A THEORY OF OBSTRUCTIONS FOR STRONGLY GRADED ALGEBRAS ^{1}

by ANTONIO M. CEGARRA

1 Introduction

The problem of strongly graded extensions resides in the classification and the construction of the manifold of all strongly graded k-algebras S = ⨁_{σ∈G} S_{σ}, the type being given group G and with 1-component S_{1} isomorphic to a given k-algebra R. Each such strongly graded extension realizes a generalized collective character of G in R, that is a group homomorphism Φ : G → P ic_{k}(R) of G into the group of isomorphism classes of invertible left R ⊗_{k} R^{∘}-modules, and this leads to a problem of obstruction. When a generalized collective character is specified, it is possible that no strongly graded extensions realizing the specified homomorphism can exist. The main result in this paper is to obtain a necessary and sufficient condition for the existence of such a strongly graded extension, formulated in terms of a certain 3-dimensional group cohomology class T (Φ), referred to here as the Teichmüller obstruction of Φ. The construction of T (Φ) is closely analogous to a construction by T. Kanzaki [13], for a description of the Chase-Harrison-Rosenberg seven term exact sequence [4] about the Brauer group. In the case where a generalized collective character Φ has an extension, the manifold of such strongly graded extensions is shown as a principal and homogeneous space under a 2nd cohomology group.

We have included some examples and possible applications. The first example concerns central strongly graded rings, where we generalize a known result of E.Dade [5]. In a second example, we show how the treatment of ring extensions by groups, as expounded by M. Hacque [10, 11], follows from our general theory of strongly graded extensions. In a third example, we apply the general results to Azumaya algebras over a Galois extension of commutative rings, in the sense of A. Hattory [12], and the fourth and last example is dedicated to exibiting a particular collective character which is obstructed, that is, such that T (Φ) ≠ 0.

1This paper has been financially supported by DGICYT:PB94-0823

This paper has been strongly influenced by the work on the classification of crossedproduct rings by M. Hacque in [10, 11], where he makes a systematic analysis of the important phenomenons bound to the existence of obstructions. Strongly graded algebras are a direct generalization of crossed product algebras and they were studied extensively by E. Dade in [5, 6], who originally called them graded Clifford systems, and C. Nastasescu and F. Van Oystaeyen in [14, 15], who can be referred to for general background.

2 Strongly graded extensions of algebras and generalized collective characters

Throughout the paper k is a commutative ring with identity and G is a group.

A strongly G-graded k-algebra S is a k-algebra with identity, also denoted by S, together with a family of k-submodules S_{σ}, σ ∈ G, such that S = ⨁_{σ∈G} S_{σ} and S_{σ}S_{τ} = S_{στ} for all σ, τ ∈ G. Of course, the product S_{σ}S_{τ} consists of all finite sums of ring products xy of elements x ∈ S_{σ} and y ∈ S_{τ}. Note that the 1-component S_{1} is a k-subalgebra of S and each σ-component S_{σ}, σ ∈ G, is a two-sided S_{1}-submodule of S.

By a strongly graded extension of a k-algebra R we mean a strongly graded kalgebra S whose 1-component S_{1} is isomorphic to R. More precisely, we have the following:

Definition 2.1. Let R be a k-algebra and G a group. A strongly G-graded extension of R is a pair (S, j), where S = ⨁_{σ∈G} S_{σ} is a strongly G-graded k-algebra and j : R ↪ S is a k-algebra embedding with j(R) = S_{1}.

If (S, j), (S′, j′) are two strongly G-graded extensions of R, by a morphism between them f : (S, j) → (S′, j′), we mean a grade-preserving k-algebra homomorphism f : S → S′ that respects the embeddings of R, that is, such that fj = j′.

From [15, I.3.9] it follows that any strongly graded extension morphism f : (S, j) → (S′, j′) is necessarily an isomorphism. Therefore the existence of a morphism is an equivalence relation between strongly G-graded extensions of R and, in this case, we usually say that the extensions are equivalent. Then

SGr(G, R)  (1)

denotes the set of equivalence classes of strongly G-graded extensions of the k-algebra R.

If (S, j) is a strongly G-graded extension of R, then each S_{σ}, σ ∈ G, is an invertible R ⊗_{k} R^{∘}-module and, for every σ, τ ∈ G, the canonical morphism S_{σ} ⊗_{R} S_{τ} → S_{στ}, x_{σ} ⊗ x_{τ} ↦ x_{σ}x_{τ}, is an R ⊗_{k} R^{∘}-isomorphism [15 I.3.6]. There is a canonical map

χ : SGr(G, R) ⟶ Hom_{Gp}(G, P ic_{k}(R)) ,  (2)

where Hom_{Gp}(G, P ic_{k}(R)) is the set of group homomorphisms of G into P ic_{k}(R), the group of isomorphim classes of invertible R ⊗_{k} R^{∘}-modules, which carries the class of a strongly G-graded extension (S, j) to the group homomorphism χ_{[S,j]} : G → P ic_{k}(R), given by

χ_{[S,j]}(σ) = [S_{σ}] , σ ∈ G .  (3)

We have the Baer notion of Kollectivcharakter in mind , and we define a generalized collective character of the group G in the k-algebra R as a group homomorphism Φ : G → P ic_{k}(R). Let us recall the exact group sequence [2 II,(5.4)],

1 → InAut(R) ⟶ Aut_{k}(R) ⟶^{δ} P ic_{k}(R),  (4)

in which δ maps a k-algebra automorphism of R, α ∈ Aut_{k}(R), to the class of the invertible R⊗_{k}R^{∘}-module R_{α}, which is the same left R-module as R with right action given by x · y = xα(y), x, y ∈ R. Then, there is a canonical embedding Out_{k}(R) ↪^{δ} P ic_{k}(R), of the group of outer automorphisms of the k-algebra R, Out_{k}(R) = Aut_{k}(R)/InAut(R), into the Picard group P ic_{k}(R). A group homomorphism Φ : G → Out_{k}(R) has been called a collective character (cf. Hacque [10, 11]; so that collective characters of G in R are those generalized ones factoring through the embedding Out_{k}(R) ↪ P ic_{k}(R). Of course, by character we understand a group homomorphism G → Aut_{k}(R).

Let us note that there are several kinds of k-algebras for which Out_{k}(R) ↪ P ic_{k}(R) is an isomorphism such as local algebras (i.e., those algebras such that module its Jacobson radical are division algebras) [6], or polynomial algebras over principal ideal domains (cf. [14]). For these algebras, every generalized collective character is a collective character.

Hence Hom_{Gp}(G, P ic_{k}(R)) is the set of generalized collective characters of G in R, and the map χ associates with each equivalence class of strongly G-graded extensions of R a generalized collective character. We refer to a collective character Φ : G → P ic_{k}(R) as realizable if it is in the image of χ, that is, if it is induced as explained above from a strongly G-graded extension of R. The map χ produces a partitioning of the set of equivalence classes of strongly G-graded extensions of R,

SGr(G, R) = ∐_{Φ} SGr_{Φ}(G, R) ,  (5)

where, for any generalized collective character Φ ∈ Hom_{Gp}(G, P ic_{k}(R)), SGr_{Φ}(G, R) = χ^{−1}(Φ) denotes the fiber of χ over Φ . Thus a generalized collective character Φ is realizable if the set SGr_{Φ}(G, R) is not empty. We refer to SGr_{Φ}(G, R) as the set of equivalence classes of realizations of the generalized collective character Φ.

3 The Teichmüller cocycle and the obstruction theorem

If R is a k-algebra, let C(R) = {r ∈ R | rx = xr , x ∈ R} denote its center. Then C(R) is a k-algebra whose group of units we denote by C(R)^{∗}.

