Hom_{k}(M,N)} ≅ Hom(κ_{M}, κ_{N}).

Consequently, Theorem 3.5 implies

Ext_{k(G∝P)}^{∗}(\underline{k}, κ_{Hom_{k}(M,N)}) ≅ Ext_{k(G∝P)}^{∗}(\underline{k}, Hom(κ_{M}, κ_{N})) ≅ Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}).

3.2. Equivariant cohomology. Before moving to the next application, we record a connection between cohomology of transporter categories and equivariant cohomology, which is perhaps known to the experts.

Proposition 3.7. The left Kan extension induces an isomorphism

λ_{M} : Ext_{k(G∝P)}^{∗}(\underline{k}, κ_{M}) ≅ H_{G}^{∗}(BP, M),

where H_{G}^{∗}(BP, M) is the equivariant cohomology group for some M ∈ kG-mod.

Proof. Take the bar resolution B_{∗}^{G∝P} → \underline{k} → 0 and consider the complex Hom_{k(G∝P)}(B_{∗}^{G∝P}, κ_{M}). The left Kan extension induces a chain map

Hom_{k(G∝P)}(B_{∗}^{G∝P}, κ_{M}) ≅ Hom_{kG}(LK_{π}B_{∗}^{G∝P}, M) ≃ Hom_{kG}(B_{∗}^{G} ⊗ ℂ_{∗}(P), M).

But the rightmost term is

Hom_{kG}(B_{∗}^{G} ⊗ ℂ_{∗}(P), M) ≅ Hom_{kG}(B_{∗}^{G}, Hom_{k}(ℂ_{∗}(P), M)),

which gives rise to H_{G}^{∗}(BP, M). □

In light of the above proposition, we may introduce the Tate cohomology of transporter categories as Tate equivariant cohomology. With Remark 3.6 in mind, one can further define negative degree Ext groups Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}).

3.3. Construction of the transfer map. Let τ : D → C be a functor between small categories. There is always a restriction res_{τ} : H^{∗}(C; \underline{k}) → H^{∗}(D; \underline{k}). However usually one cannot construct a map in the opposite direction, unless the two Kan extensions are connected by a natural transformation. In this section, based on our knowledge about representations of k(G ∝ P), we establish the Becker-Gottlieb transfer map with respect to π_{P} : G ∝ P → G. Here we provide an algebraic alternative to the construction of Becker-Gottlieb [1], and the core idea is taken from Dwyer-Wilkerson [6, 9.13]). Essentially our construction incorporates [6, 9.13] in an entirely representation-theoretic setting. The upshot is that our construction is analogous to the classical situation, see for instance [2, 3.6.17]. Keep in mind that Res_{π} and LK_{π} generalize ↓_{H}^{G} and ↑_{H}^{G}, respectively, used in group cohomology.

Theorem 3.8. Suppose Res_{π} : kG-mod → k(G ∝ P)-mod is the restriction along π and write κ_{M} = Res_{π}M for any M ∈ kG-mod. Then we have the following two maps, restriction and transfer,

Ext_{kG}^{∗}(M, N) →^{res_{P}}Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N})⟶^{tr_{P}}Ext_{kG}^{∗}(M, N),

which compose to χ(P; k) · 1, multiplication by the Euler characteristic of (the order complex of) P. When P = G/H for some subgroup H, tr_{G/H} is the usually transfer map in group cohomology.

Proof. We shall construct these two maps. Then in the sequel one can deduce the statement on their composite.

Suppose B_{∗}^{G} and B_{∗}^{G∝P} are the bar resolutions of k ∈ kG-mod and \underline{k} ∈ k(G ∝ P)-mod, respectively. Then B_{∗}^{G} ⊗ M is a projective resolution of a fixed M ∈ kG-mod. For another kGmodule N, the cochain complex Hom_{kG}(B_{∗}^{G} ⊗ M, N) computes Ext_{kG}^{∗}(M, N). The exact functor Res_{π} sends this cochain complex to Hom_{k(G∝P)}(κ_{B_{∗}^{G}⊗M}, κ_{N}). From Theorem 3.5 we obtain the composite of two chain maps, unique up to chain homotopy,

Hom_{kG}(B_{∗}^{G} ⊗ M, N) →^{Res_{π}}Hom_{k(G∝P)}(κ_{B_{∗}^{G}⊗M}, κ_{N}) →^{Π_{M}^{∗}}Hom_{k(G∝P)}(B_{∗}^{G∝P} ˆ⊗κ_{M}, κ_{N}).

