SUPPORT VARIETIES FOR TRANSPORTER CATEGORY ALGEBRAS

FEI XU

Abstract. Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-algebra kP.

We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson’s support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall-Solberg’s (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules.

1. Introduction

Let G be a finite group and P a finite G-poset. Throughout this paper, we assume k is an algebraically closed field of characteristic p, dividing the order of G. We are interested in a finite category G ∝ P, which is the Grothendieck construction on the G-poset P and which we will call a transporter category in this paper. When G = {e} is trivial, {e} ∝ P ≅ P and when P = • is trivial, G ∝ • ≅ G. A transporter category G ∝ P is the algebraic or categorical predecessor of the Borel construction EG ×_{G} BP on the finite G-CW-complex BP, in the sense that B (G ∝ P) ≃ EG×_{G} BP. Our interests in transporter categories come from the fact that the equivariant cohomology ring H_{G}^{∗}(BP, k) = H^{∗}(EG×_{G} BP, k) is Noetherian. Through an algebraic construction of the equivariant cohomology ring, we may introduce in a natural way modules over this ring and hence extend Carlson’s support variety theory for finite group algebras to one for finite transporter category algebras.

Let us recall some historical developments in support variety theory. Suppose a G-space X is compact or paracompact with finite cohomological dimension. Quillen [23, 24] proved that H_{G}^{∗}(X) is Noetherian. Following his notation, we put H_{G}(X) to be H_{G}^{∗}(X) if p = 2 or H_{G}^{ev}(X), the even part of the equivariant cohomology ring, if p > 3. When X = • is just a point fixed by G, the equivariant ring reduces to the group cohomology ring and we shall write H_{G}^{∗} = H_{G}^{∗}(•) and H_{G} = H_{G}(•). Quillen’s work began with the observation that the graded ring H_{G}(X) is commutative Noetherian. It enabled him to define an affine variety V_{G,X} as the maximal ideal spectrum MaxSpecH_{G}(X), and subsequently described it in terms of V_{E} = V_{E,•} = MaxSpecH_{E}, where E runs over the set of all elementary abelian p-subgroups of G. This is what we nowadays refer to as the Quillen stratification. Restricting to the special case of X = •, based on the fact that Ext_{kG}^{∗}(M, M) is finitely generated over H_{G}^{∗} ≅ Ext_{kG}^{∗}(k, k), Carlson [11] extended Quillen’s work by attaching to every finitely generated kG-module M a subvariety of V_{G} = V_{G,•}, denoted by V_{G}(M) = MaxSpecH_{G}/I_{G}(M), called the (cohomological) support variety of M, where I_{G}(M) is the kernel of the following map

ϕ_{M} = − ⊗_{k} M : H_{G}^{∗} ≅ Ext_{kG}^{∗}(k, k) → Ext_{kG}^{∗}(M, M).

Especially since ϕ_{k} is the identity, V_{G} = V_{G}(k). Following Carlson’s construction, Avrunin and Scott [5] quickly generalized the Quillen stratification from V_{G} to V_{G}(M). By showing that support varieties are well-behaved with respect to module operations, gradually Benson, Carlson and many others developed a remarkable theory, being a significant progress in group representations and cohomology. Since then, some other analogous support variety theories have been introduced for restricted Lie algebras [17], for finite group schemes [6, 18], for complete intersections [4] and for certain finite-dimensional algebras [21, 15, 25].

2010 Mathematics Subject Classification. Primary 16E40, 16P40, 20C05, 20J06, 55N25; Secondary 57S17.

Key words and phrases. Equivariant cohomology, transporter category algebras, support varieties, Quillen stratification.

The author 徐斐 was supported by a Beatriu de Pinós research fellowship from the government of Catalonia of Spain, as well as a Grant MTM2010-20692 “Analisis local en grupos y espacios topologicos” from the Ministry of Science and Innovation of Spain.

Quillen’s work on equivariant cohomology rings has not been fully exploited, partially because there existed no suitable modules which H_{G}^{∗}(X) (hence H_{G}(X)) acts on or maps to, as in Carlson’s theory. In this article we attempt to use category algebras to solve the problem: if X = BP comes from a finite G-poset, then we consider the category algebra k(G ∝ P) of the transporter category G ∝ P, based on which we will generalize Carlson’s theory. In fact, let \underline{k} be the trivial k (G ∝ P)- module (see Section 2.2). Then Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) is a graded commutative ring and there exists a natural ring isomorphism

Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(EG ×_{G} BP, k) = H_{G}^{∗}(BP, k).

We shall call the above ring the ordinary cohomology ring of k (G ∝ P) (instead of the equivariant cohomology ring), as opposed to the Hochschild cohomology ring of k (G ∝ P). Then we define V_{G∝P} = V_{G,BP} = MaxSpecH_{G}(BP). The virtue of having an entirely algebraic construction of the equivariant cohomology theory is that it allows us to consider

Ext_{k (G∝P)}^{∗}(𝔐, 𝔑)

for any finitely generated 𝔐, 𝔑 ∈ k (G ∝ P)-mod, and moreover construct a map

ϕ_{𝔐} = − ˆ⊗𝔐 : Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{k (G∝P)}^{∗}(𝔐, 𝔐).

Here ˆ⊗ is the tensor product in the closed symmetric monoidal category (k (G ∝ P)-mod, ˆ⊗, \underline{k}). Note that \underline{k} serves as the tensor identity. Since we have shown in [32] that Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over the ordinary cohomology ring, we may define the support variety of 𝔐 ∈ k (G ∝ P)-mod as V_{G∝P}(𝔐) = MaxSpecH_{G}(BP)/I_{G∝P}(𝔐), where I_{G∝P}(𝔐) is the kernel of ϕ_{𝔐}. Especially V_{G∝P} = V_{G∝P}(\underline{k}). When P = •, the is exactly Carlson’s construction because k(G ∝ •) ≅ kG, \underline{k} becomes the trivial kG-module k and ˆ⊗ reduces to ⊗_{k} under the circumstance.

As a surprising consequence of our investigations of transporter category algebras, we assert that Snashall-Solberg’s (Hochschild) support variety theory (for Gorenstein algebras) applies to every block of a finite transporter category algebra. Furthermore, our support variety theory is closely related with Snashall-Solberg’s as what happens in the case of group algebras and their blocks. A notable point is that the block algebras of a transporter category algebra are usually nonselfinjective and non-commutative, opposing to the cases of (selfinjective) Hopf algebras [11, 6, 18] and of commutative Gorenstein algebras [4] considered by others.

This paper is organized as follows. Section 2 recalls the definitions of a transporter category, a category algebra and the category cohomology. Various necessary constructions are recorded for the convenience of the reader. Here we show a transporter category algebra is Gorenstein and the ordinary cohomology ring of such an algebra is identified with an equivariant cohomology ring. Then in Section 3, we define the support variety for a module over a transporter category algebra. To motivate the reader, we describe the relevant works of Carlson, Linckelmann and Snashall-Solberg, before we develop some standard properties. Sections 4 and 5 contain various properties of support varieties, including the Quillen stratification for modules, as well as results related with sub-transporter categories and tensor products. Notably we propose a concept of support varieties of Mackey functors.

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.

The morphisms in a poset are customarily denoted by ≤. The dimension of a poset P, dim P, is defined to be the maximal integer n such that a chain of non-isomorphisms x_{0} < x_{1} < ⋯ < x_{n} exists in P.

Any G-set is usually regarded as a G-poset with trivial relations. One the other hand, since in a G-poset P, both Ob P and Mor P are naturally G-sets, we shall use terminologies for G-sets in our situation without further comments.

2.1. Transporter categories as Grothendieck constructions. We deem a group as a category with one object, usually denoted by •. The identity of a group G is always 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. It is equivalent to say that we have a group homomorphism G → Aut(P). The Grothendieck construction on F will be called a transporter category.

Definition 2.1.1. 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.

If (g, gx ≤ y) and (h, hy ≤ z) are two morphisms in G ∝ P, then their composite is easily seen to be (hg, (hg)x ≤ z) = (h, hy ≤ z) ∘ (g, gx ≤ y).

Remark 2.1.2. One can check directly that if Hom_{G∝P}(x, y) ≠ ∅ then both Aut_{G∝P}(x) and Aut_{G∝P}(y) act freely on Hom_{G∝P}(x, y). This simple observation is quite useful to us.

We note that for each x ∈ Ob (G ∝ P) = Ob P, Aut_{G∝P}(x) is exactly the isotropy group of x. For the sake of simplicity, we will often write G_{x} = Aut_{G∝P}(x), and [x] = Gx, the orbit of x. Note that [x] is a G-set, consisting of exactly the objects in G ∝ P that are isomorphic to x.

If P_{1} is a G_{1}-poset and P_{2} is a G_{2}-poset, then

(G_{1} × G_{2}) ∝ (P_{1} × P_{2}) ≅ (G_{1} ∝ P_{1}) × (G_{2} ∝ P_{2}).

We will utilize this in Section 5.2.

From the definition one can easily see that there is a natural embedding ι : P ↪ G ∝ P via (x ≤ y) ↦ (e, x ≤ y). On the other hand, the transporter category admits a natural functor π : G ∝ P → G, given by x ↦ • and (g, gx ≤ y) ↦ g. Thus we always have a sequence of functors

P↪^{ι}G ∝ P⟶^{π}G

such that π ∘ ι(P) is the trivial subgroup or subcategory of G. Topologically it is well known that B(G ∝ P) ≃ hocolim_{G}BP ≃ EG ×_{G} BP. Passing to classifying spaces, we obtain a fibration sequence

BP⟶^{Bι}EG ×_{G} BP⟶^{Bπ}BG.

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. This is the first instance explaining why a transporter category has anything to do with the equivariant cohomology theory.

Example 2.1.3. If G acts trivially on P, then G ∝ P = G×P. In this case for any x ∈ Ob(G×P), G_{x} = G.

Example 2.1.4. 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. Making Grothendieck constructions on transitive G-sets reveals the isotropy groups.

For an arbitrary G-poset P and x ∈ Ob P, we have a category equivalence G ∝ [x] ≃ G_{x}, see Remark 2.1.2. For instance, G acts on P = S_{p}, the poset of non-identity p-subgroup of G, by conjugation. Then G_{x} = N_{G}(x) for every x ∈ Ob(G ∝ S_{p}) = Ob S_{p}. The isotropy group of x, G_{x}, is frequently identified with the transporter category G_{x} × x ≅ G_{x} × •.

In the upcoming Section 2.2 we will see that a category equivalence D → C induces a Morita equivalence between their category algebras, kD ≃ kC, as well as a homotopy equivalence BD ≃ BC (see [28]). It means there is no essential difference between H and G ∝ (G/H) as far as we concern. Hence it makes sense if we deem transporter categories as generalized subgroups for a fixed finite group.

2.2. Category algebras and their representations. We recall some facts about category algebras. The reader is referred to [28, 30] for further details. Let C be a finite category and k a field. One can define the category algebra kC, which, as a vector space, has a basis the set of all morphisms in C. We then define a product on the base elements and extend it linearly to kC. The product α ∗ β of any two base elements α, β ∈ Mor C is defined to be αβ, if they are composable, or zero otherwise. It is a finite-dimensional associative algebra with identity 1 = ∑_{x∈Ob C} 1_{x}. The category algebra kC possesses a distinguished module \underline{k}, called the trivial module. It can be defined as \underline{k} = k Ob C. If α is a base element of kC and x ∈ Ob C, we ask α · x = y if α ∈ Hom_{C}(x, y). Otherwise we set α · x = 0.

When C is a group, kC is exactly the group algebra and \underline{k} = k. All modules we consider here are finitely generated left modules, unless otherwise specified. 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 some distinguished modules, namely \underline{k} and, in the special case of C = G ∝ P, κ_{M} which are obtained from kG-modules (to be defined shortly).

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 (see [28]), 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.

