Auslander Generators of Iterated Tilted Algebras

Let Λ be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of Λ is three in case Λ is representation infinite. Recently there has been a lot of attention to compute the representation dimension of a finite dimensional algebra Λ, a notion introduced by Auslander in [A] in an attempt to measure the complexity of the representation theory of Λ. It seems from the results obtained in the last few years that it actually measures the homological complexity of Λ, and we want to provide some further evidence. We will not need the original definition, but rather the following characterization already going back to Auslander in [A]; see also [EHIS] or [CP] for a more detailed account. For this let Λ be an arbitrary finite dimensional algebra. Let modΛ be the category of finitely generated left Λ−modules. Let M ∈ modΛ be a generatorcogenerator. So we have that ΛΛ⊕DΛΛ ∈ addM, where D is the standard duality on modΛ and addM is the full subcategory of modΛ containing the direct summands of direct sums of M. Let d be the minimum such that there is a generatorcogenerator with the following property: For each X ∈ modΛ there is an exact sequence 0 → M → · · · → M → M → X → 0 such that 0 → HomΛ(M,M) → · · · → HomΛ(M,M) → HomΛ(M,X) → 0 is exact, where M i ∈ addM for 0 ≤ i ≤ t. Then the representation dimension rep.dimΛ of Λ is d + 2. It follows from Iyama’s result [I] that the representation dimension of Λ is always finite. Trivially, if Λ is representation finite, then the representation dimension of Λ is two. We call a generator-cogenerator where the minimum is attained an Auslander generator. As the main result of this article we will show here that for an iterated tilted algebra Λ there is a generator-cogenerator M such that d = 1. We will recall the definition of this class of algebras below. In particular we obtain that an iterated Received by the editors April 7, 2009, and, in revised form, July 24, 2009. 2000 Mathematics Subject Classification. Primary 16E05, 16E10, 16G10. The results presented here were obtained while the second and third authors were visiting IME-USP. They thank their coauthor for his kind hospitality during their pleasant stay in São Paulo. The project was made possible by a grant from FAPESP, Brazil. The first author also acknowledges a grant from CNPq. c ©2010 American Mathematical Society