We will often use the following elementary fact, which is a consequence of [2, II,(3.5)]:

Lemma 3.1. If P, Q are invertible R ⊗_{k} R^{∘}-modules, then for any two R ⊗_{k} R^{∘}- isomorphisms α, β : P → Q, there exists a unique u ∈ C(R)^{∗} such that β = uα = αu (i.e., β(x) = uα(x) = α(ux) for all x ∈ P).

Proof. Given  α : P → Q,  an  R ⊗_{k} R^{∘}-isomorphism,  the  map

C(R)^{∗} → Isom_{R⊗_{k}R^{∘}}(P, Q), u ↦ uα, is bijective since it can be obtained as the composite map of the canonical group isomorphism C(R)^{∗} ≅ Aut_{R⊗_{k}R^{∘}}(R), the group isomorphism − ⊗_{R} P : Aut_{R⊗_{k}R^{∘}}(R) ≅ Aut_{R⊗_{k}R^{∘}}(P) and the bijection induced by α, α_{∗} : Aut_{R⊗_{k}R^{∘}}(P) ≅ Isom_{R⊗_{k}R^{∘}}(P, Q).

If P is any invertible R ⊗_{k} R^{∘}-module and u ∈ C(R)^{∗}, since x ↦ xu is an R ⊗_{k} R^{∘}- automorphism of P, there exists a unique element α_{P} (u) ∈ C(R)^{∗} such that α_{P} (u)x = xu for all x ∈ P. Clearly α_{P} : C(R)^{∗} → C(R)^{∗} is an automorphism and

P ic_{k}(R) ⟶^{ρ} Aut(C(R)^{∗}) ,  ρ([P]) = α_{P}  (6)

is a group homomorphism (note that ρ is the restriction to C(R)^{∗} of Bass’ homomorphism h : P ic_{k}(R) → Aut_{k}(C(R)) [2 II,(5.4)]). Hence C(R)^{∗} is a P ic_{k}(R)-module.

By composition with the homomorphism (6) we have for any group G a map

Hom_{Gp}(G, P ic_{k}(R)) ⟶^{ρ_{∗}} Hom_{Gp}(G, Aut(C(R)^{∗}))  (7)

that to each generalized collective character of G in the k-algebra R, Φ : G → P ic_{k}(R), associates a character Φ^{∗} = ρΦ : G → Aut(C(R)^{∗}) from group G in the abelian group C(R)^{∗}. Of course, the set of characters Hom_{Gp}(G, Aut(C(R)^{∗})) is the set of G-module structures on C(R)^{∗}. Hence every collective character Φ : G → P ic_{k}(R), of G in R, determines a G-module structure on C(R)^{∗} for which the corresponding G-action of an element σ ∈ G on an element u ∈ C(R)^{∗} is given by ^{σ}u = α_{P} (u) for any P ∈ Φ(σ). In particular,

xu = ^{σ}ux  (8)

for any σ ∈ G, u ∈ C(R)^{∗}, x ∈ P and P ∈ Φ(σ). We will denote by H_{Φ}^{n}(G, C(R)^{∗}), n ≥ 0, the n-th cohomology group of G with coefficients in this G-module.

We will now show how every generalized collective character Φ : G → P ic_{k}(R) has a cohomology class T (Φ) ∈ H_{Φ}^{3}(G, C(R)^{∗}) canonically associated with it, whose construction has several precedents: the Teichmuller cocycle homomorphism H^{0}(G, Br(R)) → H^{3}(G, R^{∗}) [16, 9], defined when R/k is a field Galois extension with group G; the Eilenberg-Mac Lane obstruction defined by a G-kernel, defined in [8] for the study of group extensions with a non-abelian kernel; the description by T. Kanzaki [13] of the homomorphism H^{1}(G, P ic_{R}(R)) → H^{3}(G, R^{∗}), in the Chase-Harrison-Rosenberg seven term exact sequence [4], about the Brauer group relative to a Galois extension of commutative rings R/k; the Teichmuller obstruction associated to a collective character Φ : G → Out(R), by M. Haque in [10, 11] for the study of obstructions to the existence of crossed product rings.

Let Φ : G → P ic_{k}(R) be a collective character of a group G in a k-algebra R. In each isomorphism class Φ(σ) ∈ P ic_{k}(R), choose an invertible R ⊗_{k} R^{∘}-module P_{σ} ∈ Φ(σ); in particular, select P_{1} = R. Since Φ is a homomorphism, the modules P_{σ} ⊗_{R} P_{τ} and P_{στ} must be R ⊗_{k} R^{∘}-isomorphic for each pair σ, τ ∈ G. Then we can select R ⊗_{k} R^{∘}-isomorphisms

Γ_{σ,τ} : P_{σ} ⊗_{R} P_{τ} → P_{στ}  (9)

with Γ_{σ,1}(x ⊗ r) = xr and Γ_{1,σ}(r ⊗ x) = rx, r ∈ R, x ∈ P_{σ}.

For any three elements σ, τ, γ ∈ G, the diagram

[figure: P_{σ} ⊗_{R} P_{τ} ⊗_{R} P_{γ}; P_{στ} ⊗_{R} P_{γ}; P_{σ} ⊗_{R} P_{τγ}; P_{στγ}; Γ_{σ,τ} ⊗P_{γ}; Γ_{σ,τγ}; P_{σ}⊗Γ_{τ,γ}; Γ_{στ,γ}]  (10)

need not be commutative but, by Lemma 3.1, there exists a unique element T_{σ,τ,γ}^{Φ} ∈ C(R)^{∗} such that

Γ_{στ,γ} (Γ_{σ,τ} ⊗ P_{γ}) = T_{σ,τ,γ}^{Φ} (Γ_{σ,τγ}(P_{σ} ⊗ Γ_{τ,γ})) .  (11)

Clearly T_{1,τ,γ}^{Φ} = T_{σ,1,γ}^{Φ} = T_{σ,τ,1}^{Φ} = 1 so that the choices of P_{σ} and Γ_{σ,τ} determine a normalized 3-dimensional cochain of G with coefficients in C(R)^{∗}.

Lemma 3.2. The cochain T = T^{Φ} : G^{3} → C(R)^{∗} is a 3-cocycle of G with coefficients in the G-module C(R)^{∗}.

Proof. We must prove the identity

T_{σ,τ,γ} T_{σ,τγ,δ} ^{σ}T_{τ,γ,δ} = T_{στ,γ,δ} T_{σ,τ,γδ}  (12)

for any (σ, τ, γ, δ) ∈ G^{4}. To see this, we compute the isomorphism

J = (Γ_{στγ,δ})(Γ_{στ,γ} ⊗ 1)(Γ_{σ,τ} ⊗ 1 ⊗ 1) : P_{σ} ⊗_{R} P_{τ} ⊗_{R} P_{γ} ⊗_{R} P_{δ} ⟶ P_{στγδ}

in two ways. On one hand, for all x ∈ P_{σ}, y ∈ P_{τ} , z ∈ P_{γ} and t ∈ P_{δ}, we have

J(x ⊗ y ⊗ z ⊗ t) = Γ_{στγ,δ}(Γ_{στ,δ}(Γ_{σ,τ} (x ⊗ y) ⊗ z) ⊗ t)
=^{(11)} T_{σ,τ,γ}Γ_{στγ,δ}(Γ_{σ,τγ}(x ⊗ Γ_{τ,γ}(y ⊗ z)) ⊗ t)
=^{(11)} T_{σ,τ,γ}T_{σ,τγ,δ}Γ_{σ,τγδ}(x ⊗ Γ_{τγ,δ}(y ⊗ z)) ⊗ t))
=^{(11)} T_{σ,τ,γ}T_{σ,τγ,δ}Γ_{σ,τγδ}(x ⊗ T_{τ,γ,δ}Γ_{τ,γδ}(y ⊗ Γ_{γ,δ}(z ⊗ t))
=^{(8)} T_{σ,τ,γ}T_{σ,τγ,δ} ^{σ}T_{τ,γ,δ}Γ_{σ,τγδ}(x ⊗ (Γ_{τ,γδ}(y ⊗ Γ_{γ,δ}(z ⊗ t))),

and on the other hand

\begin{aligned}J(x ⊗ y ⊗ z ⊗ t) & = & T_{στ,γ,δ}Γ_{στ,γδ}(Γ_{σ,τ} (x ⊗ y) ⊗ Γ_{γ,δ}(z ⊗ t)) \\ & =^{(11)} & T_{στ,γ,δ}T_{σ,τ,γδ}Γ_{σ,τγδ}(x ⊗ (Γ_{τ,γδ}(y ⊗ Γ_{γ,δ}(z ⊗ t))). \end{aligned}

and comparing the two expressions together with Lemma 3.1 gives (12).