It is often helpful to utilize the underlying functor structure of a module. For instance, upon the preceding category equivalence we can alternatively define the trivial module \underline{k} as a constant functor taking k as its value at every object of C. Meanwhile since V ect_{k} is a symmetric monoidal category, V ect_{k}^{C} inherits this structure. It means 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). Let α ∈ Mor C be a base element of kC. Then α acts on 𝔐 ˆ⊗𝔑 via α⊗α. Obviously \underline{k} is the tensor identity with respect to ˆ⊗ and 𝔐 ˆ⊗𝔑 ≅ 𝔑 ˆ⊗𝔐. The category kC-mod has function objects, also called internal hom [30], in the sense that, for 𝔏, 𝔐, 𝔑 ∈ kC-mod,

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

We record the basic tools for comparing category algebras and their modules. When τ : D → C is a functor between two finite categories, there are adjoint functors for comparing their representations. The functor τ usually does not induce an algebra homomorphism from kD to kC. However it does induce an exact functor, called the restriction along τ,

Res_{D}^{C} : kC-mod → kD-mod.

If we regard a kC-module as a functor, then its restriction is the precomposition with τ. If we consider the functor π : G ∝ P → G, then any kG-module M restricts to a k(G ∝ P)- module, written as κ_{M} = Res_{G∝P}^{G} M, with only one exception \underline{k} = κ_{k} = Res_{G∝P}^{G} k. It is known that Hom(κ_{M}, κ_{N}) ≅ κ_{Hom_{k}(M,N)} for any two M, N ∈ kG-mod. In this paper, for the sake of simplicity, if D → C is a functor and 𝔐 is a kC-module, when it will not cause confusions, we sometimes neglect Res_{D}^{C} and regard 𝔐 (instead of Res_{D}^{C} 𝔐) as a kD-module.

The functor Res_{D}^{C} is equipped with two adjoints: the left and right Kan extensions along τ

LK_{D}^{C}, RK_{D}^{C} : kD-mod → kC-mod.

The definition of the left and right Kan extensions depend on the so-called over-categories and under-categories, respectively. Despite their seemingly abstract definitions, they are quite computable and thus play an important role in category representations and cohomology, see [29, 31, 32], as well as Sections 2.5, 3.4 and 4.1.

Note that our notations for the restriction and Kan extensions are slightly different from earlier articles. The reason is that in this place we feel it is necessary to emphasize the two categories involved in order to make the notations more indicative.

2.3. EI categories. When a category C is an EI category, that is, every endomorphism is an isomorphism, there exists a natural partial order on the set of isomorphism classes of objects in C [28]. Indeed the partial order is given by [x] ≤ [y] if and only if Hom_{C}(x, y) ≠ ∅. Groups, posets and transporter categories are all EI categories. The upshot is that this partial order allows us to give a filtration of any kC-module 𝔐. Let us call x ∈ Ob C an 𝔐-object if 𝔐(x) ≠ 0. Assume x is minimal among all 𝔐-object, then we can define a submodule 𝔐x̂ such that 𝔐x̂(y) = 𝔐(y) unless y ≅ x in C, in which case 𝔐x̂(y) = 0. Then we have a short exact sequence

0 → 𝔐x̂ → 𝔐 → 𝔐/𝔐x̂ → 0.

Denote by 𝔐_{x} = 𝔐/𝔐x̂. This is an example of the so-called atomic modules of kC. A kC-module 𝔐 is called atomic if as a functor 𝔐 takes zero values on all but one isomorphism classes of objects. From the above analysis we see that every 𝔐 admits a filtration with atomic modules as composition factors. Obviously from any module 𝔐 and an 𝔐-object x we may define an atomic module by brutal truncation (the restriction along the inclusion [x] → C). Abusing terminology, we always write such modules as 𝔐_{x}.

Assume C is finite EI. Then we can characterize projective and injective kC-modules. Recall from [28] that each indecomposable left (resp. right) projective kC-module, up to isomorphism, is a direct summand of kHom_{C}(x, −) = kC · 1_{x} (resp. kHom_{C}(−, x) = 1_{x} · kC) for some x ∈ Ob C. The (indecomposable) right (resp. left) injective modules can be obtained as k-duality.

Lemma 2.3.1. (1) If 𝔓 is a projective (or injective) k (G ∝ P)-module, then 𝔓_{x} is a projec-

tive k(G ∝ [x])-module, for all 𝔓-objects x.

(2) A k (G ∝ P)-module 𝔐 is of finite projective (equivalently, finite injective) dimension if and only if 𝔐_{x} is a projective k(G ∝ [x])-module, for all 𝔐-objects x. Under the circumstance, both proj.dim 𝔐 and inj.dim 𝔐 are bounded by dim P.

(3) The transporter category algebra k (G ∝ P) is Gorenstein, which means as either the left or the right regular module, it has finite injective dimension.

Proof. These are direct consequences of the fact that if Hom_{G∝P}(x, y) ≠ ∅, then kHom_{G∝P}(x, y) is both free kG_{x}- and kG_{y}-module, along with the characterizations of projectives and injectives. We only prove (2).

Let 𝔓_{∗} → 𝔐 → 0 be a finite projective resolution. Then its restriction to G ∝ [x] is a finite projective resolution of 𝔐_{x}. Since G ∝ [x] ≃ G_{x}, their category algebras are Morita equivalent and thus k(G ∝ [x]) is selfinjective. Then the finite projective resolution (𝔓_{x})_{∗} → 𝔐_{x} → 0 splits and 𝔐_{x} is a projective (and injective) k(G ∝ [x])-module. On the other hand, assume 𝔐 satisfies the property that 𝔐_{x} is projective (or zero) for every object x. Take the projective cover 𝔓 → 𝔐. We immediately know that 𝔓_{x} → 𝔐_{x} splits as a k(G ∝ [x])-map. Let [y_{1}], ⋯, [y_{n}] be the isomorphism classes of all minimal objects among 𝔐-objects. Then ⨁_{i=1} n𝔓_{y_{i}} is the projective cover of 𝔐_{y} which implies ⨁_{i=1}^{n} 𝔓_{y_{i}} ≅ 𝔐_{y}. Hence if we examine the kernel 𝔐′ of 𝔓 → 𝔐, it has the property that 𝔐′_{y} is a projective k(G ∝ [y])-module for all objects y such that [y] > [y_{i}] for some 1 ≤ i ≤ n, and zero else. Repeat the same process for 𝔐′, eventually we will obtain the finite projective resolution of 𝔐. It implies proj.dim 𝔐 ≤ dim P.

As to the injective dimension, we consider the right k (G ∝ P)-module 𝔐^{∧}. It satisfies 𝔐^{∧}(x) = 𝔐(x)^{∧}. Similar to the left module situation, it has a finite projective resolution. When we dualize it, it becomes an injective resolution of 𝔐. □

Since G ∝ [x] ≃ G_{x} (see Example 2.1.4.), in the first two statements, we may replace 𝔓_{x}, 𝔐_{x}, k(G ∝ [x]) by 𝔓(x), 𝔐(x), kG_{x}, respectively.

It is helpful to give the following characterization of a transporter category algebra as a skew group algebra. Recall from [3, Chapt. III, Section 4] that if a k-algebra A is a G-algebra, then we may define the skew group algebra A[G] to be the k-vector space A ⊗_{k} kG equipped with a multiplication rule determined by

(a_{1} ⊗ g_{1}) ∗ (a_{2} ⊗ g_{2}) = a_{1}(g_{1} · a_{2}) ⊗ g_{1}g_{2},

where a_{1}, a_{2} ∈ A and g_{1}, g_{2} ∈ G. Here g_{1} · a_{2} means the image of a_{2} under the action of g_{1}. For instance we immediately have kG ≅ (k•)[G]. The reader is referred to [2] and [3] for further details and known results about skew group algebras. Proposition 2.2 of [2] says that A[G] is Gorenstein if and only if A is. Since it is easy to verify that kP is Gorenstein, with the following result we have another proof of k (G ∝ P) being Gorenstein.

Lemma 2.3.2. There is an algebra isomorphism k (G ∝ P) ≅ kP[G].

Proof. The isomorphism is given by (gx ≤ y, g) ↦ (gx ≤ y) ⊗ g with inverse (x ≤ y) ⊗ h ↦ (h(h^{−1}x) ≤ y, h) for x, y ∈ Ob P and g, h ∈ G. □

The modular representation theory of k (G ∝ P) will be studied in another place.

2.4. Category cohomology and spectrum. For any two kC-modules it makes sense to consider Ext_{kC}^{∗}(𝔐, 𝔑) = ⨁_{i≥0} Ext_{kC}^{i}(𝔐, 𝔑). Furthermore if 𝔐′ and 𝔑′ are two other modules, the tensor product ˆ⊗ induces a cup product as follows

∪ : Ext_{kC}^{i}(𝔐, 𝔑) ⊗ Ext_{kC}^{j}(𝔐′, 𝔑′) → Ext_{kC}^{i+j}(𝔐 ˆ⊗𝔐′, 𝔑 ˆ⊗𝔑′).

In particular Ext_{kC}^{∗}(\underline{k}, \underline{k}) is a graded commutative ring and we have a natural isomorphism Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k) [30]. This ring is called the ordinary cohomology ring of kC and it acts on Ext_{kC}^{∗}(𝔐, 𝔑) via the cup product. For any kC-module 𝔐, the Yoneda splice provides a ring structure on Ext_{kC}^{∗}(𝔐, 𝔐). When 𝔐 = \underline{k}, the cup product and Yoneda splice give the same ring structure on Ext_{kC}^{∗}(\underline{k}, \underline{k}). There exists a ring homomorphism whose image lies in the center of the graded ring Ext_{kC}^{∗}(𝔐, 𝔐)

− ˆ⊗𝔐 : Ext_{kC}^{∗}(\underline{k}, \underline{k}) → Ext_{kC}^{∗}(𝔐, 𝔐).

Moreover given a short exact sequence 0 → 𝔐_{1} → 𝔐_{2} → 𝔐_{3} → 0 the resulting connecting homomorphism is a morphism of Ext_{kC}^{∗}(\underline{k}, \underline{k})-modules.

In summary, finite category cohomology behaves very much like the special case of finite group cohomology, except the finite generation property. The ordinary cohomology ring of a category algebra is usually far from finitely generated, but it is so when C = G ∝ P is finite as it is isomorphic to the equivariant cohomology ring H_{G}^{∗}(BP, k), see Section 2.5 and [23, 31].

The functor Res_{D}^{C} introduced earlier leads to a restriction on cohomology

res_{D}^{C} : Ext_{kC}^{∗}(\underline{k}, \underline{k}) → Ext_{kD}^{∗}(\underline{k}, \underline{k}).

It coincides with the restriction H^{∗}(BC, k) → H^{∗}(BD, k), induced by the continuous map Bτ : BD → BC between two classifying spaces, see [30].

From now on, we assume k is algebraically closed. Throughout this paper let us denote by

H(C) = Ext_{kC}(\underline{k}, \underline{k}) = \{ \begin{matrix}Ext_{kC}^{∗}(\underline{k}, \underline{k}), & if the characteristic of k = 2; \\ Ext_{kC}^{2∗}(\underline{k}, \underline{k}) , & if the characteristic of k > 2. \end{matrix}

This graded ring is commutative. If Ext_{kC}^{∗}(\underline{k}, \underline{k}) is Noetherian, then we can consider the maximal ideal spectrum, an affine variety,

V_{C} = MaxSpecH(C).

Under the circumstance we will call V_{C} the variety of C.

Assume both Ext_{kC}^{∗}(\underline{k}, \underline{k}) and Ext_{kD}^{∗}(\underline{k}, \underline{k}) are Noetherian and there exists a functor τ : D → C. Since the preimage of a maximal ideal is still a maximal ideal, there exists a map between two varieties

ι_{D}^{C} := (res_{D}^{C})^{−1} : V_{D} → V_{C}.

These varieties and their subvarieties are our main subjects and thus it is helpful if we can handle the restriction. In various interesting cases the map is well understood. As an application of Lemma 2.3.1 we obtain the following result.

