Graded coalgebras and Morita-Takeuchi contexts
Let \(G\) be a group and \(k\) a field. The present paper deals with \(G\)-graded coalgebras \(C=\bigoplus_{g\in G}C_g\), that is, with right \(kG\)-comodule coalgebras, and with graded \(C\)-comodules, that is, comodules over the smash product \(C\rtimes kG\). The authors continue the investigation started by \textit{C. Năstăsescu} and \textit{B. Torrecillas} [Tsukuba J. Math. 17, No. 2, 461-479 (1993; Zbl 0819.16036)]. Their main result is as follows. The coalgebras \(C_1\) and \(C\rtimes kG\) are connected by a Morita-Takeuchi context in which one of the structure maps is injective. Some applications of this result are also given. It is proved that \(C\) is strongly graded iff the other structure map is injective. Finally two duality theorems are obtained, these being analogues of the Cohen-Montgomery duality theorems for groups acting and coacting on algebras.
scientific article; zbMATH DE number 895946