Theory and Applications of Categories, Vol. 11, No. 10, 2003, pp. 215–251.

SOME ALGEBRAIC APPLICATIONS OF GRADED CATEGORICAL GROUP THEORY

A.M. CEGARRA AND A.R. GARZÓN

ABSTRACT. The homotopy classification of graded categorical groups and their homomorphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts. Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or groups, or rings, or rings-groups or algebras as well as for graded Clifford systems with operators, equivariant Azumaya algebras over Galois extensions of commutative rings and for strongly graded bialgebras and Hopf algebras with operators. These specialized classifications follow from the theory of graded categorical groups after identifying, in each case, adequate systems of factor sets with graded monoidal functors to suitable graded categorical groups associated to the structure dealt with.

1. Introduction

Graded categorical groups provide a suitable categorical setting for the treatment of an extensive list of subjects with recognized mathematical interest. Let us briefly recall that, if Γ is a group, then a Γ-graded categorical group is a groupoid 𝔾 equipped with a grading functor gr : 𝔾 → Γ and with a graded monoidal structure, by graded functors ⊗ : 𝔾 ×_{Γ} 𝔾 → 𝔾 and I : Γ → 𝔾, such that for each object X, there is an object X′ with a 1-graded arrow X ⊗ X′ → I (see Section 2 for the details). These graded categorical groups were originally considered by Fröhlich and Wall in [20] to study Brauer groups in equivariant situations (see also [18, 19, 21]). Indeed, the principal examples in that context are either the Picard, or the Azumaya or the Brauer Γ-graded categorical groups defined by a commutative ring on which the group Γ acts by automorphisms. Several other examples of graded categorical groups that deal with algebraic problems are considered in this paper, but there are also interesting instances arising in algebraic topology (see [4, 6]). Furthermore, in the same way which was dealt with by Sinh the ungraded case [36], homotopy classifications theorems for graded categorical groups and their homomorphisms have been shown in [6]. From these general results derive then the utility of graded categorical group theory either in equivariant homotopy theory or group

Financially supported by DGI of Spain and FEDER, Project: BFM2001-2886 Received by the editors 2002-06-26 and, in revised form, 2003-05-07. Transmitted by Ieke Moerdijk. Published on 2003-05-22. 2000 Mathematics Subject Classification: 18D10, 20J06, 20M10, 20M50, 16S35, 16W50, 16H05,

16W30..

Key words and phrases: graded categorical group, graded monoidal functor, equivariant group cohomology, crossed product, factor set, monoid extension, group extension, ring-group extension, Clifford system, Azumaya algebra, graded bialgebra, graded Hopf algebra.

\begin{matrix}c \\ ◯ \end{matrix} A.M. Cegarra and A.R. Garzón, 2003. Permission to copy for private use granted.

extensions with operators. In this article, our goal is to state several theorems solving a selected list of classification problems of equivariant algebraic nature and to prove them by using also the above quoted abstract results on the homotopy classification of graded categorical groups and their homomorphisms.

The equivariant classification results we obtain along the paper mainly concern commutative rings on which an action by automorphisms from a group Γ is given. These so-called Γ-rings are a classical subject of investigation in Algebra as either the 14th Hilbert’s problem and the invariant theory, or even the more recent equivariant algebraic K-theory, make clear (cf. [31]). More precisely, the most striking Γ-rings we consider as a matter of study are central simple Γ-algebras over a field or, more generally, central separable Γ-algebras over a commutative ring k, whose classes are indeed the elements of the equivariant Brauer group of k (see [17, 18, 19, 20, 21]). Azumaya Γ-algebras are also in the heart of Long’s Brauer groups [28] (cf. [2, chapter VI]). If R is a commutative Γ-ring and R/k is a Galois extension, with Galois group G, splitting a central separable Γ-algebra A, then A represents, in the equivariant Brauer group, the same element than a strongly G-graded Γ-algebra whose 1-component is R, that is, a (G, R)-graded Γ-Clifford system. This is the reason why we pay attention to these last kind of Γ-rings. Moreover, in the particular case that the Picard group of R vanishes, then the structure of such Azumaya Γ-algebras is precisely that of a Γ-equivariant crossed product algebra of R by G and this justifies its separate study in Section 5. When the ground ring k = ℤ then, the possibility of offering an unified treatment of that theory together with the one of extensions of groups with operators motivates the study about Γ-RINGS-GROUPS we carry out in Section 4.

The plan of this paper is briefly as follows. In the second section we summarize what will be needed in the paper regarding graded categorical groups theory and also on the equivariant cohomology theory of groups with operators studied in [5]. Theorem 2.2, in the second section, shows a classification of the homotopy classes of graded monoidal functors between graded categorical groups with discrete domain. It is of special relevance because suitable specializations of it lead to the classification of several kinds of systems of factor sets for the diverse types of “crossed product constructions with operators” studied throughout the subsequent sections.

In the third section we deal with the general problem of equivariant perfect Rédei extensions of monoids by groups when a group of operators Γ is acting on both structures (also called equivariant monoid coextensions of groups). This problem includes that of group extensions with operators studied in [6, 37]. We develop here an equivariant factor set theory for equivariant extensions of Γ-monoids by Γ-groups. Since these equivariant factor sets become graded monoidal functors from the discrete graded categorical group associated with the group to the holomorph graded categorical group of the monoid, we can deduce the desired cohomological classification for equivariant perfect extensions of a Γ-monoid by a Γ-group from the categorical results.

SOME ALGEBRAIC APPLICATIONS OF GRADED CATEGORICAL GROUP THEORY 217

The next section is devoted to the classification of extensions of RINGS-GROUPS with operators. Our results extend to the equivariant case the theory developed by Hacque in [24, 25] and they are directly applicable both to the theory of Γ-group extensions and to the theory of crossed product Γ-rings. We explain a theory of mixed crossed products with operators that reveals how graded monoidal functors, from discrete graded categorical groups to the graded categorical groups of holomorphisms of Γ-RINGS-GROUPS, are the most suitable systems of data to construct the survey of all equivariant extensions of any RING-GROUP with operators by a group with operators.

In the following section we focus our attention on the classification of graded Clifford systems (also called strongly graded algebras) with operators, which were introduced by Dade [12, 13] to develop Clifford’s theory axiomatically. The key in obtaining that classification lies in the fact that giving a graded Γ-Clifford system is equivalent to giving a graded monoidal functor from discrete graded categorical groups to Γ-graded Picard categorical groups of Γ-algebras. Our results for equivariant graded Clifford systems are then used in three subsections, where we deal respectively with equivariant crossed product algebras, equivariant central graded Clifford systems and Azumaya algebras with operators over Galois extensions of commutative rings as well. In the two last cases, the classification theorems are reformulated in terms of certain equivariant cohomology exact sequences.

The last section is dedicated to the classification of strongly graded Γ-bialgebras and strongly graded Hopf Γ-algebras. Our results here parallel the non-equivariant case studied in [8, 3] and they are obtained, from the general theory outlined in the second section, in a similar way to that previously run for equivariant Clifford systems.

2. On graded categorical groups

The beginning of this section is devoted to recalling the definition and some basic facts concerning the cohomology of groups with operators and graded categorical groups. Then, we will derive, from general results established in [6], a classification theorem for homotopy classes of graded monoidal functors between graded categorical groups whose domain is discrete. This result is what is needed for the applications set forth in the next sections.

Hereafter Γ is a fixed group. Let us recall that a Γ-group G means a group G enriched with a left Γ-action by automorphisms, and that a Γ-equivariant module over a Γ-group G is a Γ-module A, that is, an abelian Γ-group, endowed with a G-module structure such that^{σ}(^{x}a) = ^{(^{σ}x)}(^{σ}a) for all σ ∈ Γ, x ∈ G and a ∈ A [5, Definition 2.1]. The abelian groups of Γ-equivariant derivations from a Γ-group G into equivariant G-modules define a leftexact functor Der_{Γ}(G, −) from the category of equivariant G-modules to the category of abelian groups, whose right derived functors lead to the equivariant cohomology functors H∗_{Γ}(G, −). More specifically, the cohomology groups of a Γ-group G with coefficients in an equivariant G-module A are [5, (3)]:

Hn_{Γ}(G, A) = (R^{n−1}Der_{Γ}(G, −))(A), n ≥ 1,

and H0_{Γ}(G, A) = 0.

We shall recall from [5] that the cohomology groups Hn_{Γ}(G, A), for n ≤ 3, can be computed as the cohomology groups of the Whitehead [37] truncated cochain complex

0 → C 1_{Γ}(G, A) ⟶^{∂} C 2_{Γ}(G, A) ⟶^{∂} Z 3_{Γ}(G, A) → 0 ,

in which C 1_{Γ}(G, A) consists of normalized maps c : G → A, C 2_{Γ}(G, A) consists of normalized maps c : G^{2} ∪ (G × Γ) → A and Z 3_{Γ}(G, A) consists of all normalized maps c : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) → A satisfying the following 3-cocycle conditions:

c(x, y, zt) + c(xy, z, t) = ^{x}c(y, z, t) + c(x, yz, t) + c(x, y, z) ,
σc(x, y, z) + c(xy, z, σ) + c(x, y, σ) = c(^{σ}x,^{σ}y,^{σ}z) +^{(^{σ}x)}c(y, z, σ) + c(x, yz, σ) ,
σc(x, y, τ) + c(^{τ}x,^{τ}y, σ) + c(x, σ, τ) +^{(^{στ} x)}c(y, σ, τ) = c(x, y, στ) + c(xy, σ, τ) ,
σc(x, τ, γ) + c(x, σ, τγ) = c(x, στ, γ) + c(^{γ}x, σ, τ) ,

for x, y, z, t ∈ G, σ, τ, γ ∈ Γ. For each c ∈ C 1_{Γ}(G, A), the coboundary ∂c is given by

\begin{matrix}(∂c)(x, y) = ^{x}c(y) − c(xy) + c(x) , \\ (∂c)(x, σ) = ^{σ}c(x) − c(^{σ}x) , \end{matrix}

and for c ∈ C 2_{Γ}(G, A), ∂c is given by

(∂c)(x, y, z) = ^{x}c(y, z) − c(xy, z) + c(x, yz) − c(x, y) ,
(∂c)(x, y, σ) = ^{σ}c(x, y) − c(^{σ}x,^{σ}y) −^{(^{σ}x)}c(y, σ) + c(xy, σ) − c(x, σ) ,
(∂c)(x, σ, τ) = ^{σ}c(x, τ) − c(x, στ) + c(^{τ}x, σ).

Next, we regard the group Γ as a category with exactly one object, say ∗, where the morphisms are the members of Γ and the composition law is the group operation:

∗ →^{σ} ∗ →^{τ} ∗ = ∗ →^{τσ} ∗.

A Γ-grading on a category 𝔾 [20] is a functor gr : 𝔾 → Γ. For any morphism f in 𝔾 with gr(f) = σ, we refer to σ as the grade of f, and we say that f is an σ-morphism. The grading is stable if, for any object X of 𝔾 and any σ ∈ Γ, there exists an isomorphism X →^{∼} Y with domain X and grade σ; in other words, the grading is a cofibration in the sense of Grothendieck [23]. Suppose (𝔾, gr) and (ℍ, gr) are stably Γ-graded categories. A graded functor F : (𝔾, gr) → (ℍ, gr) is a functor F : 𝔾 → ℍ preserving grades of morphisms. If F′ : (𝔾, gr) → (ℍ, gr) is also a graded functor, then a graded natural equivalence θ : F → F′ is a natural equivalence of functors such that all isomorphisms θ_{X} : F X →^{∼} F′X are of grade 1.

For a Γ-graded category (𝔾, gr), we write 𝔾 ×_{Γ} 𝔾 for the subcategory of the product category 𝔾×𝔾 whose morphisms are those pairs of morphisms of 𝔾 with the same grade; this has an obvious grading, which is stable if and only if gr is.

A Γ-graded monoidal category [20] (see [34, Chapter I, §4.5] for the general notion of fibred monoidal category) 𝔾 = (𝔾, gr, ⊗, I, a, l, r), is a stably Γ-graded category (𝔾, gr) together with graded functors

⊗ : 𝔾 ×_{Γ} 𝔾 → 𝔾 , I : Γ → 𝔾 ,

and graded natural equivalences

a : (− ⊗ −) ⊗ − ⟶^{∼} − ⊗ (− ⊗ −) ,

l : I gr(−) ⊗ − ⟶^{∼} id_{𝔾}, r : − ⊗ I gr(−) ⟶^{∼} id_{𝔾} ,

such that for any objects X, Y, Z, T ∈ 𝔾, the following two coherence conditions hold:

a_{X,Y,Z⊗T} a_{X⊗Y,Z,T} = (X ⊗ a_{Y,Z,T}) a_{X,Y ⊗Z,T} (a_{X,Y,Z} ⊗ T) ,
(X ⊗ l_{Y}) a_{X,I,Y} = r_{X} ⊗ Y .

If 𝔾, ℍ are Γ-graded monoidal categories, then a graded monoidal functor

F = (F, Φ, Φ_{∗}) : 𝔾 → ℍ ,

consists of a graded functor F : 𝔾 → ℍ, natural isomorphisms of grade 1,

Φ = Φ_{X,Y} : F X ⊗ F Y ⟶^{∼} F (X ⊗ Y) ,

and an isomorphism of grade 1 (natural with respect to the elements σ ∈ Γ)

Φ_{∗} : I ⟶^{∼} F I

(where I = I(∗)) such that, for all objects X, Y, Z ∈ 𝔾, the following coherence conditions hold:

Φ_{X,Y ⊗Z} (F X ⊗ Φ_{Y,Z}) a_{F X,F Y,F Z} = F (a_{X,Y,Z}) Φ_{X⊗Y,Z} (Φ_{X,Y} ⊗ F Z) ,  (1)
F (r_{X}) Φ_{X,I} (F X ⊗ Φ_{∗}) = r_{F X} , F (l_{X}) Φ_{I,X} (Φ_{∗} ⊗ F X) = l_{F X} .

Suppose F′ : 𝔾 → ℍ is also a graded monoidal functor. A homotopy (or graded monoidal natural equivalence) θ : F → F′ of graded monoidal functors is a graded natural equivalence θ : F →^{∼} F′ such that, for all objects X, Y ∈ 𝔾, the following coherence conditions hold:

Φ′_{X,Y} (θ_{X} ⊗ θ_{Y}) = θ_{X⊗Y} Φ_{X,Y} , θ_{I} Φ_{∗} = Φ′_{∗} .  (2)

For later use, we state here the lemma below [6, Lemma 1.1].

2.1. Lemma. Every graded monoidal functor F = (F, Φ, Φ_{∗}) : 𝔾 → ℍ is homotopic to a graded monoidal functor F′ = (F′, Φ′, Φ′_{∗}) with F′I = I and Φ′_{∗} = id_{I}.

A Γ-graded monoidal category, ℍ = (ℍ, gr, ⊗, I, a, l, r) is said to be a Γ-graded categorical group if every morphism is invertible; that is, ℍ is a groupoid, and for any object X there is an object X′ with a 1-morphism X ⊗ X′ → I. In subsequent sections several examples of Γ-graded categorical groups will be discussed. However, we shall describe now an elementary example, the discrete Γ-graded categorical group defined for any Γ-group G, which will be frequently used in this paper.