Lemma 2.4.1. Consider a transporter category G ∝ P and an object x ∈ Ob (G ∝ P). Then the inclusions G_{x} × x → G ∝ [x] → G ∝ P induce two restrictions which fit into the following commutative diagram

[figure: Ext_{k (G∝P)}^{∗}(𝔐, 𝔑); ^{res{G∝P \atop G∝[x]}}; Ext_{kG∝[x]}^{∗}(𝔐_{x}, 𝔑_{x}); ≅; Ext_{k (G∝P)}^{∗}(𝔐, 𝔑); _{res{G∝P \atop Gx×x}}; Ext_{kG_{x}}^{∗}(𝔐(x), 𝔑(x))] ,

for any two modules 𝔐, 𝔑 ∈ k (G ∝ P)-mod.

Proof. Let 𝔓 → 𝔐 → 0 be a projective resolution. Then 𝔓_{x} → 𝔐_{x} → 0 remains a projective resolution of the kG ∝ [x]-module 𝔐_{x}, by Lemma 2.3.1 (1). Hence the functor

res_{G∝[x]}^{G∝P} : k (G ∝ P) -mod → k(G ∝ [x])-mod

induces a restriction

res_{G∝[x]}^{G∝P} : Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) → Ext_{k(G∝[x])}^{∗}(𝔐_{x}, 𝔑_{x}).

Similarly we have a map res_{G_{x}×x}^{G∝P}. The commutative diagram follows directly from the natural isomorphism Ext_{k(G∝[x])}^{∗}(𝔐_{x}, 𝔑_{x}) ≅ Ext_{kG_{x}}^{∗}(𝔐(x), 𝔑(x)). □

Because the two restrictions res_{G_{x}×x}^{G∝P} : Ext_{k (G∝P)}^{∗}(𝔐, 𝔐) → Ext_{kG_{x}}^{∗}(𝔐(x), 𝔐(x)) and res_{G∝[x]}^{G∝P} : Ext_{k (G∝P)}^{∗}(𝔐, 𝔐) → Ext_{k(G∝[x])}^{∗}(𝔐_{x}, 𝔐_{x}) are ring homomorphisms with respect to the Yoneda splice, there are two maps

ι_{G_{x}×x}^{G∝P} : V_{G_{x}×x} → V_{G∝P},

induced by res_{x} for 𝔐 = 𝔑 = \underline{k}, and similarly

ι_{G∝[x]}^{G∝P} : V_{G∝[x]} → V_{G∝P}.

Note that V_{G_{x}×x} = V_{G∝[x]}.

2.5. Bar resolution and equivariant cohomology. One concept that we will refer to is the bar resolution 𝔅_{∗}^{C}, a combinatorially constructed projective resolution, of \underline{k} ∈ kC-mod. Given a functor τ : D → C, one can define ℂ_{∗}(τ/−, k), a complex of projective kC-modules, such that, for each x ∈ Ob C, τ/x is a finite category (the category over x or just an overcategory) and ℂ_{∗}(τ/x, k) is the (normalized) chain complex resulting from the simplicial k-vector spaces coming from the nerve of τ/x [29]. Since overcategories are used in the proof of a couple of results in the next section, we recall its definition. The objects of τ/x are pairs (d, α) where d ∈ Ob D and α : τ(d) → x is a morphism in C. A morphism from (d, α) to (d′, α′) is a morphism f : d → d′ in D, satisfying α = τ(f)α′. For each finite category D we define 𝔅_{∗}^{D} = ℂ_{∗}(Id_{D} /−, k). It is known that

LK_{D}^{C}𝔅_{∗}^{D} ≅ ℂ_{∗}(τ/−, k).

As an example, for a group G the only overcategory Id_{G} /• is the Cayley graph and thus 𝔅_{∗}^{G} is the bar resolution of k in group cohomology. For the convenience of the reader we recall from [31] how we obtain equivariant cohomology from category cohomology. Since we can explicitly calculate the unique overcategory as π/• ≅ Id_{G} / • ×P, it follows that

LK_{G∝P}^{G}𝔅_{∗}^{G∝P} ≅ ℂ_{∗}(π/•, k) ≃ 𝔅_{∗}^{G} ⊗ ℂ_{∗}(P, k)

Thus we have the following chain isomorphisms and homotopy, for any M ∈ kG-mod,

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

Note that ℂ_{∗}(P, k) is a finite complex and consists in each dimension of a permutation kG-module. From [10, VII.7] we see immediately Ext_{k (G∝P)}^{∗}(\underline{k}, κ_{M}) ≅ H_{G}^{∗}(BP, M) for any M ∈ kG-mod.

In [31] we also established the following isomorphism

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

Since Hom(κ_{M}, κ_{N}) ≅ κ_{Hom_{k}(M,N)}, the Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})-action on

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

can be obtained from the canonical map \underline{k} → Hom(κ_{M}, κ_{M}) ≅ κ_{End_{k}(M)}, induced by 1_{k} ↦ Id_{M}. This is analogous to the group case.

3. Support varieties for modules

Let k be an algebraically closed field. Suppose C = G ∝ P is a transporter category. We have learned that Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) is Noetherian, over which Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) is a finitely generated module. The action factors through the natural ring homomorphism

− ˆ⊗𝔐 : Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{k (G∝P)}^{∗}(𝔐, 𝔐),

and

− ˆ⊗𝔑 : Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{k (G∝P)}^{∗}(𝔑, 𝔑).

Based on these, we are about to develop a support variety theory. Since a transporter category G ∝ P is intimately related with G, we will see our theory is a generalization of Carlson’s theory. Standard references for Carlson’s theory include [7, Chapter 5] and [16, Chapters 8, 9, 10].

3.1. Basic definitions. For convenience, we shall assume G ∝ P is connected, unless otherwise specified. It is equivalent to saying that \underline{k} ∈ k (G ∝ P)-mod is indecomposable or that

Ext_{k (G∝P)}^{0}(\underline{k}, \underline{k}) = H^{0}(B (G ∝ P), k) = k.

It does not mean that P is connected, see Example 2.1.4.

Definition 3.1.1. Given a transporter category G ∝ P and modules 𝔐, 𝔑 ∈ k (G ∝ P)-mod, we write I_{G∝P}(𝔐) for the kernel of the map

− ˆ⊗𝔐 : Ext_{k (G∝P)}(\underline{k}, \underline{k}) → Ext_{k (G∝P)}(𝔐, 𝔐),

and V_{G∝P}(𝔐), the support variety of 𝔐, for the subvariety MaxSpec(H(G ∝ P)/I_{G∝P}(𝔐)) of V_{G∝P} = V_{G∝P}(\underline{k}).

Since both Ext_{k (G∝P)}^{∗}(𝔐, 𝔐) and Ext_{k (G∝P)}^{∗}(𝔑, 𝔑) act on Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) via Yoneda splice, we further define I_{G∝P}(𝔐, 𝔑) as the annihilator of Ext_{k (G∝P)}(\underline{k}, \underline{k}) on Ext_{k (G∝P)}^{∗}(𝔐, 𝔑). Then we set V_{G∝P}(𝔐, 𝔑) = MaxSpec(H(G ∝ P)/I_{G∝P}(𝔐, 𝔑)).

We say a subvariety of V_{G∝P} is trivial if it is 𝔪 = Ext_{k (G∝P)}^{+}(\underline{k}, \underline{k}), the positive part of Ext_{k (G∝P)}(\underline{k}, \underline{k}).

Since I_{G∝P}(𝔐, 𝔐) = I_{G∝P}(𝔐), we get that V_{G∝P}(𝔐, 𝔐) = V_{G∝P}(𝔐), and that

I_{G∝P}(𝔐, 𝔑) ⊂ I_{G∝P}(𝔐) + I_{G∝P}(𝔑).

The latter implies

V_{G∝P}(𝔐, 𝔑) ⊂ V_{G∝P}(𝔐) ∩ V_{G∝P}(𝔑).

Let P be a G-poset and Q an H-poset. Suppose there exists a group homomorphism ϕ : H → G as well as a functor θ : Q → P such that ϕ(h)θ = θ ∘ h for all h ∈ H. For convenience we record such a map as (ϕ, θ) : (H, Q) → (G, P). They induce a functor

Θ : H ∝ Q → G ∝ P,

which in turn gives rise to a restriction map

res_{H∝Q}^{G∝P} : Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) → Ext_{k(H∝Q)}^{∗}(\underline{k}, \underline{k})

and a map between varieties

ι_{H∝Q}^{G∝P} : V_{H∝Q} → V_{G∝P}.

For instance, Lemma 2.4.1 dealt with (i, i) : (G_{x}, x) → (G, P) and (Id_{G}, i) : (G, [x]) → (G, P), where i stands for the inclusion.

Example 3.1.2. Suppose G acts trivially on P. Then (Id_{G}, pt) : (G, P) → (G, •) induce π : G ∝ P → G and

res_{G×P}^{G} : Ext_{kG}^{∗}(k, k) → Ext_{k(G×P)}^{∗}(\underline{k}, \underline{k})

Since Ext_{k(G×P)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kG}^{∗}(k, k) ⊗ Ext_{kP}^{∗}(\underline{k}, \underline{k}) by the Künneth formula, and Ext_{kP}^{∗}(\underline{k}, \underline{k}) = H^{∗}(BP, k) is finite-dimensional, V_{G×P} = ∏_{n} V_{G}, where n is the number of connected components of P. The restriction induces a natural map V_{G×P} → V_{G}. With the assumption that G ∝ P = G × P is connected, we actually have n = 1 because G × P is connected if and only if P is.

At this point, it seems to be a good idea to compare our theory with those of Carlson, Linckelmann and Snashall-Solberg. By putting our approach into the right context, we can better understand the ideas and see what properties we may expect. Afterwards, we will present various results concerning support varieties.

3.2. Carlson’s theory. When P = • our theory is just the theory of Carlson. However, combining recent works in group and category cohomology, Carlson’s theory can be recovered in a more subtle way. To be explicit, if κ_{M} ∈ k (G ∝ P)-mod for some M ∈ kG-mod, then we have a commutative diagram [32]

\xymatrix{Ext_{kG}^{∗}(k, k) \ar[r]^{−⊗M} \ar[d]_{_{res{G \atop G∝P}}} & Ext_{kG}^{∗}(M, M) \ar[d]^{_{res{G \atop G∝P}}} \\ Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) \ar[r]_{− ˆ⊗κ_{M}} & Ext_{k (G∝P)}^{∗}(κ_{M}, κ_{M})}

One can quickly deduce that the restriction map induces

Ext_{kG}^{∗}(k, k)/I_{G}(M) → Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})/I_{G∝P}(κ_{M}),

and hence a finite (usually not surjective) map

V_{G∝P}(κ_{M}) → V_{G}(M).

If the Euler characteristic χ(P, k) is invertible in k, then by using the Becker-Gottlieb transfer map [32], both vertical maps are injective. Furthermore, if we let P = S_{p}, the poset of non-identity p-subgroups of G, the left res_{G∝S_{p}}^{G} becomes an algebra isomorphism (see [10, Chap. X, Section 7] and [14, Part I, Sections 7 and 8] for instance). Hence we get

I_{G}(M) ≅ I_{G∝S_{p}}(κ_{M}) and V_{G}(M) ≅ V_{G∝S_{p}}(κ_{M}).

It means that various properties of V_{G}(M) can be rewritten for V_{G∝S_{p}}(κ_{M}). As an example we have a tensor product formula

V_{G∝S_{p}}(κ_{M} ˆ⊗κ_{N}) = V_{G∝S_{p}}(κ_{M⊗N}) = V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N) = V_{G∝S_{p}}(κ_{M}) ∩ V_{G∝P}(κ_{N}).

Here the third equality comes from [7, Theorem 5.7.1]. One can similarly deduce other properties for V_{G∝S_{p}}(κ_{M}) but we shall leave it to the reader as they are just reformulations of known results for V_{G}(M). Our interests really lie in V_{G∝S_{p}}(𝔐), or more generally V_{G∝P}(𝔐), for modules 𝔐 ≠ κ_{M} for any M ∈ kG-mod.

