On the Representation Dimension of Finite Dimensional Algebras

Abstract

The importance of the representation finite Artin algebras for the whole representation theory of Artin algebras is well understood and there is much literature on the subject (see [1, 6, 12]). These are Artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules. Moreover there is a bijective correspondence between the class of representation finite Artin algebras and that of Artin algebras with global dimension of at most 2 and with dominant dimension of at least 2. This result was established by Auslander in [1]. Motivated by this correspondence, Auslander introduced the concept of representation dimension for Artin algebras to study the connection of arbitrary Artin algebras with representation finite Artin algebras. “It is hoped that this notion gives a reasonable way of measuring how far an Artin algebra is from being representation finite type” [1, p. 134]. Representation dimension is a Morita-invariant of Artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators. Unfortunately, little seems to be known about representation dimension. So there are a lot of essential questions on this invariant. Dealing with the computation and estimation of the representation dimension of an Artin algebra, there are a few cases known: Auslander proved that the representation dimension of an Artin algebra is two if and