ar X iv : m at h . R A / 0 20 50 34 v 1 3 M ay 2 00 2 REPRESENTATIONS OF ALGEBRAS AS UNIVERSAL LOCALIZATIONS

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra S in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is Mn(S) where n is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra. Two applications of this are then considered. Firstly, given a ring homomorphism φ : R → S, we say that S is stably flat over R if and only if Tori (S, S) = 0 for all i > 0. In a recent paper [2], the first two authors show that there is a long exact sequence in algebraic K-theory associated to a universal localization provided the localization is stably flat. We show that this construction provides many examples of universal localizations that are not stably flat since the finite dimensional algebra we localize is of global dimension 2 and stably flat universal localization cannot increase the global dimension. Secondly, the Malcolmson normal form states that every element of the localised ring can be written in the form as−1b where s : P → Q lies in