Let G be a Γ-group. Then, the discrete Γ-graded categorical group \underline{dis}^{Γ}G has the elements of G as objects and their morphisms σ : x → y are the elements σ ∈ Γ with σx = y. Composition is multiplication in Γ and the grading gr : \underline{dis}_{Γ}G → Γ is the obvious map gr(σ) = σ. The graded tensor product is given by

(x →^{σ} y) ⊗ (x′ →^{σ} y′) = (xx′ →^{σ} yy′) ,

and the graded unit I : Γ → \underline{dis}^{Γ}G by

I(∗ →^{σ} ∗) = (1 →^{σ} 1) ;

the associativity and unit isomorphisms are identities.

Each Γ-graded categorical group ℍ has two associated homotopy groups, which we shall describe below (see [6, Proposition 1.2] for details):

π_{0}ℍ = the Γ-group of 1-isomorphism classes of the objects in ℍ, where multiplication is induced by tensor product, [X] [Y] = [X ⊗ Y], and the Γ-action is defined by σ[X] = [Y] whenever there exists a morphism X → Y in ℍ of grade σ.

π_{1}ℍ = the Γ-equivariant π_{0}ℍ-module of 1-automorphism in ℍ of the unit object I, where the operation is composition, the Γ-action is ^{σ}u = I(σ) u I(σ)^{−1}, and the structure of π_{0}ℍ-module is as follows : if [X] ∈ π_{0}ℍ and u ∈ π_{1}𝔾, then ^{[X]}u is determined by the formula

δ(u) = γ(^{[X]}u),

where

\xymatrix{Aut_{1}(X) & π_{1}ℍ \ar[l]_{δ}^{∼} \ar[r]^{γ}_{∼} & Aut_{1}(X)}  (3)

are the isomorphisms defined respectively by δ(u) = r (X ⊗ u) r^{−1} and γ(u) = l (u ⊗ X) l^{−1}.

Thus, for example, if G is any Γ-group, then π_{0} \underline{dis}^{Γ}G = G, as a Γ-group, while π_{1} \underline{dis}^{Γ}G = 0.

For each Γ-group G and each Γ-graded categorical group ℍ, let

[\underline{dis}^{Γ}G, ℍ]

denote the set of homotopy classes of graded monoidal functors from \underline{dis}^{Γ}G to ℍ. There is a canonical map

π_{0} : [\underline{dis}^{Γ}G, ℍ] ⟶ Hom_{Γ}(G, π_{0}ℍ) [F] ↦ π_{0}F ,  (4)

where Hom_{Γ}(G, π_{0}ℍ) is the set of equivariant homomorphisms from the Γ-group G to the Γ-group π_{0}ℍ. We refer to a Γ-group homomorphism ρ : G → π_{0}ℍ as realizable whenever it is in the image of the above map π_{0}, that is, if ρ = π_{0}F for some graded monoidal functor F : \underline{dis}^{Γ}G → ℍ. The map (4) produces a partitioning

[\underline{dis}^{Γ}G, ℍ] = ⨆_{ρ} [\underline{dis}^{Γ}G, ℍ; ρ] ,

where, for each ρ ∈ Hom_{Γ}(G, π_{0}ℍ), [\underline{dis}^{Γ}G, ℍ; ρ] = π−1_{0}(ρ) is the set of homotopy classes of realizations of ρ.

When an equivariant homomorphism ρ : G → π_{0}ℍ is specified, it is possible that there is no graded monoidal functor that realizes ρ, and this leads to a problem of obstructions which (even in a more general form) is solved in [6, Theorem 3.2] by means of a 3dimensional equivariant group cohomology class Obs(ρ). Next we outline this theory for this particular case.

Let ρ : G → π_{0}ℍ be any given Γ-group homomorphism. Then, π_{1}ℍ is, via ρ, a Γequivariant G-module and so the equivariant cohomology groups Hn_{Γ}(G, π_{1}ℍ) are defined. The obstruction cohomology class

Obs(ρ) ∈ H3_{Γ}(G, π_{1}ℍ)  (5)

is represented by any 3-cocycle c^{ρ} ∈ Z3_{Γ}(G, π_{1}ℍ) built as follows. For each x ∈ G, let us choose an object F (x) ∈ ρ(x), with F (1) = I. Since ρ(xy) = ρ(x)ρ(y), for each x, y ∈ G, we can select a 1-morphism Φ_{x,y} : F (x) ⊗ F (y) → F (xy), with Φ_{1,y} = l and Φ_{x,1} = r. Furthermore, since ρ(^{σ}x) = ^{σ}ρ(x), for each x ∈ G and σ ∈ Γ, we can also select an σ-morphism Φ_{x,σ} : F (x) → F (^{σ}x), with Φ_{1,σ} = I(σ) and Φ_{x,1} = id. Then, the map

c^{ρ} : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) ⟶ π_{1}ℍ  (6)

is determined, for all x, y, z ∈ G and σ, τ ∈ Γ, by the commutativity of the diagrams in ℍ,

\xymatrix{(F (x) ⊗ F (y)) ⊗ F (z) \ar[d]_{a} \ar[r]^{Φ_{x,y}⊗id} & F (xy) ⊗ F (z) \ar[r]^{Φ_{xy,z}} & F (xyz) \ar@{-->}[d]^{γ(c^{ρ}(x,y,z))} \\ F (x) ⊗ (F (y) ⊗ F (z)) \ar[r]^{id⊗Φ_{y,z}} & F (x) ⊗ F (yz) \ar[r]^{Φ_{x,yz}} & F (xyz) ,}

\xymatrix{F (xy) \ar[rr]^{Φ_{xy,σ}} \ar@{-->}[d]_{γ(c^{ρ}(x,y,σ))} &  & F (^{σ}x^{σ}y) \ar[d]^{Φ_{^{σ}x,^{σ}y}^{−1}} \\ F (xy) \ar[r]^{Φ_{x,y}^{−1}} & F (x) ⊗ F (y) \ar[r]^{Φ_{x,σ}⊗Φ_{y,σ}} & F (^{σ}x) ⊗ F (^{σ}y) ,}

\xymatrix{F (x) \ar[r]^{Φ_{x,τ}} \ar@{-->}[d]_{γ(c^{ρ}(x,σ,τ))} & F (^{τ}x) \ar[d]^{Φ_{^{τ}x,σ}} \\ F (x) \ar[r]^{Φ_{x,στ}} & F (^{στ}x) ,}

where the γ’s are the isomorphisms defined in (3). The fact that c^{ρ} is an equivariant 3-cocycle, whose cohomology class

Obs(ρ) = [c^{ρ}] ∈ H3_{Γ}(G, π_{1}ℍ)

only depends on ρ, follows from [6, Theorem 3.2]. Also, as a consequence of this theorem, the following is true:

2.2. Theorem. Let G be a Γ-group and let ℍ be a Γ-graded categorical group. Then,

(i) A Γ-group homomorphism ρ : G → π_{0}ℍ is realizable, that is, [\underline{dis}^{Γ}G, ℍ; ρ] ≠ ∅, if, and only if, its obstruction Obs(ρ) vanishes.

(ii) If Obs(ρ) = 0, then there is a bijection

[\underline{dis}^{Γ}G, ℍ; ρ] ≅ H2_{Γ}(G, π_{1}ℍ).  (7)

The bijection (7), which is non-natural, can be described as follows (see [6] for details). By hypothesis, and using Lemma 2.1, there is a strictly unitary graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → ℍ realizing the equivariant homomorphism ρ. Then, once we have fixed such a (F, Φ), the bijection (7) associates to any equivariant 2-cohomology class [c] ∈ H2_{Γ}(G, π_{1}ℍ), represented by a 2-cocycle c : G^{2} ∪ (G × Γ) → π_{1}ℍ, the homotopy class of the strictly unitary graded monoidal functor (c · F, c · Φ) : \underline{dis}^{Γ}G → ℍ, where c · F is the graded functor sending a morphism σ : x → y of \underline{dis}^{Γ}G to the dotted morphism in the commutative diagram in ℍ

\xymatrix{F (x) \ar@{-->}[rr]^{(c·F)(σ)} \ar[dr]_{F (σ)} &  & F (y) \\  & F (y) , \ar[ur]_{γ(c(x,σ))}}

(γ is the isomorphism (3)), and the natural isomorphisms (c · Φ)_{x,y}, for x, y ∈ G, are the dotted morphisms in the commutative diagrams

\xymatrix{F (x) ⊗ F (y) \ar@{-->}[rr]^{(c·Φ)_{x,y}} \ar[dr]_{Φ_{x,y}} &  & F (xy) \\  & F (xy) . \ar@{-->}[ur]_{γ(c(x,y))}}

3. Equivariant extensions of monoids by groups

If M is any monoid and G is a group, then a perfect Schreier extension of M by G is a sequence of monoids and homomorphisms

E : M ↣^{i} N ↠^{p} G  (8)

which is exact in the sense that i is injective, p is surjective and i(M) = p^{−1}(1) and, moreover, such that the congruence on N defined by p is perfect, that is, p^{−1}(x)p^{−1}(y) = p^{−1}(xy), for all x, y ∈ G. Observe that, since 1 ∈ p^{−1}(x)p^{−1}(x^{−1}) for all x ∈ G, the monoid N contains at least one unit u_{x} such that p(u_{x}) = x. Hence, since the map of multiplication by u_{x} establishes a bijection i(M) ≅ p^{−1}(x), it follows that M ≅ i(M) is a normal divisor of N (cf. [22]) whose congruence on N coincides with the one defined by p. Therefore, N/M ≅ G.

The study of Schreier extensions of monoids goes back to Redei [33], and a precise classification theorem for perfect Schreier extensions of monoids by groups was stated in [7, 4.2]. Below, our goal is to study equivariant extensions of monoids by groups when a given group of operators is acting on both structures.

If Γ is a fixed group of operators, then a Γ-monoid is a monoid endowed with a Γ-action by automorphisms. An equivariant perfect Schreier extension (hereafter, an equivariant extension for short) of a Γ-monoid M by a Γ-group G is a perfect Schreier extension E as in (8) such that N is also a Γ-monoid and maps i and p are equivariant. Thus, M can be identified with an equivariant normal divisor of N and N/M ≅ G as Γ-groups. Two such extensions E and E′ are equivalent whenever there exists an equivariant isomorphism g : N → N′ such that gi = i′ and p′g = p, and we denote by

Ext^{Γ}(G, M)

the set of equivalence classes of equivariant extensions of the Γ-monoid M by the Γ-group G. This set is pointed by the class of the Γ-monoid product M × G.

Next, we shall explain an equivariant factor set theory for equivariant extensions of Γ-monoids M by Γ-groups G. Then, an equivariant factor set for an equivariant extension will be an appropriate set of data to rebuild (by an appropriate crossed product construction) the equivariant extension up to equivalence. Furthermore, as we will see later, these equivariant factor sets are essentially the same as graded monoidal functors from the discrete Γ-graded categorical group \underline{dis}^{Γ}G to the Γ-graded categorical group of holomorphims of the Γ-monoid M, \underline{Hol}^{Γ}(M). Thanks to that observation, we will able to use Theorem 2.2 in this context. To do so, we begin by describing the graded categorical group \underline{Hol}^{Γ}(M) and then we will state Theorem 3.1 below, in which proof we include the precise notion of equivariant factor set.

Let M be any Γ-monoid. Then, the group Aut(M) of all monoid automorphisms of M is a Γ-group under the diagonal Γ-action, f ↦ ^{σ}f, where ^{σ}f : m ↦ ^{σ}f(^{σ^{−1}}m), and the map

C : M^{∗} → Aut(M),

sending each unit u of M to the inner automorphism given by conjugation with u, C_{u} : m ↦ umu^{−1}, is a Γ-group homomorphism. The holomorph Γ-graded categorical group of the Γ-monoid M, \underline{Hol}^{Γ}(M), has the elements of the Γ-group Aut(M) as objects, and a morphism of grade σ ∈ Γ is a pair (u, σ) : f → g, where u ∈ M^{∗}, with ^{σ}f = C_{u}g. The composition of morphisms is given by

\xymatrix{f \ar[r]^{(u,σ)} & g \ar[r]^{(v,τ)} & h} = \xymatrix{f \ar[r]^{(^{τ}u v,τσ)} & h} ,  (9)

the graded tensor product is

\xymatrix{(f \ar[r]^{(u,σ)} & g) ⊗ (f′ \ar[r]^{(u′,σ)} & g′)} = \xymatrix{ff′ \ar[r]^{(u g(u′),σ)} & gg′} ,  (10)

and the graded unit I : Γ → \underline{Hol}^{Γ}(M) is defined by \xymatrix{I(σ) = id_{M} \ar[r]^{(1,σ)} & id_{M}} . The associativity and unit constraints are identities.

3.1. Theorem. (Equivariant crossed products theory for monoids with operators) For any Γ-monoid M and any Γ-group G, there is a bijection

Σ : [\underline{dis}^{Γ}G, \underline{Hol}^{Γ}(M)] ≅ Ext^{Γ}(G, M) ,  (11)

between the set of homotopy classes of graded monoidal functors from \underline{dis}^{Γ}G to \underline{Hol}^{Γ}(M) and the set of equivalence classes of equivariant extensions of M by G.

Proof. By Lemma 2.1, every graded monoidal functor is homotopic to a given (F, Φ, Φ_{∗}) in which F I = I and Φ_{∗} = id_{I}, and so we can restrict our attention to this kind of graded monoidal functors, (F, Φ) : dis^{Γ}G → \underline{Hol}^{Γ}(M). Then, let us start by observing that the data describing such a graded monoidal functor consist of a pair of maps (f, φ), where

f : G ⟶ Aut(M) , φ : G^{2} ∪ (G × Γ) ⟶ M^{∗} ,  (12)

such that we write F (x) = f_{x}, F ( \xymatrix{x \ar[r]^{σ} & ^{σ}x} ) = \xymatrix{f_{x} \ar[r]^{(φ(x,σ),σ)} & f_{^{σ}x}} and Φ_{x,y} = \xymatrix{f_{x}f_{y} \ar[r]^{(φ(x,y),1)} & f_{xy}} ,

for all x, y ∈ G and σ ∈ Γ. When we attempt to write the conditions of (F, Φ) being a graded monoidal functor in terms of (f, φ), then we find the following conditions for (f, φ):

f_{1} = id, φ(x, 1) = 1 = φ(1, y) ,  (13)
f_{x}f_{y} = C_{φ(x,y)} f_{xy} , ^{σ}f_{x} = C_{φ(x,σ)} f_{^{σ}x} ,  (14)
φ(x, y) φ(xy, z) = f_{x}(φ(y, z)) φ(x, yz) ,  (15)
σφ(x, y) φ(xy, σ) = φ(x, σ) f_{^{σ}x}(φ(y, σ)) φ(^{σ}x,^{σ}y) ,  (16)
φ(x, στ) = ^{σ}φ(x, τ) φ(^{τ}x, σ) ,  (17)

for all x, y, z ∈ G, σ, τ ∈ Γ. To prove this in full several verifications are needed, but they are straightforward: conditions (14) say that (φ(x, y), 1) and (φ(x, σ), σ) are, respectively, morphisms in \underline{Hol}^{Γ}M; (15) expresses the coherence condition (1), while (16) means that the isomorphisms Φ_{x,y} are natural and (17) that F preserves the composition of morphisms. The normalization condition (13) says that F preserves both identities and the unit object.