We now observe the effect of different choices of P_{σ} and Γ_{σ,τ} in the construction of the 3-cocycle T^{Φ} for a given collective character Φ : G → P ic_{k}(R).

Lemma 3.3. i) If the choice of Γ in (9) is changed, then T^{Φ} is changed to a cohomologous cocycle. By suitably changing Γ, T^{Φ} may be changed to any cohomologous cocycle.

ii) If the choice of the invertible R ⊗_{k} R^{∘}-modules P is changed, then a suitable new selection of Γ leaves cocycle T^{Φ} unaltered.

Proof. i) By Lemma 3.1, any other choice of Γ_{σ,τ} in (9) has the form Γ′_{σ,τ} = h_{σ,τ} Γ_{σ,τ} , where h : G^{2} → C(R)^{∗} is a normalized 2-cochain of G in C(R)^{∗}.

For any σ, τ, γ ∈ G we have the following expressions for the isomorphism J = Γ′_{στ,γ} (Γ′_{σ,τ} ⊗ P_{γ}) from P_{σ} ⊗_{R} P_{τ} ⊗_{R} P_{γ} onto P_{στγ}:

J(x ⊗ y ⊗ z) = Γ′_{στ,γ}(Γ′_{σ,τ} (x ⊗ y) ⊗ z)
=^{(8)} h_{στ,γ} Γ_{στ,γ}(h_{σ,τ} Γ_{σ,τ} (x ⊗ y) ⊗ z)
= h_{στ,γ} h_{σ,τ} Γ_{στ,γ}(Γ_{σ,τ} (x ⊗ y) ⊗ z)
=^{(11)} h_{στ,γ} h_{σ,τ} f_{σ,τ,γ} Γ_{σ,τγ}(x ⊗ Γ_{τ,γ}(y ⊗ z))

and

J(x ⊗ y ⊗ z) =^{(11)} T′_{σ,τ,γ} Γ′_{σ,τγ}(x ⊗ Γ′_{τ,γ}(y ⊗ z))
= T′_{σ,τ,γ} h_{σ,τγ} Γ_{σ,τγ}(x ⊗ h_{τ,γ} Γ_{τ,γ}(y ⊗ z))
=^{(8)} T′_{σ,τ,γ} h_{σ,τγ} ^{σ}h_{τ,γ} Γ_{σ,τγ}(x ⊗ Γ_{τ,γ}(y ⊗ z))

and comparing the two expressions together with Lemma 3.1 yield

T′_{σ,τ,γ} h_{σ,τγ} ^{σ}h_{τ,γ} = h_{στ,γ} h_{σ,τ} T_{σ,τ,γ},  (13)

an identity that asserts that the 3-cocycles T and T′ are cohomologous.

ii) If P′_{σ} ∈ Φ(σ), σ ∈ G, is another selection of invertible R ⊗_{k} R^{∘}-modules, then we can select R ⊗_{k} R^{∘}-isomorphisms φ_{σ} : P′_{σ} → P_{σ} and choose Γ′_{σ,τ} : P′_{σ} ⊗_{R} P′_{τ} → P′_{στ}, the isomorphism making the following diagram commutative:

[figure: P′_{σ} ⊗_{R} P′_{τ}; P′_{στ}; P_{σ} ⊗_{R} P_{τ}; P_{στ} ,; Γ′_{σ,τ}; Γ_{σ,τ}; φ_{σ}⊗φ_{τ}; φ_{στ}]  (14)

for each σ, τ ∈ G. Thus we have

φ_{στγ}(Γ′_{στ,γ}(Γ′_{σ,τ} (x ⊗ y) ⊗ z)) = Γ_{στ,γ}(φ_{στ} (Γ′_{σ,τ} (x ⊗ y)) ⊗ φ_{γ}(z))
= Γ_{στ,γ}((Γ_{σ,τ} (φ_{σ}(x) ⊗ φ_{τ} (y)) ⊗ φ_{γ}(z))
= T_{σ,τ,γ}Γ_{σ,τγ}(φ_{σ}(x) ⊗ Γ_{τ,γ}(φ_{τ} (y)) ⊗ φ_{γ}(z)))
= T_{σ,τ,γ} Γ_{σ,τγ}(φ_{σ}(x) ⊗ φ_{τγ}(Γ′_{τ,γ}(y ⊗ z))
= T_{σ,τ,γ} φ_{στγ}(Γ′_{σ,τγ}(x ⊗ Γ′_{τ,γ}(y ⊗ z))
= φ_{στγ}(T_{σ,τ,γ} Γ′_{σ,τγ}(x ⊗ Γ′_{τ,γ}(y ⊗ z))) ,

for all x ∈ P′_{σ}, y ∈ P′_{τ} and z ∈ P′_{γ}.

Hence Γ′_{στ,γ}(Γ′_{σ,τ} (x ⊗ y) ⊗ z) = T_{σ,τ,γ} Γ′_{σ,τγ}(x ⊗ Γ′_{τ,γ}(y ⊗ z)) and the 3-cocycle T is unchanged.

These lemmas show that each generalized collective character Φ : G → P ic_{k}(R) determines in invariant fashion a 3-dimensional cohomology class T (Φ) = [T^{Φ}] ∈ H_{Φ}^{3}(G, C(R)^{∗}) . We refer to the map Φ ↦ T (Φ) as the Teichmüller obstruction map (see [11] for background).

Next we prove the main objective of this paper.

Theorem 3.4. A generalized collective character Φ : G → P ic_{k}(R) is realizable if and only if its Teichmüller obstruction T (Φ) ∈ H_{Φ}^{3}(G, C(R)^{∗}) vanishes.

Proof. Suppose first that (S = ⨁_{σ∈G} S_{σ}, j) is a realization of Φ. Then, in the construction of the Teichmüller 3-cocycle T^{Φ} of G with coefficients in the G-module C(R)^{∗}, one can take just the invertible R ⊗_{k} R^{∘}-modules S_{σ}, σ ∈ G, σ ≠ 1, and the canonical R ⊗_{k} R^{∘}-isomorphisms Γ_{σ,τ} : S_{σ} ⊗_{R} S_{τ} → S_{στ}, Γ_{σ,τ} (x ⊗ y) = xy, Γ_{σ,1}(x ⊗ r) = xj(r) and Γ_{1,σ}(r ⊗ x) = j(r)x for each σ, τ ∈ G. Since multiplication in the k-algebra S is associative Γ_{στ,γ} (Γ_{σ,τ} ⊗ S_{γ}) = Γ_{σ,τγ} (S_{σ} ⊗ Γ_{τ,γ}) for all σ, τ, γ ∈ G, and then T_{σ,τ,γ}^{Φ} = 1 in (11). Therefore, T (Φ) = [T^{Φ}] is the zero cohomology class.

