On the representation dimension of artin algebras

TL;DR

This paper investigates the representation dimension of artin algebras, proving bounds for torsionless-finite cases and exploring how tensor products affect the representation dimension, leading to new examples with large dimensions.

Contribution

It provides a full proof that torsionless-finite artin algebras have representation dimension at most 3 and analyzes how tensor products influence the representation dimension.

Findings

Torsionless-finite artin algebras have representation dimension at most 3.

Tensor products of path algebras can produce algebras with arbitrarily large representation dimension.

The tensor product of n representation-infinite path algebras has representation dimension exactly n+2.

Abstract

The representation dimension of an artin algebra as introduced by M.Auslander in his Queen Mary Notes is the minimal possible global dimension of the endomorphism ring of a generator-cogenerator. The paper is based on two texts written in 2008 in connection with a workshop at Bielefeld. The first part presents a full proof that any torsionless-finite artin algebra has representation dimension at most 3, and provides a long list of classes of algebras which are torsionless-finite. In the second part we show that the representation dimension is adjusted very well to forming tensor products of algebras. In this way one obtains a wealth of examples of artin algebras with large representation dimension. In particular, we show: The tensor product of n representation-infinite path algebras of bipartite quivers has representation dimension precisely n+2.