Mathematical Proceedings of the Cambridge Philosophical Society

Representations of algebras as universal localizations

a1 Center for Mathematics and its Applications, School of Mathematical Sciences, John Dedman Building, The Australian National University, Canberra ACT 0200, Australia. e-mail:
a2 School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Edinburgh EH9 3JZ, Scotland. e-mail:
a3 School of Mathematics, University of Bristol, Bristol BS8 1TW. e-mail:

Article author query
neeman a   [Google Scholar] 
ranicki a   [Google Scholar] 
schofield a   [Google Scholar] 

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 $M_n(S)$ 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.

(Received May 8 2002)
(Revised July 9 2002)