We call an equivariant factor set, or a non-abelian 2-cocycle of the Γ-group G with coefficients in the Γ-monoid M, any pair (f, φ) as in (12) satisfying the conditions (13)- (17). We should stress that when Γ = 1 (the trivial group), a factor set is just the same as a Redei system of factor sets for a Schreier extension of the monoid M by the group G [33, 16]. In particular, when M is a group, we have precisely a Schreier factor system for a group extension [35]. Furthermore, if M and G are any two Γ-groups, then such a pair (f, φ) is a Schreier system of factor sets for an equivariant group extension of M by G [6].

Suppose now that (f′, φ′) describes another graded monoidal functor (F′, Φ′) : \underline{dis}^{Γ}G → \underline{Hol}^{Γ}M. Then, any homotopy θ : F → F′ is given by a map

g : G → M^{∗},

such that one writes θ_{x} = \xymatrix{f_{x} \ar[r]^{(g(x),1)} & f′_{x}} for all x ∈ G. In terms of map g, the conditions for θ to be a homotopy are:

g(1) = 1  (18)
f_{x} = C_{g(x)} f′_{x} ,  (19)
φ(x, y) g(xy) = g(x) f′_{x}(g(y)) φ′(x, y) ,  (20)
φ(x, σ) g(^{σ}x) = ^{σ}g(x) φ′(x, σ) ,  (21)

for all x, y ∈ G, σ ∈ Γ. Condition (19) expresses that θ_{x} is a morphism in \underline{Hol}_{Γ}(M) from f_{x} to f′_{x}, (21) is the naturalness of θ and (20) and (18) say that the coherence conditions (2) hold.

We now are ready to prove bijection (11).

Every equivariant factor set (f, φ) as in (12) gives rise to an equivariant crossed product extension of M by G,

Σ(f, φ) : M ↣^{i} M×_{f,φ}G ↠^{p} G ,  (22)

in which M×_{f,φ}G is Rédei’s Schreier product monoid defined in [33]; that is, it is the cartesian product set M × G, with multiplication given by (m, x)(n, y) = (m f_{x}(n) φ(x, y), xy). This multiplication is associative and unitary thanks to equalities (14), (16) and (13). The Γ-action on M×_{f,φ}G is given by ^{σ}(m, x) = (^{σ}m φ(x, σ),^{σ}x) which, owing to (14), (16) and (17), satisfies the required conditions in order for M×_{f,φ}G to be a Γ-monoid. The Γequivariant monoid homomorphisms i and p are given by i(m) = (m, 1) and p(m, x) = x. Since (m, xy) = (mφ(x, y)^{−1}, x)(1, y) ∈ p^{−1}(x)p^{−1}(y), we see that p^{−1}(xy) = p^{−1}(x)p^{−1}(y), and therefore the congruence is perfect. Thus, Σ(f, φ) is an equivariant extension of the Γ-monoid M by the Γ-group G.

Suppose (f′, φ′) is another equivariant factor set. If there is a Γ-monoid isomorphism, say g : M×_{f,φ}G ≅ M×_{f′,φ′}G, establishing an equivalence between the corresponding equivariant crossed product extensions of M by G, then we can write g in the form g(m, x) = g(m, 1)g(1, x) = (m, 1)(g(x), x) = (mg(x), x) for a map g : G → M^{∗}. Since g((1, x)(m, 1)) = g(f_{x}(m), x) = (f_{x}(m) g(x), x), while g(1, x) g(m, 1) = (g(x) f′_{x}(m), x), it follows that f_{x} = C_{g(x)} f′_{x}; that is, (19) holds. Analogously, we see that (20) follows from the equality g((1, x)(1, y)) = g(1, x)g(1, y) and (21) from the equality g(^{σ}(1, x)) = σg(1, x). Therefore, g defines a homotopy between (the graded monoidal functors defined by) (f′, φ′) and (f, φ). Conversely, if (f, φ) and (f′, φ′) are made homotopic by a g : G → M^{∗}, thus satisfying (18)-(21), then they lead to isomorphic equivariant crossed product extensions, just by the map g : (m, x) ↦ (mg(x), x), as we can see by retracing our steps.

Finally, we prove that any equivariant extension of M by G, E : M ↣^{i} N ↠^{p} G, has an associated equivariant factor set, that is, it is equivalent to an equivariant crossed product like (22), for some equivariant factor set (f, φ). To do so, there is no loss of generality in assuming that i is the inclusion map. Let us choose then, for each x ∈ G, a unit u_{x} ∈ p^{−1}(x) ∩ N^{∗}. We take u_{1} = 1. Hence, the maps m ↦ mu_{x} establish bijections M = p^{−1}(1) ≅ p^{−1}(x), for all x ∈ G. Since p(u_{x}u_{y}) = xy = p(u_{xy}) and p(^{σ}u_{x}) = ^{σ}x = p(u_{^{σ}x}), it follows that there are unique elements φ(x, y), φ(x, σ) ∈ M, such that u_{x} u_{y} = φ(x, y) u_{xy} and ^{σ}u_{x} = φ(x, σ) u_{^{σ}x}, for all x, y ∈ G and σ ∈ Γ. Note that φ(x, y), φ(x, σ) ∈ M^{∗} since the u_{x} are units. Moreover, each x ∈ G induces an automorphism f_{x} of M, f_{x} : m ↦ u_{x} m u_{x}^{−1}, and this pair of maps

f : G → Aut(M), φ : G^{2} ∪ (G × Γ) → M^{∗} ,

is actually an equivariant factor set. Indeed, conditions (13) and (14) follow immediately from the definition of (f, φ). To observe the remaining conditions (15)-(17), note that every element of N has a unique expression of the form m u_{x}, with m ∈ M and x ∈ G. Because u_{x} m = f_{x}(m) u_{x}, it follows that the Γ-monoid structure of N can be described, in terms of the Γ-group structure of G, the Γ-monoid structure of M and the pair (f, φ), by (m u_{x}) (m′u_{y}) = m f_{x}(m′) φ(x, y) u_{xy} and ^{σ}(m u_{x}) = ^{σ}m φ(x, σ) u_{^{σ}x}. Then, (15) follows from the associative law u_{x}(u_{y}u_{z}) = (u_{x}u_{y})u_{z} in M, (16) follows from the equality σ(u_{x}u_{y}) = ^{σ}u_{x}^{σ}u_{y}, while (17) is a consequence of the equality^{τ}(^{σ}u_{x}) = ^{τσ}u_{x}. Hence, (f, φ) defines an equivariant factor set, whose associated equivariant crossed product extension Σ(f, φ) is equivalent to E by the existence of the Γ-monoid isomorphism M×_{f,φ}G ≅ N, (m, x) ↦ m u_{x}.

Let us now observe that, for any Γ-monoid M, the homotopy groups of the Γ-graded categorical group \underline{Hol}^{Γ}(M) are:

π_{0} \underline{Hol}^{Γ}(M) = Coker(M^{∗} ⟶^{C} Aut(M)) = Aut(M)/In(M) = Out(M) , the group of outer automorphisms of M, which is a Γ-group with action ^{σ}[f] = [^{σ}f], where ^{σ}f : m ↦ ^{σ}f(^{σ^{−1}}m),

π_{1} \underline{Hol}^{Γ}(M) = Ker(M^{∗} ⟶^{C} Aut(M)) = Z(M)^{∗} the abelian group of units in the centre of the monoid M, which is a Γ-submonoid of M and a Γ-equivariant Out(N)-module with action ^{[f]}u = f(u), f ∈ Aut(M), u ∈ Z(M)^{∗}.

Then, owing to bijection (11) and map (4), each equivalence class [E] of an equivariant extension E : M ↣ N ↠ G of the Γ-monoid M by the Γ-group G, determines a Γequivariant group homomorphism

ρ_{[E]} : G → Out(M) ,

by ρ_{[E]}(x) = [f_{x}], where (f, φ) is any equivariant factor set for E. A pair (M, ρ), where M is a Γ-monoid and ρ : G → Out(M) is a Γ-group homomorphism, is termed an equivariant G-kernel for an equivariant monoid extension (or an equivariant collective character of G in M, cf. [14, 25, 7]), and an equivariant extension of an equivariant G-kernel (M, ρ) is an equivariant extension E of M by G that realizes ρ, that is, with ρ_{[E]} = ρ. We denote by

Ext^{Γ}(G, (M, ρ))

the set of equivalence classes of such equivariant extensions of (M, ρ). Hence, there is a canonical partition

Ext^{Γ}(G, M) = ⨆_{ρ} Ext^{Γ}(G, (M, ρ)) .

Since, for each equivariant homomorphism ρ : G → Out(M), the bijection (11) restricts to a bijection Σ : [\underline{dis}^{Γ}G, \underline{Hol}^{Γ}(M); ρ] ≅ Ext^{Γ}(G, (M, ρ)), Theorem 2.2 i) shows, in this particular case, a necessary and sufficient condition for the set Ext^{Γ}(G, (M, ρ)) to be non-empty in terms of a 3-dimensional equivariant cohomology class (5)

Obs(M, ρ) ∈ H3_{Γ}(G, Z(M)^{∗}) ,

of the Γ-group G with coefficients in the equivariant G-module Z(M)^{∗}, where the G-action is via ρ (i.e., ^{x}u = ^{ρ(x)}u). We refer to this invariant as the obstruction of the equivariant G-kernel, and it is not hard to see that the general construction (6) of an equivariant 3-cocycle c^{ρ}, representing in this case the cohomology class Obs(M, ρ), particularizes as follows. In each automorphism class ρ(x), x ∈ G, let us choose an automorphism f_{x} of M; in particular, we select f_{1} = id_{M}. Since f_{x}f_{y} ∈ ρ(xy) and f_{^{σ}x} ∈ ^{σ}ρ(x), for x, y ∈ G and σ ∈ Γ, we can select elements φ(x, y), φ(x, σ) ∈ M^{∗}, such that f_{x}f_{y} = C_{φ(x,y)} f_{xy} and σf_{x} = C_{φ(x,σ)} f_{^{σ}x}, with φ(x, 1) = 1 = φ(1, y) = φ(1, σ). Then, the pair of maps

f : G → Aut(M), φ : G^{2} ∪ (G × Γ) → M^{∗} ,

satisfies conditions (13) and (14), although (15)-(17) need not be satisfied. The measurement of such a lack is precisely given by the equivariant 3-cocyle

c^{ρ} : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) ⟶ Z(M)^{∗}

determined by the equations:

\begin{matrix}c^{ρ}(x, y, z) φ(x, y) φ(xy, z) = f_{x}(φ(y, z)) φ(x, yz) , \\ c^{ρ}(x, y, σ) ^{σ}φ(x, y) φ(xy, σ) = φ(x, σ) f_{^{σ}x}(φ(y, σ)) φ(^{σ}x,^{σ}y) , \\ c^{ρ}(x, σ, τ) φ(x, στ) = ^{σ}φ(x, τ) φ(^{τ}x, σ) , \end{matrix}  (23)

for x, y, z ∈ G, σ, τ ∈ Γ.

If Obs(M, ρ) = 0, so that there exists an equivariant crossed product extension Σ(f, φ) of M by G that realizes ρ, then Theorem 2.2 ii) states the existence of a bijection H2_{Γ}(G, Z(M)^{∗}) ≅ Ext^{Γ}(G, (M, ρ)). In this case, such a bijection associates to any equivariant 2-cohomology class represented by a 2-cocycle c : G^{2} ∪ (G × Γ) → Z(M)^{∗}, the equivalence class of the equivariant crossed product monoid extension Σ(f, c · φ), where (c · φ)(x, y) = c(x, y) φ(x, y) and (c · φ)(x, σ) = c(x, σ) φ(x, σ), for all x, y ∈ G and σ ∈ Γ.

In summary, we have the following classification theorem for equivariant perfect Schreier extensions of a Γ-monoid by a Γ-group.

3.2. Theorem. Let M be a Γ-monoid and G a Γ-group.

(i) There is a canonical partition Ext^{Γ}(G, M) = ⨆_{ρ} Ext^{Γ}(G, (M, ρ)), where, for each Γ-equivariant homomorphism ρ : G → Out(M), Ext_{Γ}(G, (M, ρ)) is the set of classes of equivariant extensions of the equivariant G-kernel (M, ρ).

(ii) Each equivariant G-kernel (M, ρ) invariably determines a 3-dimensional cohomology class Obs(M, ρ) ∈ H3_{Γ}(G, Z(M)^{∗}).

(iii) An equivariant G-kernel (M, ρ) is realizable, that is, Ext^{Γ}(G, (M, ρ)) ≠ ∅, if and only if its obstruction Obs(M, ρ) vanishes.

(iv) If the obstruction of an equivariant G-kernel, (M, ρ), vanishes, then there is a bijection Ext^{Γ}(G, (M, ρ)) ≅ H2_{Γ}(G, Z(M)^{∗}).

To end this section, we should remark that the treatment of group extensions with operators as expounded in [6] (cf. [37]), is particularized from the above by requiring simply that the Γ-monoid M be a Γ-group.

4. Equivariant extensions of RINGS-GROUPS

Let us briefly recall from [25] that a RING-GROUP is a pair [R, M] consisting of a unitary ring R together with a subgroup M of the group R^{∗} of all units in R. A morphism of RINGS-GROUPS f : [R, M] → [S, N] is then a ring homomorphism f : R → S such that f(M) ⊆ N. If [R, M] is any RING-GROUP and G is a group, then an extension of [R, M] by G is a pair

E = ([R, M] ↣^{i} [S, N], N ↠^{p} G) ,  (24)

in which i is an injective RING-GROUP morphism and p is a surjective group homomorphism, such that the following conditions hold:

i) The action of N on S by inner automorphisms is restricted to an action of N on R.

ii) M ↣^{i} N ↠^{p} G is an extension of groups.

iii) S decomposes as a direct sum of subgroups S = ⊕_{x∈G}S_{x}, where, for each x ∈ G, S_{x} = Rp^{−1}(x)R is the (R,R)-subbimodule of S generated by the set of elements of N mapped by p to x.

For any group M, isomorphism classes of extensions of the RING-GROUP [ℤ(M), M] by a group G are in bijection with isomorphism classes of group extensions of M by G; while for any ring R, isomorphism classes of extensions of the RING-GROUP [R, R^{∗}] by a group G are in bijection with isomorphism classes of crossed product rings of R by G [25, Example 1.6].

Here, we deal with RINGS-GROUPS with operators and their equivariant extensions by a group with the same operators. Let Γ be any fixed group. Then a Γ-RING-GROUP is a RING-GROUP [R, M] on which Γ acts by RING-GROUP automorphisms, that is, a homomorphism Γ → Aut[R, M] is given. Note that R is then a Γ-ring and M is an equivariant subgroup of the Γ-group of units R^{∗}.