These seemingly abstract maps can be spelled out explicitly, but we will leave it to the interested reader. The composite Π_{M}^{∗} ∘ Res_{π} induces a map on cohomology, which we call the restriction,

res_{P} : Ext_{kG}^{∗}(M, N) → Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}).

It is helpful to have a different characterization of the restriction. In order to do so we use a series of obvious isomorphisms to rewrite the previously mentioned complex Hom_{k(G∝P)}(B_{∗}^{G∝P} ˆ⊗κ_{M}, κ_{N}). Firstly by adjunction, it is isomorphic to

Hom_{kG}(LK_{π}(B_{∗}^{G∝P} ˆ⊗κ_{M}), N).

Here (Π_{M}^{∗} ∘Res_{π})α is mapped to (Λ_{N} ∘LK_{π})(Π_{M}^{∗} ∘Res_{π}α), where Λ_{N} : LK_{π}κ_{N} → N, determined by the counit of adjunction, is isomorphic to ϵ̄⊗ Id_{N}. The natural map ϵ̄: H_{0}(P) → k is induced by the augmentation map ϵ : ℂ_{0}(P) → k (or rather P → •), by Corollary 3.4. Secondly from the same corollary our cochain complex is canonically isomorphic to

Hom_{kG}(LK_{π}(B_{∗}^{G∝P}) ⊗ M, N).

Thirdly by Corollary 3.3 LK_{π}B_{∗}^{G∝P} ≅ B_{∗}^{G} ⊗ ℂ_{∗}(P), the above complex is

Hom_{kG}(LK_{π}(B_{∗}^{G∝P}) ⊗ M, N) ≃ Hom_{kG}(B_{∗}^{G} ⊗ ℂ_{∗}(P) ⊗ M, N)
≅ Hom_{kG}(B_{∗}^{G} ⊗ M ⊗ ℂ_{∗}(P), N).

The observations imply that res_{P} : Ext_{kG}^{∗}(M, N) → Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}) is the same as the map induced by the following chain map

Hom_{kG}(B_{∗}^{G} ⊗ M, N) → Hom_{kG}(B_{∗}^{G} ⊗ M ⊗ ℂ_{∗}(P), N),

which is given by the augmentation ϵ : ℂ_{∗}(P) → k. Now we are ready to build the transfer map. From the cochain complex Hom_{kG}(B_{∗}^{G} ⊗ M ⊗ ℂ_{∗}(P), N) we continue to establish a map which leads back to group cohomology

Θ^{∗} : Hom_{kG}(B_{∗}^{G} ⊗ M ⊗ ℂ_{∗}(P), N) → Hom_{kG}(B_{∗}^{G} ⊗ M, N).

The map Θ^{∗} is induced by the following chain map Θ : k → ℂ_{∗}(P) of Dwyer-Wilkerson [6, 9.13], as the composite of

k →^{a↦a·Id}Hom_{k}(ℂ_{∗}(P), ℂ_{∗}(P))→^{≅}ℂ_{∗}(P)^{∧} ⊗ ℂ_{∗}(P) →^{Id⊗Δ}ℂ_{∗}(P)^{∧} ⊗ ℂ_{∗}(P) ⊗ ℂ_{∗}(P) →^{ev⊗Id}ℂ_{∗}(P).

Here ℂ_{∗}(P)^{∧} = Hom_{k}(ℂ_{∗}(P), k), non-positively graded, is the k-dual of ℂ_{∗}(P), Δ is the diagonal map and ev is the evaluation map. The composite of

Hom_{k(G∝P)}(B_{∗}^{G∝P} ˆ⊗κ_{M}, κ_{N})→^{≅}Hom_{kG}(LK_{π}(B_{∗}^{G∝P} ˆ⊗κ_{M}), N)→^{Θ^{∗}}Hom_{kG}(B_{∗}^{G} ⊗ M, N)

defines a map

tr_{P} : Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N}) → Ext_{kG}^{∗}(M, N).

which is called the transfer.

