On the Abelian Galois coverings of an algebra
Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field \(k\), given by a pair \((Q,I)\), where \(Q\) is a finite oriented graph without double arrows and \(I\) is an admissible ideal of the path algebra \(kQ\) of \(Q\), \(\pi\) be its fundamental group and \({\tilde \Lambda}\) its universal Galois covering. Then each group homomorphism \(\psi\colon \pi\to G\) produces a Galois covering \(p_{\psi}\colon \Lambda_{\psi}\to \Lambda\) with group \(G\), moreover \(\psi\) induces a certain action of the group \(X\) of characters of \(G\) on \(\Lambda\). The author shows that if \(G\) is a finite abelian group, whose order is prime with the characteristic of \(k\), then the Galois covering \(p_{\psi}\) is isomorphic to a certain covering \(p'\colon P\to \Lambda\), where \(P\) is a category of indecomposable projectives over a skew group algebra \(\Lambda X\) of \(X\) over \(\Lambda\), and consequently \(\Lambda_{\psi}\) and \(\Lambda X\) are Morita equivalent. Another proof of the last fact is given by \textit{I. Reiten} and \textit{C. Riedtmann} [J. Algebra 92, 224--282 (1985; Zbl 0549.16017)].
scientific article; zbMATH DE number 3946393
scientific article