Actions of Picard groups on graded rings^{∗}

Jeremy Haefner and Angel del Río^{†}

September 2, 1998

1 Introduction

Throughout this paper, G denotes a group with identity e. If R is a G-graded ring, we write R = ⊕_{g∈G}R_{g} and we refer to R_{e} as the coefficient ring. We denote the category of graded (left) R-modules (i.e., those left R-modules M with a G-grading M = ⊕_{g∈G}M_{g} such that R_{h}M_{g} ⊆ M_{hg}) by R-gr.

The aim of this paper is to present a method to reduce the study of one strongly graded ring R to the study of another strongly graded ring R′ that is more tractable. This reduction process has two aspects: reducing the coefficient ring or reducing the grading. Since we apply this method in our study of module-theoretic properties such as semisimplicity for graded rings, we require that this reduction process preserve the category of modules R-mod and the category of graded modules R-gr. This leads us to the notion of graded equivalence.

In this paper, a Morita context is the usual tuple (A, B, P, Q, τ, μ) where we assume τ and μ are isomorphisms. A G-graded context is a Morita context (R, R′, P, Q, τ, μ), where R and R′ are G-graded rings, _{R}P_{R′} and _{R′}Q_{R} are graded bimodules (i.e., they have a grading which makes them into both graded R-modules and graded R′-modules on the corresponding side) and τ and μ are graded bimodule homomorphisms. See [14] for details about graded modules and homomorphisms. Given a functor F : R-gr → R′-gr, we say that F is a graded equivalence of categories if any of the following equivalent conditions hold (see for example [8]):

1. F ∘ T_{g} = T_{g} ∘ F, for every g ∈ G, where T_{g} denotes the g-th suspension.

2. F = Hom_{R}(P, −) where P is part of a G-graded Morita context (R, R′, P, Q, τ, μ).

3. F = Q ⊗_{R} − where Q is part of a G-graded Morita context (R, R′, P, Q, τ, μ).

4. F lifts to an equivalence of categories F′ : R-mod → R′-mod, that is, there is an equivalence of categories F′, as above, such that F′(M) = F (M) for every M ∈ R-gr.

We say that R and R′ are graded equivalent if there is a graded equivalence F : R-gr → R′-gr.

A special type of graded equivalence arises from graded isomorphism. Given G-graded rings R and R′, we say that R and R′ are graded isomorphic if there is a graded isomorphism f : R → R′; that is, a ring homomorphism f such that f(R_{g}) = R′_{g}. In case A = R_{e} = R′_{e} then R is said to be graded A-isomorphic if there is a graded isomorphism f : R → R′ that induces the identity on A. Of course, graded isomorphic rings have isomorphic categories of modules and graded modules, but this is too strong condition for our purposes.

∗Math. Subject Classification: 16W50, 16S35, 16P90, 16D60.

†The first author has been partially supported by grants from DGICYT(SAB 95-0215) and the National Security Agency and the second one by DGICYT (PB93-0515-C02-02) and DGICYES (PR 95-469)

The reduction process mentioned above can be rephased in terms of graded equivalences. The main problem of this paper is the following:

Problem A: Given a (strongly) G-graded ring, find another G-graded ring R′ that is graded equivalent to R such that either the identity component R′_{e} is simpler (e.g., R_{e} is semiperfect and R′_{e} is basic semiperfect) or the grading of R′ is simpler (e.g., R′ is a crossed product or a skew group ring).

We apply our solution of Problem A to the study of a more concrete problem:

Problem B: Characterize semisimple strongly G-graded rings.

The key to our study is the development of an action of graded Morita contexts on graded rings. For the purposes of this introduction, if C is a Morita context and R is a graded ring, then R^{C} denotes the action of C on R (see section 2 for the details). We use this action to reduce the coefficient ring. For example, we can apply a graded Morita context on a graded ring R whose coeffiecient ring is of the form M_{n}(A) to obtain a graded equivalent ring R′ whose coefficient ring is A. We also use this action to reduce the grading. Crossed products, twisted, skew and ordinary group rings are examples of gradings that are more tractable than strongly graded rings. The Cohen-Montgomery Duality is an example of how this action can reduce the grading of a strongly graded ring to that of a skew group ring. To be more specific, we introduce some notation and present one of our main results.

Let R be G-graded and let A = R_{e}. A (graded) invertible R-module is an bimodule _{R}P_{R} which is part of a (graded) Morita context (R, R, P, Q, μ, τ). The class of A-bimodules isomorphic to a given A-bimodule P is denoted by [P]. In addition, we use the following: (Here P_{e} denotes the identity component of the graded module P.)

Aut(A) = Group of automorphisms ofA
Inn(A) = Group of inner automorphisms ofA
Aut_{R}(A) = Group of automorphisms of A that extends to a graded automorphism of R
Out(A) = Aut(A)/Inn(A)
Out_{R}(A) = Aut_{R}(A)/(Inn(A) ∩ Aut_{R}(A))
Pic(A) = Picard group of A
Pic_{R}(A) = {[P_{e}] : P is a graded invertible R-bimodule}

The next theorem summarizes our results: (Here given a group G and a subgroup H of G, G/H denotes the set of either right or left H-cosets of G.)

Theorem A. Let R be a graded ring with coefficient ring A.

1. If R and R′ are strongly G-graded rings, then R and R′ are graded equivalent if and only if there exists a Morita context C̃ between A and the coefficient ring of R′, such that C̃ induces a graded Morita context C between R and R′ so that R′ and R^{C} are graded isomorphic.

2. If R is strongly graded, Pic(A)/Pic_{R}(A) parametrizes the strongly graded rings R′, with coefficient ring A, that are graded equivalent to R but not graded Aisomorphic to R.

3. If R is strongly graded, Pic(A)/Out(A) parametrizes the strongly graded rings, with coefficient ring A, that are graded equivalent to R but not graded isomorphic to R.

4. Out(A)/Out_{R}(A) ≃ Aut(A)/Aut_{R}(A) parametrizes the graded rings that are graded isomorphic to A, but not graded A-isomorphic.

See Theorem 3.2, Propositions 4.4 and 4.6. As a consequence of (1), the graded equivalence class of R is completely determined by the Morita equivalence class of its coefficient ring. Statements (2) through (4) give parametrizations of the equivalence classes of (strongly) graded rings with the same coefficient ring, under graded equivalence, graded isomorphism or graded A-isomorphism. Finally, synthesizing some results from [14] and [5], we reduce the distinction between graded equivalence and graded isomorphism to a cohomology problem (see Section 5) and we obtain the following:

Theorem B. There are graded equivalent, strongly G-graded rings R and R′, sharing coefficient ring A, and, for every g ∈ G, R_{g} ≃ R′_{g} as A-bimodules but R and R′ are not graded isomorphic.

See Theorem 5.8.

Theorem A implies that reducing the coefficient ring A is limited by the Morita equivalence class of A. As an example we obtain a result of [9]: If R is a strongly graded ring so that R_{e} is semiperfect, then R is graded equivalent to a strongly graded ring R′ such that R′_{e} is the basic ring of R_{e}. In particular, it follows that R′ is a crossed product. As a result, in the semiperfect situation, we can reduce not only the coefficient ring, but the grading as well. Skew group rings and twisted group rings are examples of gradings that are more tractable than crossed products, and so we ask whether the process can go further to one of these two cases. However, reducing to a twisted group ring is impossible if the original graded ring is not already twisted (Corollary 3.4). On the other hand, reducing to a skew group ring is possible via the Cohen-Montgomery Duality but at the cost of complicating the coefficient ring. Nonetheless, this is the best we can expect because reducing the grading to a skew group ring and keeping a tractable coefficient ring are somehow incompatible. Specifically, if R_{e} ≃ R′_{e} is basic semiperfect and R is a skew group ring, then R′ is a skew group ring as well (Proposition 6.5).

This result is helpful to the solution of Problem B. In particular, if R is semisimple, so is R_{e} and hence to solve Problem B, we may assume that R_{e} is semisimple. By Theorem A, we may reduce R to a crossed product over a finite product of division rings. Moreover, since Theorem A determines how a graded equivalence results from a Morita context of the coefficient rings, we are very specific on the resulting crossed product (see Section 7). However we cannot expect to go further (e.g.; to unskew or to untwist) via graded equivalence. Nonetheless, using the particularities of these crossed products, we can reduce the semisimplicity of the original ring to the semisimplicity of a finite set of concrete crossed products over division rings. We prove:

Theorem C: Let R be a strongly graded ring with semisimple coefficient ring A. Let B be the semisimple basic ring that is Morita equivalent to A (so B is a direct sum of division rings). Then R is graded equivalent to a crossed product R′ with coefficient ring B. Moreover, there exists a finite collection of crossed products over division rings, {D_{i}∗G_{i} : i = 1, …, n}, such that R is semisimple if and only if each D_{i}∗G_{i} is semisimple.

See Theorem 7.5. Moreover, if the original grading is already ”untwisted” (see Section 7), then we can reduce to skew group rings over division rings (Corollary 7.7). Finally, we mention that when the above mentioned division rings are fields, we can apply the results of [2] and [3] to characterize all the semisimple strongly graded rings of this form (Corollary 7.8.)

Notation: We denote ring automorphisms exponentionally; that is, the action of α ∈ Aut(R) on r ∈ R is denoted by r^{α}. Accordingly αβ means, first α, then β.

If u is a unit, then ι_{u} denotes the inner automorphism (r^{ι_{u}} = u^{−1}ru) induced by u.

Let R be a G-graded ring. For every g ∈ G, R_{g} denotes the g-th homogeneous component of R. The notation r_{g}, is normally used to emphasize that r_{g} ∈ R_{g}. R-gr denotes the category of left graded R-modules.

R is said to be strongly graded if R_{g}R_{h} = R_{gh} for every g, h ∈ G. R is said to be a crossed product if R_{g} contains a unit for every g ∈ G. Crossed products are determined by parameter sets: A parameter set of a group G over a ring A is a pair of maps (α : G → Aut(A), t : G × G → U(A) satisfying the following conditions:

α_{g}α_{h} = t_{gh}ι_{t_{g,h}} and t_{gh,k}t_{g,h}^{α_{k}} = t_{g,hk}t_{h,k}

The crossed product A ∗_{t}^{α} G defined by the parameter set (α, t) is the free right A-module with basis \{\overline{g} : g ∈ G\} with multiplication given by

r\overline{g} = \overline{g}r^{α_{g}} \overline{g}\overline{h} = \overline{gh}t(g, h)

A skew group ring is a crossed product A ∗_{t}^{α} G = A ∗^{α} G, such that t(g, h) = 1 for every g, h ∈ G. A twisted group ring is a crossed product A ∗_{t}^{α} G = A ∗_{t} G, such that α_{g} = 1 for every g ∈ G.

Whenever a map X ⊗ Y → Z is denoted by M, means that the map is given by M(x ⊗ y) = xy, where the multiplication will be clear from the context. Finally, we have frequently abuse the notation and identify X ⊗_{A} A, A ⊗_{A} X and X without explicit mention.

Finally, we would like to thank the referee for the helpful suggestions that were incorporated into this paper.

2 Morita contexts acting on graded rings.

We begin this section by defining a general action by Morita contexts. Let A and B arbitrary unital rings. By Morita context, we mean the usual Morita context 6-tuple (A, B, P, Q, τ, μ) where _{A}P_{B} and _{B}Q_{A} are bimodules and τ : P ⊗_{B} Q → A and μ : Q ⊗_{A} P → B are bimodule isomorphisms. In the literature, the condition that τ and μ are bijective is usually not required and a Morita context satisfying this condition is called a strict Morita context. But all the Morita context used in the paper are strict, so we just say ”Morita context” to mean ”strict Morita context”. In particular, a Morita context shall always induce an equivalence of categories between A-mod and B-mod.

