Tilted algebras

Source abstract

Let A A be a finite dimensional hereditary algebra over a field, with n n simple A A -modules. An A A -module T A T_A with n n pairwise nonisomorphic indecomposable direct summands and satisfying Ex t 1 ( T A , T A ) = 0 {\text {Ex}}{{\text {t}}^1}({T_A},\,{T_A}) = 0 is called a tilting module, and its endomorphism ring B B is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable B B -modules are in one-to-one correspondence to the indecomposable A 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.