Conversely, suppose that the generalized collective character Φ has a vanishing cohomology class f(Φ). Select any invertible R ⊗_{k} R^{∘}-modules P_{σ} ∈ Φ(σ), σ ∈ G, with P_{1} = R. By Lemma 3.3 i), there is a choice of R ⊗_{k} R^{∘}-isomorphisms Γ_{σ,τ} : P_{σ} ⊗_{R} P_{τ} → P_{στ} with Γ_{1,σ} and Γ_{σ,1} the canonical ones, such that the Teichmüller 3-cocycle T^{Φ} is identically 1. This means that (10) is commutative for any σ, τ, γ ∈ G. Hence, the family (P_{σ}, Γ_{σ,τ}) gives rise to a generalized crossed product algebra in the sense of Kanzaki [13] Δ = ⨁_{σ∈G} P_{σ}, where the product of elements x ∈ P_{σ} and y ∈ P_{τ} is defined by xy = Γ_{σ,τ} (x ⊗ y), which is a strongly G-graded k-algebra, extension of R by the canonical injection j : R ↪ Δ. Since χ_{[Δ,j]}(σ) = [P_{σ}] = Φ(σ), Φ is realized, that is, SGr_{Φ}(G, R) ≠ ∅.

In (5), we showed the existence of a canonical partition of the set of isomorphism classes of strongly G-graded extensions of R SGr(G, R) = ∐_{Φ} SGr_{Φ}(G, R), where for each generalized collective character Φ : G → P ic_{k}(R), SGr_{Φ}(G, R) is the set of isomorphism classes that realize Φ, which, by Theorem 3.4, is non-empty if and only if its Teichmüller obstruction T (Φ) ∈ H_{Φ}^{3}(G, C(R)^{∗}) vanishes. Now, to complete the classification of strongly G-graded extensions of a k-algebra R, we have the following result, which is essentially due to Nastasescu-Van Oystaeyen [15 I.3.18]:

Theorem 3.5. If a generalized collective character Φ : G → P ic_{k}(R) is realizable, then the set of isomorphism classes of realizations of Φ, SGr_{Φ}(G, R), is a principal homogeneous space under the abelian group H_{Φ}^{2}(G, C(R)^{∗}). In particular, there is a (non-canonical) bijection

SGr_{Φ}(G, R) ≅ H_{Φ}^{2}(G, C(R)^{∗}).

Proof. We will describe an action

H_{Φ}^{2}(G, C(R)^{∗}) × SGr_{Φ}(G, R) ⟶ SGr_{Φ}(G, R)  (15)

below.

Let h : G^{2} → C(R)^{∗} be a normalized 2-cocycle representative of an element [h] ∈ H_{Φ}^{2}(G, C(R)^{∗}) and (S = ⨁_{σ∈G} S_{σ}, j : R ≅ S_{1}) be a strongly G-graded extension of R, representative of an element [S, j] ∈ SGr_{Φ}(G, R). A new strongly G-graded extension of R, (^{h}S, j) is defined by considering the k-algebra ^{h}S which is the same G-graded k-algebra as S = ⨁_{σ∈G} S_{σ}, where the product of elements x ∈ S_{σ} and y ∈ S_{τ} is now defined by

x ⋆ y = j(h_{σ,τ})xy.

Since for any x ∈ S_{σ}, y ∈ S_{τ} and z ∈ S_{γ} we have

\begin{matrix}x ⋆ (y ⋆ z) = & x ⋆ (j(h_{σ,τ})yz) = j(h_{σ,τγ})xj(h_{τ,γ})yz \\ =^{(8)} & j(h_{σ,τγ})j(^{σ}h_{τ,γ})xyz = j(h_{στ,γ})j(h_{σ,τ})xyz \\ = & j(h_{στ,γ})(x ⋆ y)z = (x ⋆ y) ⋆ z , \end{matrix}

the multiplication is associative and so ^{h}S is a k-algebra. Furthermore, S_{σ} ⋆ S_{τ} = j(h_{σ,τ})S_{σ}S_{τ} = j(h_{σ,τ})S_{στ} = S_{στ}, since h_{σ,τ} is invertible for all σ, τ ∈ G.

Therefore (^{h}S, j) is actually a strongly G-graded k-algebra extension of R, clearly representing an element [^{h}S, j] ∈ SGr_{Φ}(G, R), which we maintain depends only on [h] and [S, j]. To see this, let us suppose that h′ is another representative of [h] and (S′, j′) is another representative of [S, j]. Then, there must exist a 1-cochain ψ : G → C(R)^{∗} such that h′_{σ,τ} ^{σ}ψ_{τ} ψ_{σ} = ψ_{στ} h_{σ,τ}, σ, τ ∈ G, and a grade-preserving isomorphism f : S → S′ such that fj = j′, from which we build the grade-preserving k-isomorphism ψf : ^{h}S → ^{h′}S′, ^{ψ}f(x) = f(j(ψ_{σ})x) if x ∈ S_{σ}. For each x ∈ S_{σ} and y ∈ S_{τ}, we have

\begin{matrix}^{ψ}f(x ⋆ y) = & f(j(ψ_{στ})j(h_{σ,τ})xy) = f(j(h′_{σ,τ})j(ψ_{σ}) j(^{σ}ψ_{τ})xy) \\ =^{(8)} & f(j(h′_{σ,τ})j(ψ_{σ})xj(ψ_{τ})y)) = j′(h′_{σ,τ}) ^{ψ}f(x) ^{ψ}f(y) \\ = & ^{ψ}f(x) ⋆ ^{ψ}f(y) , \end{matrix}

so that ^{ψ}f : (^{h}S, j) → (^{h′}S′, j′) is actually an isomorphism of strongly G-graded extensions of R, that is, [^{h}S, j] = [^{h′}S′, j′].

Therefore, ([h], [S, j]) ↦ [^{h}S, j] is a well-defined action of the abelian group H_{Φ}^{2}(G, C(R)^{∗}) on SGr_{Φ}(G, R), which furthermore is a principal one. In fact, if we suppose that [^{h}S, j] = [S, j], there must exist a grade preserving k-algebra isomorphism f : ^{h}S → S such that fj = j′. For each σ ∈ G, the restriction f_{/S_{σ}} : S_{σ} → S_{σ} is a R ⊗_{k} R^{∘}-isomorphism, and, by Lemma 3.1, there exists a unique ψ_{σ} ∈ C(R)^{∗} such that f(x) = j(ψ_{σ})x for all x ∈ S_{σ}. Thus ψ : G → C(R)^{∗} is a 1-cochain. Since f(x ⋆ y) = f(x)f(y), for any x ∈ S_{σ}, y ∈ S_{τ}, σ, τ ∈ G, we have j(ψ_{στ} h_{σ,τ})xy = j(ψ_{σ})xj(ψ_{τ})y =^{(8)} j(ψ_{σ} ^{σ}ψ_{τ})xy. Therefore, since S_{σ}S_{τ} = S_{στ}, Lemma 3.1 implies that ψ_{στ} h_{σ,τ} = ψ_{σ} ^{σ}ψ_{τ}, that is, h = ∂(ψ) represents the zero class in H_{Φ}^{2}(G, C(R)^{∗}).