For shorthand notation, all Morita contexts will be denoted by C with some modifier. For example, C′ denotes the Morita context (A′, B′, P′, Q′, τ′, μ′).

Definition 2.1 Let C and C′ be two Morita contexts. A morphism of Morita contexts from C to C′ is a 4-tuple of maps ϕ = (α, β, π, ϱ) where:

1. α : A → A′ and β : B → B′ are ring homomorphisms,

2. π : P → P′ is an α-β semilinear map,

3. ϱ : Q → Q′ is a β-α semilinear map and 4. the following diagrams are commutative:

\begin{matrix}P ⊗_{B} Q & →^{τ} & A \\ π ⊗ ϱ ↓ & & ↓ α \\ P′ ⊗_{B′} Q′ & →^{τ′} & A′ \end{matrix}  \begin{matrix}Q ⊗_{A} P & →^{μ} & B \\ ϱ ⊗ π ↓ & & ↓ β \\ Q′ ⊗_{A′} P′ & →^{μ′} & B′ \end{matrix}

As with our notation convention for Morita contexts, we shall use similar rules to denote morphisms of Morita contexts; e.g., the morphism ϕ_{1} is formed by the maps α_{1}, β_{1}, π_{1}, ϱ_{1}.

Morita contexts and the morphisms between Morita contexts define a category in a natural way.

Definition 2.2 Given two Morita contexts C and C′, with B = A′, we define the multiplication

C · C′ = (A, B′, P ⊗_{B} P′, Q′ ⊗_{B} Q,
τ(1 ⊗ τ′ ⊗ 1) : (P ⊗_{B} P′) ⊗_{B′} (Q′ ⊗_{B} Q) → A,
μ′(1 ⊗ μ ⊗ 1) : (Q′ ⊗_{B} Q) ⊗_{A} (P ⊗_{B} P′) → B′)

It is straightforward to check that this product is well-defined and associative. The next lemma shows that this product can be translated to the isomorphism classes of Morita contexts.

Lemma 2.3 The product of Morita contexts is compatible with the isomorphism of Morita contexts; i.e., if C ≃ C_{1} and C′ ≃ C′_{1} are Morita contexts so that B = A′ and B_{1} = A′_{1}, then C ·C′ ≃ C_{1} ·C′_{1}.

Proof. Let ϕ : C → C_{1} and ϕ′ : C′ → C′_{1} isomorphisms of Morita contexts. Then one checks by straightforward computations that (α, β′, π ⊗ π′, ϱ′ ⊗ ϱ) is an isomorphism of Morita contexts C · C′ ≃ C_{1} · C′_{1}.

We now turn our attention to graded rings. Let R be a G-graded ring. As we want a context C = (A, B, P, Q, τ, μ) to act on R, we need to make sure that R can be viewed as an A-bimodule. Thus, for technical reasons, it is convenient to assume that the coefficient ring R_{e} is isomorphic to A. This leads us to the following definition of an (A, G)-graded ring and the appropriate context action.

Definition 2.4 Let A be a ring and G a group. An (A, G)-graded ring is a pair (R, f) where R is a G-graded ring and f : A → R_{e} is a ring isomorphism.

Let α : A → A′ be a ring homomorphism, (R, f) an (A, G)-graded ring and (R′, f′) an (A′, G)- graded ring. A graded α-homomorphism from (R, f) to (R′, f′) is a graded ring homomorphism Φ : R → R′ which makes the diagram

\begin{matrix}A & →^{f} & R \\ α ↓ & & ↓ Φ \\ A′ & →^{f′} & R′ \end{matrix}

commutative.

(A, G)-graded rings and the 1_{A}-morphisms of (A, G) define a category in a natural way. The graded 1_{A}-isomorphism class of an (A, G)-graded ring (R, f) is denoted by [R, f]. Now we define an action of Morita contexts on graded rings.

Definition 2.5 Given a Morita context C and an (A, G)-graded ring (R, f) we define (R, f)^{C} = (R^{C}, f^{C}), where 1. R^{C} = Q ⊗_{A} R ⊗_{A} P = ⊕_{g∈G}Q ⊗_{A} R_{g} ⊗_{A} P with the product defined by

(q ⊗ r ⊗ p)(q′ ⊗ r′ ⊗ p′) = (q ⊗ rτ(p ⊗ q′)r′ ⊗ p′)

2. f^{C} = (1 ⊗ f ⊗ 1)μ^{−1}.

Next lemma shows that this action induces an action of the isomorphism classes of Morita contexts on the class of (graded) isomorphism classes of graded rings.

Lemma 2.6 Let (R, f) be an (A, G)-graded ring, (R′, f′) an (A′, G)-graded ring, ϕ = (α, β, π, ρ) : C → C′ a morphism of Morita contexts and Φ : (R, f) → (R′, f′) a graded α-morphism of rings. Then Φ^{ϕ} = ϱ ⊗ Φ ⊗ π : R^{C} → R′^{C′} is a graded β-morphism. In particular, if ϕ : C ≃ C′ and (R, f) is α-isomorphic to (R′, f′), then (R, f)^{C} is β-isomorphic to (R′, f′)^{C′}.

Proof. We leave to the reader to check that Φ^{ϕ} is well defined. We check that Φ^{ϕ} is a ring homomorphism. Given q, q′ ∈ Q, r, r′ ∈ R and p, p′ ∈ P, then

Φ^{ϕ}((q ⊗ r ⊗ p)(q′ ⊗ r′ ⊗ p′)) = Φ^{ϕ}(q ⊗ rτ(p ⊗ q′)r′ ⊗ p′)
= ϱ(q) ⊗ Φ(rτ(p ⊗ q′)r′) ⊗ π(p′)
= ϱ(q) ⊗ Φ(r)(Φfτ)(p ⊗ q′)Φ(r′) ⊗ π(p′)
= ϱ(q) ⊗ Φ(r)(f′ατ)(p ⊗ q′)Φ(r′) ⊗ π(p′)
= ϱ(q) ⊗ Φ(r)(f′τ′(π ⊗ ϱ))(p ⊗ q′)Φ(r′) ⊗ π(p′)
= ϱ(q) ⊗ Φ(r)τ′(π(p) ⊗ ϱ(q′))Φ(r′) ⊗ π(p′)
= (ϱ(q) ⊗ Φ(r) ⊗ π(p))(ϱ(q′)) ⊗ Φ(r′) ⊗ π(p′))
= Φ^{ϕ}(q ⊗ r ⊗ p)Φ^{ϕ}(q′ ⊗ r′ ⊗ p′)

It is straightforward to see that Φ^{C} is graded. Finally we prove that Φ^{ϕ} is a β-homomorphism. Since the diagrams

\begin{matrix}A & →^{f} & R \\ α ↓ & & ↓ Φ \\ A′ & →^{f′} & R′ \end{matrix} and \begin{matrix}Q ⊗_{A} P & →^{μ} & B \\ ϱ ⊗ π ↓ & & ↓ β \\ Q′ ⊗_{A′} P′ & →^{μ′} & B′ \end{matrix}

are commutative, the diagram

B →^{f^{C}} Q ⊗_{A} R ⊗_{A} P
β ↓  ↓ Φ^{ϕ}
B′ →^{f′^{C′}} Q′ ⊗_{A′} R′ ⊗_{A′} P′

is also commutative.

We close this section by showing that the action of (isomorphism classes ) of Morita contexts on the (isomorphism classes) of graded rings is multiplicative.

Lemma 2.7 Let (R, f) be an (A, G)-graded ring and C and C′ Morita contexts so that B = A′. Then (R, f)^{C·C′} is isomorphic to ((R, f)^{C})^{C′}.

Proof. This follows by showing that the classical isomorphism

(Q′ ⊗_{A′} Q) ⊗_{A} R ⊗_{A} (P ⊗_{A′} P′) → Q′ ⊗_{A′} (Q ⊗_{A} R ⊗_{A} P) ⊗_{A′} P′

is an isomorphism of (B′, G)-graded rings.

Notation 2.8 We denote the action of the isomorphism class of the Morita context C on the isomorphism class [R, f] of the (A, G)-graded ring (R, f) by [R, f]^{C}.

3 Graded equivalences

The moral of the previous section is that we may consider the equivalence classes of Morita contexts (partially) acting multiplicatively on graded rings. In this section, we show that this action completely characterizes graded equivalences for strongly graded rings.

The key to graded contexts is that they induce Morita contexts for the coefficient ring when the ring R is strongly graded. This is the essence of the following lemma.

Lemma 3.1 Let (R, f) be a strongly (A, G)-graded ring and (R′, f′) a strongly (A′, G)-graded ring. If C = (R, R′, X, Y, τ′, μ′) is a graded context, then C = (A, A′, P = X_{e}, Q = Y_{e}, τ = f^{−1}τ′_{e}, μ = f′^{−1}μ′_{e}) is a Morita context, where τ′_{e} : P ⊗_{A′} Q → A and μ′_{e} : Q ⊗_{A} P → A′ are given by τ′_{e}(p ⊗ q) = τ′(p × q) and μ′_{e}(q ⊗ q) = μ′(q × p).

Proof. Since τ′ is surjective, there are x_{i} ∈ X and y_{i} ∈ Y (i = 1, 2, …, n) so that τ′(∑_{i=1}^{n} x_{i}⊗y_{i}) = 1. We may assume that every x_{i} is homogeneous, say of degree g_{i}, and that every y_{i} is homogeneous of degree g_{i}^{−1}. For every i = 1, 2, …, n, choose r_{ij} ∈ R′_{g_{i}^{−1}} and s_{ij} ∈ R′_{g_{i}} (j = 1, 2, …, k_{i}) so

that ∑_{j} r_{ij}s_{ij} = 1. Then τ(∑_{i} ∑_{j} x_{i}r_{ij} ⊗ s_{ij}y_{i}) = f^{−1}τ′(∑_{i} ∑_{j} x_{i}r_{ij} ⊗ s_{ij}y_{i}) = f^{−1}τ′(∑_{i}(x_{i} ⊗ ∑_{j}(r_{ij}s_{ij})y_{i}) = f^{−1}(τ′(∑_{i=1}^{n} x_{i} ⊗ y_{i}) = f^{−1}(1) = 1. This shows that τ is surjective. By symmetry, μ is also surjective.

We can now present our characterization of graded equivalences using our context action.

Theorem 3.2 Let (R, f) be an (A, G)-graded ring and let (R′, f′) be a (A′, G)-graded ring.

1. If there exists a Morita context C such that [R′, f′] = [R, f]^{C}, then R and R′ are graded equivalent.

2. If R and R′ are strongly graded, then the converse of (1) holds; that is, if R and R′ are strongly graded and graded equivalent, then there exists a Morita context C such that [R′, f′] = [R, f]^{C}.

Proof. 1. It is enough to show that R and R^{C} are graded equivalent. Let Q ⊗_{A} − : R-gr → R^{C}-gr be the functor defined as follows: If M ∈ R-gr, then Q ⊗_{A} − maps M to Q ⊗_{A} M with the grading (Q ⊗_{A} M)_{g} = Q ⊗_{A} M_{g} and the left multiplication by elements of R^{C} is defined by

(q ⊗ r ⊗ p)(q′ ⊗ m) = qrτ(p ⊗ q′) ⊗ m

The action of Q ⊗_{A} − on morphisms is the natural one. It is straightforward to see that Q ⊗_{A} − commutes with the suspension functor T_{g} and that Q ⊗_{A} − is a category equivalence. It follows that Q ⊗_{A} − is a graded equivalence from our discussion in the introduction. For further details, see [8].

