Representations of non-splittable extension algebras
Let \(A\) be a basic finite dimensional algebra over a field \(k\) whose ordinary quiver contains no oriented cycles. Denote by \(\text{mod\,}A\) the category of finitely generated left \(A\)-modules, by \(\text{mod}_ pA\) the full subcategory of \(\text{mod\,}A\) formed by modules without projective direct summands and by \(D:\text{mod\,}A\rightleftharpoons\text{mod\,}A^{op}\) a self-duality. Moreover, let \(T\) be an extension over \(A\) with kernel \(DA\) and \(A\ltimes DA\) be the trivial extension of \(A\) by \(DA\). It is shown that \(\text{mod}_ pT\) is equivalent to \(\text{mod}_ pA\ltimes DA\), and the factor algebra \(T/soc T\) is isomorphic to the trivial extension \(A\ltimes(DA/\text{soc\,}DA)\). Moreover, the Auslander-Reiten quivers of \(T\) and \(A\ltimes DA\) are isomorphic. As a consequence one obtains that \(T\) is of finite representation type iff so is \(A\ltimes DA\).
categories of finitely generated left modules
scientific article; zbMATH DE number 4049819
scientific article