Tilted Algebras

Abstract

Let $A$ be a finite dimensional hereditary algebra over a field, with $n$ simple $A$-modules. An $A$-module $T_A$ with $n$ pairwise nonisomorphic indecomposable direct summands and satisfying ${\text {Ex}}{{\text {t}}^1}({T_A}, {T_A}) = 0$ is called a tilting module, and its endomorphism ring $B$ is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable $B$-modules are in one-to-one correspondence to the indecomposable $A$-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.