2. Let C = (R, R′, X, Y, τ′, μ′) be graded context and assume that R and R′ are strongly graded. By Lemma 3.1 C = (A, A′, P = X_{e}, Q = Y_{e}, τ = f^{−1}τ′_{e}, μ = f′^{−1}μ′_{e}) is a Morita context. Let Φ : R^{C} = Q ⊗_{A} R ⊗_{A} P → R′ be defined by Φ(q ⊗ r ⊗ p) = μ′(qr ⊗ p).

We first show that Φ is a ring homomorphism.

Φ((q ⊗ r ⊗ p)(q′ ⊗ r′ ⊗ p′)) = Φ(q ⊗ rτ(q ⊗ q′)r′ ⊗ p′)

= μ′(qrτ′(p ⊗ q′)r′ ⊗ p′) = μ′(μ′(qr ⊗ p)q′r′ ⊗ p′) = μ′(qr ⊗ p)μ′(q′r′ ⊗ p′) = Φ(q ⊗ r ⊗ p)Φ(q′ ⊗ r′ ⊗ p′).

It is clear that Φ is graded.

Let r′_{g} ∈ R′_{g}. Since μ′ is surjective, there are x_{i} ∈ X and y_{i} ∈ Y (i = 1, …, n) so that r′_{g} = μ′(∑_{i} x_{i} ⊗ y_{i}). We may assume that every x_{i} is homogeneous (of degree gh_{i}) and y_{i} is homogeneous (of degree h_{i}^{−1}). Then r′_{g} = Φ(∑_{ij} x_{i}a′_{ij} ⊗ b′_{ij}a_{ik} ⊗ b_{ik}y_{i}), where a_{ik} ∈ R_{h_{i}^{−1}}, b_{ik} ∈ R_{h_{i}}, a′_{ij} ∈ R_{(gh_{i})^{−1}}, b′_{ij} ∈ R_{gh_{i}} and ∑_{k} a_{ik}b_{ik} = 1 = ∑_{j} a′_{ij}b′_{ij}, for every i. This shows that Φ is surjective.

To prove that Φ is injective, we first prove μ′ restricts to a bijection μ_{g} : Y_{g} ⊗_{A} P → R′_{g}, for every g ∈ G. Indeed, since μ = f^{−1}μ_{e}, μ_{e} is a bijection. On the other hand the following diagram is commutative

R′_{g^{−1}} ⊗_{A} Y_{g} ⊗_{A} P →^{1⊗μ_{g}} R′_{g^{−1}} ⊗_{A} R′_{g}
M ⊗ 1 ↓  ↓ M
Q ⊗_{A} P  →^{μ_{e}}  R′_{e}

and so 1 ⊗ μ_{g} is a bijection. Since R_{g^{−1}} is faithfully flat as right A-module, μ_{g} is a bijection.

Now assume that Φ(∑_{i} q_{i} ⊗ r_{i} ⊗ p_{i}) = 0. Then μ′(∑_{i} q_{i}r_{i} ⊗ p_{i}) = 0. We may assume that every r_{i} is homogeneous and that they all have the same degree, say g. Then μ_{g}(∑_{i} q_{i}r_{i} ⊗ p_{i}) = 0 and hence ∑_{i} q_{i}r_{i} ⊗ p_{i} = 0 in X_{g} ⊗_{A} P. Therefore, ∑_{i} q_{i} ⊗ r_{i} ⊗ p_{i} = 0 in Q ⊗_{A} R_{g} ⊗_{A} P.

It only remains to show that the following diagram is commutative

\begin{matrix}A′ & →^{f^{C}} & R^{C} \\ 1 ↓ & & ↓ Φ \\ A′ & →^{f′} & R′ \end{matrix}

However Φf^{C}μ(q ⊗ p) = Φ(1 ⊗ f ⊗ 1)(q ⊗ p) = Φ(q ⊗ 1 ⊗ p) = μ′(q ⊗ p) = f′μ(q ⊗ p).

The following corollary is a direct consequence of Theorem 3.2.

Corollary 3.3 If R is a strongly graded ring and A is Morita equivalent to R_{e}, then R is graded equivalent to a strongly graded ring R′ such that R′_{e} = A.

Since a strongly graded ring R is a twisted group ring if and only if the homogeneous components of R are isomorphic to R_{e} as R_{e}-bimodules, we have the following corollary.

Corollary 3.4 If R is a twisted group ring, then every ring graded equivalent to R is also a twisted group ring.

Remark 3.5 We note that the results of [12] and [13] are similar to some of the results found in this and the previous section. For example, Theorem 1.1 of [13] compares to Lemma 2.3 and Lemma 2.7, which shows that Morita contexts, as well as their action on graded rings, are multiplicative. Moreover, Theorem 3.4 of [12] is similar in flavor to Theorem 3.2, which shows that graded equivalences arise from Morita contexts of the coefficient ring.

4 The Picard group acting on graded rings.

The philosophy of this section is that, using Corollary 3.3, we have reduced the coefficient ring (via graded equivalence) as much as possible and, consequently, we now fix the coefficient ring A. That is, we are interested in all the G-graded rings R, with R_{e} = A and our goal is to identify the graded equivalent classes. In this case, the action of Morita contexts on graded rings reduces to an action of Pic(A) and the graded equivalent classes turn out to be the orbits under this action. For technical reasons, it is more convenient to consider all the G-graded rings R such that R_{e} is isomorphic to A. This does not change our study but rather simplifies the proofs.

Notation 4.1 Pic(A) denotes the Picard group of A. If P is an invertible A-bimodule, [P] ∈ Pic(A) denotes the class of invertible A-bimodules isomorphic to P.

Given an automorphism σ of A, A^{σ} is the A-bimodule defined as follows: A^{σ} = A as a left Amodule and right multiplication is given by: p · a = pσ(a) (a ∈ A, p ∈ A^{σ}). Similarly, we define ^{σ}A. It is well known that A^{σ} is invertible and that the map α ↦ [A^{σ}] defines a group homomorphism Aut(A) → Pic(A) whose kernel is Inn(A), the set of inner automorphisms of A. Accordingly we identify Out(A) = Aut(A)/Inn(A) with the image of this map.

Now let R be a G-graded ring. Then Picgr(R) denotes the group of graded isomorphism classes of invertible R-bimodules P that occur in a graded context (R, R, P, Q, τ, μ).

Let (R, f) be an (A, G)-graded ring. If σ is a graded automorphism of R, then R^{σ} ∈ Picgr(R). Moreover, σ induces an automorphism σ_{e} = f^{−1}σf in A. We denote the set of graded automorphisms of R by Autgr(R) and the set of inner automorphisms of R induced by an invertible element of A by Inn_{A}(R). Set

Pic_{R}(A) = {[P] ∈ Pic(A) : [P] = [Q_{e}] for some Q ∈ Picgr(R)}
Aut_{R}(A) = {σ ∈ Aut(A) : f^{−1}σf extends to a graded automorphism of R}

and

Out_{R}(A) = Aut_{R}(A)/Inn(A)

If (R, f) is a strongly (A, G)-graded ring, then G acts on the center of A via the Miyashita action σ : G → Aut(Z(A)). That is, if g ∈ G and a ∈ Z(A), then a^{σ_{g}} is defined by

ar_{g} = r_{g}a^{σ_{g}} (r_{g} ∈ R_{g})

In this paper, the coboundary, cocycle and cohomology groups B^{n}(G, U(Z(A))), Z^{n}(G, U(Z(A))) and H^{n}(G, U(Z(A))) are considered with respect to this action.

Proposition 4.2 If (R, f) is a strongly G-graded ring, there is a commutative diagram

1  1  1
↓  ↓  ↓
1 → B^{1}(G, U(Z(A))) →^{α_{1}} Inn_{A}(R) →^{β_{1}} Inn(A) → 1
j ↓ j ↓ j ↓
1 → Z^{1}(G, U(Z(A))) →^{α_{2}} Autgr(R) →^{β_{2}} Aut_{R}(A) → 1
T ↓ T ↓ T ↓
1 → H^{1}(G, U(Z(A))) →^{α_{3}} Picgr(R) →^{β_{3}} Pic_{R}(A) → 1

↓ 1

with exact rows and columns of group homomorphisms, and all the vertical homomorphisms are canonical.

Proof. We first define the morphisms α and β. If c ∈ Z^{1}(G, U(Z(A))), then α_{2}(c) : R → R is given by α_{2}(c)(r_{g}) = r_{g}c(g) (g ∈ G and r_{g} ∈ R_{g}). α_{1} is the restriction of α_{2} to B^{1}(G, U(Z(A))). So the upper left square is commutative and then α_{3} is the only group homomorphism making the lower left square commutative.

β_{1} and β_{2} are the restriction maps and β_{3} is the map given by β_{3}([P]) = [P_{e}]. Note that by Lemma 3.1, β_{3} maps Picgr(R) into Pic(A).

Commutativity of the diagram is straightforward to check. We only have to prove that β_{3} is a group homomorphism and that the rows are exact.

For the remainder of the proof g denotes an arbitrary element of G and x_{i} ∈ R_{g^{−1}} and y_{i} ∈ R_{g} are so that ∑_{i} x_{i}y_{i} = 1.

Let P, P′ ∈ Picgr(R) and consider the maps Φ : P_{e} ⊗_{A} P′_{e} → (P ⊗_{R} P′)_{e} the inclusion map and Ψ : (P ⊗_{R} P′)_{e} → P_{e} ⊗_{A} P′_{e} given by Ψ(p ⊗ p′) = ∑_{i} px_{i} ⊗ y_{i}p′, for p ⊗ p′ ∈ P_{g} ⊗ P′_{g}.

We have to check that Ψ is well defined. Let a_{i} ∈ R_{g^{−1}} and b_{i} ∈ R_{g} such that ∑_{i} a_{i}b_{i} = 1. Let p × p′ ∈ P_{g} ⊗ P′_{g}. Then

∑_{i} px_{i} ⊗ y_{i}p′ = ∑_{i} p ∑_{j} a_{j}b_{j}x_{i} ⊗ y_{i}p′
= ∑_{i} ∑_{j} pa_{j} ⊗ b_{j}x_{i}y_{i}p′
= ∑_{j} pa_{j} ⊗ b_{j}p′

By straightforward computations, one shows that Φ and Ψ are inverse A-bimodule isomorphisms. This shows that β_{3} is a group homomorphism.

We leave the details of checking that the first row is exact. For the second, let c ∈ Ker α_{2}, then r_{g}c(g) = r_{g}, for every r_{g} ∈ R_{g}. Since R_{g} is faithful as right A-module, c = 1. Thus α_{2} is injective. It is clear that Im α_{2} ⊆ Ker β_{2}. Let σ ∈ Ker β_{2}. We claim that for every g ∈ G, there is a unit c(g) ∈ A, such that σ(r_{g}) = r_{g}c(g), for every r_{g} ∈ R_{g}. Let c(g) = f^{−1}(∑_{i} x_{i}σ(y_{i})). Then σ(r_{g}) = σ(r_{g} ∑_{i} x_{i}y_{i}) = r_{g}c(g). If a ∈ A, then r_{g}ac(g) = σ(r_{g}a) = σ(r_{g})a = r_{g}c(g)a. Therefore c(g) is central and, since r_{g} ↦ r_{g}c(g) is an automorphism of _{A}R_{g}, c(g) is invertible in A. Moreover, for every r_{g} ∈ R_{g} and s_{h} ∈ R_{h}, r_{g}s_{h}c(gh) = r_{g}c(g)s_{h}c(h) = r_{g}s_{h}c(g)^{σ_{h}}c(h) and hence c ∈ Z^{1}(G, U(Z(A))). Thus σ ∈ Im α_{2}.