Given a Γ-RING-GROUP [R, M] and a Γ-group G, then an equivariant extension of [R, M] by G is an extension E of the RING-GROUP [R, M] by the group G, say (24), in which [S, N] is endowed with a Γ-action, that is, [S, N] is a Γ-RING GROUP, such that both maps i : R ↣ S and p : N ↠ G are equivariant. We denote by

Ext^{Γ}(G, [R, M])

the set of isomorphism classes of Γ-equivariant extensions of [R, M] by G.

Next we show a cohomological solution for the problem of classifying all equivariant RING-GROUP extensions for any prescribed pair (G, [R, M]). The treatment is parallel to the known theory [24, 25] in the case in which Γ = 1, the trivial group. However, the proofs here bear only an incidental similarity with those by Hacque, since we derive the results on equivariant RING-GROUP extensions from the results obtained on the classification of graded categorical groups, that is, from Theorem 2.2.

Our conclusions can be summarized as follows.

If [R, M] is a Γ-RING-GROUP, then the group Aut[R, M] of all RING-GROUP automorphisms of [R, M] is also a Γ-group under the diagonal Γ-action, ^{σ}f : r ↦ ^{σ}f(^{σ^{−1}}r), and the map C : M → Aut[R, M] sending each element m ∈ M into the inner automorphism given by conjugation with m, C_{m} : r ↦ mrm^{−1}, is a Γ-group homomorphism. Then, the centre-group [24, Definition 2.3] of [R, M], Z[R, M] = Ker(C), and the group of outer automorphisms Out[R, M] = Aut[R, M]/In[R, M] = Coker(C), are both Γ-groups. Furthermore, Z[R, M] is a Γ-equivariant Out[R, M]-module with action ^{[f]}m = f(m).

If (24) is any equivariant extension of the Γ-RING-GROUP [R, M] by the Γ-group G, then the assignment to each n ∈ N of the operation of conjugation by n in S restricted to R, that is, the mapping n ↦ ρ_{n} ∈ Aut[R, M] such that iρ_{n}(r) = n i(r) n^{−1}, r ∈ R, induces an equivariant homomorphism (which depends only on the isomorphism class of E),

ρ_{[E]} : G → Out[R, M].

A pair ([R, M], ρ : G → Out[R, M]), where [R, M] is a Γ-RING-GROUP and ρ is an equivariant homomorphism of groups, is termed an equivariant collective character of the Γ-group G in [R, M] (cf. [25, Section 2]).

We state the following theorem:

4.1. Theorem. Let [R, M] be a Γ-RING-GROUP and let G be a Γ-group. (i) There is a canonical partition

Ext^{Γ}(G, [R, M]) = ⨆_{ρ} Ext^{Γ}(G, [R, M]; ρ) ,

where, for each equivariant collective character ρ : G → Out[R, M], Ext^{Γ}(G, [R, M]; ρ) is the set of isomorphism classes of equivariant extensions E that realize ρ, that is, such that ρ_{[E]} = ρ.

(ii) Each equivariant collective character invariably determines a 3-dimensional equivariant cohomology class

T(ρ) ∈ H3_{Γ}(G, Z[R, M]) ,

of G with coefficients in the centre-group of the RING-GROUP (with respect to the Γequivariant G-module structure on Z[R, M] obtained via ρ). This invariant is called the “Teichmüller obstruction” of ρ.

(iii) An equivariant collective character ρ is realizable, that is, Ext^{Γ}(G, [R, M]; ρ) ≠ ∅, if and only if its obstruction vanishes, that is, T(ρ) = 0.

(iv) If the obstruction of an equivariant collective character ρ : G → Out[R, M] vanishes, then there is a bijection

Ext^{Γ}(G, [R, M]; ρ) ≅ H2_{Γ}(G, Z[R, M]) .

As we will show later, Theorem 4.1 is a specialization of Theorem 2.2 for the particular Γ-graded categorical group ℍ = \underline{Hol}^{Γ}[R, M], the holomorph graded categorical group of the Γ-RING-GROUP [R, M], which is defined below similarly to the holomorph graded categorical group of a Γ-monoide (see Section 3). The objects of \underline{Hol}^{Γ}[R, M] are the elements of the Γ-group Aut[R, M]. A morphism of grade σ ∈ Γ from f to g is a pair (m, σ) : f → g, where m ∈ M, with ^{σ}f = C_{m} g. The composition of morphisms is given by equality (9), the graded tensor product is given by equality (10) and the graded

unit I : Γ → \underline{Hol}^{Γ}[R, M] is defined by I(σ) = id_{R} ⟶^{(1,σ)} id_{R}. The associativity and unit constraints are identities.

We develop next the device of mixed crossed products for equivariant extensions of RINGS-GROUPS by groups, such as Hacque did in [25] in the non-equivariant case. These constructions allow us to show how the graded monoidal functors \underline{dis}^{Γ}G → \underline{Hol}^{Γ}[R, M] are the appropriate systems of data to construct the manifold of all equivariant extensions of [R, M] by G.

4.2. Theorem. (Structure of equivariant extensions of RINGS-GROUPS) For any Γ-RING-GROUP [R, M] and Γ-group G, there is a bijection

Σ : [\underline{dis}^{Γ}G, \underline{Hol}^{Γ}[R, M]] ≅ Ext^{Γ}(G, [R, M]) ,  (25)

between the set of homotopy classes of graded monoidal functors from \underline{dis}^{Γ}G to \underline{Hol}^{Γ}[R, M] and the set of equivalence classes of equivariant extensions of [R, M] by G.

Proof. This is quite parallel to the proof of Theorem 3.1, to which we refer for some details. First, observe that the data describing a strictly unitary graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → \underline{Hol}^{Γ}[R, M] consist of a pair of maps (f, φ), where

f : G → Aut[R, M], φ : G^{2} ∪ (G × Γ) → M,

such that we write F (σ : x → y) = (φ(x, σ), σ) : f_{x} → f_{y} and Φ_{x,y} = (φ(x, y), 1) : f_{x}f_{y} → f_{xy}, for all x, y ∈ G and σ ∈ Γ. Then, the conditions of (F, Φ) being a graded monoidal functor in terms of (f, φ), are precisely those given by the five equations (13)-(17). A pair (f, φ) describing a graded monoidal functor from \underline{dis}^{Γ}G to \underline{Hol}^{Γ}[R, M], is what we call an equivariant factor set, or a non-abelian 2-cocycle of a Γ-group G with coefficients in a Γ-RING-GROUP [R, M]. Further, if (f′, φ′) is another factor set describing the graded monoidal functor (F′, Φ′) : \underline{dis}^{Γ}G → \underline{Hol}^{Γ}[R, M], then any homotopy θ : F → F′ is given by a map g : G → M, such that one writes θ_{x} = (g(x), 1) : f_{x} → f′_{x}, for all x ∈ G. In terms of this map g, the conditions for θ being a homotopy are those described by the four equations (18)-(21).

Let us stress that when Γ = 1, the trivial group, then a factor set is exactly a Hacque’s mixed system of factor sets for a RING-GROUP extension of [R, M] by G [25]. Particularly, if M = R^{∗}, then such a pair (f, φ) is just a factor set for a crossed product ring of R by G (cf. [29]). Furthermore, for M and G any two Γ-groups, if one considers the Γ-RING-GROUP [R, M] = [ℤ(M), M], then a pair (f, φ) is a Schreier system of factor sets for an equivariant group extension of M by G (cf. [6]).

Every equivariant factor set (f, φ) provides a Γ-equivariant RING-GROUP extension of [R, M] by G,

Σ(f, φ) : ([R, M] ↣^{i} [R⋆_{f,φ}G, M×_{f,φ}G], M×_{f,φ}G ↠^{p} G) ,  (26)

which we term an equivariant mixed crossed product extension. In it, the crossed product ring R⋆_{f,φ}G = ⊕_{x} R×{x} is the free left R-module with basis {(1, x), x ∈ G}, with multiplication according to the rule (r, x) (s, y) = (r f_{x}(s) φ(x, y), xy), and the crossed product group M×_{f,φ}G is the subgroup M × G ⊆ (R⋆_{f,φ}G)^{∗}. The Γ-action on [R⋆_{f,φ} G, M×_{f,φ}G] is given by ^{σ}(r, x) = (^{σ}r φ(x, σ),^{σ}x) and the equivariant homomorphisms i and p are defined by i(r) = (r, 1) and p(r, x) = x, respectively. Note that, for each x ∈ G, Rp^{−1}(x)R = R × {x}.

If (f′, φ′) is another equivariant factor set such that there is an equivariant RING-GROUP isomorphism, say g : [R⋆_{f′,φ′}G, M×_{f′,φ′}G] → [R⋆_{f,φ}G, M×_{f,φ}G], establishing an isomorphism between the corresponding equivariant mixed crossed product extensions Σ(f′, φ′) and Σ(f, φ), then we can write g(r, x) = (rg(x), x) for a map g : G → M satisfying the equalities (18)-(21). And conversely, the existence of a map g : G → M satisfying such equations implies that Σ(f′, φ′) and Σ(f, φ) are isomorphic equivariant RING-GROUP extensions, just by the map g : (r, x) ↦ (r g(x), x). Hence, the equivariant mixed crossed product construction induces an injective map Σ : [\underline{dis}^{Γ}G, \underline{Hol}^{Γ}[R, M]] ⟶ Ext^{Γ}(G, [R, M]) .

To complete the proof of the theorem, it only remains to prove that every equivariant extension of [R, M] by G, say E = ([R, M] ↣^{i} [S, N], N ↠^{p} G), has an associated equivariant factor set, that is, it is isomorphic to an equivariant mixed crossed product extension Σ(f, φ) for some (f, φ) as above. To do so, we assume that i is the inclusion map, and, for each x ∈ G, let us choose a representative u_{x} ∈ N, with u_{1} = 1. Then, there are unique elements φ(x, y), φ(x, σ) ∈ M, such that u_{x} u_{y} = φ(x, y) u_{xy} and ^{σ}u_{x} = φ(x, σ) u_{^{σ}x}. Moreover, each x ∈ G induces an automorphism f_{x} of [R, M], f_{x} : r ↦ u_{x} r u_{x}^{−1}. The pair of maps so defined (f : G → Aut[R, M], φ : G^{2} ∪ (G × Γ) → M) is actually an equivariant factor set, which we recognize as an equivariant factor set for the given equivariant RING-GROUP extension by the existence of the Γ-equivariant RING-GROUP isomorphism g : [R⋆_{f,φ}G, M×_{f,φ}G] → [S, N], ∑_{x}(r_{x}, x) ↦ ∑_{x} r_{x} u_{x}. It is straightforward to see that g is an equivariant RING-GROUP homomorphism. To prove that it is actually an isomorphism, let us recall that S = ⊕_{x}S_{x} with S_{x} = Rp^{−1}(x)R. Since for each x ∈ G, p^{−1}(x) = u_{x}M = Mu_{x}, we have S_{x} = Ru_{x}MR = u_{x}RMR = u_{x}R = Ru_{x}. It follows that the set {u_{x}, x ∈ G} is a basis of S as a left R-module and therefore the map g : R⋆_{f,φ}G → S is a bijection, which clearly restricts to a bijection between M×_{f,φ}G and N.

The bijection (25) is all one needs to obtain, from Theorem 2.2, the classification of equivariant RING-GROUP extensions as stated in Theorem 4.1. Indeed, for any Γ-RING-GROUP [R, M], the homotopy groups of the Γ-graded categorical group \underline{Hol}^{Γ}[R, M] are

π_{0} \underline{Hol}^{Γ}[R, M] = Coker(M ⟶^{C} Aut[R, M]) = Out[R, M],

π_{1} \underline{Hol}^{Γ}[R, M] = Ker(M ⟶^{C} Aut[R, M]) = Z[R, M],

and the Teichmüller obstruction T(ρ) ∈ H3_{Γ}(G, Z[R, M]) in part (ii) of Theorem 4.1, of any equivariant collective character ρ : G → Out[R, M], is defined to be precisely the cohomology class Obs(ρ) in (5) for the particular graded categorical group here considered ℍ = \underline{Hol}^{Γ}[R, M].

Moreover, for any factor set (f, φ) describing a graded monoidal functor F : \underline{dis}^{Γ}G → \underline{Hol}^{Γ}[R, M], we have π_{0}F = ρ_{[Σ(f,φ)]}. Therefore, bijection (25) provides, by restriction, bijections Σ : [\underline{dis}^{Γ}G, \underline{Hol}^{Γ}[R, M]; ρ] ≅ Ext^{Γ}(G, [R, M]; ρ), for any equivariant collective character ρ : G → Out[R, M]. Hence, parts (iii) and (iv) of Theorem 4.1 follow from Theorem 2.2.

For any given equivariant collective character ρ : G → Out[R, M], the general construction of a representative equivariant 3-cocycle c^{ρ} of the Teichmüller obstruction T(ρ) in (6) works in this case as follows. In each automorphism class ρ(x), x ∈ G, let us choose an automorphism f_{x} of [R, M]; in particular, select f_{1} = id_{R}. Then, for x, y ∈ G and σ ∈ Γ, we can select elements φ(x, y), φ(x, σ) ∈ M such that f_{x}f_{y} = C_{φ(x,y)} f_{xy} and σf_{x} = C_{φ(x,σ)} f_{^{σ}x}, with φ(x, 1) = 1 = φ(1, y) = φ(1, σ). Then, the equivariant 3-cocycle

c^{ρ} : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) ⟶ Z[R, M]

is determined by the three equations (23).

If T(ρ) = 0, so that there exists an equivariant mixed crossed product extension Σ(f, φ) that realizes ρ, then the bijection Ext^{Γ}(G, [R, M]; ρ) ≅ H2_{Γ}(G, Z[R, M]) associates to any 2-cohomology class represented by a 2-cocycle c : G^{2} ∪ (G × Γ) → Z[R, M], the isomorphism class of the equivariant mixed crossed product Σ(f, c·φ), where (c·φ)(x, y) = c(x, y) φ(x, y) and (c · φ)(x, σ) = c(x, σ) φ(x, σ), for all x, y ∈ G and σ ∈ Γ.

As a final comment here, we shall stress that, similarly as in the non-equivariant case (cf. [25]), the theory presented of equivariant RING-GROUP extensions particularizes on the one hand to the theory of equivariant extensions of groups with operators as stated in [6] (just by considering the RINGS-GROUPS with operators [ℤ(M), M] for any group with operators M), and, on the other hand, to the theory of crossed product rings with operators (simply by considering the RINGS-GROUPS with operators [R, R^{∗}] for any ring with operators R).

5. Graded Clifford systems with operators

Throughout this section, k is a unitary commutative ring and all algebras are unitary kalgebras. Further, Γ is a fixed group of operators and a Γ-algebra is an algebra provided with a Γ-action by automorphisms (σ, r) ↦ ^{σ}r.