In fact we have the following commutative diagram of cochain complexes

\xymatrix{Hom_{kG}(B_{∗}^{G} ⊗ M, N) \ar[r] \ar[d]_{=} & Hom_{k(G∝P)}(B_{∗}^{G∝P} ˆ⊗κ_{M}, κ_{N}) \ar[r] \ar[d]_{≅} & Hom_{kG}(B_{∗}^{G} ⊗ M, N) \ar[d]^{=} \\ Hom_{kG}(B_{∗}^{G} ⊗ M, N) \ar[r] & Hom_{kG}(B_{∗}^{G} ⊗ M ⊗ ℂ_{∗}(P), N) \ar[r] & Hom_{kG}(B_{∗}^{G} ⊗ M, N).}

Upon passing to cohomology, both rows give rise to

Ext_{kG}^{∗}(M, N) →^{res_{P}}Ext_{k(G∝P)}^{∗}(κ_{M}, κ_{N})→^{tr_{P}}Ext_{kG}^{∗}(M, N).

In the end we prove that the composite res_{P} ∘tr_{P} = χ(P)·1. We directly compute the following composite

k → ℂ_{∗}(P) → k

and show it is a scalar multiplication by χ(P). Write the natural basis of ℂ_{n}(P) as {c_{n}^{i}}_{i=1}^{d_{n}} for d_{n} = dim_{k} ℂ_{n}(P) (see Section 2.2). We have

1 ↦ Id_{ℂ_{∗}(P)}
↦ ∑_{n=0}^{d}(−1)^{n}{∑_{i=1}^{d_{n}}(c_{n}^{i})^{∧} ⊗ c_{n}^{i}}
↦ ∑_{n=0}^{d}(−1)^{n}{∑_{i=0}^{d_{n}}(c_{n}^{i})^{∧} ⊗ [c_{n}^{i} ⊗ t(c_{n}^{i})]}
= ∑_{n=0}^{d}(−1)^{n}{∑_{i=1}^{d_{n}}[(c_{n}^{i})^{∧} ⊗ c_{n}^{i}] ⊗ t(c_{n}^{i})}
↦ ∑_{n=0}^{d}(−1)^{n}{∑_{i=1}^{d_{n}} t(c_{n}^{i})}
↦ ∑_{n=0}^{d}(−1)^{n}d_{n}
= χ(P).

Here t(c_{n}^{i}) ∈ ℂ_{0}(P) denotes the last object, i.e. the target, of the n-chain of morphisms c_{n}^{i} ∈ ℂ_{n}(P). □

It is well-known that, in cohomology of algebras, if ρ : A → B is an algebra homomorphism, and V, W are two B-modules, then one can construct a natural map res_{ρ} : Ext_{B}^{∗}(V, W) → Ext_{A}^{∗}(Res_{ρ}V, Res_{ρ}W). This cannot be used in our situation because the functor π : G ∝ P → G does not induce a homomorphism between their category algebras.

Remark 3.9. In fact, by Remark 3.6, the above restriction and transfer coincide with

Ext_{kG}^{∗}(k, Hom_{k}(M, N)) →^{res_{P}}Ext_{k(G∝P)}^{∗}(\underline{k}, κ_{Hom_{k}(M,N)})⟶^{tr_{P}}Ext_{kG}^{∗}(k, Hom_{k}(M, N)).

When M = N = k, our construction is exactly the Becker-Gottlieb transfer ([1],[6]), because Ext_{k(G∝P)}^{∗}(\underline{k}, κ_{M}) ≅ H_{G}^{∗}(BP, M).

We emphasize that if either M ∈ kG-mod is not acted trivially by kG or if G sends a connected component of P to a different one, then the constantly valued κ_{M} ∈ k(G ∝ P)-mod is not truly constant since H^{0}(G ∝ P; κ_{M}) ≅ \underleftarrow{lim}_{G∝P} κ_{M} ≅ \underleftarrow{lim}_{G} \underleftarrow{lim}_{P} κ_{M} ≅ (M ⊗ H^{0}(P))^{G}.

4. Transfer for Hochschild cohomology

Using exactly the same method we can build a transfer map between Hochschild cohomology. Consider the following commutative diagram of functors

\xymatrix{ & (G ∝ P)^{e} \ar[r]^{π^{e}} & G^{e} \\ F (P) \ar[d]_{t} \ar[r] & F (G ∝ P) \ar[d]_{t} \ar[r]^{F (π)} \ar[u]^{∇} & F (G) \ar[d]^{t} \ar[u]_{∇} \\ P \ar[r] & G ∝ P \ar[r]_{π} & G}

Recall that in last section, based on the bottom row, we were able to establish a transfer map between ordinary cohomology. Since the three target functors all induce homotopy equivalences, the middle row may as well give a transfer map Ext_{kF (G∝P)}^{∗}(−, −) → Ext_{kF (G)}^{∗}(−, −), for suitable modules. Since furthermore the upper right square demonstrates connections between the ordinary cohomology over kF (G ∝ P) and kF (G) with the Hochschild cohomology of kF (G ∝ P) and kF (G), respectively, we automatically wonder if there would be a transfer between the Hochschild cohomology of k(G ∝ P) and kG? The answer is yes, and a predecessor of this construction is due to Linckelmann [8]. For G a group and H a subgroup, he developed a method over symmetric algebras and defined a transfer map from HH^{∗}(kH) to HH^{∗}(kG). Recall that H can be recovered as the transporter category G ∝ (G/H), our construction generalizes his.