For the third row, let c ∈ Z^{1}(G, U(Z(A))). If c + B^{1}(G, U(Z(A))) ∈ Ker α_{3}, then α_{2}(c) ∈ Inn_{A}(R). Thus, there exists u a unit in Z(A), such that r_{g}c(g) = ur_{g}u^{−1} = r_{g}u^{σ_{g}}u^{−1}, for every g ∈ G and every r_{g} ∈ R_{g}. Consequently, c(g) = u^{σ_{g}}u^{−1} and hence c ∈ B^{1}(G, U(Z(A))). This shows that α_{3} is injective. Since Im α_{2} = Ker β_{2} and T : Z^{1}(G, U(Z(A))) → H^{1}(G, U(Z(A))) is surjective, Im α_{3} ⊆ Ker β_{3}. Assume that [P] ∈ Ker β_{3}; this means that P_{e} is isomorphic to A as A-bimodule and hence P ≃ R ⊗_{A} P_{e} is isomorphic to R as graded left R-module. It follows that [P] = [A^{σ}] for some σ ∈ Ker β_{2}. Using that the middle exact sequence is exact, it is now easy to prove that [P] ∈ Im α_{3}.

Notation 4.3 Set

Gr(A, G) = {[R, f] : (R, f) is a (A, G)-graded ring}

and

StGr(A, G) = {[R, f] ∈ Gr(A, f) : R is strongly graded}

Every element [P] ∈ Pic(A) canonically defines a Morita context

C(P) = (A, A, P, P^{−1} = Hom(_{A}P ,_{A}A), τ, μ),

where τ(p⊗f) = (p)f and p′μ(f ⊗p) = (p′)fp. Moreover if P and P′ are two invertible A-modules, then [P] ≃ [P′] if and only if there is an isomorphism ϕ : C(P) ≃ C(P′) of Morita contexts so that α = β = 1.

By the results from the previous section, we define the following action of Pic(A) on Gr(A, G):

[R, f]^{[P]} = [R, f]^{C(P)}

It is clear that StGr(A, G) is invariant under this action.

We can use Theorem 3.2 to compute the orbit of an [R, f] ∈ StGr(A, G).

Proposition 4.4 Let [R, f] ∈ StGr(A, G).

1. The orbit of [R, f] in StGr(A, G) by Pic(A) is

Orb_{Pic(A)}([R, f]) = {[R′, f′] ∈ StGr(A, G) : R′ is graded equivalent to R}.

2. The stabilizer of [R, f] by Pic(A) is Pic_{R}(A).

3. There is a one-to-one correspondence between the set of isomorphism classes of strongly (A, G)-graded rings, graded Morita equivalent to R and the set of cosets Pic(A)/Pic_{R}(A).

Proof. (1) This is a consequence of Theorem 3.2.

(2) Lemma 3.1 shows that Pic_{R}(A) is contained in the stabilizer of (R, f) in Pic(A).

Assume now that [P] belongs to the stabilizer of (R, f) in Pic(A). Set C = C(P) and Φ : R → R^{C} a graded isomorphism making commutative the diagram

\begin{matrix}A & →^{f} & R \\ 1 ↓ & & ↓ Φ \\ A & →^{f^{C}} & R^{C} \end{matrix}

Let Ψ be the composition P ⊗_{A} R →^{1⊗Φ} P ⊗_{A} Q ⊗_{A} R ⊗_{A} P →^{τ⊗1} R ⊗_{A} P. Using this isomorphism, we can endow X = R ⊗_{A} P with an R-bimodule structure by defining the left multiplication in the canonical way and the right multiplication by: xr = Ψ(Ψ^{−1}(x)r). We check that these two multiplications make R ⊗_{A} P into an R-bimodule. Assume that τ^{−1}(1) = ∑_{i} p_{i} ⊗ q_{i} and that Φ(r′) = ∑_{j} x_{j} ⊗ s_{j} ⊗ y_{j}. Then

(r ⊗ p)r′ = Ψ(Ψ^{−1}(r ⊗ p)r′)
= Ψ((1 ⊗ Φ^{−1})(∑_{i} p_{i} ⊗ q_{i} ⊗ r ⊗ p)r′)
= Ψ(∑_{i} p_{i} ⊗ Φ^{−1}(q_{i} ⊗ r ⊗ p)r′)
= (τ ⊗ 1)(∑_{i} p_{i} ⊗ (q_{i} ⊗ r ⊗ p)Φ(r′))
= (τ ⊗ 1)(∑_{i} p_{i} ⊗ (q_{i} ⊗ r ⊗ p) ∑_{j} x_{j} ⊗ s_{j} ⊗ y_{j})
= (τ ⊗ 1)(∑_{i} p_{i} ⊗ ∑_{j} q_{i} ⊗ rτ(p ⊗ x_{j})s_{j} ⊗ y_{j})
= ∑_{j}(∑_{i} τ(p_{i} ⊗ q_{i})rτ(p ⊗ x_{j})s_{j} ⊗ y_{j})
= ∑_{j}(rτ(p ⊗ x_{j})s_{j} ⊗ y_{j})
= r((1 ⊗ p)r′)

Therefore r((r_{1} ⊗p)r′) = r((1⊗p)r′) = (rr_{1} ⊗p)r′ = (r(r_{1} ⊗p))r′. We make X into a graded left Rmodule by using the grading of R. In a similar way, we make P ⊗_{A} R into a graded right R-module. Since Ψ is a graded isomorphism of abelian groups, X and P ⊗_{A} R are graded R-bimodules, so that Ψ is a graded isomorphism of graded R-bimodules. Similarly, Y = Q ⊗_{A} R and R ⊗_{A} Q are isomorphic R-bimodules.

Now X ⊗_{R} Y ≃ P ⊗_{A} R ⊗_{R} Q ⊗_{A} R ≃ P ⊗_{A} R ⊗_{R} R ⊗_{A} Q ≃ P ⊗_{A} R ⊗_{A} Q ≃ P ⊗_{A} Q ⊗_{A} R ≃ A ⊗_{A} R ≃ R. Similarly Y ⊗_{R} X ≃ R and these isomorphisms are graded. Since [P] ∈ Pic(A) and P ⊗_{A} R ∈ Picgr(R), then we have shown that [P] ∈ Pic_{R}(A). Moreover P ≃ X_{e} as A-bimodules.

(3) This follows from 1 and 2.

The following fact is elementary: Every element of Gr(A, G) has a representative of the form [R, 1]; i.e., a representative, where R_{e} = A and f = 1. Thus we have the following corollary.

Corollary 4.5 Given a ring A, there is a one-to-one correspondence between the graded equivalence classes of strongly G-graded rings R such that R_{e} = A and the orbits of Pic(A) in StGr(A, G).

The above descriptions of the orbit require strongly graded rings. However, when we restrict our attention to the action induced by Out(A) = Aut(A)/Inn(A), we can describe the orbits even for non-strongly graded rings. The key observation is that the homomorphisms β_{1} and β_{2} can be defined even if R is not strongly graded. In that case, we set Out^{gr}(R) = Aut^{gr}(R)/Inn_{A}(R) and so there is a homomorphism β′_{3} : Out^{gr}(R) → Out_{R}(A) making the diagram (with exact rows and columns)

1  1
↓  ↓
Inn_{A}(R) →^{β_{1}} Inn(A) → 1
↓  ↓
Autgr(R) →^{β_{2}} Aut_{R}(A) → 1
↓  ↓
Outgr(R) →^{β′_{3}} Out_{R}(A) → 1
↓  ↓
1  1

commutative.

Proposition 4.6 Let [R, f] ∈ Gr(A, G)

1. The orbit of [R, f] by Out(A) is

Orb_{Out(A)}([R, f]) = {[R′, f′] ∈ Gr(A, G) : R′ is graded isomorphic to R}.

2. The stabilizer of [R, f] by Out(A) is Aut_{R}(A).

3. There is a one-to-one correspondence between the set of isomorphism classes of (A, G)-graded rings graded A-isomorphic to R and the set of cosets Out(A)/Out_{R}(A) (or equivalently Aut(A)/Aut_{R}(A)).

Proof. (1) Let σ be an automorphism of A. Then C(A^{σ}) (see Notation 4.3) is isomorphic to the following Morita context C = (A, A, A^{σ},^{σ}A, τ, μ_{σ}), where τ(a⊗b) = ab and μ_{σ}(a⊗b) = σ(ab). The map

\begin{matrix}^{σ}A ⊗_{A} R ⊗_{A} A^{σ} & →^{Ψ^{σ}} & R \\ a ⊗ r ⊗ b & ↦ & arb \end{matrix}

is a graded ring isomorphism. This proves that Orb_{Out(A)}([R, f]) is embedded in the graded isomorphism class of R.

Assume now that (R′, f′) is an (A, G)-graded ring and Φ : R → R′ is a graded ring isomorphism. Let σ = f′^{−1}Φ_{e}f ∈ Aut(A). Then the following diagram is commutative and consists of ring isomorphisms

A →^{μ_{σ}^{−1}} ^{σ}A ⊗_{A} A ⊗_{A} A^{σ} →^{1⊗f⊗1} ^{σ}A ⊗_{A} R_{e} ⊗_{A} A^{σ}
σ^{−1} ↓  ↓ Ψ_{e}^{σ}
A ⟶^{f} R_{e}
σ ↓  ↓ Φ_{e}
A  ⟶^{f′}  R′_{e}

Therefore, [R′, f′] = [R, f]^{σ} and so (1) holds.

(2) The automorphism σ stabilizes [R, f] if and only if there is a graded isomorphism Φ : R → R^{C} which makes the diagram

A  ⟶^{f}  R
1 ↓  ↓ Φ
A →^{μ_{σ}^{−1}} ^{σ}A ⊗_{A} A ⊗_{A} A^{σ} →^{1⊗f⊗1} ^{σ}A ⊗_{A} R ⊗_{A} A^{σ}

commutative. But this will hold if and only if the diagram

A  →^{f}  R_{e}
σ^{−1} ↓  ↓ Φ_{e}
A →^{M^{−1}} ^{σ}A ⊗_{A} A ⊗_{A} A^{σ} →^{1⊗f⊗1} ^{σ}A ⊗_{A} R_{e} ⊗_{A} A^{σ}
1 ↘  ↓ M  ↓ M
A  →^{f}  R_{e}

is commutative. Thus σ stabilizes [R, f] if and only if fσf^{−1} extends to a graded automorphism of R; in other words if σ ∈ Aut_{R}(A).

(3) is a consequence of (1) and (2).

Corollary 4.7 Given a ring A, there is a one-to-one correspondence between the graded isomorphism classes of G-graded rings R, with R_{e} = A and the orbits of Out(A) (or Aut(A)) in Gr(A, G).

Our last remark of this section shows that Corollary 4.7 is a generalization of [11, Proposition 2].

Remark 4.8 By [14, Section I.1.3], the isomorphism classes of strongly graded rings R such that R_{e} = A can be given in terms of a group homomorphism G → Pic(A) (g ↦ [R_{g}]) and a set of bimodule homomorphism R_{g} ⊗_{A} R_{h} → R_{gh} satisfying certain conditions. Crossed products are determined by parameter sets. In the notation of [11], two crossed products R and R′, with R_{e} = R′_{e} = A are A-isomorphic if and only if [R, 1] = [R′, 1]. Further, two parameter sets define A-isomorphic crossed products if and only if they are equivalent in the sense of [11]. Let CP(A, G) be the subset of StGr(A, G) formed by the classes that contain a crossed product. The action of Aut(A) (or Out(A)) on StGr(A, G) restricts to an action on CP(A, G) and hence this action can be translated to an action on the set of equivalence classes of parameter sets. Therefore Corollary 4.7 generalizes [11, Proposition 2].