In the above arguments, S_{p} may be replaced by various G-subposets which are G-homotopy equivalent to it, see [7, Section 6.6]. A typical example is E_{p}, the G-poset of all elementary abelian p-subgroups of G. However we want to emphasize that most of our results are established without specifying a poset, see for instance Sections 4 and 5.

3.3. Varieties in blocks. A group algebra can be written as a direct product of (indecomposable) block algebras ∏_{i} b_{i}. (Here for convenience we denote by b, instead of kGb, a block algebra.) Each indecomposable kG-module belongs to exactly one of these blocks in the sense that all but one block act as zero on it. The block that k belongs to is called the principal block, denoted by b_{0}.

In [20], Linckelmann introduced to each block algebra b a Noetherian graded commutative ring H^{∗}(b), called the block cohomology ring. Then he showed that there exists a natural injective homomorphism

H^{∗}(b) → HH^{∗}(b),

and thus H^{∗}(b) acts on Ext_{b}^{∗}(M, M) via HH^{∗}(b) = Ext_{b^{e}}^{∗}(b, b) (this action is explained in Section 3.4), if M ∈ b-mod. Particularly if b_{0} is the principal block of a group algebra kG, then H^{∗}(b_{0}) is isomorphic to H^{∗}(G, k) and the above injection coincides with the composite of two canonical maps

H^{∗}(b_{0}) ≅ H^{∗}(G, k) → HH^{∗}(kG) → HH^{∗}(b_{0}).

Based on these, he was able to define support varieties for modules of a block algebra as above in a natural way [21], being a refinement of Carlson’s theory. Most significantly Linckelmann’s work brought Hochschild cohomology into the theory of support varieties, which was taken up by Snashall and Solberg to develop a new support variety theory using Hochschild cohomology rings, see Section 3.4. Recently [22] Linckelmann had demonstrated that, for a block algebra b, the inclusion H^{∗}(b) → HH^{∗}(b) induces an isomorphism upon quotient out nilpotent elements. It implies that the two support variety theories, of Lincklemann based on the block cohomology ring and of Snashall-Solberg defined over the Hochschild cohomology ring of a block of a finite group, are identical. See [8] as well.

3.4. Snashall-Solberg’s theory. Snashall and Solberg [25] developed a support variety theory for certain finite-dimensional algebras using Hochschild cohomology rings. Let A be a finitedimensional algebra, and M, N two finitely generated A-modules. Then there exists a natural action of the Hochschild cohomology ring on Ext groups via the following homomorphism

ϕ_{M} = − ⊗_{A} M : Ext_{A^{e}}^{∗}(A, A) → Ext_{A}^{∗}(M, M).

Based on Yoneda splice, one can introduce an action on Ext_{A}^{∗}(M, N) for any two A-modules.

For technical reasons, now let A be an indecomposable algebra. Consequently Z(A) becomes a commutative local algebra. Let HH(A) = Ext_{A^{e}}(A, A) be defined in the same fashion as H(C) in Section 2.4. Assume (Fg.1) there is a graded Noetherian subalgebra H ⊂ Ext_{A^{e}}(A, A) with H^{0} = Ext_{A^{e}}^{0}(A, A) = Z(A);

and (Fg.2) for any M, N ∈ A-mod, Ext_{A}^{∗}(M, N) is finitely generated over H.

Under the above assumptions, Snashall-Solberg considered the maximal ideal spectrum V_{H} = MaxSpecH. Since H acts on Ext_{A}^{∗}(M, N) for any two A-modules, assuming I_{H}(M, N) is the annihilator they defined a subvariety by

V_{H}(M, N) = MaxSpec(H/I_{H}(M, N)).

Write I_{H}(M) = I_{H}(M, M). Then the (Hochschild) support variety of M ∈ A-mod is given by

V_{H}(M) = MaxSpec(H/I_{H}(M)).

They showed that V_{H}(M) = V_{H}(M, A/ Rad A) = V_{H}(A/ Rad A, M) and V_{H}(A/ Rad A) = V_{H}. Various satisfactory properties were obtained in [25, 15], see Theorem 3.4.4 below for a summary. For future reference, we record the following definition.

Definition 3.4.1. A subvariety of V_{H} is called trivial if it is ⟨Rad H^{0}, H^{+}⟩, where H^{+} consists of all the positive degree elements in H.

Unfortunately the above conditions (Fg.1) and (Fg.2) are not met by all finite-dimensional algebras, see [29]. Indeed they put restrictions on the algebras that one may consider. For example, two necessary conditions are that the algebra A has to be Gorenstein [15], and that Ext_{A^{e}}^{∗}(A, A) itself must be Noetherian.

Although many algebras do not satisfy (Fg.1) and (Fg.2), we can show that Snashall-Solberg theory works for block algebras of a transporter category algebra k (G ∝ P). From [31] we know that Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) is a Noetherian graded commutative ring such that Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over it, for any pair of 𝔐, 𝔑 ∈ k (G ∝ P)-mod. It was also showed there that Ext_{k (G∝P)^{e}}^{∗}(k (G ∝ P), k (G ∝ P)) is Noetherian. We shall prove that Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})- and Ext_{k (G∝P)^{e}}^{∗}(k (G ∝ P), k (G ∝ P))-actions on Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) are compatible and hence it implies that Ext_{k (G∝P)}^{∗}(𝔐, 𝔑) is finitely generated over the Hochschild cohomology ring as well.

Theorem 3.4.2. Let 𝔐 ∈ kC-mod. We have a commutative diagram

\xymatrix{Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{≅} \ar[d] & Ext_{kC}^{∗}(\underline{k}, \underline{k}) \ar[d]^{− ˆ⊗𝔐} \\ Ext_{kC^{e}}^{∗}(kC, kC) \ar[r]_{−⊗_{kC}𝔐} & Ext_{kC}^{∗}(𝔐, 𝔐)}

with the left vertical map an injective algebra homomorphism.

This actually generalizes [29, Theorem A]. Let us first recall some other necessary constructions from [29]. For any category C there is a category of factorizations in C, written as F (C). The objects are the morphisms in C. When a morphism α ∈ Mor C is regarded as an object in F (C), we will denote it by [α] to distinguish their roles. If [α], [β] are two objects in F (C), then a morphism [α] → [β] is a pair (μ, γ), μ, γ ∈ Mor C, such that β = μαγ (that is, α is a factor of β). When C is a group, F (C) plays the role of the diagonal subgroup △G ⊂ G × G. Indeed, there is a category equivalence △G ≃ F (G).

Given a morphism α in C, we denote by t(α) and s(α) the target and source of α. They induce two functors t : F (C) → C and ∇ = (t, s) : F (C) → C^{e} = C × C^{op}, fitting into the following commutative diagram

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

where p is the projection. By definition t and s send [α] to the target and source of α, respectively. In [29], we investigated the left Kan extensions LK_{F (C)}^{C} : kF (C)-mod → kC-mod and LK_{F (C)}^{C^{e}} : kF (C)-mod → kC^{e}-mod, proving LK_{F (C)}^{C}\underline{k} ≅ \underline{k} and LK_{F (C)}^{C^{e}}\underline{k} ≅ kC. Furthermore LK_{F (C)}^{C} induces an isomorphism Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) → Ext_{kC}^{∗}(\underline{k}, \underline{k}), while LK_{F (C)}^{C^{e}} induces an injective algebra homomorphism Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) → Ext_{kC^{e}}^{∗}(kC, kC). Especially F (C) is connected if and only if C is. At last LK_{C^{e}}^{C} : kC^{e}-mod → kC-mod is explicitly expressed as LK_{C^{e}}^{C} ≅ − ⊗_{kC} \underline{k}, where \underline{k} is the trivial left kC-module.

When 𝔐 = \underline{k}, the lower horizontal map becomes the split surjection in [29, Theorem A].

Proof. Consider the bar resolution 𝔅_{∗}^{F (C)} → \underline{k} → 0, and a map f : 𝔅_{n}^{F (C)} = ℂ_{n}(Id_{F (C)} /−, k) → \underline{k} representing an element ξ ∈ Ext_{kF (C)}^{n}(\underline{k}, \underline{k}). We need to prove LK_{F (C)}^{C^{e}}ξ ⊗_{kC} 𝔐 = LK_{F (C)}^{C}ξ ˆ⊗𝔐 ∈ Ext_{kC}^{n}(𝔐, 𝔐). We do it by explicit calculations.

Firstly, LK_{F (C)}^{C^{e}}f : LK_{F (C)}^{C^{e}}𝔅_{n}^{F (C)} = ℂ_{n}(∇/−, k) → LK_{F (C)}^{C^{e}}\underline{k} = kC is given on each (x, y) ∈ Ob C^{e} as

(LK_{F (C)}^{C^{e}}f)_{(x,y)} : ℂ_{n}(∇/(x, y), k) → kC(x, y) = kHom_{C}(y, x)

by

([α_{∗}], (β_{∗}, γ_{∗})) ↦ f_{[α_{∗}]}β_{n}α_{n}γ_{n}

Here we denote by ([α_{∗}], (β_{∗}, γ_{∗})) = ([α_{0}], (β_{0}, γ_{0})) → ⋯ → ([α_{n}], (β_{n}, γ_{n})) a base element of ℂ_{n}(∇/(x, y), k), where (β_{i}, γ_{i}) : ∇([α_{i}]) → (x, y) is a morphism in C^{e}. From ([α_{∗}], (β_{∗}, γ_{∗})) we can extract a base element of ℂ_{n}(Id_{F (C)} /[β_{n}α_{n}γ_{n}], k), written as [α_{∗}] = [α_{0}] → ⋯ → [α_{n}], so our definition makes sense. Note that β_{0}α_{0}γ_{0} = ⋯ = β_{n}α_{n}γ_{n}.

Secondly, in a similar fashion, LK_{F (C)}^{C}f : LK_{F (C)}^{C}𝔅_{n}^{F (C)} = ℂ_{n}(t/−, k) → LK_{F (C)}^{C}\underline{k} = \underline{k} is given on each x ∈ Ob C as

(LK_{F (C)}^{C}f)_{x} : ℂ_{n}(t/x, k) → \underline{k}(x) = k

by

([α_{∗}], μ_{∗}) ↦ f_{[α_{∗}]}.

Here we denote by ([α_{∗}], μ_{∗}) = ([α_{0}], μ_{0}) → ⋯ → ([α_{n}], μ_{n}) a base element of ℂ_{n}(t/x, k) and [α_{∗}] the encoded base element of ℂ_{n}(Id_{F (C)} /[μ_{n}α_{n}], k).

Thirdly, LK_{F (C)}^{C^{e}}ξ ⊗_{kC} 𝔐 is represented by LK_{F (C)}^{C^{e}}f ⊗_{kC} Id_{𝔐}, while LK_{F (C)}^{C}ξ ˆ⊗𝔐 is represented by some (LK_{F (C)}^{C}f ˆ⊗ Id_{𝔐})∘Φ_{n} provided that P_{n} → 𝔐 → 0 is a projective resolution and Φ_{∗} : P_{n} →

LK_{F (C)}^{C}𝔅_{∗}^{F (C)} ˆ⊗𝔐 is a lifting of the identity map of 𝔐. Let us take P_{∗} = LK_{F (C)}^{C^{e}}𝔅_{∗}^{F (C)} ⊗_{kC} 𝔐 and construct Φ_{∗}, which has to be unique up to homotopy. To this end, for any x ∈ Ob C and n ≥ 0 we define

Φ_{n}^{x} : 1_{x} · LK_{F (C)}^{C^{e}}𝔅_{n}^{F (C)} ⊗_{kC} 𝔐 → 1_{x} · LK_{F (C)}^{C}𝔅_{∗}^{F (C)} ⊗ 𝔐(x)

by

([α_{∗}], (β_{∗}, γ_{∗})) ⊗ (m_{w}) ↦ ([α_{∗}], β_{∗}) ⊗ (β_{n}α_{n}γ_{n}) · m_{y}.