Lemma 4.1. Let kG ∈ kG^{e}-mod and k(G ∝ P) ∈ k(G ∝ P)^{e}-mod. Then the kF (G ∝ P)-module Res_{F (π)}Res_{∇}kG contains Res_{∇}k(G ∝ P) as a submodule.

Proof. Let (x, y) ∈ Ob(G ∝ P)^{e}. Then

k(G ∝ P)(x, y) = kHom_{G∝P}(y, x)
= k{(h, hy ≤ x) | h ∈ G}
⊂ kG
= (Res_{π^{e}}kG)(x, y).

From here we can verify that k(G ∝ P) is a k(G ∝ P)^{e}-submodule of Res_{π^{e}}kG. Hence the result follows. □

If P = G/H, then G ∝ (G/H) ≃ H and F (G ∝ (G/H)) ≃ H. The above inclusion says that the kH-module kG ↓_{H}^{G} contains kH as a submodule. Here kG and kH are acted by G and H, respectively, by conjugation.

Lemma 4.2. Consider the functor F (π) : F (G ∝ P) → F (G). Then for any [g] ∈ Ob F (G), F (π)/[g] ≅ F (P)_{[g]} × Id_{F (G)}/[g], where F (P)_{[g]} denotes a poset, indexed by [g] ∈ Ob F (G), which is canonically isomorphic to F (P). Consequently

ℂ_{∗}(F (π)/[g]) ≅ ℂ_{∗}(F (P)_{[g]} × Id_{F (G)}/[g])

as complexes of kF (G)-modules. Since F (G) ≃ G, for the identity element e ∈ G,

ℂ_{∗}(F (π)/[e]) ≅ ℂ_{∗}(F (P)_{[e]} × Id_{F (G)}/[e]) ≃ ℂ_{∗}(F (P)) ⊗ B_{∗}^{G}

is a projective resolution of the complex of kG-modules ℂ_{∗}(F (P)).

Proof. Suppose [g] is an object in F (G). The the overcategory F (G)/[g] has objects {([(h, hx ≤ y)], (h_{1}, h_{2}))}, in which [(h, hx ≤ y)] ∈ Ob F (G ∝ P). It implies g = h_{1}hh_{2}. As a consequence, hx ≤ y is equivalent to h_{2}^{−1}x ≤ g^{−1}h_{1}y. Morphisms ((v_{1}, v_{1}y ≤ y′), (v_{2}, v_{2}x′ ≤ x)) are given by

\xymatrix{[h] = F (π)([(h, hx ≤ y)]) \ar[dr]^{(h_{1},h_{2})} \ar[dd]_{(v_{1},v_{2})=F (π)((v_{1},v_{1}y≤y′),(v_{2},v_{2}x′≤x))} \\  & [g] \\ [h′] = F (π)([(h′, h′x′ ≤ y′)]) \ar[ur]_{(h′_{1},h′_{2})}}

It implies various identities: g = h_{1}hh_{2} = h′_{1}h′h′_{2}, h_{1} = h′_{1}v_{1} and h_{2} = v_{2}h′_{2}. We define a functor F (π)/[g] → F (P) × Id_{F (G)}/[g] such that on objects

([(h, hx ≤ y)], (h_{1}, h_{2})) ↦ ([h_{2}^{−1}x ≤ g^{−1}h_{1}y], ([h], (h_{1}, h_{2}))),

and on morphisms ((v_{1}, v_{1}y ≤ y′), (v_{2}, v_{2}x′ ≤ x)) ↦ ((e, e), (v_{1}, v_{2})), because v_{1}y ≤ y′ implies g^{−1}h_{1}y ≤ g^{−1}h′_{1}y′ while v_{2}x′ ≤ x implies h′_{2}^{−1}x′ ≤ h_{2}^{−1}x. The inverse of this functor is defined by ([x ≤ y], ([h], (h_{1}, h_{2}))) ↦ ([(h, h(h_{2}x) ≤ h_{1}^{−1}gy)], (h_{1}, h_{2})) on objects, and on morphisms is defined by ((e, e), (v_{1}, v_{2})) ↦ ((v_{1}, v_{1}h_{1}^{−1}gy ≤ h′_{1}^{−1}gy′), (v_{2}, v_{2}h′_{2}x′ ≤ h_{2}x)). Thus we obtain an isomorphism