5 Strongly graded rings with isomorphic components

The aim of this section is to show that graded equivalent strongly graded rings with the same homogeneous components are not necessarily graded isomorphic; see Theorem 5.8. We begin with some notation.

Notation 5.1 Fix a unital ring A, a group G, and a strongly (A, G)-graded ring (R, f). Let Pic_{(R,f)}(A) denote the centralizer of {[R_{g}] : g ∈ G}; i.e.,

Pic_{(R,f)}(A) = {[P] ∈ Pic(A) : [P][R_{g}] = [R_{g}][P] for every g ∈ G}

Let StGr_{(R,f)}(A, G) = {[R′, f′] ∈ StGr(A, G) : [R_{g}] = [R′_{g}] for every g ∈ G}

The next lemma is an obvious consequence of Theorem 3.2.

Lemma 5.2 There is a map Σ : Pic_{(R,f)}(A) → StGr_{(R,f)}(A, G) given by Σ([P]) = [R, f]^{P}, whose image is the set of the [R′, f′] ∈ StGr_{(R,f)}(A, G) such that R′ is graded equivalent to R.

The significance of Lemma 5.2 is that Pic_{(R,f)}(A)/Pic_{R}(A) parametrizes the class of strongly graded rings that are graded equivalent to R but not graded A-isomorphic to R and that have homogeneous components A-isomorphic to those of R. However, there is another well-documented way to parametrize StGr_{(R,f)}(A, G) via the cohomology group H^{2}(G, U(Z(A))).

Definition 5.3 Given c ∈ Z^{2}(G, U(Z(A))), R^{c} denotes the strongly graded ring such that R^{c} = R as an additive group with the multiplication in R^{c} defined by

r_{g} · s_{h} = r_{g}s_{h}c(g, h) (g, h ∈ G, r_{g} ∈ R_{g}, s_{h} ∈ R_{h})

Recall that we have reserved the letter M to denote multiplication maps. To distinguish different multiplication maps induced by an (A, G)-graded ring (R, f), we denote M_{g,h} = M : R_{g} ⊗_{A} R_{h} → R_{gh}.

Theorem 5.4 [14, Theorem A.I.3.16] Let (R, f) be a strongly G-graded ring. There is a bijection Ψ : StGr_{(R,f)}(A, G) → H^{2}(G, U(Z(A))) given as follows: Ψ^{−1}(\overline{c}) = [R^{c}, f] and Ψ([R′, f′]) = \overline{c_{(R′,f′)}} (the homology induced by c_{(R′,f′)}) where the following map is right multiplication by c_{(R′,f′)}(g, h), for every g, h ∈ G,

R_{gh} →^{M_{g,h}^{−1}} R_{g} ⊗_{A} R_{h} →^{δ_{g}⊗δ_{h}} R′_{g} ⊗_{A} R′_{h} →^{M_{g,h}} R′_{gh} →^{δ_{gh}^{−1}} R_{gh}

and δ_{g} : R_{g} → R′_{g} is an isomorphism of A-bimodules for every g ∈ G.

The key to the main result of this section (Theorem 5.8) requires an example computed in [5] and the map ϕ defined in [6]. Recall that if R is a strongly graded ring, then σ : G → Aut(Z(R_{e})) denotes the Miyashita action induced by R (see Notation 4.1).

Notation 5.5 Every [P] ∈ Pic(A) defines an automorphism α_{P} of Z(A) defined by pz = α_{P} (z)p for all p ∈ P and z ∈ Z(A). This gives rise to an action α : Pic(A) → Aut(Z(A)) via [P] ↦ α_{P}. If [P] ∈ Pic_{(R,f)}(A), then α_{P} commutes with σ_{g} for every g ∈ G. Therefore, P induces an automorphism β_{P} of H^{2}(G, Z(A)) and so we have another action β : Pic_{(R,f)}(A) → Aut(H^{2}(G, Z(A)) via [P] ↦ β_{P}. We denote the image of x ∈ Pic_{(R,f)}(A) under β by β_{x}.

Proposition 5.6 [5] There is an exact sequence

1 → H^{1}(G, U(Z(A))) →^{α_{3}} Pic^{gr}(R) →^{β_{3}} Pic_{(R,f)}(A) →^{ϕ} H^{2}(G, U(Z(R_{e})))

where α_{3} and β_{3} are the group homomorphisms from Proposition 4.2 and for every x, y ∈ Pic_{(R,f)}(A), ϕ(xy) = ϕ(x)β_{x}(ϕ(y))

Now, to use [5], we must show that the maps ϕ and Ψ ∘ Σ : Pic_{(R,f)}(A) → H^{2}(G, U(Z(A))) are strongly related. This is precisely what the next lemma does.

Lemma 5.7 ϕ ∘ (−)^{−1} = Ψ ∘ Σ; i.e., the following diagram

Pic_{(R,f)}(A) →^{(−)^{−1}}  Pic_{(R,f)}(A)
Σ ↓  ↓ ϕ
StGr_{(R,f)}(A, G) →^{Ψ} H^{2}(G, U(Z(A)))

is commutative ((−)^{−1} denotes the inverse map).

Proof. Let [P] ∈ Pic_{(R,f)}(A) and (A, A, P, Q, τ, μ) be a Morita context. We need to show that ϕ([P]) = (ΨΣ)([Q]).

Fix g, h ∈ G.

There is a A-bimodule isomorphism γ_{g} : P → R_{g^{−1}} ⊗_{A} P ⊗_{A} R_{g}. Following the proof of Proposition 5.6 in [5] one can obtain ϕ([P]) in terms of the map γ_{g}. More precisely, ϕ([P]) = \overline{c} where γ_{gh}^{−1}(M_{h^{−1},g^{−1}} ⊗ 1 ⊗ M_{g,h})(1 ⊗ γ_{g} ⊗ 1)γ_{h} is left multiplication by c(g, h).

Let δ_{g} = (ρ_{g}⊗1)(1⊗γ_{g}⊗1)(1⊗τ^{−1}) where ρ_{g} : R_{g}⊗_{A}R_{g^{−1}}⊗P → P is given by ρ_{g}(r_{g}⊗s_{g^{−1}}⊗p) = (r_{g}s_{g^{−1}})p.

By Theorem 5.4, (Ψ ∘ Σ)([Q]) = \overline{d} where for every g, h ∈ G, δ_{gh}^{−1}M_{g,h}(δ_{g} ⊗ δ_{h})M_{g,h}^{−1} is right multiplication by d(g, h). Fix r ∈ R_{g} and s ∈ R_{h}. We show that rsd(g, h) = rsc(g, h) and so it will follow that c = d.

We are going to use a Sweedler-like sigma notation. Given a map f : X → Y ⊗ Z and x ∈ X, the element f(x) is going to be denoted by ∑_{f} x_{(Y)} ⊗ x_{(Z)}. If Y = R_{g}, then x_{(R_{g})} is denoted x_{(g)}. For example τ^{−1}(a) = ∑_{τ^{−1}} a_{(P)} ⊗ a_{(Q)}, for a ∈ A and γ_{g}(p) = ∑_{γ_{g}} p_{(g^{−1})} ⊗ p_{(P)} ⊗ p_{(g)}, for p ∈ P.

Then

δ_{g}(r) = (ρ_{g} ⊗ 1)(1 ⊗ γ_{g} ⊗ 1)(1 ⊗ τ^{−1})(r)
= (ρ_{g} ⊗ 1)(1 ⊗ γ_{g} ⊗ 1)(r ⊗ ∑_{τ^{−1}} 1_{(P)} ⊗ 1_{(Q)})
= (ρ_{g} ⊗ 1)(r ⊗ ∑_{τ^{−1},γ_{g}} 1_{(P)}_{(g^{−1})} ⊗ 1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)} ⊗ 1_{(Q)})
= ∑_{τ^{−1},γ_{g}} r1_{(P)}_{(g^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)} ⊗ 1_{(Q)}).

Similarly, δ_{h}(s) = ∑_{τ^{−1},γ_{h}} s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(h)} ⊗ 1_{(Q)}. Thus

M_{g,h}(δ_{g} ⊗ δ_{h})M_{g,h}^{−1}(rs) =

M_{g,h}(δ_{g} ⊗ δ_{h})(r ⊗ s) =

M_{g,h}(∑_{τ^{−1},γ_{g}} r1_{(P)}_{(g^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)} ⊗ 1_{(Q)} ⊗ ∑_{τ^{−1},γ_{h}} s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(h)} ⊗ 1_{(Q)}) = ∑_{τ^{−1},γ_{g},τ^{−1},γ_{h}} r1_{(P)}_{(g^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)}μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)})1_{(P)}_{(h)} ⊗ 1_{(Q)}).

Let 1 = ∑_{k} x_{k}y_{k} where x_{k} ∈ R_{h} and y_{k} ∈ R_{h^{−1}}. Since γ_{gh}^{−1}(M_{h^{−1},g^{−1}} ⊗ 1 ⊗ M_{g,h})(1 ⊗ γ_{g} ⊗ 1)γ_{h} is left multiplication by c(g, h), then

(1 ⊗ γ_{gh}^{−1} ⊗ 1)(ρ_{gh}^{−1} ⊗ 1)M_{g,h}(δ_{g} ⊗ δ_{h})M_{g,h}^{−1}(rs) = (1 ⊗ γ_{gh}^{−1} ⊗ 1)(ρ_{gh}^{−1} ⊗ 1) (∑_{τ^{−1},γ_{g},τ^{−1},γ_{h}} r1_{(P)}_{(g^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)}μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)})1_{(P)}_{(h)} ⊗ 1_{(Q)}) = (1 ⊗ γ_{gh}^{−1} ⊗ 1) (∑_{τ^{−1},γ_{g},τ^{−1},γ_{h},k} rx_{k} ⊗ y_{k}1_{(P)}_{(g^{−1})} ⊗ 1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)}μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)})1_{(P)}_{(h)} ⊗ 1_{(Q)}) = ∑_{τ^{−1},γ_{g},τ^{−1},γ_{h},k} rx_{k} ⊗ c(g, h)γ_{h}^{−1}(y_{k}γ_{g}^{−1}(1_{(P)}_{(g^{−1})} ⊗ 1_{(P)}_{(P)} ⊗ 1_{(P)}_{(g)}))⊗ μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)})1_{(P)}_{(h)}) ⊗ 1_{(Q)} = ∑_{τ^{−1},τ^{−1},γ_{h},k} rx_{k} ⊗ c(g, h)γ_{h}^{−1}(y_{k} ⊗ 1_{(P)} ⊗ μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)})1_{(P)}_{(h)}) ⊗ 1_{(Q)} = ∑_{τ^{−1},τ^{−1},γ_{h},k} rx_{k} ⊗ c(g, h)γ_{h}^{−1}(y_{k} ⊗ 1_{(P)}μ(1_{(Q)} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)}) ⊗ 1_{(P)}_{(h)}) ⊗ 1_{(Q)} = ∑_{τ^{−1},τ^{−1},γ_{h},k} rx_{k} ⊗ c(g, h)γ_{h}^{−1}(y_{k} ⊗ τ(1_{(P)} ⊗ 1_{(Q)})s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)}) ⊗ 1_{(P)}_{(h)}) ⊗ 1_{(Q)} = ∑_{τ^{−1},γ_{h},k} rx_{k} ⊗ c(g, h)γ_{h}^{−1}(y_{k} ⊗ s1_{(P)}_{(h^{−1})}1_{(P)}_{(P)} ⊗ 1_{(P)}_{(h)}) ⊗ 1_{(Q)} = ∑_{τ^{−1},γ_{h},k} rx_{k} ⊗ y_{k}sc(g, h)γ_{h}^{−1}(1_{(P)}_{(h^{−1})} ⊗ 1_{(P)}_{(P)} ⊗ 1_{(P)}_{(h)}) ⊗ 1_{(Q)} = rsc(g, h) ⊗ ∑_{τ^{−1}} 1_{(P)} ⊗ 1_{(Q)}.