A graded Clifford system (over k) is a triple (S, {S_{x}}_{x∈G}, G), where S is an algebra, G is a group and {S_{x}}_{x∈G} is a family of k-submodules of S, one for each x ∈ G, such that S = ⨁_{x∈G} S_{x} and S_{x}S_{y} = S_{xy} for all x, y ∈ G. A morphism of graded Clifford systems is a pair (h, α) : (S, {S_{x}}_{x∈G}, G) → (S′, {S′_{x}}_{x∈G′}, G′), where h : S → S′ is a homomorphism of algebras and α : G → G′ is a homomorphism of groups, such that h(S_{x}) ⊆ S′_{α(x)} for all x ∈ G. Graded Clifford systems were introduced and applied by E. Dade [12] to develop Clifford’s theory axiomatically, but they have also been extensively studied with the terminology of strongly graded algebras [13] or with that of generalized crossed product algebras [32]. Our goal in this section is the classification of graded Clifford systems \underline{S} on which a group of operators Γ acts by automorphisms; that is, a homomorphism Γ → Aut(S, {S_{x}}_{x∈G}, G) is given. In such a case, we refer to (S, {S_{x}}_{x∈G}, G) as a graded Γ-Clifford system.

Whenever (S, {S_{x}}_{x∈G}, G) is a graded Γ-Clifford system, then G is a Γ-group and S is a Γ-algebra. Furthermore, the 1-component S_{1} is a Γ-subalgebra of S and then it is natural to see S as an equivariant extension of the Γ-algebra S_{1} by the Γ-group G. More precisely, we establish the notion below:

5.1. Definition. Let R be a Γ-algebra and G a Γ-group. A (G, R)-graded Γ-Clifford system is a pair (\underline{S}, j_{S}), where \underline{S} = (S, \{S_{x}\}_{x∈G}, G) is a graded Γ-Clifford system such that the Γ-action on G is the given one, that is, ^{σ}S_{x} = S_{^{σ}x}, and j_{S} : R ≅ S_{1} is an isomorphism of Γ-algebras.

The most basic example is the group algebra R(G) = ⨁_{x∈G} R × {x} , which is a Γ-algebra with the action ^{σ}(r, x) = (^{σ}r, ^{σ}x), but also, equivariant crossed products of Γ-algebras by Γ-groups (considered in Subsection 5.4), yield examples of (G, R)-graded Γ-Clifford systems.

If (\underline{S}, j_{S}) and (\underline{T} , j_{T}) are two (G, R)-graded Γ-Clifford systems, by a morphism h : (\underline{S}, j_{S}) → (\underline{T} , j_{T}) we mean a Γ-equivariant homomorphism of algebras h : S → T, so that h(^{σ}s) = ^{σ}h(s) for σ ∈ Γ and s ∈ S, which is grade-preserving, in the sense that h(S_{x}) ⊆ T_{x} for all x ∈ G, and such that hj_{S} = j_{T}. It follows from [13, Corollary 2.10] that such a morphism is always an isomorphism and

Cliff_{k}^{Γ}(G, R)

will denote the set of isomorphism classes of (G, R)-graded Γ-Clifford systems, for any prescribed Γ-algebra R and Γ-group G.

The general results about abstract graded categorical groups can be used now to obtain an appropriate treatment of the equivariant graded Clifford systems. The key for such a treatment lies in the observation we will detail below that giving a (G, R)-graded Γ-Clifford system is essentially the same as giving a Γ-graded monoidal functor from the discrete Γ-graded categorical group \underline{dis}^{Γ}G to the Γ-graded Picard categorical group, of the Γ-algebra R, denoted here by

\underline{Pic}_{k}^{Γ}(R) ,

which is an example of Γ-graded categorical group originally considered by Fröhlich and Wall in [20]. This graded categorical group, canonically built from any Γ-algebra R, has, as objects, the invertible (R, R)-bimodules over k. A morphism P → P′ in \underline{Pic}_{k}^{Γ}(R), of grade σ ∈ Γ, is a pair (φ, σ), where φ : P →^{∼} P′ is an isomorphism of k-modules with φ(rp) = ^{σ}r φ(p) and φ(pr) = φ(p)^{σ}r for all r ∈ R and p ∈ P. The composition of (φ, σ) with another morphism (φ′, σ′) : P′ → P′′ is given by (φ′, σ′)(φ, σ) = (φ′φ, σ′σ). The graded monoidal product ⊗ : \underline{Pic}_{k}^{Γ}(R) ×_{Γ} \underline{Pic}_{k}^{Γ}(R) → \underline{Pic}_{k}^{Γ}(R) is defined by the usual tensor product of bimodules

\xymatrix{(P \ar[r]^{(φ,σ)} & P′) ⊗ (Q \ar[r]^{(ψ,σ)} & Q′)} = \xymatrix{(P ⊗_{R} Q \ar[r]^{(φ⊗_{R}ψ,σ)} & P′ ⊗_{R} Q′) ,}

and the graded unit I : Γ → \underline{Pic}_{k}^{Γ}(R) is defined by I(σ) = (σ, σ) : R → R, r ↦ ^{σ}r. The associativity and unit 1-graded constraints are the usual isomorphisms for the tensor product of bimodules.

Now, suppose that a Γ-group G and a Γ-algebra R are given. Then, a strictly unitary Γ-graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R) consists of: i) a family of invertible (R, R)-bimodules F_{x}, x ∈ G; ii) isomorphisms of k-modules

F_{x,σ} : F_{x} →^{∼} F_{^{σ}x}, σ ∈ Γ, x ∈ G,  (27)

such that F_{x,σ}(r a) = ^{σ}r F_{x,σ}(a) and F_{x,σ}(a r) = F_{x,σ}(a)^{σ}r, for any r ∈ R and a ∈ F_{x}; and iii) isomorphisms of (R, R)-bimodules

Φ_{x,y} : F_{x} ⊗_{R} F_{y} →^{∼} F_{xy}, x, y ∈ G,  (28)

such that the following conditions hold:

F_{1} = R, F_{x,1} = id_{F_{x}}, F_{1,σ} = id_{R}, Φ_{x,1} : a ⊗ r ↦ a r, Φ_{1,y} : r ⊗ a ↦ r a,
F_{x,τσ} = F_{^{σ}x,τ} F_{x,σ} ,
Φ_{x,yz} (id_{F_{x}} ⊗_{R} Φ_{y,z}) = Φ_{xy,z} (Φ_{x,y} ⊗_{R} id_{F_{z}}) ,
F_{xy,σ} Φ_{x,y} = Φ_{^{σ}x,^{σ}y} (F_{x,σ} ⊗_{R} F_{y,σ}) .

Thus, such a graded monoidal functor should be termed an equivariant factor set, since when Γ = 1, this is exactly a factor set in the sense of Kanzaki [27]. Indeed, for any graded monoidal functor (F, Φ), the family of the invertible (R, R)-bimodules F_{x}, x ∈ G, together with the isomorphisms of bimodules Φ_{x,y}, x, y ∈ G, is actually a Kanzaki factor set and so (F, Φ) gives rise to a generalized crossed product algebra in his sense; that is, Σ(F, Φ) = ⨁_{x∈G} F_{x}, where the product of the elements a ∈ F_{x} and b ∈ F_{y} is defined by a b = Φ_{x,y}(a ⊗ b). Moreover, thanks to the isomorphisms F_{x,σ}, the group Γ acts on Σ(F, Φ) by the rule ^{σ}a = F_{x,σ}(a) if a ∈ F_{x}. In this way, \underline{Σ}(F, Φ) = (Σ(F, Φ), \{F_{x}\}, G), together with the identity R = F_{1}, is a (G, R)-graded Γ-Clifford system, which we refer to as the generalized crossed product (G, R)-graded Γ-Clifford system with equivariant factor set (F, Φ).

Suppose now that F′ = (F′, Φ′) : \underline{dis}_{Γ}G → \underline{Pic}_{k}^{Γ}(R) is another strictly unitary graded monoidal functor. Then, a homotopy θ : (F, Φ) → (F′, Φ′) consists of a family of (R, R)- bimodule isomorphisms θ_{x} : F_{x} →^{∼} F′_{x}, x ∈ G, satisfying that θ_{1} = id_{R}, θ_{^{σ}x} F_{x,σ} = F′_{x,σ} θ_{x} and θ_{xy} Φ_{x,y} = Φ′_{x,y} (θ_{x} ⊗_{R} θ_{y}), for all x, y ∈ G and σ ∈ Γ. Therefore, it is clear that giving such a homotopy is equivalent to giving an isomorphism of Γ-algebras, Σ(F, Φ) ⟶^{∼} Σ(F′, Φ′), a ↦ θ_{x}(a), a ∈ F_{x}, which establishes an isomorphism of (G, R)- graded Γ-Clifford systems θ : (\underline{Σ}(F, Φ), id_{R}) ⟶^{∼} (\underline{Σ}(F′, Φ′), id_{R}).

Moreover, any (G, R)-graded Γ-Clifford system (\underline{S}, j_{S}) , \underline{S} = (S, \{S_{x}\}_{x∈G}, G), is actually isomorphic to a generalized crossed product (G, R)-graded Γ-Clifford system for a certain equivariant factor set (F, Φ). Indeed, up to an isomorphism, we can assume that R = S_{1} and j_{S} = id_{R}. By [32, Proposition 2.5], for any x, y ∈ G, the canonical multiplication map Φ_{x,y} : S_{x} ⊗_{R} S_{y} ≅ S_{xy}, a ⊗ b ↦ a b, is a (R, R)-bimodule isomorphism and each component S_{x} is an invertible (R, R)-bimodule. Furthermore, the Γ-action on S determines, for all x ∈ G and σ ∈ Γ, isomorphisms of k-modules F_{x,σ} : S_{x} ≅ S_{^{σ}x}, a ↦^{σ}a, and it is straightforward to see that, in this way, we have a strictly unitary graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R), with F_{x} = S_{x} for each x ∈ G, whose associated generalized crossed product (G, R)-graded Γ-Clifford system (\underline{Σ}(F, Φ), id_{R}) is exactly (\underline{S}, id_{R}).

All in all, we state the following:

5.2. Theorem. For any Γ-algebra R and any Γ-group G, the mapping carrying any strictly unitary graded monoidal functor to the associated generalized crossed product (G, R)-graded Γ-Clifford system, induces a bijection

\underline{Σ} : [\underline{dis}^{Γ}G, \underline{Pic}_{k}^{Γ}(R)] ≅ Cliff_{k}^{Γ}(G, R) ,  (29)

between the set of homotopy classes of graded monoidal functors from \underline{dis}^{Γ}G to \underline{Pic}_{k}^{Γ}(R) and the set of isomorphism classes of (G, R)-graded Γ-Clifford systems.

Now, let us remark that, for any Γ-algebra R, the homotopy groups of the Γ-graded categorical group \underline{Pic}_{k}^{Γ}(R) are:

π_{0} \underline{Pic}_{k}^{Γ}(R) = Pic_{k}(R), the Γ-group of isomorphism classes [P] of invertible (R, R)- bimodules over k, where the multiplication is induced by the tensor product, [P] [Q] = [P ⊗_{R} Q], and the Γ-action is given by

σ[P] = [_{σ}P_{σ}] ,  (30)

where _{σ}P_{σ} denotes the invertible (R, R)-bimodule which is the same k-module as P with R-actions r · p = ^{σ^{−1}}r p and p · r = p ^{σ^{−1}}r, for r ∈ R and p ∈ P;

π_{1} \underline{Pic}_{k}^{Γ}(R) = Z(R)^{∗}, the Γ-equivariant Pic_{k}(R)-module of all units in the center of R. The Γ-action is given by restriction of the one on R and the action of Pic_{k}(R) on Z(R)^{∗} is given by the restriction to Z(R)^{∗} of Bass’ homomorphism Pic_{k}(R) → Aut_{k}(Z(R)) [1], that is, for each [P] ∈ Pic_{k}(R) and u ∈ Z(R)^{∗}, ^{[P]}u is the element in Z(R)^{∗} uniquely determined by the equality ^{[P]}u p = p u for all p ∈ P.

Every (G, R)-graded Γ-Clifford system (\underline{S} = (S, \{S_{x}\}_{x∈G}, G), j_{S}) induces a Γ-equivariant homomorphism (which depends only on its isomorphism class [\underline{S}])

χ_{[\underline{S},j_{S}]} : G → Pic_{k}(R), x ↦ [S_{x}] , x ∈ G .

Hence, there is a canonical map

χ : Cliff_{k}^{Γ}(G, R) ⟶ Hom_{Γ}(G, Pic_{k}(R)) ,  (31)

where Hom_{Γ}(G, Pic_{k}(R)) is the set of Γ-equivariant group homomorphisms of G into Pic_{k}(R), which clearly makes this diagram commutative:

\xymatrix{[\underline{dis}^{Γ}G, \underline{Pic}_{k}^{Γ}(R)] \ar[rr]^{\underline{Σ}} \ar@{-->}[dr]_{π_{0}} &  & Cliff_{k}^{Γ}(G, R) \ar[dl]^{χ} \\  & Hom_{Γ}(G, Pic_{k}(R)) .}  (32)

We say that any Γ-equivariant group homomorphism ρ : G → Pic_{k}(R) is an equivariant generalized collective character of the Γ-group G in the Γ-algebra R, and we say that a (G, R)-graded Γ-Clifford system (\underline{S}, j_{S}) realizes such an equivariant generalized collective character ρ whenever χ_{[\underline{S},j_{S}]} = ρ. The map χ provides a partitioning

Cliff_{k}^{Γ}(G, R) = ⨆_{ρ} Cliff_{k}^{Γ}(G, R; ρ) ,

where Cliff_{k}^{Γ}(G, R; ρ) is the set isoclasses of (G, R)-graded Γ-Clifford systems that realize each Γ-equivariant generalized collective character ρ : G → Pic_{k}(R). Due to the commutativity of triangle (32), bijection (29) restricts to bijections \underline{Σ} : [\underline{dis}^{Γ}G, \underline{Pic}_{k}^{Γ}(R); ρ] ≅ Cliff_{k}^{Γ}(G, R; ρ), and then Theorem 2.2 i) shows here a necessary and sufficient condition for the set Cliff_{k}^{Γ}(G, R; ρ) to be non-empty in terms of a 3-dimensional equivariant cohomology class (5)

T(ρ) ∈ H3_{Γ}(G, Z(R)^{∗})  (33)

of the Γ-group G with coefficients in the equivariant G-module Z(R)^{∗}, where the G-action is via ρ (i.e., ^{x}u = ^{ρ(x)}u). We refer to this invariant as the Teichmüller obstruction of ρ since its construction, outlined below, has a clear precedent in the Teichmüller cocycle map Br_{G}(R) → H^{3}(G, R^{∗}), defined for any finite Galois field extension R/k with group G. The general construction (6) of an equivariant 3-cocycle c^{ρ} representing the cohomology class T(ρ) particularizes as follows. In each isomorphism class ρ(x), x ∈ G, let us choose an invertible (R, R)-bimodule P_{x}; in particular, select P_{1} = R. Since ρ is a Γ-equivariant homomorphism, we can select (R, R)-bimodule isomorphisms Φ_{x,y} : P_{x} ⊗_{R} P_{y} ≅ P_{xy} and Φ_{x,σ} : _{σ}(P_{x})_{σ} ≅ P_{^{σ}x} for every x, y ∈ G and σ ∈ Γ, chosen as the canonical ones whenever either x, y or σ are identities. For any three elements x, y, z ∈ G, the isomorphisms of (R, R)-bimodules Φ_{xy,z}(Φ_{x,y} ⊗_{R} id_{P_{z}}) and Φ_{x,yz}(id_{P_{x}} ⊗_{R} Φ_{y,z}) from P_{x} ⊗_{R} P_{y} ⊗_{R} P_{z} to P_{xyz} need not coincide, but then there is (cf. [9, Lemma 3.1] for example) a unique element c^{ρ}(x, y, z) ∈ Z(R)^{∗} such that

c^{ρ}(x, y, z) Φ_{xy,z}(Φ_{x,y}(p_{x} ⊗ p_{y}) ⊗ p_{z}) = Φ_{x,yz}(p_{x} ⊗ Φ_{y,z}(p_{y} ⊗ p_{z})),

for all p_{x} ∈ P_{x}, p_{y} ∈ P_{y} and p_{z} ∈ P_{z}. Analogously, for any x, y ∈ G and σ, τ ∈ Γ, there are unique elements c^{ρ}(x, y, σ) and c^{ρ}(x, σ, τ) in Z(R)^{∗} such that

c^{ρ}(x, y, σ) Φ_{^{σ}x,^{σ}y}(Φ_{x,σ}(p_{x}) ⊗ Φ_{y,σ}(p_{y})) = Φ_{xy,σ}(Φ_{x,y}(p_{x} ⊗ p_{y}))
c^{ρ}(x, σ, τ) Φ_{x,στ}(p_{x}) = Φ_{^{τ}x,σ}(Φ_{x,τ}(p_{x})).