F (π)/[g] ≅ F (P)_{[g]} × Id_{F} (G)/[g],

where F (P)_{[g]} denotes a copy of F (P) indexed by the object [g]. If (l_{1}, l_{2}) : [g] → [g′] is a morphism in F (G). Then it induces a functor F (π)/[g] → F (π)/[g′] given by

([(h, hx ≤ y)], (h_{1}, h_{2})) ↦ ([(h, hx ≤ y)], (l_{1}h_{1}, h_{2}l_{2}))

and

((v_{1}, v_{1}y ≤ y′), (v_{2}, v_{2}x′ ≤ x)) ↦ ((v_{1}, v_{1}y ≤ y′), (v_{2}, v_{2}x′ ≤ x))

Using the isomorphisms F (π)/[g] ≅ F (P)_{[g]} × Id_{F (G)}/[g] and F (π)/[g′] ≅ F (P)_{[g′]} × Id_{F (G)}/[g′], one can see it induces an isomorphism F (P)_{[g]} → F (P)_{[g′]}.

Finally since G is isomorphic to the automorphism group of [e] in the groupoid F (G), we have an equivalence F (G) ≃ G. Thus ℂ_{∗}(F (P)_{[e]} × Id_{F (G)}/[e]) ≃ ℂ_{∗}(F (P)) ⊗ ℂ_{∗}(Id_{F (G)}/[e]) is a complex of projective kG-modules. Furthermore because ℂ_{∗}(Id_{F (G)}/[e]) = B_{∗}^{F (G)}([e]), it has to be a projective resolution of the trivial kG-module k. Hence we get the chain homotopy equivalence as stated. □

This lemma allows us to describe the transfer map in Theorem 3.8 in terms of factorization categories:

tr_{P} : Ext_{kF (G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{kF (G)}^{∗}(\underline{k}, \underline{k}).

It makes sense because from Section 2.4, for any C, the functor t : F (C) → C induces an isomorphism Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}). Indeed by Lemma 2.5 we have a commutative diagram

\xymatrix{F (G ∝ P) \ar[r]^{F (π)} \ar[d]_{t} & F (G) \ar[d]^{t} \\ G ∝ P \ar[r]_{π} & G}

Thus we get a commutative diagram of cochain complexes

\xymatrix{Hom_{kF (G∝P)}(B_{∗}^{F (G∝P)}, \underline{k}) \ar[r]^{≅} \ar[d]_{≅} & Hom_{kF (G)}(LK_{F (π)}B_{∗}^{F (G∝P)}, \underline{k}) \ar[d]^{≅} \\ Hom_{k(G∝P)}(LK_{t}B_{∗}^{F (G∝P)}, \underline{k}) \ar[r]_{≅} \ar[d]_{≃} & Hom_{kG}(LKπ̃B_{∗}^{F (G∝P)}, k) \ar[d]^{≃} \\ Hom_{k(G∝P)}(B_{∗}^{G∝P}, \underline{k}) \ar[r]_{≃} & Hom_{kG}(ℂ_{∗}(P) ⊗ B_{∗}^{G}, k) \ar[d]^{Θ^{∗}} \\  & Hom_{kG}(B_{∗}^{G}, k) \ar[d]^{≃} \\  & Hom_{kG}(LK_{t}B_{∗}^{F (G)}, k) \ar[d]^{≅} \\  & Hom_{kF (G)}(B^{F (G)}, \underline{k})}

The isomorphisms are given by adjunctions and the chain homotopy equivalences are induced by changing projective resolutions. From the upper left corner to the lower right corner is the transfer that we want to describe. It factors through Θ^{∗}, induced by the chain map of Dwyer-Wilkerson for M = N = k, which gives rise to the transfer constructed in Theorem 3.8.