Finally,

rsd(g, h) = δ_{gh}^{−1}M_{g,h}(δ_{g} ⊗ δ_{h})M_{g,h}^{−1}(rs)
= (1 ⊗ τ)(1 ⊗ γ_{gh}^{−1} ⊗ 1)(ρ_{gh}^{−1} ⊗ 1)M_{g,h}(δ_{g} ⊗ δ_{h})M_{g,h}^{−1}(rs)
= (1 ⊗ τ)(rsc(g, h) ⊗ ∑_{τ^{−1}} 1_{(P)} ⊗ 1_{(Q)})
= rsc(g, h).

Now we are ready for the main theorem of this section.

Theorem 5.8 There are strongly G-graded rings R and R′ satisfying

1. R_{e} = R′_{e}.

2. For every g ∈ G, R_{g} and R′_{g} are isomorphic as R_{e}-bimodules.

3. R and R′ are graded equivalent.

4. R and R′ are not graded isomorphic.

Proof. By [5], there is a strongly graded ring R (actually a skew group ring) for which the map ϕ is not trivial. By Lemma 5.7, ΨΣ ≠ 1 which together with Lemma 5.2 proves the theorem.

A specific example of the above theorem now follows:

Example 5.9 Let A be a unital ring, G a group and α : G → Aut(A) a group homomorphism. Let w ∈ Aut(A) and β : G → Aut(A) be given by β(g) = w^{−1}α(g)w. Let R = A_{α}G the skew group ring associated to the action α and R′ = A_{β}G the skew group ring associated to the action β. Then [R′, 1] = [R, 1]^{A^{w}}, so R and R′ are graded equivalent. Moreover, if [A^{w}] ∈ Pic_{(R,f)}(A), then [R′_{g}] = [R_{g}] for every g ∈ G. However the following example from [5] shows that they may be non graded isomorphic.

Let K = Q(ϵ, α) where ϵ is primitive 3th root of unity and α^{3} = d ∈ Q−Q(ϵ)^{3} (for example, let α be the real cube root of 3). Let G =Gal(K/Q) be the Galois group of this extension. G = ⟨π, τ⟩ ≃ S_{3} where π and τ are given by

ϵ^{π} = ϵ, α^{π} = ϵα, ϵ^{τ} = ϵ^{2} and α^{τ} = α

Let π′, τ′ ∈ Aut(K^{2}) be given by (a, b)^{π′} = (a^{π}, b^{π}) and (a, b)^{τ′} = (b, a). Let A be the skew polynomial ring K^{2}[X; π′]. Let y be the inner automorphism of A given by (α, α^{2}) and w the automorphism of A which acts as τ′ on K^{2} and X^{w} = X.

Let G = ⟨y, w⟩ ⊆ Aut(A), the group of automorphisms of A generated by y and w. Let α : G → Aut(A) the inclusion map, R = A_{α}G, β : G → Aut(A) be given by β(g) = w^{−1}α(g)w and R′ = A_{β}G. Clearly Aw = A^{w} and [R′, 1] = Σ([A^{w}]). By [5] [Aw] ∈ Pic_{(R,1)}(A) and ϕ([Aw]) ≠ 1. Thus, R′ and R are graded equivalent and R_{g} and R′_{g} are isomorphic as A-bimodules. However, by Lemma 5.7, Ψ([R′, 1]) = (Ψ ∘ Σ)(A^{w^{−1}}) ≠ 1. Finally, by Theorem 5.4, R′ is not graded isomorphic to R.

6 Applications I. Strongly graded rings graded equivalent to a crossed product

In this section, we complete our study of Problem A from the introduction by using the results of the previous sections to show that graded equivalence is a viable tool for reducing the study of strongly graded rings.

By the structure of projective modules over semiperfect rings (see [2, Theorem 27.11]), if A is a basic semiperfect ring, then Pic(A) = Out(A). Therefore, strongly graded rings with basic semiperfect coefficient rings are crossed products. Corollary 3.3 now implies the following result which first appeared in [9]:

Corollary 6.1 [9] If R is a strongly graded ring and R_{e} is semiperfect, then R is graded equivalent to a crossed product R′ whose coefficient ring is the basic ring of R_{e}.

In order to give our solution to Problem B in the next section, we need to describe the parameter set of the crossed product from the above corollary. Let R be a strongly graded ring, so that R_{e} = A is semiperfect. Let e be a basic idempotent of A and C = (A, B = eAe, P = Ae, Q = eA, τ, μ), the Morita context, where both τ and μ are multiplication maps. Since R^{C} is a crossed product, for every g ∈ G, Q ⊗_{A} R_{g} ⊗_{A} P ≃ B as right B-modules and so Q ⊗_{A} R_{g} ≃ Q as right A-modules. Let Ψ_{g} : Q ⊗_{A} R_{g} ≃ Q be an isomorphism of right A-modules for every g ∈ G. Then there is an automorphism α_{g} of B such that Ψ_{g} is an isomorphism of B-A-bimodules Q ⊗_{A} R_{g} ≃ ^{α_{g}}Q [6, Theorem 55.12]. On the other hand, for every g, h ∈ G, the composition of the following isomorphisms

δ_{g,h} : Q →^{Ψ_{h}^{−1}} Q ⊗_{A} R_{h} →^{Ψ_{g}^{−1}⊗1} Q ⊗_{A} R_{g} ⊗_{A} R_{h} →^{1⊗ρ_{g,h}} Q ⊗_{A} R_{gh} →^{Ψgh} Q

is an isomorphism of right A-modules. Therefore there exists a unit t(g, h) of B so that δ_{g,h}(q) = t(g, h)q, for every q ∈ Q.

Lemma 6.2 With the above notation, (α, t) is a parameter set of G over B and R^{C} is graded isomorphic to B ∗_{t}^{α} G.

Proof. To check that (α, t) is a parameter set, it suffices to check that the multiplication in the crossed product R ∗_{t}^{α} G is associative [15, Lemma 1.1]. So to prove the lemma, we need only to show that there is a bijection R^{C} → R ∗_{t}^{α} G that preserves addition and multiplication.

Let Γ : R^{C} → B ∗_{c}^{α} G be defined by Γ(q ⊗ r_{g} ⊗ p) = \overline{g}μ(Ψ_{g}(q ⊗ r_{g}) ⊗ p)) and extended linearly. Γ is an additive group isomorphism and we check that it preserves multiplication.

Γ((q ⊗ r_{g} ⊗ p)(q′ ⊗ r′_{h} ⊗ p′)) = \overline{gh}μ(Ψ_{gh} ⊗ 1)(q ⊗ r_{g}τ(p ⊗ q′)r′_{h} ⊗ p′)
= \overline{gh}μ(Ψ_{gh}(q ⊗ r_{g}τ(p ⊗ q′)r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}Ψ_{h}(Ψ_{g} ⊗ 1)(1 ⊗ ρ_{g,h}^{−1})(q ⊗ r_{g}τ(p ⊗ q′)r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}Ψ_{h}(Ψ_{g} ⊗ 1)(q ⊗ r_{g} ⊗ τ(p ⊗ q′)r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}Ψ_{h}(Ψ_{g}(q ⊗ r_{g}) ⊗ τ(p ⊗ q′)r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}Ψ_{h}(Ψ_{g}(q ⊗ r_{g})τ(p ⊗ q′) ⊗ r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}Ψ_{h}(μ(Ψ_{g}(q ⊗ r_{g}) ⊗ p)q′ ⊗ r′_{h}) ⊗ p′)
= \overline{gh}μ(δ_{g,h}μ(Ψ_{g}(q ⊗ r_{g}) ⊗ p)^{α_{h}}Ψ_{h}(q′ ⊗ r′_{h}) ⊗ p′)
= \overline{gh}t(g, h)μ(Ψ_{g}(q ⊗ r_{g}) ⊗ p)^{α_{h}}μ(Ψ_{h}(q′ ⊗ r′_{h}) ⊗ p′)
= \overline{g}μ(Ψ_{g}(q ⊗ r_{g}) ⊗ p)\overline{h}μ(Ψ_{h}(q′ ⊗ r′_{h}) ⊗ p′)
= Γ(q ⊗ r_{g} ⊗ p)Γ(q′ ⊗ r′_{h} ⊗ p′)

Our next reduction application comes from a specific family of strongly graded rings and crossed products studied by Saorín [16] and Jespers and Okniński [7].

Remark 6.3 Saorín [16] proved that if R = A ∗ G is a left perfect crossed product such that R_{e}/J(R_{e}) is a finite direct product of finite-dimensional simple algebras over an algebraic closed field, then G is finite. Actually his result is stated for strongly graded rings but, as it was pointed out by Jespers and Okniński [7], Saorín’s proof is not correct for strongly graded rings because [16, Lemma 7] is based in a false statement in [14]. Jespers and Okniński gave a correct proof in [7] for strongly graded rings. However relying on Saorín’s proof for crossed products, one can easily extend the result for strongly graded rings using Corollary 6.1.

Corollary 6.4 [7] Let R be a perfect strongly G-graded ring such that R_{e}/J(R_{e}) is a finite direct product of finite-dimensional simple algebras over an algebraic closed field. Then G is necessarily finite.

Proof. Let A be the basic algebra of R_{e}. By Corollary 6.1, R is graded equivalent to a crossed product A ∗ G and A/J(A) is a finite direct product of copies of an algebraic closed field. Now by [14], G is finite.

Another reduction application using graded equivalences appears in [10], in which the reduction process is used to simplify the study of finite representation type for orders.

The results above indicate that it is possible, using graded equivalence, to reduce from strongly graded rings to crossed products. If the crossed product obtained is not a twisted group ring we cannot expect to reduce to another twisted group ring (see Corollary 3.4). As our final analysis of Problem A, we wish to investigate when we can make a further reduction to skew group rings. It is well known that, using the Cohen-Montgomery Duality theory, every G-graded ring R with G finite is graded equivalent to the skew group ring (R#G) ∗ G (see [8]). But while this simplifies the grading, it complicates the coefficient ring by making it larger. Consequently, we close this section by considering the question: When is a strongly graded ring R graded equivalent to a skew group ring R′ so that R_{e} ≃ R′_{e}? This question has a very general negative answer.

Proposition 6.5 Let A be a basic semiperfect ring and R and R′ be graded equivalent strongly graded rings with R_{e} ≃ R′_{e} ≃ A. If R is a skew group ring, then so is R′.

Proof. Consider R and R′ as (A, G)-graded rings via the isomorphisms f : A ≃ R_{e} and f′ : A ≃ R′_{e}. By Proposition 4.4, there is [P] ∈ Pic(A) so that [R′, f′] = [R, f]^{[P]}. If A is basic semiperfect, then P = A^{β} for some β ∈ Aut(A). Since R is a skew group ring, there is a group homomorphism α : G → Aut(A) so that R ≃ A ∗^{α} G. For every g ∈ G, let ḡ = 1 ⊗ g ⊗ 1 ∈ A^{β^{−1}} ⊗_{A} Ag ⊗ A^{β}. Then ḡh̄ = (1 ⊗ g ⊗ 1)(1 ⊗ h ⊗ 1) = 1 ⊗ gh ⊗ 1. This shows that R^{[P]} is a skew group ring and hence so is R′.

7 Application II. Strongly Graded Semisimple rings

In this final section, we analyze Problem B from the introduction. Our goal is to characterize when a strongly graded ring R is semisimple; see Theorem 7.5. Particular cases appear in Corollaries 7.7 and 7.8.