Finally, we observe that action (15) is transitive. Let (S, j),(S′, j′) be any two strongly G-graded extensions of R representing elements in SGr_{Φ}(G, R). Since S_{σ},S′_{σ} ∈ Φ(x) for any σ ∈ G, there must exist R ⊗_{k} R^{∘}-isomorphisms f_{σ} : S_{σ} → S′_{σ}, σ ∈ G, with f_{1} = j′j^{−1}. For each pair σ, τ ∈ G, the square

[figure: S_{σ} ⊗_{R} S_{τ}; S_{στ}; S′_{σ} ⊗_{R} S′_{τ}; S′_{στ} ,; f_{στ}; f_{σ}⊗f_{τ}]

where the horizontal arrows represent the canonical isomorphisms x ⊗ y ↦ xy, need not to be commutative. But, by Lemma 3.1, there exists a unique h_{σ,τ} ∈ C(R)^{∗} such that f_{στ} (xy) = j′(h_{σ,τ})f_{σ}(x)f_{τ} (y) for all x ∈ S_{σ}, y ∈ S_{τ}. Thus h : G → C(R)^{∗} is a normalized 2-cochain. For any x ∈ S_{σ}, y ∈ S_{τ} and y ∈ S_{γ}, we have f_{στγ}(xyz) = j′(h_{στ,γ})f_{στ} (xy)f_{γ}(z) = j′(h_{στ,γ}h_{σ,τ})f_{σ}(x)f_{τ} (y)f_{γ}(z), and analogously, f_{στγ}(xyz) = j′(h_{σ,τγ})f_{σ}(x)f_{τγ}(z) =

j′(h_{σ,τγ})f_{σ}(x)j′(h_{τ,γ})f_{τ} (y)f_{γ}(z) =^{(8)}j′(h_{σ,τγ}^{σ}h_{τ,γ})f_{σ}(x)f_{τ} (y)f_{γ}(z). Lemma 3.1 im-

plies that h_{στ,γ}h_{σ,τ} = h_{σ,τγ}^{σ}h_{τ,γ}, that is, h is a 2-cocycle of G on C(R)^{∗}. Clearly f = ⨁_{σ∈G} f_{σ} establishes a G-graded extension isomorphism (S, j) → (^{h}S′, j′), and so action (15) is transitive.

4 Examples and applications

4.1 Central strongly graded rings

This example is motivated by Dade’s classification of strongly graded algebras (i.e. Clifford systems) over a field k for which the 1-component is 1-dimensional over k. In [5, Section 14], he shows that such a strongly G-graded extension of a field k is essentially the same as a central group extension of k^{∗} by group G. In other words, there is a bijection H^{2}(G, k^{∗}) ≅ SGr(G, k) between the set of isomorphism clases of strongly G-graded extensions of the k-algebra k and the 2nd cohomology group of G with coefficients in the trivial G-module k^{∗}.

If k is an arbitrary commutative ring, a strongly G-graded ring S = ⨁ S_{σ}, σ ∈ G, such that S_{1} is isomorphic to k as a ring and is central in S, it is called a central strongly G-graded extension of ring k. Clearly, this is equivalent to saying that S is a strongly G-graded extension of the k-algebra k, and so SGr(G, k) also represents the set of isomorphism classes of central strongly G-graded extensions of ring k.

Let us note that SGr(G, k) is an abelian group under the operation [S, j][S′, j′] = [S ⨂ S′, j ⨂ j′], where

S ⨂ S′ = ⨁_{σ∈G}(S_{σ} ⊗_{k} S′_{σ}),

with multiplication (x_{σ} ⊗ x′_{σ})(y_{σ} ⊗ y′_{σ}) = x_{σ}y_{σ} ⊗ x′_{σ}y′_{σ} for all x_{σ}, y_{σ} ∈ S_{σ} , x′_{σ}, y′_{σ} ∈ S′_{σ}, σ ∈ G, and j ⨂ j′ is the composition k ≅ k ⊗_{k} k ≅ S_{1} ⊗_{k} S′_{1}.

Next, we generalize the mentioned Dade theorem as follows.

Proposition 4.1. Let k be a commutative ring and G a group. There exists an exact sequence of abelian groups

0 → H^{2}(G, k^{∗}) ⟶^{Δ} SGr(G, k) ⟶^{χ} Hom(G, P ic_{k}(k)) ⟶^{T} H^{3}(G, k^{∗})  (16)

in which the homomorphism Δ is defined by means of the crossed product construction, the homomorphism χ is the generalized collective character map and T is the Teichmüler cocycle map.

Proof. The exactness in the two first terms is a consequence of Theorem 3.5, and that in the third of Theorem 3.4.

When k is a field, but also when k is any commutative ring with P ic_{k}(k) = 0 (a principal ideal domain, a local ring,...), then SGr(G, k) ≅ H^{2}(G, k^{∗}). Note that the construction of the above exact sequence (16) is completely similar to the corresponding part of the well-known seven-term exact sequence of Chase-Harrison-Rosenberg about the Auslander-Goldman-Brauer group relative to a Galois extension of commutative rings [4].

4.2 Extensions of algebras by groups

In this example we focus on that class of rings known as crossed-product group kalgebras. According to [11], an extension of a k-algebra R by a group G is a mixed sequence written as

{0, 1} → [R; R^{∗}] →^{j} [S, N] → G → {1} ,  (17)

in which R^{∗} refers to the group of units of R, j : R ↪ S is a k-algebra embedding, N ⊆ S^{∗} is a subgroup of the group of units of S such that the action of N on S, by inner automorphisms, restricts to an action of N on R, G = N/R^{∗} is the factor group of N by its normal subgroup R^{∗}, and the following condition is satisfied: For any σ ∈ G, let N_{σ} be the set of elements in N whose class in the quotient group G is σ and let S_{σ} be the R ⊗_{k} R^{∘}-submodule of S generated by the elements in N_{σ}. Then S decomposes as a direct sum, S = ⨁_{σ∈G} S_{σ}.

Next we remark on the relationship between such kinds of extensions and strongly graded extensions. If we have an extension of a k-algebra R by a group G (17) as above, then for any σ ∈ G and u_{σ} ∈ N_{σ} we have S_{σ} = RN_{σ}R = Ru_{σ}R^{∗}R = u_{σ}RR^{∗}R = u_{σ}R = Ru_{σ}. Hence S_{1} = j(R) and for any σ, τ ∈ G, S_{σ}S_{τ} = u_{σ}Ru_{τ} R = u_{σ}u_{τ} R = S_{στ} and so (S, j) is a strongly G-graded extension of R. Moreover, in a strongly G-graded algebra if a unit is in a σ-component, σ ∈ G, then its inverse lies in the σ^{−1}-component. Therefore, if for any σ ∈ G we choose a u_{σ} ∈ N_{σ}, then S^{∗} ∩ S_{σ} = S^{∗} ∩ u_{σ}R = u_{σ}(S^{∗} ∩ R) = u_{σ}R^{∗} = N_{σ} and therefore N = ⋃_{σ∈G} N_{σ} = ⋃_{σ∈G}(S^{∗} ∩ S_{σ}) is just the group of homogeneous units of S. It follows that extension (17) of R by G is completely determined by the strongly G-graded extension of R which it defines. Conversely, let (S = ⨁_{σ∈G} S_{σ}, j) be a strongly Ggraded extension of R such that for any σ ∈ G S^{∗} ∩ S_{σ} is nonempty. The set theoretical union N = ⋃_{σ∈G}(S^{∗} ∩ S_{σ}) is a subgroup of S^{∗} and the degree map N → G is a surjective group homomorphism with kernel S^{∗} ∩ j(R) = R^{∗}1_{S}. Furthermore, for any u_{σ} ∈ S^{∗} ∩ S_{σ}, we have u_{σ}R = S_{σ} = Ru_{σ}, since u_{σ} is a unit of S, i.e., u_{σ}Ru_{σ}^{−1} = R1_{S} and also R(S^{∗} ∩ S_{σ})R = R(S^{∗} ∩ u_{σ}R)R = Ru_{σ}(S^{∗} ∩ j(R))R = Ru_{σ}R^{∗}R = u_{σ}R = S_{σ}. Therefore, we have defined an extension of R by G like (17), which clearly defines the given strongly G-graded extension of R.