Theorem 4.3. There exists a transfer map

htr_{P} : Ext_{k(G∝P)^{e}}^{∗}(k(G ∝ P), k(G ∝ P)) → Ext_{kG^{e}}^{∗}(kG, kG),

which is compatible with the transfer map tr_{P} in Theorem 3.8, in the sense that there exists a commutative diagram

\xymatrix{Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{^{tr^{P}}} \ar[d]_{injection} & Ext_{kG}^{∗}(k, k) \ar[d]^{injection} \\ Ext_{k(G∝P)^{e}}^{∗}(k(G ∝ P), k(G ∝ P)) \ar[r]_{_{htr_{P}}} & Ext_{kG^{e}}^{∗}(kG, kG)}

Proof. Using Lemma 4.2 we have cochain maps

Hom_{kF (G∝P)}(B_{∗}^{G∝P}, Res_{F (π)}Res_{∇}kG) ≅ Hom_{kF (G)}(LK_{F (π)}B_{∗}^{G∝P}, Res_{∇}kG)
≅ Hom_{kF (G)}(ℂ_{∗}(F (π)/−), Res_{∇}kG)
≅ Hom_{kG}(ℂ_{∗}(F (π)/[e]), kG)
≃ Hom_{kG}(ℂ_{∗}(F (P)) ⊗ B_{∗}^{G}, kG)
→ Hom_{kG}(B_{∗}^{G}, kG)

Here the third isomorphism is true because F (G) is a groupoid whose skeleton is isomorphic to the automorphism group of [e], which is exactly G. The module kG in Hom_{kG}(−, kG) is acted by kG via conjugations. The last map is induced by k → ℂ_{∗}(F (P)), a chain map constructed in the same way as the one in the proof of Theorem 3.8 k → ℂ_{∗}(P). Note the functor t : F (P) → P induces a homotopy equivalence BF (P) ≃ BP, and thus a chain homotopy equivalence ℂ_{∗}(F (P)) ≃ ℂ_{∗}(P).

By Lemma 4.1 we also have a chain map

Hom_{kF (G∝P)}(B_{∗}^{G∝P}, Res_{∇}kF (G ∝ P)) → Hom_{kF (G∝P)}(B_{∗}^{G∝P}, Res_{F (π)}Res_{∇}kG)

induced by the inclusion Res_{∇}kF (G ∝ P) → Res_{F (π)}Res_{∇}kG. Hence altogether we obtain a chain map

Hom_{kF (G∝P)}(B_{∗}^{G∝P}, Res_{∇}kF (G ∝ P)) → Hom_{kG}(B_{∗}^{G}, kG).

Passing to cohomology we get a map between Hochschild cohomology (see Section 2.4)

\xymatrix{Ext_{kF (G∝P)}^{∗}(\underline{k}, Res_{∇}kF (G ∝ P)) \ar[r]^{^{htr^{P}}} \ar[d]_{≅} & Ext_{kG}^{∗}(k, kG) \ar[d]^{≅} \\ Ext_{k(G∝P)^{e}}^{∗}(kF (G ∝ P), kF (G ∝ P)) \ar[r]_{_{htr_{P}}} & Ext_{kG^{e}}^{∗}(kG, kG)}

The above map deserves to be called a transfer since \underline{k} | Res_{∇}kF (G ∝ P), k | kG (kG acting on kG via conjugation c_{g} = g(−)g^{−1}, see [3, Theorem 2.11.2]) and \underline{k} | Res_{∇}kF (G). In the construction of htr_{P} if we replace the second module in all Hom_{−}(−, −) and Ext_{−}^{∗}(−, −) by \underline{k} then it is exactly the transfer created in Theorem 3.8 and reinterpreted before Theorem 4.3. In other words, we have a commutative diagram

\xymatrix{Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{^{tr^{P}}} \ar[d]_{≅} & Ext_{kG}^{∗}(k, k) \ar[d]^{≅} \\ Ext_{kF (G∝P)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{_{tr_{P}}} \ar[d]_{injection} & Ext_{kF (G)}^{∗}(\underline{k}, \underline{k}) \ar[d]^{injection} \\ Ext_{kF (G∝P)}^{∗}(\underline{k}, Res_{∇}kF (G ∝ P)) \ar[r] \ar[d]_{≅} & Ext_{kF (G)}^{∗}(\underline{k}, Res_{∇}kF (G)) \ar[d]^{≅} \\ Ext_{k(G∝P)^{e}}^{∗}(k(G ∝ P), k(G ∝ P)) \ar[r]_{_{htr_{P}}} & Ext_{kG^{e}}^{∗}(kG, kG)}

□

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[16] F. Xu, On local categories of finite groups, Math. Z., to appear.

Departament of Mathematics, Shantou University, Shantou, Guangdong 515063, China.

  E-mail address: fxu@stu.edu.cn