It is well known that if R is semisimple, then R_{e} is semisimple. Thus, we assume, for the remainder of this section, that R_{e} is a direct product of finite matrix rings over division rings.

Our strategy is the following. First we use Lemma 6.2 to compute a crossed product graded equivalent to R. Then we use ideas from [8] to reduce the study to the case of crossed products over division rings. Finally, in the case when R_{e} is a direct product of matrix rings over fields, we can reduce to crossed products over fields and then use the results of [4] to give specific conditions for the semisimplicity of these crossed products.

Notation 7.1 Given an automorphism α ∈ D, (D a ring) and positive integers, n, m, there exists a group automorphism of M_{n,m}(D), given by (a_{i,j}) ↦ (a_{i,j}^{α}). We abuse the notation and denote this map also by α. It is clear that if a ∈ M_{n,m}(D) and b ∈ M_{m,l}(D), then (ab)^{α} = a^{α}b^{α}.

Given an element x in a direct product ∏_{i} X_{i}, and i ∈ I, x(i) stands for the i-th coordinate of x.

Now we describe, up to graded isomorphisms, all the strongly graded rings R, such that R_{e} is semisimple. This characterization is essentially based on the discussion in [14, A.I.3] and the use of a factor set, but our description is more explicit.

Fix a semisimple ring A = ∏_{i=1}^{k} M_{n_{i}}(D_{i}) where n_{i} is a positive integer and D_{i} is a division ring for every i. Assume that if D_{i} and D_{j} are isomorphic, then they are equal.

Let P be the subgroup of permutations σ of {1, 2, …, n} such that D_{i} = D_{σ(i)}, for every i. P acts on ∏_{i=1}^{k} Aut(D_{i}) by permuting the coordinates, i.e. σ(α)(i) = α(σ^{−1}(i)), for α ∈ ∏_{i=1}^{k} Aut(D_{i}). Let H be the semidirect product induced by this action, i.e. H = ∏_{i=1}^{k} Aut(D_{i})×P as a set and the product is given by (α, σ)(β, τ) = (ασ(β), στ).

For every (α, σ) ∈ H, let A(α, σ) = ∏_{i=1}^{k} M_{n_{σ(i)},n_{i}}(D_{i}), with the following bimodule structure:

(ap)(i) = a(σ(i))^{α(σ(i))}p(i),  (pa)(i) = p(i)a(i)  (a ∈ A, p ∈ A(σ, α))

Lemma 7.2 For every (α, σ) ∈ H, [A(α, σ)] ∈ Pic(A) and the map f : H → Pic(A), (α, σ) ↦ [A(α, σ)], is a group epimorphism whose kernel is ∏_{i=1}^{k} Inn(D_{i}) × 1.

Proof. Let Φ : A(α, σ)⊗_{A}A(β, τ) → A(ασ(β), στ) be the map given by Φ(p⊗q)(i) = p(τ(i))^{β(τ(i))}q(i). To prove that A(α, σ) is invertible and f is a group homomorphism it is enough to show that Φ is a bimodule isomorphism. First we check that it is well defined:

\begin{matrix}Φ(pa ⊗ q)(i) = (pa)(τ(i))^{β(τ(i))}q(i) = (p(τ(i))a(τ(i)))^{β(τ(i))}q(i) \\ = p(τ(i))^{β(τ(i))}a(τ(i))^{β(τ(i))}q(i) = p(τ(i))^{β(τ(i))}(aq)(i) = Φ(p ⊗ aq)(i) \end{matrix}

Next we check that Φ is a bimodule homomorphism:

Φ(ap ⊗ q)(i) = (ap)(τ(i))^{β(τ(i))}q(i)
= (a(στ(i))^{α(στ(i))}p(τ(i)))^{β(τ(i))}q(i)
= a(στ(i))^{α(στ(i))β(τ(i))}p(τ(i))^{β(τ(i))}q(i)
= a(στ(i))^{(ασ(β))(στ(i))}Φ(p ⊗ q)(i)
= (aΦ(p ⊗ q))(i)
Φ(p ⊗ qa)(i) = p(τ(i))^{β(τ(i))}(qa)(i)
= p(τ(i))^{β(τ(i))}q(i)a(i)
= Φ(p ⊗ q)(i)a(i)
= (Φ(p ⊗ q)a)(i)

Now we prove that f is surjective. Let e_{1}, e_{2}, …, e_{k} be the primitive central idempotents of A. Let [P] ∈ Pic(A). For every i = 1, 2, …, k, (P e_{i})_{Ae_{i}} ≃ M_{m_{i},n_{i}}(D_{i}) for some m_{i} ∈ N. Then

A = End(P) ≃^{ϕ} ∏_{i=1}^{k} End((P e_{i})_{Ae_{i}}) ≃ ∏_{i=1}^{k} M_{m_{i}}(D_{i}). Let f_{1}, f_{2}, …, f_{k} be the primitive central idempotents of ∏_{i=1}^{k} M_{m_{i}}(D_{i}). Then there is a permutation σ ∈ P, such that m_{i} = n_{σ(i)} and ϕ(e_{i}) = f_{σ(i)} for every i. Moreover, ϕ restricts to an automorphism α′(i) of M_{n_{i}}(D_{i}). By the Skolem-Noether theorem, there is an automorphism α(i) of D_{i}, such that α′(i)α(i)^{−1} is inner. It follows that α(i) induces an isomorphism of A-bimodules P ≃ A(α, σ).

Finally we show that Ker f = ∏_{i=1}^{k} Inn(D_{i})×1. Assume that (α, σ) ∈ Ker f. Then e_{i}A(α, σ)e_{j} = A(α, σ)e_{σ(i)}e_{j} and so σ = 1. This implies that _{A}Ae_{i} ≃ _{A}A(α, 1)e_{i} ≃ ^{α(i)}(Ae_{i}) for every i, and hence α(i) is inner (in M_{n_{i}}(D_{i})) for every i. But this implies that α(i) is inner.

To define a strongly graded ring we need a notion a bit more complicated than a parameter set.

Definition 7.3 Let A as above and G a group. A factor set of G in A is a triple of maps

(β : G → ∏_{i=1}^{k} Aut(D_{i}), σ : G → P, t : G ⊗ G → ∏_{i=1}^{k} D_{i}^{∗})

satisfying the following conditions for every g, h, k ∈ G and i = 1, 2, …, k (the images of g by β, σ and t are denoted by β_{g}, σ_{g} and t_{g,h}):

1. σ is a group homomorphism.

2. β_{gh}ι_{t_{g,h}} = β_{g}σ_{g}(β_{h}).

3. t_{gh,k}(i)t_{g,h}(σ_{k}(i))^{β_{k}(σ_{k}(i))} = t_{g,hk}(i)t_{h,k}(i)

Given a factor set (β, σ, t), we define the G-graded ring A(β, σ, t) = ⊕_{g∈G}A(β_{g}, σ_{g}) where the product is given by

(r_{g}r_{h})(i) = t_{g,h}(i)r_{g}(σ_{h}(i))^{β_{h}(σ_{h}(i))}r_{h}(i)

Two factor sets (β, σ, t) and (β′, σ′, t′) are said to be equivalent if σ = σ′ and there exists a map u : G → ∏_{i=1}^{k} D_{i}^{∗} such that

β_{g} = ι_{u_{g}}β′_{g} and u(gh)(i)t_{g,h}(i) = t′_{g,h}(i)u(g)(σ_{h}(i))^{β_{h}(σ_{h}(i))}u_{h}(i)

for every g, h ∈ G and i = 1, 2, …, n.

Proposition 7.4 Every strongly G-graded ring R, with R_{e} = A, is A-isomorphic to a ring of the form A(β, σ, t) for some factor set (β, σ, t). Moreover, two factor sets give rise to graded Aisomorphic rings if and only if they are equivalent.

Proof. By Lemma 7.2, if R is a strongly graded ring so that R_{e} = A, then R_{g} is isomorphic as A-bimodule to A(β_{g}, σ_{g}) for some (β_{g}, σ_{g}) ∈ H. We will assume that R_{g} = A(β_{g}, σ_{g}). If ρ_{g,h} : A(β_{g}, σ_{g}) ⊗_{A} A(β_{h}, σ_{h}) → A(β_{gh}, σ_{gh}) is the multiplication map and Φ_{g,h} : A(β_{g}, σ_{g}) ⊗_{A} A(β_{h}, σ_{h}) → A(β_{g}σ_{g}(β_{h}), σ_{g}σ_{h}) is the isomorphism defined in the proof of Lemma 7.2, then T_{g,h} = ρ_{g,h}Φ_{g,h}^{−1} : A(β_{g}σ_{g}(β_{h}), σ_{g}σ_{h}) → A(β_{gh}, σ_{gh}) is an isomorphism of A-bimodules. By Lemma 7.2, σ is a group homomorphism and there is a unit t_{g,h} so that β_{g}σ_{g}(β_{h}) = β_{gh}ι_{t_{g,h}}. Moreover, one may assume that T_{g,h}(x) = t_{g,h}x for every x. Therefore, the multiplication in R is given by

(r_{g}r_{h})(i) = t_{g,h}(i)r_{g}(σ_{h}(i))^{β_{h}(σ_{h}(i))}r_{h}(i)

Now it is a matter of computation to show that this multiplication is associative if and only if (β, σ, t) is a factor set. This follows from the following computations:

((r_{g}r_{h})r_{k})(i) = t_{gh,k}(i)(r_{g}r_{h})(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{k}(i)
= t_{gh,k}(i)(t_{g,h}(σ_{k}(i))r_{g}(σ_{hk}(i)))^{β_{h}(σ_{hk}(i))}r_{h}(σ_{k}(i)))^{β_{k}(σ_{k}(i))}r_{k}(i)
= t_{gh,k}(i)t_{g,h}(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{g}(σ_{hk}(i))^{β_{h}(σ_{hk}(i))β_{k}(σ_{k}(i))}r_{h}(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{k}(i)
= t_{gh,k}(i)t_{g,h}(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{g}(σ_{hk}(i))^{(β_{h}σ_{)}(β_{k})(σ_{hk}(i))}r_{h}(σ_{k}(i)^{β_{k}(σ_{k}(i))}r_{k}(i)

and

(r_{g}(r_{h}r_{k}))(i) = t_{g,hk}(i)r_{g}(σ_{hk}(i))^{β_{hk}(σ_{hk}(i))}(r_{h}r_{k})(i)
= t_{g,hk}(i)r_{g}(σ_{hk}(i))^{β_{hk}(σ_{hk}(i))}t_{h,k}(i)r_{h}(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{k}(i)
= t_{g,hk}(i)t_{h,k}(i)r_{g}(σ_{hk}(i))^{(β_{h}σ(h)(β(k))(σ_{hk}(i))}r_{h}(σ_{k}(i))^{β_{k}(σ_{k}(i))}r_{k}(i)

Assume now that (β, σ, t) and (β′, σ′, t′) are two factor sets and Φ : A(β, σ, t) → A(β′, σ′, t′) is a graded A-isomorphism. Then, for every g ∈ G, the restriction Φ_{g} of Φ to A(β_{g}, σ_{g}) → A(β′_{g}, σ′_{g}) is a bimodule isomorphism. By Lemma 7.2, σ = σ′ and hence A(β_{g}, σ_{g}) and A(β′_{g}, σ′_{g}) coincide as left A-modules. Therefore, there exist units u(g) (g ∈ G) of A, such that Φ_{g}(r_{g})(i) = u(g)(i)r_{g}(i), for every r_{g} ∈ A(β_{g}, σ_{g}). By straightforward computations, one proves that, if u(g) is a unit for every g, then the map u : A(β, σ, t) → A(β′, σ′, t′), given by u(r_{g})(i) → u_{g}(i)r_{g}(i) is a A-isomorphism if and only if β_{g} = ι_{u_{g}}β′_{g} and u(gh)(i)t_{g,h}(i) = t′_{g,h}(i)u(g)(σ_{h}(i))^{β_{h}(σ_{h}(i))}u_{h}(i). Since the matrices having only 0 and 1 as entries are fixed by β_{g} and β′_{g}, one concludes that u(g)(i) is a scalar matrix, for every i and hence we may assume that u(g)(i) ∈ D_{i}^{∗}

The significance of Proposition 7.4 is that we need only study the strongly graded rings of the form A(β, σ, t) for a factor set (β, σ, t).