Thus extensions of a k-algebra R by a group G are the same as strongly G-graded extensions of R satisfying the condition that in any component there is at least one unit. As in Hacque’s paper [11], let Ext(G, R) denote the set of isomorphism classes of extensions of a k-algebra R by a group G. Then Ext(G, R) ⊆ SGr(G, R), and we shall characterize this subset of SGr(G, R) by means of collective characters as in the following proposition, where we take into account the canonical group embedding Out_{k}(R) ↪^{δ} P ic_{k}(R) induced by the group exact sequence (4), whose image is [2, chap. II prop. (5.3)]

Img(δ) = {[P] ∈ P ic_{k}(R) | P ≅ R as left R − module}.

Proposition 4.2. For any ring k-algebra R and group G there is a cartesian square

[figure: Ext(G, R); SGr(G, R); Hom_{Gp}(G, Out_{k}(R)); Hom_{Gp}(G, P ic_{k}(R)); δ_{∗}; χ]  (18)

that is, Ext(G, R) = χ^{−1}(Hom_{Gp}(G, Out_{k}(R)) is the set of classes of those strongly G-graded extensions of R which realize collective characters (in the sense of [10, 11]).

Proof. Let (S = ⨁_{σ∈G} S_{σ}, j) be a strongly G-graded extension of R such that for any σ ∈ G, there exists u_{σ} ∈ S^{∗} ∩ S_{σ}, that is, an extension of R by G. Right multiplication by u_{σ} is an isomorphism of left R-modules R →^{∼} Ru_{σ} = S_{σ}, σ ∈ G, and therefore the generalized collective character realized by [S, j], χ_{[S,j]} : G → P ic_{k}(R), χ_{[S,j]}(σ) = [S_{σ}] factors through Out_{k}(R). Conversely, suppose (S = ⨁_{σ∈G} S_{σ}, j) is a strongly G-graded extension of R such that χ_{[S,j]} = δΦ for some Φ : G → Out_{k}(R). Then, if we choose any k-automorphism f(σ) ∈ Φ(σ) for each σ ∈ G, there must exist an R ⊗_{k} R^{∘}-isomorphism φ_{σ} : R_{f(σ)} ≅ S_{σ}. If u_{σ} = φ_{σ}(1), then S_{σ} = Ru_{σ} = u_{σ}R. From S_{σ}S_{σ^{−1}} = R1_{S} = S_{σ^{−1}}S_{σ}, it follows that u_{σ}Ru_{σ^{−1}} = R1_{S} = u_{σ^{−1}}Ru_{σ}. Then there exist a, b ∈ R such that 1 = u_{σ}au_{σ^{−1}} = u_{σ^{−1}}bu_{σ} so that u_{σ} ∈ S^{∗} ∩ S_{σ} and therefore (S, j) represents an extension of R by G.

From the general results about strongly graded extensions of algebras, we deduce the following group cohomology classification of extensions of an algebra by a group, which was firstly proved by Hacque in [11].

Corollary 4.3. Let G be a group and R be a k-algebra.

i) Each collective character of G in R, Φ : G → Out_{k}(R) determines in an invariant fashion a three-dimensional cohomology class T (Φ) ∈ H_{Φ}^{3}(G, C(R)^{∗}) of G with coefficients in the G-module (via Φ) of all units in the center of R.

ii) There is a canonical partition of the set of equivalence classes of extensions of R by G,

Ext(G, R) = ∐_{Φ} Ext_{Φ}(G, R) ,

where, for any collective character Φ : G → Out_{k}(R), Ext_{Φ}(G, R) is the set of equivalence classes of those extensions realizing Φ.

iii) A collective character Φ : G → Out_{k}(R) is realizable, that is, Ext_{Φ}(G, R) ≠ ∅ if and only if its obstruction vanishes.

iv) If the obstruction of a collective character Φ : G → Out_{k}(R) vanishes, then Ext_{Φ}(G, R) is a principal homogeneous space under H_{Φ}^{2}(G, C(R)^{∗}). In particular, there is a bijection

Ext_{Φ}(G, R) ≅ H^{2}(G, C(R)^{∗}).  (19)

4.3 Azumaya algebras over Galois extensions

Throughout this paragraph R ⊇ k is a Galois extension of commutative rings with finite Galois group G ⊆ Aut_{k}(R).

An R/k-Azumaya algebra [12] is a pair (S, j) of an Azumaya k-algebra (i.e. central separable) S and a maximal commutative embedding j : R ↪ S. A morphism of such algebras is a k-algebra morphism f : S → S′which respects the embeddings of R, that is , such that fj = j′. Note that Ker(f) is a two-sided ideal of S and is therefore of the form αS for some ideal α of k. Since fj = j′, it follows that α = 0 and f is a monomorphism. Then we have S′ = f(S) ⊗_{k} C, with C the commutant of f(S) in S′. In particular, the elements of C commute with those of j′(R), so that C ⊆ j′(R). It follows immediately that C = k and that f is onto. Hence, any morphism of R/k-Azumaya algebras is necessarily an isomorphism; so that the existence of a morphism is an equivalence relation between R/k-Azumaya algebras. Let

Az(R/k)

denote the corresponding set of equivalence classes.

The following two propositions, which essentially are due to T. Kanzaki (cf. [13, Prop. 2 and 3]), pertain to on the relationship between R/k-Azumaya algebras and strongly G-graded extensions of R.

Proposition 4.4. i) If (S = ⨁_{σ∈G} S_{σ}, j) is a strongly G-graded extension of R such that

σ(x) s_{σ} = s_{σ} x  (20)

for all σ ∈ G, x ∈ R and s_{σ} ∈ S_{σ}, then (S, j) is an R/k-Azumaya algebra.

ii) Conversely, if (S, j) is an R/k-Azumaya algebra, then S has a unique structure of strongly G-graded k-algebra S = ⨁_{σ∈G} S_{σ} such that S_{1} = j(R) and (20) holds for all σ ∈ G, x ∈ R and s_{σ} ∈ S_{σ}. Hence (S, j) is a strongly G-graded extension of R.

Proof. i) Since S is a finitely generated k-module, to conclude that S is separable over k, it suffices to observe that S ⊗_{k} k_{m} is separable over the local ring k_{m} for each maximal ideal m of k. But S ⊗_{k} k_{m} = ⨁_{σ∈G}(S_{σ} ⊗_{k} k_{m}) is a strongly G-graded extension of the k_{m}-algebra R ⊗_{k} k_{m}, which is a finite Galois extension of k_{m} with the same Galois group G. Therefore, we may assume that k is local, so R is semilocal and every invertible R-module is a rank 1 free module. We can write S_{σ} = Ru_{σ} = u_{σ}R, σ ∈ G, for some u_{σ} ∈ S_{σ}, with u_{1} = 1_{S}. From S_{σ}S_{σ^{−1}} = R1_{S} = S_{σ^{−1}}S_{σ}, it follows that u_{σ}Ru_{σ^{−1}} = R1_{S} = u_{σ^{−1}}Ru_{σ}, so that u_{σ} is a unit of S for each σ ∈ G. Let h : R ⊗_{k} S^{∘} ⟶ End_{R}(S), h(x ⊗ s)(t) = xts, be the canonical R-algebra homomorphism. To show that h is onto, we note that End_{R}(S) is a free R-module generated by all e_{σ,τ}, σ, τ ∈ G, where e_{σ,τ} (u_{γ}) = δ_{σ,γ}u_{τ}. Since R/k is a Galois extension, there exist x_{1}, ⋯, x_{n}, y_{1}, ⋯ y_{n} in R that satisfy ∑_{i=0}^{n} σ(x_{i})τ(y_{i}) = δ_{σ,τ} for all σ, τ ∈ G. Let s_{σ,τ} = ∑_{i} σ(x_{i}) ⊗ y_{i}u_{σ}^{−1}u_{τ} ∈ R ⊗_{k} S, σ, τ ∈ G; then,

h(s_{σ,τ})(u_{γ}) = ∑_{i} σ(x_{i})u_{γ}y_{i}u_{σ}^{−1}u_{τ} = ∑_{i} σ(x_{i})γ(y_{i})u_{γ}u_{σ}^{−1}u_{τ} = δ_{σ,γ}u_{τ} ,

that is, h(s_{σ,τ}) = e_{σ,τ}. Since R ⊗_{k} S and End_{R}(S) are both free R-modules of rank |G|^{2}, h being onto implies h is an isomorphism. Thus R ⊗_{k} S^{∘} is a separable R-algebra. It follows that S is a separable k-algebra.

