Liu, Guohua; Wang, Shuanhong Graded Morita theory for group coring and graded Morita-Takeuchi theory. (English) Zbl 1259.16049 Taiwanese J. Math. 16, No. 3, 1041-1056 (2012). 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 S. Caenepeel, K. Janssen and 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. 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. Reviewer: Blas Torrecillas (Almeria) MSC: 16W50 Graded rings and modules (associative rings and algebras) 16T15 Coalgebras and comodules; corings 16D90 Module categories in associative algebras Keywords:graded Morita contexts; group corings; Hopf group coalgebras; graded Morita-Takeuchi contexts Citations:Zbl 1145.16017 PDFBibTeX XMLCite \textit{G. Liu} and \textit{S. Wang}, Taiwanese J. Math. 16, No. 3, 1041--1056 (2012; Zbl 1259.16049) Full Text: DOI Link