Generalized standard Auslander-Reiten components without oriented cycles
Let \(A\) be an Artin algebra and denote by \(\text{rad}^ \infty(\text{mod }A)\) the infinite radical of \(\text{mod } A\), the category of finitely generated right \(A\)-modules. A connected component \({\mathcal C}\) of the Auslander-Reiten quiver \(\Gamma (A)\) of \(A\) is called generalized standard if \(\text{rad}^ \infty (X,Y) =0\) for all \(X\), \(Y\) in \({\mathcal C}\). The author studies generalized standard components without oriented cycles. In the first theorem he gives several characterizations of a generalized standard component \({\mathcal C}\) without oriented cycles. The precise statements would need some preparations. Roughly some of them say that \({\mathcal C}\) is generalized standard exactly if there are no nonzero homomorphisms between certain well defined modules in \({\mathcal C}\). In the next part components \({\mathcal C}\) with a section \(\Sigma\) are considered. It is shown that \({\mathcal C}\) is generalized standard for example if \(\Hom (X,\tau Y)=0\) for \(X,Y\in \Sigma\) and \(\tau\) the Auslander-Reiten translation. Moreover \(\Sigma\) is finite. If \({\mathcal C}\) is a component in \(\Gamma(G)\) with a finite section \(\Gamma\), then the annihilator \(I\) of \({\mathcal C}\) coincides with the annihilator of \(\Sigma\) (Lemma 3), hence \({\mathcal C}\) can be considered as a component in \(\Gamma (A/I)\) with a faithful section. From this the author deduces that an Artin algebra \(A\) is tilted if and only if it admits a faithful generalized standard component containing a section. These results are applied in the last part of the paper to obtain new characterizations of concealed algebras.
category of finitely generated right \(A\)-modules
scientific article; zbMATH DE number 579210