On the radical of a module category and tilting theory

C. CHAIO; GUAZZELLI VICTORIA

Luminy

Conferencia; Advances in representation theory of algebras; 2017

Let $A$ be an a finite dimensional $k$-algebra over an algebraically closed field $k$, and $mbox{mod},A$ thecategory of finitely generated left $A$-modules. We denote by $Re(mbox{mod},A)$ the Jacobson radical of $mbox{mod},A$.It is well-known,by a result of M. Auslander, that an artin algebra is of finite representation typeif and only if there is a positive integer $n$ such that $Re^{n}(mbox{mod},A)=0.$ In the case that $A$ is a finite dimensional algebra over an algebraically closed field of finite representation type,the minimal lower bound $m geq 1$ such that $Re^{m}(mbox{mod},A)$ vanishes was found in terms of the left and right degrees of particular irreducible morphisms (see [2]). The aim of this work is todetermine such minimal boundin case $A$ is a representation-finite string algebra. This bound is given in terms of strings and taking into account their respective ordinary quivers. Furthermore, we also show how to read the degree of any irreducible morphism in such algebras.