Now, we show that j embeds R as a maximal commutative subalgebra of S, for if s = ∑_{σ∈G} s_{σ}, s_{σ} ∈ S_{σ}, is an element of S such that sx = xs for all x ∈ R, then ∑ xs_{σ} = ∑ σ(x)s_{σ} and therefore xs_{σ} = σ(x)s_{σ} for all x ∈ R, σ ∈ G. Then s_{σ} = ∑_{i} x_{i}y_{i}s_{σ} = ∑_{i} x_{i}σ(y_{i})s_{σ} = 0 for σ ≠ 1. Hence we have s = s_{1} ∈ R1_{S} = j(R).

Finally, the center of S is k, for if x ∈ R lies in the center, then xs_{σ} = s_{σ}x = σ(x)s_{σ} for all s_{σ} ∈ S_{σ} and all σ ∈ G. Since each S_{σ} is a faithful R-module, x = σ(x) for every σ ∈ G and so x ∈ k = R^{G}.

ii) Since S, and therefore S^{∘}, is split by R, we have an isomorphism of R-algebras, h : R ⊗_{k} S^{∘} ≅ End_{R}(S), h(x ⊗ s)(t) = xts, and Morita equivalence N ↦ N ⊗_{R} S between the category of R-modules and the category of R ⊗_{k} S^{∘}-modules, with quasiinverse the functor M ↦ M^{R} = {m ∈ M | xm = mx ∀x ∈ R} ≅ Hom_{R⊗_{k}S^{∘}}(S, M). Hence, for any R ⊗_{k} S^{∘}-module M, there is an R ⊗_{k} S^{∘}-isomorphism M^{R} ⊗_{R} S ≅ M, m ⊗ s ↦ ms. If M is an R ⊗_{k} S^{∘}-module, let ^{σ}M be the R ⊗_{k} S^{∘}-module equal to M as right S-module with (x ⊗ s) · m = σ(x)ms for all x ∈ R, s ∈ S, m ∈ M, and let M_{σ} = (^{σ}M)^{R} = {m ∈ M | σ(x)m = mx ∀x ∈ R}. Then M_{σ} ⊗_{R} S ≅ ^{σ}M as R ⊗_{k} S^{∘}- modules for each σ ∈ G.

We shall show that the family of k-submodules {S_{σ} = (^{σ}S)^{R}, σ ∈ G} defines the announced strongly G-graded k-algebra structure on S. Since j(R) is a maximal commutative subalgebra of S, then S_{1} = S^{R} = j(R), and it is immediately evident that S_{σ}S_{τ} ⊆ S_{στ} for any σ, τ ∈ G. From the following chain of natural R⊗_{k}S^{∘}-isomorphisms,

S_{στ} ⊗_{R} S ≅ ^{στ} S ≅ ^{τ} ( ^{σ}S) ≅ ^{τ} (S_{σ} ⊗_{R} S) ≅ ^{τ} S_{σ} ⊗_{R} ^{τ} S ≅ ( ^{τ} S_{σ} ⊗_{R} S_{τ}) ⊗_{R} S ,

we deduce that the canonical R-homomorphism ^{τ} S_{σ} ⊗_{R} S_{τ} → S_{στ}, s ⊗ t ↦ st, is an isomorphism, and therefore that S_{σ}S_{τ} = S_{στ} for all σ, τ ∈ G. Next we note that S_{σ} ⋂ S_{τ} = 0 for each σ ≠ τ, for if s ∈ S_{σ} ⋂ S_{τ}, then σ(x)s = sx = τ(x)s for all x ∈ R and therefore s = ∑_{i=1}^{n} σ(x_{i})σ(y_{i})s = ∑_{i=1}^{n} σ(x_{i})τ(y_{i})s = 0. Finally, we form S′ = ∑_{σ∈G} S_{σ}. Then (S′, j) is clearly a strongly G-graded extension of R which satisfies condition (20), and therefore, by i), is an R/k-Azumaya algebra. The inclusion map S′ ↪ S being a morphism of R/k-Azumaya algebras implies tha it is onto, so that S′ = S, completing the proof.

From the above proposition it follows that Az(R/k) ⊆ SGr(G, R). Next we shall characterize this subset of SGr(G, R) by means of the generalized collective characters Φ : G → P ic_{k}(R) as follows:

There is a split exact sequence of groups [2 II, (5.4)],

[figure: 1; P ic_{R}(R); in; P ic_{k}(R); h; Aut_{k}(R); δ; 1 ,]

where for [P] ∈ P ic_{k}(R), h([P]) = α_{P} is defined by α_{P} (x)s = sx for all s ∈ P, x ∈ R, and for α ∈ Aut_{k}(R), δ(α) = [R_{α}] is the class of the invertible R⊗_{k}R^{∘}-module R_{α}, which is the same left R-module as R with right action given by x · y = xα(y). The abelian group P ic_{R}(R) is a G-module with action ^{σ}[P] = [ ^{σ}P], where ^{σ}P ≅ R_{σ} ⊗_{R} P ⊗_{R} R_{σ^{−1}} is the same k-module as P with x · p = σ^{−1}(x)p = pσ^{−1}(x) = p · x for x ∈ R, p ∈ P.

Note that if P is an invertible R ⊗_{k} R^{∘}-module and σ ∈ G, then σ(x)s = sx for all x ∈ R if and only if h([P]) = σ. Accordingly, if Φ : G → P ic_{k}(R), Φ(σ) = [S_{σ}], is a generalized collective character of G in R, we have σ(x)s_{σ} = s_{σ}x for all σ ∈ G, x ∈ R and s_{σ} ∈ S_{σ} if and only if hΦ = id_{G}. Such a generalized collective character of G in R will be called normal. The set of all normal generalized collective characters Φ : G → P ic_{k}(R) is in one-to-one correspondence with the set Der(G, P ic_{R}(R)) of all crossed homomorphisms from G into P ic_{R}(R). From Proposition 4.4 we deduce the following

Proposition 4.5. There is a cartesian square

[figure: Az(R/k); SGr(G, R); Der(G, P ic_{R}(R)); Hom_{Gp}(G, P ic_{k}(R)),; χ]

that is, Az(R/k) = χ^{−1}Der(G, P ic_{R}(R)), where χ is the generalized collective character realization map (2).