Let R = A(β, σ, t) be such a strongly graded ring. Let

C = (A, B = ∏_{i=1}^{k} D_{i}, P = ∏_{i=1}^{k} M_{n_{i}⊗1}(D_{i}), Q = ∏_{i=1}^{k} M_{1⊗n_{i}}(D_{i}), μ, τ)

be the obvious Morita context. Then R^{C} is a crossed product over B. We use Lemma 6.2 to compute the parameter set for this crossed product. For every g ∈ G, let Ψ_{g} : Q ⊗_{A} R_{g} → Q be the map given by Ψ_{g}(q ⊗ r)(i) = q(σ_{g}(i))^{β_{g}(σ_{g}(i))}r(i). Ψ_{g} is an isomorphism of right A-modules. Let α_{g} : A → A be the map given by a^{α_{g}}(i) = a(σ_{g}(i))^{β_{g}(σ_{g}(i))}. Then α_{g} is an automorphism of A and Φ_{g} is an isomorphism of A-bimodules from Q ⊗_{A} R_{g} to ^{α_{g}}Q. So α is the required action.

We show that t_{g,h} is the required cocycle. Indeed if q ∈ Q, r_{g} ∈ R_{g} and r_{h} ∈ R_{h}, then

(Ψ_{gh}(1 ⊗ ρ_{g,h})(q ⊗ r_{g} ⊗ r_{h}))(i) = Ψ_{gh}(q ⊗ r_{g}r_{h})(i)
= q(σ_{gh}(i))^{β_{gh}(σ_{gh}(i))}(r_{g}r_{h})(i)
= q(σ_{gh}(i))^{β_{gh}(σ_{gh}(i))}t_{g,h}(i)r_{g}(σ_{h}(i))^{β_{h}(σ_{h}(i))}r_{h}(i)
= t_{g,h}(i)q(σ_{gh}(i))^{(β_{g}σ_{g}(β_{h}))(σ_{gh}(i))}r_{g}(σ_{h}(i))^{β_{h}(σ_{h}(i))}r_{h}(i)
= t_{g,h}(i)(q(σ_{gh}(i))^{β_{g}(σ_{gh}(i))}r_{g}(σ_{h}(i)))^{β_{h}(σ_{h}(i))}r_{h}(i)
= t_{g,h}(i)(Ψ_{g}(q ⊗ r_{g})(σ_{h}(i))^{β_{h}(σ_{h}(i))}r_{h}(i)
= t_{g,h}(i)Ψ_{h}((Ψ_{g}(q ⊗ r_{g}) ⊗ r_{h})(i)
= t_{g,h}(i)Ψ_{h}(Ψ_{g} ⊗ 1)(q ⊗ r_{g} ⊗ r_{h})(i)

Therefore, if δ_{g,h} is the map from Section 5, δ_{g,h}(q) = Ψ_{gh}(1 ⊗ ρ_{g,h})(Ψ_{g}^{−1} ⊗ 1)Ψ_{h}^{−1}(q) = t_{g,h}q.

Let (α, t) be a parameter set in a product of rings ∏_{i∈I} R_{i}. If J is a subset of I and H a subgroup of G, such that (∏_{j∈J} R_{j})^{α_{h}} = ∏_{j∈J} R_{j} for every h ∈ H, (α^{(J)}, t^{(J)}) denotes the parameter set of

H over ∏_{i∈J} R_{i} given by x^{α_{h}^{(J)}}(j) = x^{α_{h}}(j) and t_{g,h}^{(J)}(j) = t_{g,h}(j), for every x ∈ ∏_{i∈J} R_{i}, j ∈ J and h ∈ H. If j ∈ I, α^{(j)} (resp. t^{(j)}) stands for α^{({j})} (resp. t^{({j})}).

Now we are ready to state the main theorem of this section.

Theorem 7.5 Let A = ∏_{i=1}^{k} M_{n_{i}}(D_{i}) be a semisimple artinian ring, where every D_{i} is a division ring and D_{i} = D_{j}, if they are isomorphic. Let B = ∏_{i=1}^{k} D_{i} and (β, σ, t) be a factor set of a group G over A. For every g ∈ G, let α_{g} be the automorphism of B given by a^{α_{g}}(i) = a(σ_{g}(i))^{β_{g}(σ_{g}(i))}. Let j_{1}, j_{2}, …, j_{n} be representatives of the orbits of the action σ of G on {1, 2, …, k}. For every i = 1, …, k, let J_{i} be the orbit of j_{i} and G_{i} the stabilizer of j_{i}. Then (α, t) defines a parameter set of G over B and the following assertions are equivalent:

1. A(β, σ, t) is semisimple.

2. B ∗_{t}^{α} G is semisimple.

3. (∏_{i∈J_{i}} D_{j_{i}}) ∗_{t^{(J_{i})}}^{α^{(J_{i})}} G is semisimple for every i = 1, 2, …, n.

4. D_{j_{i}} ∗_{t^{(j_{i})}}^{β^{(j_{i})}} G_{i} is semisimple for every i = 1, 2, …, n

Proof. The fact that (α, t) is a parameter set is a consequence of Lemma 6.2. (1) ⇔ (2) is a consequence of the fact that A(β, σ, t) and B ∗_{t}^{α} G are graded equivalent.

To prove (2) ⇔ (3) it is enough to realize that B ∗_{t}^{α} G = ∏_{i=1}^{n}(∏_{i∈J_{i}} D_{j_{i}}) ∗_{t^{(J_{i})}}^{α^{(J_{i})}} G.

To simplify the proof of (3) ⇔ (4), we may assume that n = 1, i.e. the action of G on {1, 2, …, k} induced by σ is transitive. Let D = D_{1} and H be the stabilizer of 1 by σ. Then, using ideas from [7] we prove that R = B ∗_{t}^{α} G is isomorphic to M_{k}(D ∗_{t^{(1)}}^{β^{(1)}} H).

Let e_{1}, e_{2}, …, e_{k} be the primitive idempotents of B. Since the action σ of G on {1, 2, …, k} is transitive, for every i, there exists g ∈ G, such that e_{1}ḡ = ḡe_{i}. Thus the map x ↦ xḡ defines an isomorphism _{R}Re_{1} ≃ _{R}Re_{i}. Therefore _{R}R ≃ (_{R}Re_{1})^{k} and so R ≃ End(_{R}R) ≃ M_{n}(End(_{R}Re_{1})) ≃ M_{n}(e_{1}Re_{1}). But e_{1}Re_{1} ≃ D ∗_{t^{(1)}}^{β^{(1)}} H.

Proposition 7.6 A skew group ring B∗^{α}G where B is a direct product of division rings is semisimple if and only if G is finite and there exists b ∈ B, such that ∑_{g∈G} b^{α_{g}} = 1.

Proof. A very classical argument shows that if R = B ∗^{α} G is semisimple, then G is finite. Then R is semisimple if and only if it is Von Neumann regular. Since B is abelian regular (i.e. every idempotent is central), the result is a consequence of [1].

As a direct consequence of Theorem 7.5 and Proposition 7.6, we have the following.

Corollary 7.7 Let A = ∏_{i=1}^{k} M_{n_{i}}(D_{i}) be a semisimple artinian ring, where every D_{i} is a division ring and D_{i} = D_{j}, if they are isomorphic. Let (β, σ, t) be a factor set of a group G over A. Assume that t_{g,h} = 1 for every g, h. Then the following are equivalent:

1. A is semisimple.

2. G is finite and there exists b ∈ B, such that ∑_{g∈G} b_{σ_{g}(i)}^{β_{g}(σ_{g}(i))} = 1, for every 1 ≤ i ≤ k.

3. If j_{1}, j_{2}, …, j_{i} is a set of representatives of the action σ of G on {1, 2, …, n}, then the stabilizer G_{i} of j_{i} by this action is finite and for every 1 ≤ i ≤ n, there exists x_{i} ∈ D_{j_{i}}, so that ∑_{g∈G_{i}} b^{β_{g}(j_{i})} = 1.

Finally we consider the case where A is a direct product of matrix rings over fields and the grading group is finite. Using Theorem 7.5, the semisimplicity of a strongly graded ring over A reduces to the semisimiplicity of crossed products over fields. This case has been studied recently by Aljadeff and Robinson [4].

Let R = K ∗_{t}^{α} G be a crossed product over a field of characteristic p ≠ 0 and assume that G = ∏_{i=1}^{r} G_{i} with G_{i} cyclic of order p^{k_{i}}. Then, from the discussion in [4], H^{2}(G, K^{∗}) ≃ ⊕_{i=1}^{r}K^{∗}/(K^{∗})^{p^{k_{i}}}, so every cocycle t of G over K is represented by an r-tuple (a_{1}(K^{∗})^{p^{k_{1}}} , a_{2}(K^{∗})^{p^{k_{2}}} , …, a_{r}(K^{∗})^{p^{kr}}).

Corollary 7.8 Let A = ∏_{i=1}^{k} M_{n_{i}}(K_{i}), where every K_{i} is a field of characteristic p_{i} (where p_{i} could be 0) and K_{i} = K_{j}, if they are isomorphic. Let (β, σ, t) be a factor set of a group G over A and j_{1}, j_{2}, …, j_{n} be representatives of the orbits of the action σ of G on {1, 2, …, k}. For every i = 1, 2, …, n, let G_{i} be the stabilizer of j_{i} under this action and let H_{i} = {g ∈ G_{i} : β_{g} = 1}. If p_{i} ≠ 0, let P_{i} be a Sylow p_{i}-subgroup of H_{i}. The following assertions are equivalent:

1. A(β, σ, t) is semisimple.

2. K_{j_{i}} ∗_{t^{(j_{i})}}^{β^{(j_{i})}} G_{i} is semisimple for every i = 1, 2, …, n.

3. K_{j_{i}} ∗_{t^{(j_{i})}} H_{i} is semisimple for every i = 1, 2, …, n.

4. For every i = 1, 2, …, n, either p_{i} does not divide the order of H_{i} or the following conditions hold:

(a) | H′_{i} | is prime to p_{i}, so that P_{i} is abelian, say isomorphic to ∏_{i=1}^{r} C_{i} with C_{i} cyclic of order p^{k_{i}}.

(b) The restriction of t^{j_{i}} to P_{i} is represented by an r-tuple

(a_{1}(K_{i}^{∗})^{p_{i}^{k_{1}}} , a_{2}(K_{i}^{∗})^{p_{i}^{k_{2}}} , …, a_{r}(K_{i}^{∗})^{p_{i}^{kr}})

such that {a_{1}, a_{2}, …, a_{r}} is p-independent over K_{i}^{p_{i}}.

Proof. The equivalence of (1) and (2) comes from Theorem 7.5. (2) ⇔ (3) is a consequence of [3, Corollary 4.2] and (3) ⇔ (4) is a consequence of [4, Theorem 2].

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Department of Mathematics  Departamento de Matemáticas
University of Colorado  Universidad de Murcia
Colorado Springs C0  30071. Murcia.
60933 U.S.A.  Spain
haefner@vision.uccs.edu  adelrio@fcu.um.es