Here we write each element in 𝔐 = ⨁_{w∈Ob C} 𝔐(w) as ∑_{w} m_{w}, in which m_{w} ∈ 𝔐(w). Directly from the definition one can verify LK_{F (C)}^{C^{e}}f ⊗_{kC} Id_{𝔐} = (LK_{F (C)}^{C}f ˆ⊗ Id_{𝔐})∘Φ. It means LK_{F (C)}^{C^{e}}ξ⊗_{kC} 𝔐 = LK_{F (C)}^{C}ξ ˆ⊗𝔐 ∈ Ext_{kC}^{n}(𝔐, 𝔐). □

To consider Snashall-Solberg’s theory, we want the algebra in question to be indecomposable. Suppose k (G ∝ P) = ∏_{i} b_{i} is a decomposition into (indecomposable) block algebras, see for instance [1, Section 13]. Since k (G ∝ P) is Gorenstein, so are its blocks. Let b be a block of k (G ∝ P). The above theorem implies that Ext_{b}^{∗}(𝔐, 𝔑) is a finitely generated module over the Noetherian ring Ext_{b^{e}}^{∗}(b, b), if 𝔐, 𝔑 ∈ b-mod. It means Snashall-Solberg’s theory works perfectly for blocks of finite transporter category algebras.

Here we record some standard properties from Snashall-Solberg’s theory. For convenience, we write

V_{b}(𝔐, 𝔑) = V_{HH(b)}(𝔐, 𝔑)

if b is a block of k (G ∝ P) and 𝔐, 𝔑 ∈ b-mod. To be consistent, write V_{b}(𝔐) = V_{b}(𝔐, 𝔐). Some terminologies are recalled first.

Definition 3.4.3. Let A be a finite-dimensional algebra and P_{∗} → M → 0 the minimal projective resolution of M. Then the complexity of M is

c_{A}(M) = min{s ∈ ℕ | r ∈ ℝ such that dim_{k} P_{n} ≤ rn^{s−1}, for n ≫ 0}

Let (−)^{∧} = Hom_{k}(−, k) : kC-mod → kC^{op}-mod be the k-dual functor. Recall that BC ≃ BC^{op} and thus Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC^{op}}^{∗}(\underline{k}, \underline{k}). It implies that Ext_{kC}^{∗}(\underline{k}, \underline{k}) also acts on Ext_{kC^{op}}^{∗}(𝔑^{∧}, 𝔐^{∧}), for any 𝔐, 𝔑 ∈ k (G ∝ P)-mod. Some of these constructions pass to every block algebra of kC.

The following statements are taken from Snashall-Solberg [25] and Erdmann-Holloway-Snashall-Solberg-Taillefer [15]. Note that a block of a transporter category is Gorenstein, but in general neither selfinjective nor symmetric.

Theorem 3.4.4. Let G be a finite group, P a finite G-poset, and k an algebraically closed field of characteristic p dividing the order of G. Suppose b is a block of k (G ∝ P) and 𝔐, 𝔑 are two finitely generated modules of b. Then

(1) V_{b}(𝔐) = ⋃_{𝔖} V_{b}(𝔐, 𝔖) = ⋃_{𝔖} V_{b}(𝔖, 𝔐), where 𝔖 runs over the set of all simple b-modules.

(2) If 0 → 𝔐_{1} → 𝔐_{2} → 𝔐_{3} → 0 is an exact sequence, then V_{b}(𝔐_{i}) ⊂ V_{b}(𝔐_{j}) ∪ V_{b}(𝔐_{l}) for {i, j, l} = {1, 2, 3}.

(3) V_{b}(𝔐_{1} ⊕ 𝔐_{2}) = V_{b}(𝔐_{1}) ∪ V_{b}(𝔐_{2}).

(4) V_{b}(𝔐) = V_{b}(Ω^{n}(𝔐)) for any integer n such that Ω^{n}(𝔐) ≠ 0.

(5) V_{b}(𝔐, 𝔑) = V_{b^{op}}(𝔑^{∧}, 𝔐^{∧}). Particularly V_{b}(𝔐) = V_{b^{op}}(𝔐^{∧}).

(6) dim V_{b}(𝔐) = c_{b}(𝔐).

(7) V_{b}(𝔐) is trivial if and only if 𝔐 has finite projective dimension if and only if 𝔐 has finite injective dimension.

(8) Let 𝔞 be a homogeneous ideal in HH(b) = Ext_{b^{e}}(b, b). Then there exists a module 𝔐_{𝔞} ∈ bmod such that V_{b}(𝔐_{𝔞}) = V (𝔞).

(9) If V_{b}(𝔐) ∩ V_{b}(𝔑) is trivial, then Ext_{b}^{i}(𝔐, 𝔑) = 0 for i > inj.dimb.

(10) If V_{b}(𝔐) = V_{1} ∪ V_{2} for some homogeneous non-trivial varieties V_{1} and V_{2} with V_{1} ∩ V_{2} trivial, then 𝔐 = 𝔐_{1} ⊕ 𝔐_{2} with V_{b}(𝔐_{1}) = V_{1} and V_{b}(𝔐_{2}) = V_{2}.

Proof. The first five properties follow directly from the definition of support variety for a module of Snashall-Solberg, and can be found in [25].

The rest come from [15, Theorem 2.5, Theorem 4.4, Proposition 7.2, Theorem 7.3]. □

4. Standard properties of V_{G∝P}(𝔐)

Snashall-Solberg’s theory on a block algebra of a transporter category algebra is quite satisfactory in many ways. However, Hochschild cohomology rings do not behave well comparing with ordinary cohomology rings. For example, since an algebra homomorphism does not necessarily induce a homomorphism between their Hochschild cohomology rings, certain important properties in Carlson’s theory are not expected to exist in Snashall-Solberg’s theory. This is one of the reasons why we believe V_{G∝P}(𝔐) has various advantages over V_{b}(𝔐) which we try to demonstrate in the rest of this paper.

4.1. Principal block. Let k (G ∝ P) be a transporter category algebra. Remember that we assume G ∝ P is connected, which is equivalent to saying that \underline{k} is indecomposable. We pay special attention to a special block of the transporter category algebra, closely related to our support variety theory.

Definition 4.1.1. Assume C is a finite connected category. We call a block of kC the principal block if the (indecomposable) trivial module \underline{k} belongs to it, and consequently name the block b_{0}.

Since one can take a minimal projective resolution of \underline{k} consisting of projective modules belonging to b_{0}, Ext_{kC}^{∗}(\underline{k}, \underline{k}) = Ext_{b_{0}}^{∗}(\underline{k}, \underline{k}) is an invariant of the principal block, comparable to the group case, see for example [20].

Return to transporter category algebras. We claim

(i) Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) is a (usually proper) subring of Ext_{b_{0}^{e}}^{∗}(b_{0}, b_{0}); (ii) Snashall-Solberg’s theory is valid for the subring

ℍ := ⟨Z(b_{0}), Ext_{k (G∝P)}(\underline{k}, \underline{k})⟩ ↪ Ext_{b_{0}^{e}}(b_{0}, b_{0});

(iii) Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) ↪ ⟨Z(b_{0}), Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})⟩ induces an isomorphism after quotient out

nilpotent elements. The ordinary cohomology ring is known to be finitely generated [23, 30], and thus so are

the rings ⟨Z(b_{0}), Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})⟩ and ℍ. Assuming these claims, along with Theorem 3.4.2, it guarantees that Snashall-Solberg’s theory can be implemented to ℍ ⊂ HH(b_{0}) for the principal block of k (G ∝ P).

Claim (ii) comes from Theorem 3.4.2. Claim (iii) is easy to verify since the commutative local algebra Z(b_{0}) quotients out the unique maximal ideal, that is, its nilradical, is exactly k for k is algebraically closed. To establish Claim (i), we have the following proposition.

Proposition 4.1.2. Let C be a finite connected category and b_{0} the principal block of kC. The injective homomorphism

Ext_{kC}^{∗}(\underline{k}, \underline{k}) ↪ Ext_{kC^{e}}^{∗}(kC, kC)

induces an injective homomorphism

Ext_{kC}^{∗}(\underline{k}, \underline{k}) ↪ Ext_{b_{0}^{e}}^{∗}(b_{0}, b_{0}).

Proof. As a kC^{e}-module, kC = ⊕_{i}b_{i}. Since LK_{C^{e}}^{C}kC ≅ \underline{k} and LK_{C^{e}}^{C} preserves direct sums, we see LK_{C^{e}}^{C}b_{0} = b_{0} ⊗_{kC} \underline{k} = \underline{k}, and LK_{C^{e}}^{C}b_{i} = 0 if b_{i} is not principal. It implies that the split surjection in [29]

Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC}^{∗}(\underline{k}, \underline{k})

induced by LK_{C^{e}}^{C}, restricts to a split surjection Ext_{b_{0}^{e}}^{∗}(b_{0}, b_{0}) → Ext_{kC}^{∗}(\underline{k}, \underline{k}). Hence we are done. □

From the preceding discussions, we get

V_{G∝P}(𝔐) = V_{ℍ}(𝔐)

for all 𝔐 ∈ b_{0}-mod. Note that, for any 𝔐 ∈ b_{0}-mod, there exists a finite surjective map

V_{b_{0}}(𝔐) → V_{G∝P}(𝔐).

We do not know yet when it becomes an isomorphism.

4.2. Standard properties. A subvariety of V_{G∝P} is trivial if it is the ideal consisting of all positive degree elements of Ext_{k (G∝P)}(\underline{k}, \underline{k}).

Theorem 4.2.1. Let G be a finite group, P a finite G-poset, and k an algebraically closed field of characteristic p dividing the order of G. Suppose 𝔐, 𝔑 are two finitely generated modules of k (G ∝ P). Then

(1) V_{G∝P}(𝔐) = ⋃_{𝔖} V_{G∝P}(𝔐, 𝔖) = ⋃_{𝔖} V_{G∝P}(𝔖, 𝔐), where 𝔖 runs over the set of all simple k (G ∝ P)-modules.

(2) If 0 → 𝔐_{1} → 𝔐_{2} → 𝔐_{3} → 0 is an exact sequence, then V_{G∝P}(𝔐_{i}) ⊂ V_{G∝P}(𝔐_{j}) ∪ V_{G∝P}(𝔐_{l}) for {i, j, l} = {1, 2, 3}.

(3) V_{G∝P}(𝔐_{1} ⊕ 𝔐_{2}) = V_{G∝P}(𝔐_{1}) ∪ V_{G∝P}(𝔐_{2}).

(4) V_{G∝P}(𝔐) = V_{G∝P}(Ω^{n}(𝔐)) for any integer n such that Ω^{n}(𝔐) ≠ 0.

(5) V_{G∝P}(𝔐, 𝔑) = V_{(G∝P)}^{op}(𝔑^{∧}, 𝔐^{∧}). Particularly V_{G∝P}(𝔐) = V_{(G∝P)}^{op}(𝔐^{∧}).

(6) dim V_{G∝P}(𝔐) = c_{k (G∝P)}(𝔐).

(7) V_{G∝P}(𝔐) is trivial if and only if 𝔐 has finite projective dimension if and only if 𝔐 has finite injective dimension.

(8) Let 𝔞 be a homogeneous ideal in Ext_{k (G∝P)}(\underline{k}, \underline{k}). Then there exists a module 𝔐_{𝔞} ∈ k (G ∝ P)-mod such that V_{G∝P}(𝔐_{𝔞}) = V (𝔞).

(9a) If V_{G∝P}(κ_{Hom_{k}(M,N)}) is trivial, then Ext_{k (G∝P)}^{i}(κ_{M}, κ_{N}) = 0 for i > dim P.

