Graded Morita theory for group coring and graded Morita-Takeuchi theory.
Let \(G\) be a group and \(A\) an associative unital algebra over a field \(k\). A \(G\)-group \(A\)-coring \(\mathbf C\) is a family \((C_\alpha)_{\alpha\in G}\) of \(A\)-bimodules together with a family of bimodule maps \(\Delta_{\alpha,\beta}\colon C_{\alpha\beta}\to C_\alpha\otimes C_\beta\) and \(\varepsilon\colon C_e\to A\) (\(e\) is the unit of \(G\)), such that \((\Delta_{\alpha,\beta}\otimes_A\text{id}_{C_\gamma})\circ\Delta_{\alpha\beta,\gamma}=(\text{id}_{C_\alpha}\otimes_A\Delta_{\beta,\gamma})\circ\Delta_{\alpha,\beta\gamma}\) and \((\text{id}_{C_\alpha}\otimes_A\varepsilon)\circ\Delta_{\alpha,e}=\text{id}_{C_\alpha}=(\varepsilon\otimes_A\text{id}_{A_\alpha})\circ\Delta_{e,\alpha}\). This notion was introduced by \textit{S. Caenepeel, K. Janssen} and \textit{S. H. Wang} [Appl. Categ. Struct. 16, No. 1-2, 65-96 (2008; Zbl 1145.16017)] as a generalization of group coalgebras and Hopf group coalgebras.NEWLINENEWLINE In this paper, the authors construct a graded Morita context asociated to a comodule over a group coring. This Morita context connects the dual graded ring of a group coring and the graded endomorphism ring of any group coring comodule. In the last section, a graded version of Morita-Takeuchi's equivalence theorem is obtained.
scientific article; zbMATH DE number 6062764