On the radical of the module category of an endomorphism algebra

TL;DR

This paper investigates the relationship between the radical of morphisms in module categories of a finite-dimensional algebra and its endomorphism algebra of a tilting module, providing bounds for nilpotency indices in specific algebra types.

Contribution

It introduces bounds for the nilpotency index of the radical in module categories of iterated tilted algebras of Dynkin type, linking morphism radicals across related categories.

Findings

Established an upper bound for the nilpotency index in Dynkin type cases.

Compared powers of radicals in module categories and endomorphism algebras.

Analyzed the impact of tilting modules on the radical structure.

Abstract

Given a finite dimensional algebra $A$ over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra $A$ and the induced morphism in the module category of the endomorphism algebra of a tilting $A$-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.