(9b) If 𝔐, 𝔑 ∈ b_{0}-mod and V_{G∝P}(𝔐) ∩ V_{G∝P}(𝔑) is trivial, then Ext_{k (G∝P)}^{i}(𝔐, 𝔑) = 0 for

i > inj.dimb_{0}. (10a) If χ(P) ≡ 1 (mod p) and V_{G∝P}(κ_{M}) = V_{1} ∪ V_{2} for some homogeneous non-trivial vari-

eties V_{1} and V_{2} with V_{1} ∩ V_{2} trivial, then κ_{M} = κ_{M_{1}} ⊕ κ_{M_{2}} with V_{G∝P}(κ_{M_{1}}) = V_{1} and

V_{G∝P}(κ_{M_{2}}) = V_{2}.

(10b) If 𝔐 ∈ b_{0}-mod such that V_{G∝P}(𝔐) = V_{1}∪V_{2} for some homogeneous non-trivial varieties V_{1} and V_{2} with V_{1}∩V_{2} trivial, then 𝔐 = 𝔐_{1}⊕𝔐_{2} with V_{G∝P}(𝔐_{1}) = V_{1} and V_{G∝P}(𝔐_{2}) = V_{2}.

Proof. The first five properties follow directly from the definition of V_{G∝P}(𝔐, 𝔑). The proofs are entirely analogous to the group case, see [7, 16].

The proof of (6) is exactly the same as that for groups, see for example [7, Proposition 5.7.2]. As for (7), one direction is straightforward. Also by Lemma 2.3.1 (2), it is equivalent to saying that 𝔐 is of finite injective dimension. Now let us assume V_{G∝P}(𝔐) is trivial. Then by (6) c_{k (G∝P)}(𝔐) = 0. It forces the minimal projective resolution of 𝔐 to be finite.

Part (8) follows from V_{G∝P}(𝔐) = V_{ℍ}(𝔐) if 𝔐 ∈ b_{0}-mod. Let 𝔞 be a homogeneous ideal in Ext_{k (G∝P)}(\underline{k}, \underline{k}). Then we can define a homogeneous ideal 𝔞′ = ⟨Rad Z(b_{0}), 𝔞⟩ of ℍ. From Snashall-Solberg’s theory [15, Theorem 4.4], the general form of Theorem 3.4.4 (8) for b_{0}, there exists a b_{0}-module 𝔐_{𝔞} such that V_{ℍ}(𝔐_{𝔞}) = V (𝔞′). But V (𝔞′) is identified with V (𝔞) under the isomorphism V_{ℍ} → V_{G∝P}, and V_{G∝P}(𝔐_{𝔞}) = V_{ℍ}(𝔐_{𝔞}).

Since (9b) and (10b) are Theorem 3.4.4 (9) and (10) specialized to the principal block, we shall prove only (9a) and (10a).

To prove (9a), we notice that Ext_{k (G∝P)}^{i}(κ_{M}, κ_{N}) ≅ Ext_{k (G∝P)}^{i}(\underline{k}, κ_{Hom_{k}(M,N)}) (see Section 2.5). From (7), the assumption implies that κ_{Hom_{k}(M,N)} has finite injective dimension. Hence the statement follows from Lemma 2.3.1.

As to (10a), we recall from Section 3.2 that if χ(P) ≡ 1 (mod p) then there exists a finite surjective map V_{G∝P}(κ_{M}) → V_{G}(M) for M ∈ kG-mod. Under the circumstance, let V′_{i} be the images of V_{i}, for i = 1, 2, we see that V_{G}(M) = V′_{1}∪V′_{2} with V′_{1}∩V′_{2} trivial. By [7, Theorem 5.12.1], M = M_{1} ⊕ M_{2} satisfying V_{G}(M_{1}) = V′_{1} and V_{G}(M_{2}) = V′_{2}. Then κ_{M} = κ_{M_{1}} ⊕ κ_{M_{2}} and hence V_{G∝P}(κ_{M}) = V_{G∝P}(κ_{M_{1}}) ∪ V_{G∝P}(κ_{M_{2}}). Moreover since the preceding map between varieties restricts to finite surjective maps V_{G∝P}(κ_{M_{i}}) → V_{G}(M_{i}) for i = 1, 2. It implies V_{i} = V_{G∝P}(κ_{M_{i}}) for i = 1, 2. □

By Theorem 4.2.1 (7) and Lemma 2.3.1, V_{G∝P}(κ_{M}) is trivial if and only if κ_{M} has finite projective dimension if and only if M is a projective kG_{x}-module for all x ∈ Ob (G ∝ P). If P = E_{p}, the poset of all elementary abelian p-subgroups of G, by Chouinard’s theorem [7, Theorem 5.2.4] it is equivalent to saying that M is a projective kG-module.

It is not known yet whether in (9a) and (10a) we may replace the constant modules with arbitrary modules.

4.3. Consequences of module filtrations. Recall that since G ∝ P is a finite EI-category, every k (G ∝ P)-module 𝔐 is constructed from atomic modules 𝔐_{x}, where 𝔐_{x}(y) ≅ 𝔐(x) if y ≅ x or zero otherwise. The following result says that only 𝔐-objects contribute to the variety V_{G∝P}(𝔐). Moreover the non-isomorphisms do not play a big role.

Proposition 4.3.1. We have V_{G∝P}(𝔐) = ⋃_{[x]} V_{G∝P}(𝔐_{x}) = ⋃_{[x]} ι_{G∝[x]}^{G∝P}V_{G∝[x]}(𝔐_{x}), where [x] runs over the set of isomorphism classes of 𝔐-objects.

Particularly V_{G∝P}(κ_{M}) = ⋃_{[x]} ι_{G∝[x]}^{G∝P}V_{G∝[x]}(κ_{M}), for any M ∈ kG-mod.

Proof. The containment V_{G∝P}(𝔐) ⊂ ⋃_{[x]∈Is (G∝P)} V_{G∝P}(𝔐_{x}) follows from Theorem 4.2.1 (2). We establish the equality. Firstly we note that Ext_{k (G∝P)}^{∗}(𝔐_{x}, 𝔐_{x}) ≅ Ext_{k(G∝[x])}^{∗}(𝔐_{x}, 𝔐_{x}) naturally. Moreover we have a commutative diagram

\xymatrix{Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{− ˆ⊗𝔐} \ar[d]_{_{res{G∝P \atop G∝[x]}}} & Ext_{k (G∝P)}^{∗}(𝔐, 𝔐) \ar[d]^{_{res{G∝P \atop G∝[x]}}} \\ Ext_{k(G∝[x])}^{∗}(\underline{k}, \underline{k}) \ar[r]_{− ˆ⊗𝔐_{x}} & Ext_{k(G∝[x])}^{∗}(𝔐_{x}, 𝔐_{x})}

To establish the above commutative diagram, we can represent cohomology classes by n-fold extensions and notice that both restrictions and − ˆ⊗− are exact.

The Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k})-action on Ext_{k (G∝P)}^{∗}(𝔐_{x}, 𝔐_{x}) factors through the action by Ext_{G∝[x]}^{∗}(\underline{k}, \underline{k}). Hence we have V_{G∝P}(𝔐_{x}) = ι_{G∝[x]}^{G∝P}V_{G∝[x]}(𝔐_{x}). Based on the same diagram we see that I_{G∝P}(𝔐) kills Ext_{k (G∝P)}^{∗}(𝔐_{x}, 𝔐_{x}). It means V_{G∝P}(𝔐_{x}) ⊂ V_{G∝P}(𝔐). □

We remind the reader that since G ∝ [x] as a category is equivalent to G_{x}, V_{G∝[x]}(𝔐_{x}) can be identified with V_{G_{x}×x}(𝔐(x)).

Corollary 4.3.2. We have V_{G∝P}(𝔐) = ⋃_{[x]} ι_{G_{x}×x}^{G∝P}V_{G_{x}×x}(𝔐(x)), where [x] runs over the set of isomorphism classes of 𝔐-objects, and hence V_{G∝P}(κ_{M}) = ⋃_{[x]} ι_{G_{x}×x}^{G∝P}V_{G_{x}×x}(M) for any M ∈ kGmod.

If G acts trivially on a connected poset P, then V_{G×P}(𝔐) = ⋃_{x} ι_{G_{x}×x}^{G∝P}V_{G}(𝔐(x)). Especially V_{G×P}(κ_{M}) = V_{G}(M) for any M ∈ kG-mod.

The reader may go back and have another look at Example 3.1.2. We note that if x ≅ y in G ∝ P, then there exists an element g ∈ G inducing an isomorphism by conjugation G_{x} → G_{y}. It implies that V_{G_{x}×x}(𝔐(x)) ≅ V_{G_{y}×y}(𝔐(y)) ≅ V_{G∝[x]}(𝔐_{x}) and

ι_{G_{x}×x}^{G∝P}V_{G_{x}×x}(𝔐(x)) = ι_{G_{y}×y}^{G∝P}V_{G_{y}×y}(𝔐(y)) = ι_{G∝[x]}^{G∝P}V_{G∝[x]}(𝔐_{x}) = V_{G∝P}(𝔐_{x}).

Corollary 4.3.3. If 𝔐 ∈ k (G ∝ P)-mod and H ⊂ G is a subgroup, then ι_{H∝P}^{G∝P}V_{H∝P}(𝔐) ⊂ V_{G∝P}(𝔐).

Proof. By the preceding result, we have V_{H∝P}(𝔐) = ⋃_{x} ι_{H_{x}×x}^{H∝P}V_{H_{x}×x}(𝔐(x)), and V_{G∝P}(𝔐) = ⋃_{x} ι_{G_{x}×x}^{G∝P}V_{G_{x}×x}(𝔐(x)), where x runs over the set of all 𝔐-objects.

Since ι_{H∝P}^{G∝P}ι_{H_{x}×x}^{H∝P} = ι_{G_{x}×x}^{G∝P}ι_{H_{x}×x}^{G_{x}×x} and ι_{H_{x}×x}^{G_{x}×x}V_{H_{x}×x}(𝔐(x)) ⊂ V_{G_{x}×x}(𝔐(x)) is known in Carlson’s theory, our claim follows. □

In a similar fashion we can show if Q is a G-subposet of P then

ι_{G∝Q}^{G∝P}V_{H∝Q}(𝔐) ⊂ V_{G∝P}(𝔐)

for any 𝔐 ∈ k (G ∝ P)-mod.

4.4. Varieties of Mackey functors. There are various ways to construct modules for a transporter category algebra. In [31] we examined a context of three categories

\xymatrix{ & G ∝ P \ar[dl]_{π} \ar[dr]^{ρ} \\ G &  & C,}

where C is a quotient category of G ∝ P. An illuminating example is that, when P is a poset of subgroups and G acts via conjugation, we may define an orbit category C = O_{P}(G) as the quotient category of G ∝ P such that Hom_{O_{P} (G)}(H, K) = K\Hom_{G∝P}(H, K) = K\N_{G}(H, K). Thus every kO_{P}(G)-module (resp. kO_{P}^{op}(G))-module) can be restricted to a k (G ∝ P)-module (resp. k (G ∝ P)^{op}-module). If S is the poset of all subgroups of G, a kO_{S}^{op}(G)-module is often called a coefficient system (for equivariant cohomology of G-spaces). The natural examples are the Mackey functors [13, 26, 27]. A Mackey functor may be defined as a kO_{S}(G)-bimodule enjoying certain extra properties. However after restriction (brutal truncation) we have no trouble to get a kO_{P}(G)-bimodule for an arbitrary G-subposet P ⊂ S. Thus every Mackey functor gives rise to a k (G ∝ P)-bimodule. In light of this it is tempting to introduce the following concept.

Definition 4.4.1. Let 𝔐 be a Mackey functor and P a G-poset consisting of some subgroups of G. The variety of the Mackey functor is V_{G∝P}(𝔐), where 𝔐 is regarded as a (either left or right) module over k (G ∝ P).

The reason why choosing the left or right k (G ∝ P)-module structure for a Mackey functor shall not make a difference is that, as far as we concern, only its values on objects matter, see Proposition 4.3.1.

