Constructing the preprojective components of an algebra
Let \(k\) be an algebraically closed field, \(Q\) be a finite connected quiver without oriented cycles and consider the finite dimensional algebra \(A=kQ/I\), where \(I\) is an admissible ideal of \(kQ\). The Auslander-Reiten quiver \(\Gamma_A\) of \(A\) has as vertices representatives of isoclasses of finite dimensional indecomposable \(A\)-modules, and there are as many arrows from \(X\) to \(Y\) in \(\Gamma_A\) as \(\dim_k(\text{rad}(X,Y)/\text{rad}^2(X,Y))\). A component \(\mathcal P\) of \(\Gamma_A\) is said to be preprojective if \(\mathcal P\) has no oriented cycles and each module \(X\in{\mathcal P}\) has only finitely many predecessors in the path order of \(\mathcal P\). The importance of having such components is that for each \(X\) on them, it holds that \(\text{Ext}^i_A(X,X)=0\), for \(i>0\), \(\text{End }X\) is local and the isomorphism class of \(X\) is uniquely determined by the composition factors. Several classes of algebras have preprojective components, such as algebras with separation condition and quasitilted algebras. In this paper, the authors give an algorithmic procedure to construct all preprojective components in \(\Gamma_A\).
finite dimensional indecomposable modules
scientific article; zbMATH DE number 868021