Graded Morita theory (Q1322515)

Let \(G\) be a group and let \(R\) be a \(G\)-graded ring. Thus \(R\) is a direct sum \(R = \oplus_{\sigma \in G} R_ \sigma\) of additive subgroups \(R_ \sigma\) indexed by \(G\) such that \(R_ \sigma R_ \gamma \leq R_{\sigma\gamma}\) for all \(\sigma, \gamma \in G\). If \(R_ \sigma R_ \gamma = R_{\sigma\gamma}\) for all \(\sigma,\gamma \in G\) then \(R\) is said to be fully (or strongly) graded. If every component \(R_ \sigma\) contains a unit of \(R\), then \(R\) is called a crossed product. If there exists a collection \(\{x_ \sigma \in R_ \sigma\mid \sigma \in G\}\) of units of \(R\) such that \(x_ \sigma x_ t = x_{\sigma t}\) for all \(\sigma, t \in G\), then \(R\) is said to be a skew group ring. For \(X \subseteq G\), set \(R_ X = \oplus_{\sigma \in X} R_ \sigma\). If \(H\) is a subgroup of \(G\), then \(R_ H\) is an \(H\)-graded ring. Also \(R_ H\) is fully graded if and only if \(R_{\sigma^{-1}} R_ \sigma = R_ 1\) for all \(\sigma \in H\). Thus there exists a unique largest subgroup \(G_ R\) of \(G\) for which \(R_{G_ R}\) is fully graded. Also, there is a well-defined action of \(G_ R\) on (1) \(C_ R(R_ 1)\) (2) \(\text{Hom}_{R_ 1}(U,V)\) for any \(R\)-modules \(U\), \(V\) and (3) on the ideals of \(R_ 1\). An \(R\)-module \(M\) is graded if \(M = \oplus_{\sigma \in G} M_ \sigma\) of additive subgroups indexed by \(G\) such that \(R_ \sigma M_ \gamma \leq M_{\sigma\gamma}\) for all \(\sigma, \gamma \in G\). If \(R_ \sigma M_ \gamma = M_{\sigma \gamma}\) for all \(\sigma, \gamma \in G\), then \(M\) is said to be fully graded. Let \((R, R', M, M', \tau, \mu)\) be a Morita context. Suppose also that \(R\) and \(R'\) are \(G\)-graded rings, \(M\) is a graded \((R',R)\)-bimodule \((R'_ \alpha M_ \beta R_ \gamma \leq M_{\alpha \beta \gamma}\) for all \(\alpha, \beta, \gamma \in G)\), \(M'\) is a graded \((R,R')\)-bimodule, \(\tau : M' \otimes_{R'} M \to R\) is a graded \((R,R)\)-bimodule homomorphism \((\tau(M_ \alpha' \otimes_{R'} M_ \beta) \leq R_{\alpha\beta}\) for all \(\alpha, \beta \in G\)) and \(\mu : M \otimes_ R M' \to R'\) is a graded \((R',R')\) bimodule homomorphism. Then \((R, R', M, M', \tau, \mu)\) is called a \(G\)-graded Morita context (Definition 3.1). Clearly, in this case, if \(H\) is a subgroup of \(G\), then \((R_ H, R_ H', M_ H, M_ H', \tau_ H \equiv \text{Res}(\tau)\), \(\mu_ H \equiv \text{Res}(\mu))\) is an \(H\)-graded Morita context. The main result of this paper (Theorem 3.2) presents a graded analog of the Morita I Theorem in the presence of a graded Morita context in which \(\tau_ 1\) and \(\mu_ 1\) are surjective. Also Lemma 3.3 demonstrates that if \(M\) is a graded module-\(R\) such that \(M_ 1\) is a finitely generated generator in \(\text{mod-}R_ 1\) and if \(M\) is a direct summand of a direct sum of copies of \(R\) in \(\text{Gr mod-}R\), then \((R,E = \text{End}(M_ R), M,M^* = \text{Hom}_ R(M_ R,R),\tau : M^* \otimes_ E M \to R,\mu : M \otimes_ R M^* \to E)\), where \(\tau\) and \(\mu\) are the obvious maps, forms a graded Morita context. The last two sections of this paper present various examples and applications of these results. For instance, suppose that \(k\) is a field, \(G\) is a finite group and \(R\) is a fully \(G\)-graded \(k\)-algebra. Then \(P \equiv R \otimes_{R_ 1} R\) with \(\sigma\)-component \(P_ \sigma = R \otimes_{R_ 1} R_ \sigma\) for all \(\sigma \in G\) satisfies the hypotheses of Lemma 3.3 and \(E \equiv \text{End}(P_ R)\) is a skew group algebra (Theorem 4.1). Other applications include results on symmetric algebras, block induction, vertices of indecomposable modules and smash products.