Example 4.4.2. Let us now consider the poset E_{p} and the Mackey functor ℌ_{M}^{n} := H^{n}(−, M), where n is a non-negative integer and M is a kG-module. Then we have

V_{G∝E_{p}}(ℌ_{M}^{n}) = ∪_{E}ι_{N_{G}(E)}^{G∝E_{p}}V_{N_{G}(E)}(H^{n}(E, M)).

Particularly when n = 0 and M = k, ℌ_{k}^{0} = \underline{k}. Based on discussions in Section 3.2 we get

V_{G∝E_{p}}(ℌ_{k}^{0}) = V_{G}.

It shall be interesting to understand the varieties of various Mackey functors.

5. Further properties

In this section we shall deal with comparing varieties of categories. The main results are the generalized Quillen stratification and its consequences.

Bear in mind that, for the sake of simplicity, if D → C is a functor and 𝔐 is a kC-module, when it will not cause confusions, we often regard 𝔐 (instead of writing Res_{D}^{C} 𝔐) as a kD-module, even though its underlying vector space structure usually changes.

5.1. Quillen stratification. In [23, 24] Quillen worked with G-spaces and equivariant cohomology rings. Here we are interested in G-spaces which are classifying spaces of finite G-posets. In order to make a smooth transition from G-spaces and equivariant cohomology to G-posets and transporter category cohomology, we first recall some of the original constructions and then restrict to our case.

Given a G-space X and a H-space Y, a morphism from (H, Y) → (G, X) is a pair (ϕ, F) such that ϕ : H → G is a group homomorphism and θ : Y → X is a continuous map satisfying the condition that

ϕ(h)θ(y) = θ(hy), ∀h ∈ H, y ∈ Y.

Such a morphism induces a continuous map Θ : EH ×_{H} Y → EG ×_{G} X and thus a restriction map between equivariant cohomology

res_{H,Y}^{G,X} : H_{G}^{∗}(X) → H_{H}^{∗}(Y).

One can compare these constructions with those introduced before Example 3.1.2.

In Quillen’s papers [23, 24], it is proved that if X is a G-space, either compact or paracompact with finite cohomological dimension, then there exists an F-isomorphism

𝔮 = 𝔮_{G,X} : H_{G}^{∗}(X) → \underleftarrow{lim}_{A_{p}(X)}H^{∗}(E)

where A_{p}(X) is called the Quillen category and its objects will be named Quillen pairs for the G-space X in this article. More explicitly, the objects in A_{p}(X) are of the form (E, C), where E is an elementary abelian p-subgroup of G and C is a (non-empty) connected component of X^{E}. A morphism from (E′, C′) → (E, C) is a pair (ϕ, 𝔦) with ϕ = c_{g} : E′ → E for some element g ∈ G and 𝔦 : gC → C′ an inclusion. Particularly if g can be chosen to be the identity element of G, then we call (E′, C′) a Quillen subpair of (E, C), and write (E′, C′) ≤ (E, C). In this way, all Quillen pairs form a poset Q_{p}, with obvious G-action.

Remark 5.1.1. Another way to construct the Quillen category is to define it as a quotient category of the transporter category G ∝ Q_{p}, where Q is the G-poset of all Quillen pairs for X, through

Hom_{A_{p}(X)}((E′, C′), (E, C)) = Hom_{G∝Q_{p}}((E′, C′), (E, C))/C_{G}(E′, C′),

where C_{G}(E′, C′) = {g ∈ G | gC′ = C′, ghg^{−1} = h for all h ∈ E} is called the centralizer of (E′, C′). We also denote by N_{G}(E′, C′) = G_{(E′,C′)} the normalizer of (E′, C′) and W_{G}(E′, C′) = N_{G}(E′, C′)/C_{G}(E′, C′) the Weyl group of (E′, C′).

Quillen’s map 𝔮 is induced by (E, •) → (E, C) → (G, X), where • is sent into C. Since E acts trivially on C, the choice of an embedding • → C does not matter. In fact in any case this map induces a surjective map H_{E}^{∗}(C) → H_{E}^{∗}(•) which becomes an isomorphism after quotient out nilpotents in both rings. Consequently MaxSpecH_{E}(C) = MaxSpecH_{E}(•), that is, V_{E,C} = V_{E,•} = V_{E}.

Let V_{G,X} = MaxSpecH_{G}(X). The geometric version of Quillen’s map is

V_{G,X} = ⋃_{(E,C)} ι_{E,C}^{G,X}V_{E,C} = ⋃_{(E,C)} ι_{E,C}^{G,X}ι_{E,•}^{E,C}V_{E,•},

where ι_{E,C}^{G,X} : V_{E,C} → V_{G,X} is induced by res_{E,C}^{G,X}. Based on the observation that each morphism (E′, C′) → (E, C) in the category A_{p}(X) induces a ring homomorphism H_{E}^{∗} → H_{E′}^{∗}, which is compatible with the two maps H_{G}^{∗}(X) → H_{E′}^{∗} and H_{G}^{∗}(X) → H_{E}^{∗} coming from (E′, C′) → (G, X) and (E, C) → (G, X), Quillen then continued to demonstrate that V_{G,X} is a disjoint union of some locally closed subvarieties V_{E,C}^{G,X}^{+} by examining more closely the following diagram

\xymatrix{V_{E′} = V_{E′,C′} \ar[rr]^{ι_{E′}^{E}} \ar[dr]_{ι_{E′,C′}^{G,X}} &  & V_{E,C} = V_{E} \ar[dl]^{ι_{E,C}^{G,X}} \\  & V_{G,X} & .}

Here the horizontal map is ι_{E′}^{E} (corresponding to H_{E}^{∗} → H_{E′}^{∗}), not the senseless ι_{E′,C′}^{E,C}, since a morphism in A_{p}(X) is different from the morphisms introduced in the second paragraph of this section.

Theorem 5.1.2 (Quillen). The dimension of V_{G,X} equals the maximum p-rank of an elementary abelian p-subgroup from Quillen pairs.

We shall come back to Quillen’s results shortly after we sort out all terminologies.

Now we turn to the case of X = BP for a finite G-poset P. Let us remind the reader of several relevant results.

(a) Suppose a G-space X is either compact or paracompact with finite cohomological dimension. If H^{∗}(X) is finite-dimensional then so is H^{∗}(C) for all possible C from Quillen pairs, [23, Corollary 4.3].

(b) (BP)^{g} = B(P^{g}) (fixed points) for any g ∈ G, [12, Proposition 66.3]; (c) Let S_{p} and E_{p} be the G-posets of non-identity p-subgroups and of elementary abelian p-subgroups of G. Then obviously any elementary abelian p-subgroup E has the property that both (BS_{p})^{E} and (BE_{p})^{E} are non-empty.

When X = BP for a G-poset P and M ∈ kG-mod, we know (Id_{G}, pt) : (G, BP) → (G, •) induces the following restriction map

res : Ext_{kG}^{∗}(k, k) = H^{∗}(G) → H_{G}^{∗}(BP) = Ext_{k (G∝P)}^{∗}(\underline{k}, \underline{k}),

which is injective if the Euler characteristic χ(P) = χ(BP) is invertible in the base field, by invoking the Becker-Gottlieb transfer [32]. A typical example of such G-posets is S_{p}, the poset of non-identity p-subgroups in G, since Brown’s theorem [7, Corollary 6.7.4] says that χ(S_{p}) ≡ 1 (mod p). There are several well known subposets that are G-homotopy equivalent to it, and thus possess the same property.

Consider the canonical map (Id_{E}, pt) : (E, X) → (E, •), where E is an elementary abelian p-group. A more involved result says that the restriction res : H_{E}^{∗}(•) → H_{E}^{∗}(X) being injective is equivalent to saying that X^{E} is non-empty, [19, IV.1, Corollary 1]. If X = BP such that χ(P) ≡ 1 (mod p), by [9, Capter III, Theorem 4.3] we get χ(P^{E}) ≡ 1 (mod p). This implies that BP^{E} ≠ ∅ and that res is injective, matching the observation we made using the Becker-Gottlieb transfer.

From now on we focus on the case X = BP for a finite G-poset P. By doing so, we restrict to a special case of Quillen in order to get rid of topology and unveil the underlying algebraic/categorical constructions. Most importantly we gain the freedom to work with varieties of modules.

Let E be an elementary abelian p-subgroup of G. Because BP^{E} = B(P^{E}), a Quillen pair for the G-space BP is of the form (E, BC) in which C is a connected component of the poset P^{E}. Thus under the circumstance, we may write each Quillen pair as (E, C) with C a connected subposet of P^{E}, and call (E, C) a Quillen pair for the G-poset P.

Quillen stratification in our setting says (to be consistent with terminologies in Section 3.1 we choose to write ι_{E×C}^{G∝P} for ι_{E,BC}^{G,BP} and accordingly V_{G∝P} for V_{G,BP} etc.)

V_{G∝P} = ⋃_{(E,C)} ι_{E×C}^{G∝P}V_{E×C} = ⋃_{(E,C)} ι_{E×C}^{G∝P}ι_{E×•}^{E×C}V_{E×•}.

Note that E ∝ C = E × C is a subcategory of G ∝ P, and V_{E×C} ≅ V_{E×•}.

Remark 5.1.3. The theorem of Alperin-Avrunin-Evens [16, Theorem 8.3.1] says that

V_{G}(M) = ⋃_{E≤G} ι_{E}^{G}V_{E}(M),

where E runs over the set of all elementary abelian subgroups of G and M is a kG-module. Since G_{x} is the isotropy group of x ∈ Ob P, we know x ∈ Ob P^{E} if and only if E ⊂ G_{x}. Thus if (E, C) is a Quillen pair, any object x ∈ Ob C satisfies the condition that E ⊂ G_{x}. Thus

ι_{E×x}^{G_{x}×x}V_{E×x} ⊂ V_{G_{x}×x} = V_{G∝[x]}

Based on the Alperin-Avrunin-Evens’ theorem, we can rewrite Proposition 4.3.1 and Corollary 4.3.2.

V_{G∝P}(𝔐) = ⋃_{[x]} ι_{G∝[x]}^{G∝P}V_{G∝[x]}(𝔐_{x})
= ⋃_{y} ι_{G_{y}×y}^{G∝P}V_{G_{y}×y}(𝔐(y))
= ⋃_{y,E⊂G_{y}} ι_{E×y}^{G∝P}V_{E×y}(𝔐(y))
= ⋃_{(E,C)} ι_{E×C}^{G∝P}V_{E×C}(𝔐).

These shall be useful for us in the sequel. We note that if (E, C) ≅ (^{g}E, gC) for some g ∈ G, that is, two Quillen pairs are isomorphic in G ∝ P, then ι_{E×C}^{G∝P}V_{E×C}(𝔐) = ι_{^{g}E×gC}^{G∝P}V_{^{g}E×gC}(𝔐).

Let (E, C) be a Quillen pair and E′ a subgroup of E. Then we denote by C|E′ the connected component of P^{E′} which contains C, determining a unique Quillen subpair (E′, C|E′) ≤ (E, C). One can easily verify

ι_{E′}^{E}ι_{E′×x}^{E′×(C|E′)} = ι_{E×x}^{E×C}ι_{E′×x}^{E×x} : V_{E′} = V_{E′×x} → V_{E×x} = V_{E},

for every x ∈ Ob C.

Lemma 5.1.4. For any Quillen pair (E, C) and 𝔐 ∈ k(E × C)-mod

ι_{E′}^{E}V_{E′×(C|E′)}(𝔐) = {ι_{E′}^{E}V_{E′×(C|E′)}} ∩ V_{E×C}(𝔐).

