GRADED MORITA THEORY FOR GROUP CORING AND GRADED MORITA-TAKEUCHI THEORY

Abstract

A Graded Morita context is constructed for any comodule of a group coring. For any right $G$-$\underline{\mathcal{C}}$-comodule $\underline{M}$ with dual graded ring $R$, we define a graded ring $T=HOM^{G,\underline{\mathcal{C}}}(M,M) = \bigoplus_{g \in G} HOM^{G,\underline{\mathcal{C}}}(M, M)_g$, and a $G$-graded $R$-$T$ bimodule $Q = \bigoplus_{g \in G} Q^g$, where $Q^g$ is a family of right $A$-linear maps $q^g_{\alpha};\,M_{\alpha} \rightarrow R_{g\alpha}$ in $\mathcal{M}_A$. We construct a graded Morita context $M = (T,\,\,\,R,\,\,\,\bigoplus_{\alpha\in G}M_{\alpha},\,\,\,Q, \,\,\,\tau,\,\,\,\mu)$ with connecting homomorphisms $\tau:\,\, _{T}(\bigoplus_{\alpha\in G}M_{\alpha})\otimes_R Q_T\rightarrow T,\,\,m\otimes q\mapsto mq(-)$, $\,\mu:\, _{R}Q\otimes_T (\bigoplus_{\alpha\in G}M_{\alpha})_{R}\rightarrow R,\,\,q\otimes m\mapsto q(m)$, which generalized the Morita context in [3, 5-7, 10, 13]. Meanwhile, we prove the graded Morita-Takeuchi theory as a generalization of Morita-Takeuchi theory which characterize the equivalence of comodule over field.