On the dimensions of path algebras

Abstract

1. IntroductionThe notion of representation dimension of an artin algebra Λ, denoted rep.dimΛ, hasbeen introduced by Auslander in his Queen Mary Notes in 1971. His hope was to provide“a reasonable way of measuring how far the category of finitely generated modules over anartin algebra is from being of finite representation type”. Recall that an artin algebra Λis called of finite representation type if, up to isomorphisms, there are only finitely manyfinitely generated indecomposable Λ-modules. Such algebras are called representation finite.The importance of this class of artin algebras for the whole representation theory of artinalgebras is well understood, see e.g. [Au], [Ga] and [Ta].Auslander proved that Λ is of finite representation type if and only if its representationdimension is at most two. But it turned out to be hard to compute the actual value ofthe representation dimension of a given algebra. Moreover, all the algebras that one couldcompute explicitly their representation dimensions, had representation dimension at mostthree. So a natural question arises: are there artin algebras of representation dimensiongreater than three. Of course such algebras are of infinite representation type. On the otherhand, another question was whether this invariant is always finite. These questions has beenanswered positively, at almost the same time. Iyama [I] in 2003 proved that it is alwaysfinite and Rouquier [Rq1] in 2005 showed that any natural number n, except 1, can occuras the representation dimension of some algebra. In fact, he showed that the representationdimension of the exterior algebra Λk