Proof. By Corollary 4.3.2, V_{E×C}(𝔐) = ⋃_{x∈Ob C} ι_{E×x}^{E×C}V_{E×x}(𝔐(x)) because the isotropy subgroup of every x ∈ C is exactly E. Then

{ι_{E′}^{E}V_{E′×(C|E′)}} ∩ ι_{E×x}^{E×C}V_{E×x}(𝔐(x)) = ι_{E×x}^{E×C}{[ι_{E′×x}^{E×x}V_{E′×x}] ∩ V_{E×x}(𝔐(x))}
= ι_{E×x}^{E×C}ι_{E′×x}^{E×x}V_{E′×x}(𝔐(x))
= ι_{E′}^{E}ι_{E′×x}^{E′×(C|E′)}V_{E′×x}(𝔐(x))

for every x ∈ Ob C. The second equality uses [7, Proposition 5.7.7]. Consequently we get

{ι_{E′}^{E}V_{E′×(C|E′)}} ∩ V_{E×C}(𝔐) = {ι_{E′}^{E}V_{E′×(C|E′)}} ∩ {⋃_{x∈Ob C} ι_{E×x}^{E×C}V_{E×x}(𝔐(x))}
= ⋃_{x∈Ob C}{ι_{E′}^{E}ι_{E′×x}^{E′×(C|E′)}V_{E′×x}(𝔐(x))}
= ι_{E′}^{E}V_{E′×(C|E′)}(𝔐).

The last equality is true since 𝔐(y) = 0 if y ∈ Ob(C|E′) − Ob C, by definition. □

We set

V_{E×C}^{+} = V_{E×C} − ⋃_{E′⫋E} ι_{E′}^{E}V_{E′×(C|E′)}

V_{E×C}(𝔐)^{+} = V_{E×C}^{+} ∩ V_{E×C}(𝔐)

V_{E×C}^{G∝P}^{+} = ι_{E×C}^{G∝P}V_{E×C} − ⋃_{E′⫋E} ι_{E′×(C|E′)}^{G∝P}V_{E′×(C|E′)}

V_{E×C}^{G∝P}(𝔐)^{+} = ι_{E×C}^{G∝P}V_{E×C}(𝔐)^{+}

The following is established by Quillen for arbitrary P and 𝔐 = \underline{k} [24] and by Avrunin-Scott for P = • and arbitrary modules [5]. Our generalization is based on both of these special cases.

Theorem 5.1.5 (Stratification). Let G ∝ P be a finite transporter category, k an algebraically closed field of characteristic p dividing the order of G and 𝔐 ∈ k (G ∝ P)-mod. Then

V_{G∝P}(𝔐) = ⨄_{(E,C)} V_{E×C}^{G∝P}(𝔐)^{+},

where the index runs over the set of isomorphism classes of Quillen pairs. Moreover we have a homeomorphism

V_{E,C}(𝔐)^{+}/W_{G}(E, C) ≃ V_{E×C}^{G∝P}(𝔐)^{+}.

Proof. We know for 𝔐 = \underline{k} the Quillen stratification

V_{G∝P} = ⨄_{(E,C)} V_{E×C}^{G∝P}^{+},

with

V_{E×C}^{+}/W_{G}(E, C) ≃ V_{E×C}^{G∝P}^{+}.

From Remark 5.1.3 we have

V_{G∝P}(𝔐) = ⋃_{(E,C)} ι_{E×C}^{G∝P}V_{E×C}(𝔐).

Since Lemma 5.1.4 implies

V_{E×C}(𝔐)^{+} = V_{E×C}(𝔐) − ⋃_{E′⫋E} ι_{E′}^{E}V_{E′×(C|E′)}(𝔐),

we get

V_{G∝P}(𝔐) = ⋃_{(E,C)} ι_{E×C}^{G∝P}V_{E×C}(𝔐)^{+}.

It follows from Quillen’s original result that, if (E, C) runs over the set of isomorphism classes of Quillen pairs,

V_{G∝P}(𝔐) = ⨄_{(E,C)} V_{E×C}^{G∝P}(𝔐)^{+},

and that there is a homeomorphism

V_{E,C}(𝔐)^{+}/W_{G}(E, C) ≃ V_{E×C}^{G∝P}(𝔐)^{+}.
□

5.2. Subcategories and tensor products. Here we record a few more expected interesting results as consequences of the Quillen stratification.

Theorem 5.2.1. Suppose P is a G-poset and H is a subgroup of G. Then, for any 𝔐 ∈ k (G ∝ P)- mod,

ι_{H∝P}^{G∝P}^{−1}V_{G∝P}(𝔐) = V_{H∝P}(𝔐).

Proof. By Corollary 4.3.3 ι_{H∝P}^{G∝P}V_{H∝P}(𝔐) ⊂ V_{G∝P}(𝔐). Given a Quillen pair (E, C) for H ∝ P, (E, C) is also a Quillen pair for G ∝ P with W_{H}(E, C) ⊂ W_{G}(E, C). By Theorem 5.1.5, there is a commutative diagram

\xymatrix{V_{E×C}(𝔐)^{+}/W_{H}(E, C) \ar@{->>}[r] \ar[d]_{≃} & V_{E×C}(𝔐)^{+}/W_{G}(E, C) \ar[d]^{≃} \\ V_{E×C}^{H∝P}(𝔐)^{+} \ar[r]_{ι_{H∝P}^{G∝P}} & V_{E×C}^{G∝P}(𝔐)^{+}}

Since the upper horizontal map is surjective and the vertical maps are homeomorphisms, the lower horizontal map is surjective. We are done. □

We could not find a way to generalize the above statement for an arbitrary sub-transporter category of G ∝ P. However we have the following.

Proposition 5.2.2. Suppose P is a G-poset and Q is a G-subposet. Then, for any M ∈ kG-mod,

ι_{G∝Q}^{G∝P}^{−1}V_{G∝P}(κ_{M}) = V_{G∝Q}(κ_{M}).

Proof. The proof is similar to the previous one. If (E, C) is a Quillen pair for G ∝ Q, then there exists a unique Quillen pair (E, D) for G ∝ P. We note that V_{E×C}(κ_{M})^{+} = V_{E×D}(κ_{M})^{+}. Also by definition if gC = C then gD = D as well, for any g ∈ G. It implies that W_{G}(E, C) ⊂ W_{G}(E, D). □

Recall that for any category D there is a diagonal functor △ : D → D × D. Then image of D is written as △D. If C is a connected poset, then so is C × C. Suppose G ∝ P is a transporter category and (E, C) is a Quillen pair for the G-poset P. Then (△E, C × C) is a Quillen pair for the (G × G)-poset P × P. Note that from Remark 2.1.2 (G × G) ∝ (P × P) ≅ (G ∝ P) × (G ∝ P).

Corollary 5.2.3. Suppose G is a finite group and P is a finite G-poset. If H is a subgroup of G and Q is a H-subposet of P, then ι_{H∝Q}^{G∝P}^{−1}V_{G∝P}(κ_{M}) = V_{H∝Q}(κ_{M}). Particularly for any k(G × G)-module N

ι_{G∝P}^{(G×G)∝(P×P)}^{−1}V_{(G×G)∝(P×P)}(κ_{N}) = V_{G∝P}(κ_{N}).

Proof. We apply Theorem 5.2.1 and Proposition 5.2.2 to the following three transporter categories H ∝ Q ⊂ H ∝ P ⊂ G ∝ P.

For the special case, we consider G ∝ P ≅ △G ∝ △P ⊂ △G ∝ (P×P) ⊂ (G×G) ∝ (P×P). □

We need a technical result for the last main result.

Proposition 5.2.4. Suppose D and C are two finite categories with finitely generated ordinary cohomology rings. If 𝔐, 𝔐′ ∈ kD-mod and 𝔑, 𝔑′ ∈ kC-mod, then

Ext_{k(D×C)}^{∗}(𝔐 ⊗ 𝔑, 𝔐′ ⊗_{k} 𝔑′) ≅ Ext_{kD}^{∗}(𝔐, 𝔐′) ⊗_{k} Ext_{kC}^{∗}(𝔑, 𝔑′).

Proof. Let 𝔓_{∗}^{D} → 𝔐 → 0 and 𝔓_{∗}^{C} → 𝔑 → 0 be two projective resolutions. Then 𝔓_{∗}^{D} ⊗_{k} 𝔓_{∗}^{C} → 𝔐 ⊗_{k} 𝔑 → 0 is a projective resolution of the k(D × C) = kD ⊗_{k} kC-module 𝔐 ⊗_{k} 𝔑. The isomorphism follows from

Hom_{kD⊗kC}(𝔓_{∗}^{D} ⊗_{k} 𝔓_{∗}^{C}, 𝔐′ ⊗_{k} 𝔑′) ≅ Hom_{kD}(𝔓_{∗}^{D}, 𝔐′) ⊗_{k} Hom_{kC}(𝔓_{∗}^{C}, 𝔑′),

and the Künneth formula. □

When we examine transporter categories, the above result has the following consequences.

Corollary 5.2.5. Let G_{1} ∝ P_{1} and G_{2} ∝ P_{2} be two transporter categories. Then there is a natural isomorphism

V_{(G_{1}×G_{2})∝(P_{1}×P_{2})} = V_{(G_{1}∝P_{1})×(G_{2}∝P_{2})} = V_{G_{1}∝P_{1}} × V_{G_{2}∝P_{2}}.

If M ∈ kG_{1}-mod and N ∈ kG_{2}-mod, under the above isomorphism we have furthermore

V_{(G_{1}×G_{2})∝(P_{1}×P_{2})}(κ_{M⊗N}) = V_{(G_{1}∝P_{1})×(G_{2}∝P_{2})}(κ_{M} ⊗_{k} κ_{N}) = V_{G_{1}∝P_{1}}(κ_{M}) × V_{G_{2}∝P_{2}}(κ_{N}).

Proof. From Remark 2.1.2 we have (G_{1} × G_{2}) ∝ (P_{1} × P_{2}) ≅ (G_{1} ∝ P_{1}) × (G_{2} ∝ P_{2}). The statements on varieties are true, because

Ext_{k[(G_{1}×G_{2})∝(P_{1}×P_{2})]}^{∗}(κ_{M⊗N}, κ_{M⊗N}) ≅ Ext_{k[(G_{1}∝P_{1})×(G_{2}∝P_{2})]}^{∗}(κ_{M} ⊗ κ_{N}, κ_{M} ⊗ κ_{N})
≅ Ext_{k(G_{1}∝P_{1})}^{∗}(κ_{M}, κ_{M}) ⊗ Ext_{k(G_{2}∝P_{2})}^{∗}(κ_{N}, κ_{N}).
□

Based on the preceding results, we can extend the tensor product formula V_{G}(M ⊗_{k} N) = V_{G}(M) ∩ V_{G}(N) of Avrunin-Scott [5].

Theorem 5.2.6. Let G be a finite group and P a finite G-poset. Suppose M, N are two kGmodules. Then V_{G∝P}(κ_{M} ˆ⊗κ_{N}) = V_{G∝P}(κ_{M}) ∩ V_{G∝P}(κ_{N}).

Proof. Let us consider the functor △ : G ∝ P → (G ∝ P) × (G ∝ P) and the restriction along it. Because

κ_{M} ˆ⊗κ_{N} = Res_{G∝P}^{(G∝P) × (G∝P)}(κ_{M} ⊗ κ_{N}),

by Corollary 5.2.3 we have

V_{G∝P}(κ_{M} ˆ⊗_{k}κ_{N}) = ι_{G∝P}^{(G∝P) × (G∝P)}^{−1}V_{(G∝P) × (G∝P)}(κ_{M} ⊗_{k} κ_{N}).

From (G × G) ∝ (P × P) ≅ (G ∝ P) × (G ∝ P) and Corollary 5.2.5 the right-hand-side is

ι_{G∝P}^{(G∝P) × (G∝P)}^{−1}{V_{G∝P}(κ_{M}) × V_{G∝P}(κ_{N})},

which exactly is V_{G∝P}(κ_{M}) ∩ V_{G∝P}(κ_{N}). □

With Theorem 5.2.6, taking Theorem 4.2.1 (5) in account one can see there is some relation between the conditions in (9a) and (9b) in Theorem 4.2.1.

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Espanya.

   E-mail address: xu@mat.uab.cat