Infinite radicals in standard Auslander-Reiten components
Let \(A\) be a finite dimensional algebra over an algebraically closed field \(k\) and \(\text{mod }A\) the category of finitely generated left \(A\)- modules. Denote by \(\text{rad}^ \infty(\text{mod }A)\) the infinite power of the Jacobson radical \(\text{rad}(\text{mod }A)\) of \(\text{mod }A\) and by \(\Gamma_ A\) the Auslander-Reiten quiver of \(A\). Recall that a component \(\mathcal C\) of \(\Gamma_ A\) is called standard if the full subcategory of \(\text{mod }A\) formed by modules from \(\mathcal C\) is equivalent to the mesh-category \(k({\mathcal C})\) of \(\mathcal C\) [see \textit{C. M. Ringel}, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)], and \(\mathcal C\) is called generalized standard if \(\text{rad}^ \infty (X,Y) = 0\) for all modules \(X\) and \(Y\) in \(\mathcal C\) [see \textit{A. Skowroński}, J. Math. Soc. Japan 46, 517-543 (1994)]. In the paper, the author proves that every standard component \(\mathcal C\) of \(\Gamma_ A\) is generalized standard. In particular, one obtains that if \(\Gamma_ A\) is standard then \(A\) is of finite representation type. We note also that it has been proved in [\textit{A. Skowroński}, Bull. Pol. Acad. Sci. 42, 157-163 (1994)] that every generalized standard component \(\mathcal C\) of \(\Gamma_ A\) containing not both a projective and an injective module is standard.
category of finitely generated left \(A\)- modules
scientific article; zbMATH DE number 614226