On the nilpotency index of the radical of a module category

CHAIO CLAUDIA, GUAZZELLI VICTORIA, PAMELA SUAREZ.

Buenos Aires

Congreso; International Conference of representation of Algebras; 2022

Abstract: Let A be a finite dimensional algebra over an algebraically closed field and $mbox{mod}, A$ be the category of finitely generated $A$-modules. The radical of $mbox{mod}, A$ is the ideal generated by all non-isomorphisms between indecomposable $A$-modules. For $n geq 2$, the powers of the radical are defined inductively.An important research direction towards understanding the structure of a module categoryis the study of the compositions of irreducible morphisms in relation with the powers of theradical of their module categories.In case we deal with a representation finite algebra, it iswell-known by a result of M. Auslander that the radical of the module category is nilpotent.Moreover, it is also known that such a bound is the lengthof the longest non-zero path from the projective in a vertex a to the injective in the samevertex going through the simple in a.The aim of this talk is to determine which vertices of $Q_A$ are sufficient to be consider inorder to determine the nilpotency index of the radical of the module category of an algebra.Furthermore, we compute the nilpotency index of the radical ofsome representation-finite tree algebras with zero-relations not overlapped.ReferencesC. Chaio, V. Guazzellli, P. Suarez. On the nilpotency index of the radical of a module category. preprint (Arxiv)(2020).