A Morita-Takeuchi context for graded coalgebras (Q2762034)

Let \(C\) be a \(G\)-graded coalgebra and \(H\) be a finite subgroup of \(G\). The smash coproduct of \(C\) as \(kG\)-comodule algebra is denoted by \(C\rtimes kG\) and if \(HG\) is the set of right cosets modulo \(H\) we also can construct the smash coproduct \(C\rtimes k(HG)\). The main result establishes that \((A=C\rtimes k(HG),B=(C\rtimes kG)\rtimes(kH)^*,C\rtimes kG,C\rtimes kG,f,g)\) forms a strict Morita-Takeuchi context, where \(f\colon A\to(C\rtimes kG)\square_B(C\rtimes kG)\) is defined by \(f(c\rtimes H\sigma_l)=\sum_{h\in H}(c_1\rtimes h\sigma_l(\deg c_2)^{-1})\otimes(c_2\rtimes h\sigma_l)\), \((\sigma_l)\) denotes a system of representatives for the right cosets modulo \(H\), and \(g\colon B\to(C\rtimes kG)\square_A(C\rtimes kG)\) is defined by \(g((c\rtimes x)\rtimes p_h)=\sum(c_1\rtimes x(\deg c_2)^{-1})\otimes(c_2\rtimes h^{-1}x)\). The author proves now two applications: a new proof of the duality theorem proved by \textit{S. Dăscălescu, C. Năstăsescu, F. Van Oystaeyen} and the reviewer [Commun. Algebra 25, No. 1, 159-175 (1997; Zbl 0873.16025)] using Hopf-Galois theory for coalgebras and it is shown that if \(C\) is a strongly graded coalgebra, \(C_H=\bigoplus_{h\in H}C_h\), \(A\) and \(B\) are Morita-Takeuchi equivalent coalgebras.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].