The resulting map is exactly the equivariant 3-cocycle

c^{ρ} : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) ⟶ Z(R)^{∗}.  (34)

If T(ρ) = 0, then Theorem 2.2 ii) states the existence of a bijection

H2_{Γ}(G, Z(R)^{∗}) ≅ Cliff_{k}^{Γ}(G, R; ρ) ,  (35)

that can be described as follows. Once we have chosen any (G, R)-graded Γ-Clifford system ((S, {S_{x}}_{x∈G}, G), j_{S}) that realizes ρ, then the bijection associates to any equivariant 2cohomology class represented by a 2-cocycle c : G^{2} ∪ (G × Γ) → Z(R)^{∗} the class of the (G, R)-graded Γ-Clifford system ((c⋆S, {S_{x}}_{x∈G}, G), j_{S}), where c⋆S is the same k-module as S with product s_{x} ⋆ s_{y} = c(x, y)s_{x}s_{y} and Γ-action σ ⋆ s_{x} = c(x, σ)^{σ}x, for all s_{x} ∈ S_{x}, s_{y} ∈ S_{y} and σ ∈ Γ.

In summary, we have the following classification theorem:

5.3. Theorem. Let R be a Γ-algebra and G a Γ-group.

(i) There is a canonical partition Cliff_{k}^{Γ}(G, R) = ⨆_{ρ} Cliff_{k}^{Γ}(G, R; ρ), where ρ varies in the range of the set of Γ-equivariant generalized collective characters of G in R.

(ii) Each Γ-equivariant generalized collective character ρ invariably determines a 3dimensional cohomology class T(ρ) ∈ H3_{Γ}(G, Z(R)^{∗}) of G with coefficients in the group of units of the centre of R (with respect to the equivariant G-module structure on Z(R)^{∗} obtained via ρ).

(iii) An equivariant generalized collective character ρ : G → Pic_{k}(R) is realizable, that is, Cliff_{k}^{Γ}(G, R; ρ) ≠ ∅, if, and only if, its Teichmüller obstruction vanishes.

(iv) If the obstruction of an equivariant generalized collective character, ρ, vanishes, then there is a bijection

Cliff_{k}^{Γ}(G, R; ρ) ≅ H2_{Γ}(G, Z(R)^{∗}) .

In the three next subsections we shall emphasize the interest of Theorem 5.3 by explaining some of its applications.

5.4. Equivariant crossed product algebras. If R is any Γ-algebra and G is a Γ-group, then a Γ-equivariant crossed product algebra of R by G is a (G, R)-graded Γ-Clifford system ((S, {S_{x}}_{x∈G}, G), j : R ≅ S_{1}) such that every component S_{x}, x ∈ G, is a free left R-module of rank one. Let

Ext_{k}^{Γ}(G, R)

denote the set of isomorphism classes of equivariant crossed product algebras of R by G. Thus, Ext_{k}^{Γ}(G, R) ⊆ Cliff_{k}^{Γ}(G, R), and we shall characterize this subset of Cliff_{k}^{Γ}(G, R) by means of equivariant collective characters as follows.

Let us recall Bass’ group exact sequence [1, Chap. II, (5.4)]

1 → Z(R)^{∗} → R^{∗} →^{C} Aut_{k}(R) →^{δ} Pic_{k}(R) ,  (36)

in which C maps each unit u of R to the inner automorphism C_{u} : r ↦ u r u^{−1}, and δ carries each algebra automorphism f of R to the class of the invertible (R, R)-bimodule R_{f}, the underlying left R-module R with the right R-action via f, that is, r · r′ = r f(r′). Actually, since R is a Γ-algebra, sequence (36) is of Γ-groups, where Γ acts both on Z(R)^{∗} and R^{∗} by restriction, on Aut_{k}(R) by the diagonal action, that is, ^{σ}f : r ↦ ^{σ}(f(^{σ^{−1}}r)), and the Γ-action on Pic_{k}(R) is the one considered before, that is, ^{σ}[P] = [_{σ}P_{σ}], where _{σ}P_{σ} is the same k-module as P with R-actions r · p = ^{σ^{−1}}r p and p · r = p ^{σ^{−1}}r. To see that δ is a Γ-equivariant map, let us observe that the map r ↦ ^{σ^{−1}}r establishes an isomorphism of (R, R)-bimodules R_{^{σ}f} ≅ _{σ}(R_{f})_{σ}, for all f ∈ Aut_{k}(R) and σ ∈ Γ, and so, δ(^{σ}f) = ^{σ}δ(f). Then, there is an induced Γ-group embedding

Out_{k}(R) ↪^{δ} Pic_{k}(R) ,

of the group of outer automorphisms of the algebra R, Out_{k}(R) = Aut_{k}(R)/In(R), into the Picard group Pic_{k}(R). By [1, Chap. II, (5.3)] we know the image of δ, namely

Im(δ) = {[P] ∈ Pic_{k}(R) | P ≅ R as left R-modules} ,

and therefore we have a cartesian square

\xymatrix{Ext_{k}^{Γ}(G, R) \ar@{^{(}->}[r]^{in} \ar[d] & Cliff_{k}^{Γ}(G, R) \ar[d]_{χ} \\ Hom_{Γ}(G, Out_{k}(R)) \ar@{^{(}->}[r]^{δ_{∗}} & Hom_{Γ}(G, Pic_{k}(R)) ,}  (37)

where χ is the map (31). If any Γ-equivariant homomorphism

ρ : G → Out_{k}(R)

is termed a Γ-equivariant collective character of the Γ-group G in the Γ-algebra R, then equivariant crossed product algebras of R by G are exactly those (G, R)-graded Γ-Clifford systems that realize equivariant collective characters. The equivariant collective character

χ_{[\underline{S},j]} : G → Out_{k}(R)

realized by any equivariant crossed product algebra of R by G, ((S, {S_{x}}_{x∈G}, G), j : R ≅ S_{1}), is built as follows. Since each component S_{x} is an invertible (R, R)-bimodule isomorphic to R as left R-module, there must exist a f_{x} ∈ Aut_{k}(R) and an isomorphism of (R, R)-bimodules over k, ψ_{x} : R_{f_{x}} ≅ S_{x}, and therefore χ_{[\underline{S},j]}(x) = [f_{x}]. Observe that if we write u_{x} = ψ_{x}(1), then S_{x} = Ru_{x} = u_{x}R. Further, from S_{x}S_{x^{−1}} = R 1 = S_{x^{−1}}S_{x} it follows that u_{x}Ru_{x^{−1}} = R 1 = u_{x^{−1}}Ru_{x} and therefore there exist a, b ∈ R such that 1 = u_{x}au_{x^{−1}} = u_{x^{−1}}bu_{x}, whence each u_{x} is a unit of S. Since r u_{x} = ψ_{x}(r) = ψ_{x}(1 · f_{x}^{−1}(r)) = u_{x} f_{x}^{−1}(r), we conclude that f_{x}(r) = u_{x} r u_{x}^{−1} for all r ∈ R and x ∈ G. Hence, the equivariant collective character χ_{[\underline{S},j]} maps every x ∈ G to the class of the automorphism of R given by conjugation by u_{x} in S.

Equivariant factor sets, or strictly unitary Γ-graded monoidal functors (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R), for equivariant crossed product algebras of R by G have a particulary simple reformulation (well-known for Γ = 1, cf. [29]), namely, they consist of pairs of maps (f, φ), where

f : G ⟶ Aut_{k}(R) , φ : G^{2} ∪ (G × Γ) ⟶ R^{∗} ,  (38)

subject to the conditions (13)-(17). Moreover, two such factor sets (f, φ) and (f′, φ′) are equivalent, that is, they define homotopic graded monoidal functors if, and only if, there is a map g : G → R^{∗} such that the equalities (18)-(21) hold. By the bijection (29), the equivariant crossed product algebra defined by a factor set (38) is R⋆_{f,φ}G = ⨁_{x} R×{x}, the free left R-module with basis {(1, x), x ∈ G}, with multiplication according to the rule (r, x) (s, y) = (r f_{x}(s) φ(x, y), xy), on which Γ acts by ^{σ}(r, x) = (^{σ}r φ(x, σ),^{σ}x).

Since the square (37) is cartesian, Theorem 5.3 gives the following classification theorem for equivariant crossed product algebras.

5.5. Theorem. Let R be a Γ-algebra and G a Γ-group.

(i) There is a canonical partition Ext_{k}^{Γ}(G, R) = ⨆_{ρ} Ext_{k}^{Γ}(G, R; ρ), where, for each Γequivariant collective character ρ : G → Out_{k}(R), Ext_{k}^{Γ}(G, R; ρ) is the set of classes of equivariant crossed products of R by G that realize ρ.

(ii) Each equivariant collective character ρ : G → Out_{k}(R) determines a 3-dimensional cohomology class T(ρ) ∈ H3_{Γ}(G, Z(R)^{∗}), of G with coefficients in the equivariant G-module of the units in the centre of R (with respect to the Γ-equivariant G-module structure on Z(R)^{∗} obtained via ρ). This invariant is called the “Teichmüller obstruction” of ρ.

(iii) An equivariant collective character ρ is realizable, that is, Ext_{k}^{Γ}(G, R; ρ) ≠ ∅, if and only if its obstruction vanishes.

(iv) If the obstruction of an equivariant collective character, ρ, vanishes, then there is a bijection Ext_{k}^{Γ}(G, R; ρ) ≅ H2_{Γ}(G, Z(R)^{∗}).

Remarks.

i) The equivariant action of the Γ-group Out_{k}(R) on the Γ-module Z(R)^{∗} in Theorem 5.5 is given by ^{[f]}u = f(u). Indeed, since ^{δ[f]}u 1 = ^{[R_{f}]}u 1 = 1 · u = 1 f(u) = f(u), the action is the one obtained via the embedding δ : Out_{k}(R) ↪ Pic_{k}(R).

ii) The construction (34) of an equivariant 3-cocycle c^{ρ} ∈ T(ρ), for any given equivariant generalized collective character, particularizes to any equivariant collective character as follows. Let us choose an automorphism f_{x} ∈ ρ(x) for each x ∈ G, with f_{1} = id_{R}. Then, we select normalized maps (f : G → Aut_{k}(R), φ : G^{2} ∪ (G × Γ) → R^{∗}) such that f_{x}f_{y} = C_{φ(x,y)} f_{xy} and ^{σ}f_{x} = C_{φ(x,σ)} f_{^{σ}x}, for all x, y ∈ G and σ ∈ Γ. The equivariant 3-cocycle

c^{ρ} : G^{3} ∪ (G^{2} × Γ) ∪ (G × Γ^{2}) ⟶ Z(R)^{∗}

is then determined by the three equations (23).

Finally, we should comment that for any Γ-ring R and any Γ-group G,

Ext_{ℤ}^{Γ}(G, R) = Ext^{Γ}(G, [R, R^{∗}]) ,

as one can easily check by seeing that an equivariant factor set for an equivariant crossed product ring of R by G is the same as an equivariant factor set for an equivariant RING-GROUP extension of [R, R^{∗}] by G. Hence, Theorem 5.5 for k = ℤ is the same as Theorem 4.1 restricted to those RINGS-GROUPS in which the group is the full group of units.

5.6. Central graded Clifford systems with operators. This example is motivated by Dade’s classification of graded Clifford systems over a field k for which the 1-component is 1-dimensional over k. In [12, Section 14], he shows that such a (G, k)- graded Clifford system is essentially the same as a central group extension of k^{∗} by group G. In other words, there is a natural bijection H^{2}(G, k^{∗}) ≅ Cliff_{k}(G, k) between the 2nd cohomology group of G with coefficients in the trivial G-module k^{∗} and the set of isomorphism classes of (G, k)-graded Clifford systems over k.

Suppose R to be any commutative Γ-ring and G a Γ-group. A (G, R)-graded Γ-Clifford system (\underline{S}, j) = ((S, \{S_{x}\}_{x∈G}, G), j : R ≅ S_{1}) is termed central whenever S_{1} is central in S. Let

Cliff_{cen}^{Γ}(G, R)

denote the subset of Cliff_{ℤ}^{Γ}(G, R) consisting of isomorphism classes of central (G, R)-graded Γ-Clifford systems. Observe that, when Γ = 1, Cliff_{cen}^{1}(G, R) = Cliff_{R}(G, R).

The set Cliff_{cen}^{Γ}(G, R) is actually an abelian group under the operation

[\underline{S}, j][\underline{S}′, j′] = [(S ⊗ S′, \{S_{x} ⊗_{R} S′_{x}\}_{x∈G}, G), j ⊗ j′] ,

where S ⊗ S′ = ⨁_{x∈G}(S_{x} ⊗_{R} S′_{x}) , with multiplication (s_{x} ⊗ s′_{x})(s_{y} ⊗ s′_{y}) = s_{x}s_{y} ⊗ s′_{x}s′_{y} and Γ-action ^{σ}(s_{x} ⊗ s′_{x}) = ^{σ}s_{x} ⊗ ^{σ}s′_{x}, and j ⊗ j′ is the composition R ≅ R ⊗_{R} R ≅ S_{1} ⊗_{R} S′_{1}. The class of the group ring R(G) = ⨁_{x∈G} R × {x}, on which Γ-acts by σ(r, x) = (^{σ}r,^{σ}x), is the identity element of Cliff_{cen}^{Γ}(G, R), and the inverse is given by [\underline{S}, j]^{−1} = [(S^{op}, \{S_{x^{−1}}\}_{x∈G}, G), j : R ≅ S_{1}], where S^{op} is the opposite ring of S.