From the general results about strongly graded extensions of algebras, we deduce the following group cohomology classification of Azumaya algebras over a Galois extension of commutative rings,

Corollary 4.6. Let R ⊇ k be a Galois extension of commutative rings with finite Galois group G ⊆ Aut_{k}(R).

i) Each normal generalized collective character of G in R, Φ : G → P ic_{k}(R), or equivalently each derivation Φ : G → P ic_{R}(R), determines in an invariant fashion a three-dimensional cohomology class T (Φ) ∈ H^{3}(G, R^{∗}) of G with coefficients in the G-module of units of R.

ii) There is a canonical partition of the set of equivalence (i.e., R-isomorphism) classes of R/k-Azumaya algebras

Az(R/k) ≅ ∐_{Φ} Az_{Φ}(R/k)

where, for any normal generalized collective character Φ : G → P ic_{k}(R), Az_{Φ}(R/k) is the set of equivalence classes that realize Φ.

iii) A normal generalized collective character Φ of G in R is realizable, that is, Az_{Φ}(R/k) ≠ ∅ if and only if its obstruction vanishes.

iv) If the obstruction of a normal generalized collective character Φ vanishes, then Az_{Φ}(R/k) is a principal homogeneous space under H^{2}(G, R^{∗}). In particular, there is a bijection

Az_{Φ}(R/k) ≅ H^{2}(G, R^{∗}).  (21)

Remark 4.7. When P ic_{R}(R) = 0, then Az(R/k) = Br(R/k) the Brauer group of equivalence classes of Azumaya k-algebras split by R, G ↪ Aut_{k}(R) ≅^{δ} P ic_{k}(R) is the unique normal character of G in R, and bijection (21) is just the well-known cohomological description Br(R/k) ≅ H^{2}(G, R^{∗}) by Auslander and Goldman [1, Theorem A.15]. Furthermore, it is easy to deduce from the above corollary that the following sequence, which is the last part of the Chase-Harrison-Rosenberg seven term sequence, is exact (cf. [13]).

H^{0}(G, P ic_{R}(R)) → H^{2}(G, R^{∗}) → Br(R/k) → H^{1}(G, P ic_{R}(R)) → H^{3}(G, R^{∗}).

4.4 An obstructed collective character

It is very easy to find unobstructed generalized collective characters. Of course any strongly graded algebra yields one of them. In this example we shall exibit a nonrealizable collective character, that is, a group homorphism Φ : G → P ic_{k}(R), for particular group G and k-algebra R, such that there is no strongly G-graded extension of R, (S = ⨁_{σ∈G} S_{σ}, j : R ≅ S_{1}) such that Φ(σ) = [S_{σ}], σ ∈ G.

For consider G = C_{2} =< t; t^{2} = 1 >, the cyclic group of order two, k = 𝔽_{5}, the Galois field with five elements and R = 𝔽_{5}[D_{10}], the group 𝔽_{5}-algebra of the dihedral group D_{10} =< r, s; r^{10} = 1 = s^{2}, srs = r^{−1} >.

Let β : 𝔽_{5}[D_{10}] ⟶^{∼} 𝔽_{5}[D_{10}] be the algebra automorphism defined by β(r) = r^{7} and β(s) = r^{5}s. Since β^{2}(r) = r^{−1} = srs and β^{2}(s) = sr^{5}(r^{5})^{7} = s, the automorphism β^{2} is simply conjugation by s. Therefore, the equations Φ(1) = 1 and Φ(t) = [β] determine a homomorphism

Φ : C_{2} ⟶ Out(𝔽_{5}[D_{10}]) ⊆^{δ} P ic(𝔽_{5}[D_{10}])  (22)

of the cyclic group C_{2} into the Picard group of 𝔽_{5}[D_{10}], that is, a collective character of C_{2} in 𝔽_{5}[D_{10}].

Proposition 4.8. The Teichmüller obstruction T (Φ) ∈ H_{Φ}^{3}(C_{2}, C(𝔽_{5}[D_{10}])^{∗}) is nonzero.

Proof. First, let us observe that in this case, a (normalized) n-cochain h : C_{2} × ⋯ × C_{2} ⟶ C(𝔽_{5}[D_{10}])^{∗} is determined by a single constant h(t, …, t) = h ∈ C(𝔽_{5}[D_{10}])^{∗}, whose coboundary is δh = β(h)h^{−1}. Since we easily see that the Teichmüller 3-cocycle is T^{Φ} = r^{5}, the proof of the proposition amounts to checking that there is no unit h in the center of 𝔽_{5}[D_{10}] such that β(h) = r^{5}h.

Note that if β(h) = r^{5}h, then β(h^{2}) = h^{2}, and that we can reduce the proof of the non-existence of such a unit h ∈ C(𝔽_{5}[D_{10}]) to the case in which h^{2^{n}} = 1 for some n ≥ 1 (write h = h_{1}h_{2} with h_{1}^{2^{n}} = 1 = h_{2}^{m} for some odd number m ≥ 1, then β(h^{m}) = (r^{5})^{m}h^{m} and therefore β(h_{1}^{m}) = r^{5}h_{1}^{m} with (h_{1}^{m})^{2^{n}} = 1).

Now, since the center of 𝔽_{5}[D_{10}] can be described as the 8-dimensional space over 𝔽_{5} generated by the elements

c_{1} = 1, c_{2} = r + r^{9}, c_{3} = r^{2} + r^{8}, c_{4} = r^{3} + r^{7}, c_{5} = r^{4} + r^{6},
c_{6} = r^{5}, c_{7} = (c_{1} + c_{3} + c_{5})s, c_{8} = (c_{2} + c_{4} + c_{6})s

we have made the following observations and computations (using the Maple V release 5 program (Waterloo Maple Inc. 1981-1987) for ii) and v)):

i) The subspace {h | β(h) = h} ⊆ C(𝔽_{5}[D_{10}]) is generated by the following 5 elements

c_{1}, c_{2} + c_{4}, c_{3} + c_{5}, c_{6}, c_{7} + c_{8}.

ii) The group {h | β(h) = h, h^{8} = 1} ⊆ C(𝔽_{5}[D_{10}])^{∗} consists of the 16 elements described below.

\begin{matrix}c_{1} & 2c_{1} & 3c_{1} & 4c_{1} \\ c_{6} & 2c_{6} & 3c_{6} & 4c_{6} \\ c_{1} + 2c_{6} & c_{1} + 3c_{6} & 2c_{1} + c_{6} & 2c_{1} + 4c_{6} \\ 3c_{1} + c_{6} & 3c_{1} + 4c_{6} & 4c_{1} + 2c_{6} & 4c_{1} + 3c_{6} \end{matrix}  (23)

iii) Every element h in table (23) satisfies h^{4} = 1. Therefore (23) lists all the elements in the 2-component of the group C(𝔽_{5}[D_{10}])^{∗} which are β-invariant. In particular 4 is an exponent of all these elements.

iv) The subspace {h | β(h) = c_{6}h} ⊆ C(𝔽_{5}[D_{10}]) is generated by the following 5 elements

c_{1} + c_{6}, c_{2} + c_{3}, c_{4} + c_{5}, c_{7}, c_{8}.

v) The subset {h | β(h) = c_{6}h, h^{8} = 1} ⊆ C(𝔽_{5}[D_{10}]) is empty.

Hence the set

{h ∈ C(𝔽_{5}[D_{10}])| β(h) = c_{6}h, h^{2^{n}} = 1, n ≥ 0} = {h ∈ C(𝔽_{5}[D_{10}])| β(h) = c_{6}h, h^{8} = 1}

is empty, and therefore T (Φ) ≠ 0.

Remark 4.9. If we consider the Galois field 𝔽_{2} (or 𝔽_{3}, respectively) instead of 𝔽_{5}, the resulting collective character (22), Φ : C_{2} → P ic(𝔽_{2}[D_{10}]) (respectively, Φ : C_{2} → P ic(𝔽_{3}[D_{10}])), defined similarly by Φ(t) = [β], where β is the correspondig algebra automorphism determined by β(r) = r^{7} and β(s) = r^{5}s, is unobstructed. In this case the Teichmüller cocycle T^{Φ} = r^{5} = δ(h), is the coboundary of the order 4 element h = c_{2} + c_{3} + c_{7} ∈ C(𝔽_{2}[D_{10}])^{∗} (respectively, h = c_{4} + c_{5} + c_{7} + 2c_{8} ∈ C(𝔽_{3}[D_{10}])^{∗}), and therefore T (Φ) = 0.

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Departamento de Álgebra Universidad de Granada 18071-Granada, Spain e-mail:acegarra@ ugr.es