Piecewise hereditary one-point extensions
Throughout this review, by category is meant an Abelian locally finite \(k\)-category, where \(k\) is a field. If \(\mathcal A\) is such a category, we denote by \(D^b({\mathcal A})\) the bounded derived category of complexes over \(\mathcal A\). A finite-dimensional \(k\)-algebra \(A\) is said to be piecewise hereditary of type \(\mathcal H\) if there exists a hereditary category \(\mathcal H\) such that \(D^b(A)\) is triangle equivalent to \(D^b({\mathcal H})\). Examples of piecewise hereditary algebras are quasitilted algebras. While there are many results on quasitilted algebras, not much is known so far about the properties of piecewise hereditary algebras and their module categories. The author investigates those piecewise hereditary algebras which arise as one-point extensions of connected wild hereditary algebras. It is shown that if \(A\) is a wild hereditary algebra and \(M\) is an \(A\)-module such that \(A[M]\) is piecewise hereditary, then either (1) \(M\) is preprojective or preinjective and then \(A[M]\) is tilted, or (2) \(M\) is regular, indecomposable and quasisimple and then there exists \(r_0\in\mathbb{Z}\) such that \(A[\tau^r M]\) is quasitilted of type \(\mathcal H\) for all \(r\geq r_0\). The structure of the Auslander-Reiten quiver of piecewise hereditary algebras occurring in case (2) and for which \({\mathcal H}=H\text{-mod}\) for \(H\) a connected wild hereditary algebra is also investigated (note that case (1) is well-known). In fact it is shown that the connected components of the Auslander-Reiten quiver have the following structure: (1) There is one preprojective component which is the preprojective component of \(A\). (2) There is one preinjective component which is the preinjective component of a wild concealed factor algebra \(C\) of \(A[M]\). (3) The remaining components are stable of type \(\mathbb{Z} A_\infty\). This situation is similar to the one observed for tilted algebras, the only difference being the absence of a connecting component.
scientific article; zbMATH DE number 1441795