Below, as a consequence of Theorem 5.3, we obtain a generalization of the aforementioned Dade’s theorem. Let Hn_{Γ}(G, R^{∗}) denote the nth cohomology group of the Γ-group G with coefficients in the Γ-equivariant trivial G-module R^{∗} and recall from [5, Theorem 3.3] that H2_{Γ}(G, R^{∗}) is isomorphic to the abelian group of classes of equivariant central extensions of R^{∗} by G.

5.7. Theorem. Let R be a commutative Γ-ring and G a Γ-group. There exists an exact sequence of abelian groups

\xymatrix{0 \ar[r] & H2_{Γ}(G, R^{∗}) \ar[r]^{Σ} & Cliff_{cen}^{Γ}(G, R) \ar[r]^{χ} & Hom_{Γ}(G, Pic_{R}(R)) & T \ar[r] & H3_{Γ}(G, R^{∗}) ,} (39)

in which the homomorphism Δ is defined by means of the equivariant crossed product construction, that is,

Σ[φ] = [(R ⋆_{φ} G, {R × {x}}_{x∈G}, G), j : R ≅ R × {1}] ,

where R⋆_{φ}G is the twisted group ring R⋆_{φ}G = ⨁_{x∈G} R×{x} whose multiplication is given by (r, x)(r′, x′) = (rr′ψ(x, x′), xx′) and on which Γ-acts by ^{σ}(r, x) = (^{σ}rψ(x, σ),^{σ}x). The homomorphisms χ and T are given, respectively, by restricting the equivariant generalized collective character map (31) and the Teichmüller obstruction map (33).

Proof. It is straightforward to check that the maps Σ, χ and T in the sequence are certainly group homomorphisms. As for the exactness in the first two points, let us consider the bijection (35) when one chooses the Γ-group ring R(G), in the bijection, as a representative (G, R)-graded Γ-Clifford system that realizes the trivial equivariant

generalized collective character. In this case, the bijection simply says that H2_{Γ}(G, R^{∗}) ≅^{Σ} Cliff_{ℤ}^{Γ}(G, R; 0). In addition, since a (G, R)-graded Γ-Clifford system that realizes the zero collective character is necessarily central, Cliff_{ℤ}^{Γ}(G, R; 0) = Ker(χ), whence the exactness in the first two points of the sequence follows. Part (iii) of Theorem 5.3 directly gives the exactness in the remaining point.

When R is a field, but also when R is any commutative algebra with Pic_{R}(R) = 0 (for instance a principal ideal domain, a local algebra, …), then Cliff_{cen}^{Γ}(G, R) ≅ H2_{Γ}(G, R^{∗}), and Dade’s result follows. We should also remark that the construction of the above exact sequence (39) is quite similar to the corresponding part of the seven-term exact sequence of Chase-Harrison-Rosenberg about the Auslander-Goldman-Brauer group relative to a Galois extension of commutative rings [11] (cf. with exact sequence (45) in the next subsection, also with Miyashita’s generalized seven term exact sequences [30]).

5.8. Equivariant Azumaya algebras over Galois extensions with operators. Throughout this paragraph, R is a commutative k-algebra such that R ⊇ k is a Galois extension with finite Galois group G ⊆ Aut_{k}(R) and, in addition, we assume that R is a Γ-algebra through a given homomorphism Γ → G. The Galois group G is then a Γ-group by the diagonal action (σ, x) ↦ ^{σ}x, where ^{σ}x(r) = ^{σ}(x(^{σ^{−1}}r)).

An R/k-Azumaya algebra [26] is a pair (A, j) consisting of an Azumaya k-algebra (i.e. central separable) A and a maximal commutative embedding j : R ↪ A. When A is endowed with a Γ-action by algebra automorphisms such that j is Γ-equivariant, then the pair (A, j) is called a Γ-equivariant R/k-Azumaya algebra. Two such algebras, (A, j) and (A′, j′) are isomorphic whenever there exists a Γ-equivariant isomorphism of algebras ϕ : A ≅ A′ which respects the embeddings of R, that is, ϕj = j′. Let

B̂r^{Γ}(R/k)

denote the corresponding set of isomorphism classes.

If (A, j) is any equivariant R/k-Azumaya algebra and, for each x ∈ G, we let

A_{x} = {a ∈ A | x(r) a = a r for all r ∈ R} .  (40)

Then, by [27, Proposition 3] or [2, Theorem I 2.15], it holds that A = ⊕_{x∈G}A_{x}, A_{x}A_{y} = A_{xy} for all x, y ∈ G and A_{1} = j(R) ≅ R. Moreover, for any σ ∈ Γ and x ∈ G,

σA_{x} = {^{σ}a | x(r) a = a r, r ∈ R} = {a | x(r)^{σ^{−1}}a = ^{σ^{−1}}a r, r ∈ R}
= {a | x(^{σ^{−1}}r)^{σ^{−1}}a = ^{σ^{−1}}a^{σ^{−1}}r, r ∈ R} = {a | ^{σ}(x(^{σ^{−1}}r)) a = a r, r ∈ R}
= {a | ^{σ}x(r) a = a r, r ∈ R} = A_{^{σ}x} .

Hence, ((A, {A_{x}}_{x∈G}, G), j : R ≅ A_{1}) is a (G, R)-graded Γ-Clifford system over k.

And conversely, if ((A, {A_{x}}_{x∈G}, G), j : R ≅ A_{1}) is any (G, R)-graded Γ-Clifford system over k such that

x(r) a_{x} = a_{x} r for all x ∈ G, r ∈ R and a_{x} ∈ A_{x} ,  (41)

then it follows from [27, Proposition 2] that (A, j) is an equivariant R/k-Azumaya algebra.

Thus, equivariant R/k-Azumaya algebras are the same as (G, R)-graded Γ-Clifford systems such that equalities (41) hold. Next we shall characterize the subset B̂r^{Γ}(R/k) ⊆ Cliff_{k}^{Γ}(G, R) by means of the generalized collective character map (31). To do so, let us recall Bass’ split exact sequence of groups [1, II, (5.4)]

\xymatrix{1 → Pic_{R}(R) \ar[r]^{in} & Pic_{k}(R) \ar[r]^{h} & Aut_{k}(R) → 1 \ar[l]^{δ}} ,  (42)

in which δ : f ↦ [R_{f}] is the homomorphism already recalled in sequence (36), and h maps the class of any invertible (R, R)-bimodule [P] ∈ Pic_{k}(R) to the automorphism h([P]) = h_{P} ∈ Aut_{k}(R) such that h_{P} (r) p = p r for all r ∈ R and p ∈ P. Let

ξ : Pic_{k}(R) ⟶ Pic_{R}(R), [P] ↦ [ξP] ,  (43)

be the map that assigns to each class of an invertible (R, R)-bimodule P the class of the (R, R)-bimodule ξP which is the same P as left R-module but with right R-action p · r = rp. Thus, ξ[P] = [P ⊗_{R} R_{h_{P}^{−1}}] = [P][R_{h_{P}}]^{−1} = [P] (δh([P]))^{−1}.

Sequence (42) is actually of Γ-groups. Indeed, when we recalled sequence (36), we showed that δ is a Γ-equivariant homomorphism. Homomorphism h is also equivariant since, from equalities h_{P} (^{σ^{−1}}r) p = p ^{σ^{−1}}r in P, it follows that the equalities^{σ}h_{P} (^{σ^{−1}}r)·p = p·r hold in _{σ}P_{σ}, which means that ^{σ}h_{P} = h_{_{σ}P_{σ}}. Thus, ^{σ}h([P]) = h(^{σ}[P]) according to the Γ-action (30) on Pic_{k}(R). Hence, the abelian group Pic_{R}(R) is a Γ-equivariant Aut_{k}(R)- module, on which Γ acts by (30), and Aut_{k}(R) acts in a similar way by

f[P] = [_{f}P_{f}] ,

where_{f}P_{f} ≅ R_{f} ⊗_{R}P ⊗_{R}R_{f^{−1}} is the same k-module as P with r·p = f^{−1}(r) p = p f^{−1}(r) = p · r, for r ∈ R and p ∈ P. In particular, since G is a Γ-subgroup of Aut_{k}(R), Pic_{R}(R) is also a Γ-equivariant G-module.

Now, if ρ : G → Pic_{k}(R), say ρ(x) = [A_{x}], is any equivariant generalized collective character of the Γ-group G in the Γ-algebra R, then the equalities (41) hold if and only if hρ = in : G ↪ Aut_{k}(R). Moreover, since map (43) induces an isomorphism of abelian groups [5, Proposition 2.5]

{ρ ∈ Hom_{Γ}(G, Pic_{k}(R)) | hρ(x) = x, x ∈ G} ≅ Der_{Γ}(G, Pic_{R}(R)), ρ ↦ ξ ρ ,

where Der_{Γ}(G, Pic_{R}(R)) is the abelian group of Γ-equivariant derivations from G into Pic_{R}(R), we conclude that there is a cartesian square

\xymatrix{B̂r^{Γ}(R/k) \ar@{^{(}->}[r]^{in} \ar[d]_{χ̂} & Cliff_{k}^{Γ}(G, R) \ar[d]^{χ} \\ Der_{Γ}(G, Pic_{R}(R)) \ar@{^{(}->}[r] & Hom_{Γ}(G, Pic_{k}(R)) ,}  (44)

where χ is the map (31) and χ̂ : [A, j] ↦ χ̂_{[A,j]}, where χ̂_{[A,j]}(x) = [ξA_{x}] for each x ∈ G. Hence, equivariant R/k-Azumaya algebras are the same as (G, R)-graded Γ-Clifford systems over k that realize equivariant derivations from G into Pic_{R}(R).

The set B̂r^{Γ}(R/k) has a canonical structure of abelian group under the operation

[A, j][A′, j′] = [A ⊗ A′, j ⊗ j′] ,

where A ⊗ A′ = ⨁_{x∈G} ξA_{x} ⊗_{R} A′_{x}, in which, for each x ∈ G, A_{x} and A′_{x} are defined by (40) and ξA_{x} by (43), with multiplication given by the chain of isomorphisms:

ξA_{x} ⊗_{R} A′_{x} ⊗_{R} ξA_{y} ⊗_{R} A′_{y} ≅ ξA_{x} ⊗_{R} _{x}(ξA_{y})_{x} ⊗_{R} A′_{x} ⊗_{R} A′_{y}
≅ ξ(A_{x}⊗_{R}A_{y}) ⊗_{R} A′_{x} ⊗_{R} A′_{y} ≅ ξA_{xy} ⊗_{R} A′_{xy} ,

that is, (a_{x}⊗a′_{x})(a_{y}⊗a′_{y}) = a_{x}a_{y}⊗a′_{x}a′_{y}. The Γ-action is given by^{σ}(a_{x}⊗a′_{x}) = ^{σ}a_{x}⊗^{σ}a′_{x}, and j⊗j′ is the composition R ≅ R⊗_{R}R ≅ A_{1}⊗_{R}A′_{1} = ξA_{1}⊗_{R}A′_{1}. The class of the skew group algebra R⋆G = ⨁_{x∈G} R×{x} , whose multiplication is given by (r, x)(r′, y) = (r x(r′), xy)

and on which Γ-acts by ^{σ}(r, x) = (^{σ}r, ^{σ}x), gives the identity element of B̂r^{Γ}(R/k), and the inverse is given by

[A, j]^{−1} = [A^{op}, j] ,

where A^{op} is the opposite Γ-algebra of A (observe that, for any x ∈ G, A_{x}^{op} = A_{x^{−1}} with r · a_{x^{−1}} = a_{x^{−1}} r = x^{−1}(r) a_{x^{−1}} and a_{x^{−1}} · r = r a_{x^{−1}}, and then, A ⊗ A^{op} ≅ R ⋆ G by the map ∑_{x} a_{x} ⊗ b_{x^{−1}} ↦ ∑_{x}(a_{x}b_{x^{−1}}, x)).

Then, we have:

5.9. Theorem. Let R/k be a Galois extension of commutative rings with finite Galois group G ⊆ Aut_{k}(R) and suppose that an action of a group Γ on R by automorphisms in G is given. Then, there exists an exact sequence of abelian groups

\xymatrix{0 \ar[r] & H2_{Γ}(G, R^{∗}) \ar[r]^{Σ} & B̂r^{Γ}(R/k) \ar[r]^{χ̂} & Der_{Γ}(G, Pic_{R}(R)) & T^{} \ar[r] & ˆ H3_{Γ}(G, R^{∗}) ,}  (45)

in which the homomorphism Σ is defined by means of the equivariant crossed product construction, that is

Σ[φ] = [R ⋆_{in,φ} G, j : R ≅ R × {1}] ,

where R ⋆_{in,φ} G is the crossed product algebra R ⋆_{in,φ} G = ⨁_{x∈G} R × {x}, whose multiplication is given by (r, x)(r′, x′) = (r x(r′) φ(x, x′), xx′) and on which Γ-acts by ^{σ}(r, x) = (^{σ}r φ(x, σ),^{σ}x). The homomorphism χ̂ is that given in (44) and T̂ is the Teichmüller obstruction map (33) restricted to Der_{Γ}(G, Pic_{R}(R)); that is, for any Γ-derivation d : G → Pic_{R}(R), T̂(d) = T(ρ_{d}), where ρ_{d} : G → Pic_{k}(R) is the equivariant homomorphism given by ρ_{d}(x) = d(x)[R_{x}].

Proof. To check that maps Σ, χ̂ and T̂ are group homomorphisms is routine. The exactness in the two first points means that Σ establishes a bijection between H2_{Γ}(G, R^{∗}) and Ker(χ̂), but this is just bijection (35),

H2_{Γ}(G, R^{∗}) ≅^{Δ} Cliff_{k}^{Γ}(G, R; ρ_{0}) = χ^{−1}(ρ_{0}) = Ker(χ̂),

when one chooses the skew Γ-group algebra R ⋆ G, in the bijection, as the representative (G, R)-graded Γ-Clifford system that realizes the equivariant generalized collective character ρ_{0} corresponding to the zero derivation 0 : G → Pic_{k}(R). Part (iii) of Theorem 5.3 directly gives the exactness in the remaining point.

Exact sequence (45) in the above theorem extends the sequence in [10, Theorem 2.1].

6. Strongly graded bialgebras and Hopf algebras with operators

Throughout this section, k is a unitary commutative ring and all algebras and coalgebras are over k. Recall that a bialgebra (R, Δ, ϵ) is an algebra R enriched with a coalgebra structure such that the comultiplication Δ : R → R ⊗_{k} R and the counit ϵ : R → k are algebra maps.

A strongly graded bialgebra [8, Definition 6.4]) ((S, Δ, ϵ), {S_{x}}_{x∈G}, G) is a graded Clifford system over k in which S is endowed with a bialgebra structure such that every component S_{x} is a subcoalgebra. A morphism of strongly graded bialgebras is a morphism (h, α) between the underlying graded Clifford systems such that h is also a coalgebra homomorphism. Strongly graded bialgebras were classified in [8, Section 6] as a specific application of the general treatment of graded monoidal categories.

Hereafter, Γ is a fixed group of operators and a Γ-bialgebra means a bialgebra provided with a Γ-action by bialgebra automorphisms (σ, r) ↦ ^{σ}r. Our goal in this section is the classification of strongly graded bialgebras on which a group of operators Γ acts by automorphisms, that is, the classification of strongly graded Γ-bialgebras.

Similarly to what happens with graded Γ-Clifford systems, if ((S, Δ, ϵ), {S_{x}}_{x∈G}, G) is a strongly graded Γ-bialgebra, then G is a Γ-group, (S, Δ, ϵ) is a Γ-bialgebra and its 1-component S_{1} is a Γ-subbialgebra. Then, we shall regard ((S, Δ, ϵ), {S_{x}}_{x∈G}, G) as an equivariant extension of the Γ-bialgebra (S_{1}, Δ, ϵ) by the Γ-group G. More precisely, if (R, Δ, ϵ) is any Γ-bialgebra and G is a Γ-group, then a strongly (G, (R, Δ, ϵ))-graded Γ-bialgebra is a pair (\underline{S}, j_{S}), where \underline{S} = ((S, Δ, ϵ), \{S_{x}\}_{x∈G}, G) is a strongly graded Γbialgebra, such that the Γ-action on G is the given one, that is, ^{σ}S_{x} = S_{^{σ}x} for all x ∈ G and σ ∈ Γ, and j_{S} : (R, Δ, ϵ) ≅ (S_{1}, Δ, ϵ) is an isomorphism of Γ-bialgebras.

The most striking example is, for (R, Δ, ϵ) any given Γ-bialgebra, the group bialgebra R(G) = ⨁_{x∈G} R × {x} (where the multiplication is given by (r, x)(r′, x′) = (rr′, xx′) and the comultiplication, using Sweedler’s notation, by Δ(r, x) = ∑_{(r)}(r_{(1)}, x)⊗(r_{(2)}, x), which is a Γ-bialgebra with the action ^{σ}(r, x) = (^{σ}r,^{σ}x). However, equivariant crossed products of (R, Δ, ϵ) and G also yield examples of equivariant strongly graded bialgebras. Note that the underlying graded algebra of a strongly graded Γ-bialgebra is a graded Γ-Clifford system as defined in Section 5.

Two strongly (G, (R, Δ, ϵ))-graded Γ-bialgebras, (\underline{S}, j_{S}) and (\underline{T} , j_{T}), are isomorphic if there exists a Γ-equivariant isomorphism of Γ-bialgebras h : (S, Δ, ϵ) ≅ (T, Δ, ϵ) which is grade-preserving. We denote by

Ext_{k}^{Γ}(G, (R, Δ, ϵ))

the set of isomorphism classes of strongly (G, (R, Δ, ϵ))-graded Γ-bialgebras.

The classification of strongly graded Γ-bialgebras parallels the non-equivariant case (i.e., when Γ is the trivial group) studied in [8]. Let (R, Δ, ϵ) be any Γ-bialgebra. Then, the group

Pic_{k}(R, Δ, ϵ)

of isomorphism classes of invertible (R, R)-coalgebras [8, Definition 6.3] is a Γ-group with action given by^{σ}[P] = [_{σ}P_{σ}], where_{σ}P_{σ} is the invertible (R, R)-coalgebra which is the same coalgebra as P but with (R, R)-actions r·p = ^{σ^{−1}}r p and p·r = p ^{σ^{−1}}r for r ∈ R and p ∈ P. Note that, since every invertible (R, R)-coalgebra is an invertible (R, R)-bimodule, there is an obvious equivariant homomorphism, from the Γ-group Pic_{k}(R, Δ, ϵ) to the Γ-group Pic_{k}(R) of the underlying Γ-algebra as considered in Section 5, which is neither injective nor surjective in general. Moreover, the centre of the bialgebra Z(R, Δ, ϵ) [8, Definition 6.2], which is the multiplicative submonoid of R consisting of all group-like elements of R as a coalgebra that belong to the centre of R as an algebra, that is,

Z(R, Δ, ϵ) = {u ∈ R | Δ(u) = u ⊗ u, ϵ(u) = 1, and ur = ru for all r ∈ R} ,

is a Γ-monoid in the obvious way. Therefore, the group of units of Z(R, Δ, ϵ), denoted by Z(R, Δ, ϵ)^{∗}, is a Γ-module. In fact, Z(R, Δ, ϵ)^{∗} is actually a Γ-equivariant Pic_{k}(R, Δ, ϵ)- module, where the action of Pic_{k}(R, Δ, ϵ) on Z(R, Δ, ϵ)^{∗} (see [8]) is determined by the equalities ^{[P]}u p = p u, for any invertible (R, R)-coalgebra P, any u ∈ Z(R, Δ, ϵ)^{∗} and p ∈ P.

Every strongly (G, (R, Δ, ϵ))-graded Γ-bialgebra (\underline{S} = ((S, Δ, ϵ), \{S_{x}\}_{x∈G}, G), j_{S}) induces a Γ-equivariant homomorphism (which depends only on its isomorphism class [\underline{S}, j_{S}])

χ_{[\underline{S},j_{S}]} : G → Pic_{k}(R, Δ, ϵ), x ↦ [S_{x}] , x ∈ G ,

and so there is a canonical map

χ : Ext_{k}^{Γ}(G, (R, Δ, ϵ)) ⟶ Hom_{Γ}(G, Pic_{k}(R, Δ, ϵ)) .

We call any Γ-equivariant group homomorphism ρ : G → Pic_{k}(R, Δ, ϵ) an equivariant generalized collective character of the Γ-group G in the Γ-bialgebra (R, Δ, ϵ). We also say that a strongly (G, (R, Δ, ϵ))-graded Γ-bialgebra (\underline{S}, j_{S}) realizes such an equivariant generalized collective character ρ whenever χ_{[\underline{S},j_{S}]} = ρ. Map χ provides a partitioning

Ext_{k}^{Γ}(G, (R, Δ, ϵ)) = ⨆_{ρ} Ext_{k}^{Γ}(G, (R, Δ, ϵ); ρ) ,

where, for each Γ-equivariant homomorphism ρ : G → Pic_{k}(R, Δ, ϵ), Ext_{k}^{Γ}(G, (R, Δ, ϵ); ρ) is the set of isoclasses of strongly (G, (R, Δ, ϵ))-graded Γ-bialgebras that realize ρ.

Now we state the following:

6.1. Theorem. (Classification of strongly graded bialgebras with operators) Let (R, Δ, ϵ) be a Γ-bialgebra and G a Γ-group.

(i) Each Γ-equivariant generalized collective character ρ invariably determines a 3dimensional cohomology class T(ρ) ∈ H3_{Γ}(G, Z(R, Δ, ϵ)^{∗}) of G with coefficients in the equivariant G-module of all group-like central units of (R, Δ, ϵ) (with respect to the Gmodule structure on Z(R, Δ, ϵ)^{∗} obtained via ρ). This invariant is called the obstruction of ρ.

(ii) An equivariant generalized collective character ρ is realizable, that is, the set Ext_{k}^{Γ}(G, (R, Δ, ϵ); ρ) is non-empty if, and only if, its obstruction vanishes.

(iii) If the obstruction of an equivariant generalized collective character, ρ, vanishes, then there is a bijection

Ext_{k}^{Γ}(G, (R, Δ, ϵ); ρ) ≅ H2_{Γ}(G, Z(R, Δ, ϵ)^{∗}) .

As in the examples studied in the previous sections, this classification theorem for the strongly graded bialgebras with operators will follow from the general theory in Section 2, once we are able to identify, up to isomorphisms, strongly graded Γ-bialgebras with Γ-graded monoidal functors from the discrete Γ-graded categorical group \underline{dis}^{Γ}G to the Γ-graded Picard categorical group of a Γ-bialgebra (R, Δ, ϵ)

\underline{Pic}_{k}^{Γ}(R, Δ, ϵ) ,

which is defined in a similar way to the Γ-graded categorical group \underline{Pic}_{k}^{Γ}(R) of the underlying Γ-algebra R. The objects of \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) are the invertible (R, R)-coalgebras and the morphisms are pairs (φ, σ) where φ is an isomorphism of coalgebras. Composition and the graded tensor product are defined much as for \underline{Pic}_{k}^{Γ}(R), in such a way that, omitting the coalgebra structure, one has a graded monoidal functor \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) → \underline{Pic}_{k}^{Γ}(R).

Then we have:

6.2. Theorem. For any Γ-bialgebra (R, Δ, ϵ) and any Γ-group G, there is a bijection

\underline{Σ} : [\underline{dis}^{Γ}G, \underline{Pic}_{k}^{Γ}(R, Δ, ϵ)] ≅ Ext_{k}^{Γ}(G, (R, Δ, ϵ)) ,  (46)

between the set of homotopy classes of graded monoidal functors from \underline{dis}^{Γ}G to \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) and the set of isomorphism classes of strongly (G, (R, Δ, ϵ))-graded Γ-bialgebras.

Proof. It is parallel to the proof for Theorem 5.2 and so we will omit some straightforward details. First we observe that a strictly unitary Γ-graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) is the same as a strictly unitary Γ-graded monoidal functor (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R) in which every F_{x} is an invertible (R, R)-coalgebra and the isomorphisms (27) and (28) are of (R, R)-coalgebras. Such a graded monoidal functor (F, Φ) is now termed an equivariant factor set of the Γ-group G in the Γ-bialgebra (R, Δ, ϵ) and it gives rise to a generalized crossed product bialgebra Σ(F, Φ) = ⨁_{x∈G} F_{x} which is the direct sum (= coproduct) coalgebra of the coalgebras F_{x}, where the product of the elements a ∈ F_{x} and b ∈ F_{y} is defined by a b = Φ_{x,y}(a ⊗ b). Further, owing to the isomorphisms F_{x,σ}, group Γ acts on Σ(F, Φ) by the rule ^{σ}a = F_{x,σ}(a) if a ∈ F_{x}. In this way, \underline{Σ}(F, Φ) = ((Σ(F, Φ), Δ, ϵ), \{F_{x}\}, G), together with the identity R = F_{1}, is a strongly (G, (R, Δ, ϵ))-graded Γ-bialgebra.

If F′ = (F′, Φ′) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) is another strictly unitary graded monoidal functor, then it is straightforward to see that giving a homotopy θ : (F, Φ) → (F′, Φ′) is equivalent to giving an isomorphism of Γ-bialgebras, Σ(F, Φ) ⟶^{∼} Σ(F′, Φ′), a ↦ θ_{x}(a), a ∈ F_{x}, which establishes an isomorphism of strongly (G, (R, Δ, ϵ))-graded Γ-bialgebras θ : (\underline{Σ}(F, Φ), id_{R}) ⟶^{∼} (\underline{Σ}(F′, Φ′), id_{R}).

Moreover, any strongly (G, (R, Δ, ϵ))-graded Γ-bialgebra is actually isomorphic to a generalized crossed product (G, (R, Δ, ϵ))-graded Γ-bialgebra for a certain equivariant factor set (F, Φ). In fact, given any (\underline{S}, j_{S}), \underline{S} = ((S, Δ, ϵ), \{S_{x}\}_{x∈G}, G), we can assume, up to isomorphism, that R = S_{1} and j_{S} = id_{R}. For any x, y ∈ G, the canonical multiplication map Φ_{x,y} : S_{x} ⊗_{R} S_{y} → S_{xy}, a ⊗ b ↦ a b is a (R, R)-coalgebra isomorphism and each component S_{x} is an invertible (R, R)-coalgebra [8, Proposition 6.4 and Corollary 6.1]. Furthermore, the Γ-action on S determines, for all x ∈ G and σ ∈ Γ, isomorphisms of coalgebras F_{x,σ} : S_{x} → S_{^{σ}x}, a ↦^{σ}a, and it is plain to see that (F, Φ) : \underline{dis}^{Γ}G → \underline{Pic}_{k}^{Γ}(R, Δ, ϵ), with F_{x} = S_{x} for each x ∈ G, is a strictly unitary graded monoidal functor whose associated generalized crossed product (G, (R, Δ, ϵ))-graded Γ-bialgebra (\underline{Σ}(F, Φ), id_{R}) is exactly (\underline{S}, id_{R}).

The classification of strongly (G, (R, Δ, ϵ))-graded bialgebras with operators stated in Theorem 6.1 is now a consequence of Theorem 2.2 and the above bijection (46). Indeed, for any Γ-bialgebra (R, Δ, ϵ), the homotopy groups of the Γ-graded categorical group \underline{Pic}_{k}^{Γ}(R, Δ, ϵ) are π_{0}\underline{Pic}_{k}^{Γ}(R, Δ, ϵ) = Pic_{k}(R, Δ, ϵ) and π_{1}\underline{Pic}_{k}^{Γ}(R, Δ, ϵ) = Z(R, Δ, ϵ)^{∗} and the obstruction T(ρ) ∈ H3_{Γ}(G, Z(R, Δ, ϵ)^{∗}) in part (i) of Theorem 6.1, of any equivariant generalized collective character ρ : G → Pic_{k}(R, Δ, ϵ), is defined in the same way as (33) for equivariant generalized collective characters of Γ-groups in Γ-algebras.

Furthermore, for any equivariant factor set (F, Φ) of the Γ-group G in the Γ-bialgebra (R, Δ, ϵ), π_{0}F = χ_{[\underline{Σ}(F,Φ),id_{R}]}, and then bijection (46) provides, by restriction, bijections \underline{Σ} : [\underline{dis}^{Γ}G, \underline{Pic}_{k}^{Γ}(R, Δ, ϵ); ρ] ≅ Ext_{k}^{Γ}(G, (R, Δ, ϵ); ρ) for any equivariant generalized character ρ : G → Pic_{k}(R, Δ, ϵ). Hence, parts (ii) and (iii) of Theorem 6.1 follow from Theorem 2.2.

By [3, Theorem 1.1], Theorem 6.1 applies to the classification of strongly graded Hopf Γ-algebras over a field k with 1-component of finite dimension, that is, strongly graded Γ-bialgebras ((S, Δ, ϵ), {S_{x}}_{x∈G}, G) where (S, Δ, ϵ) is a Hopf algebra whose antipode map ξ : S → S satisfies that ξ(S_{x}) ⊆ S_{x^{−1}} for all x ∈ G, and S_{1} is finite dimensional over k. Indeed, when (R, Δ, ϵ) is any finite dimensional Hopf Γ-algebra, then every strongly graded Γ-bialgebra whose 1-component is isomorphic to (R, Δ, ϵ) is necessarily a strongly graded Hopf Γ-algebra and, therefore, for any Γ-group G, Ext_{k}^{Γ}(G, (R, Δ, ϵ)) is exactly the set of isomorphism classes of strongly (G, (R, Δ, ϵ))-graded Hopf Γ-algebras.

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Dep. Álgebra. Univ. Granada. 18071. Granada. Spain

Email: acegarra@ugr.es
      agarzon@ugr.es

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