Qurves and quivers

In this paper we associate to a k-qurve A (formerly known as a quasi-free algebra [J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995) 251-289] or formally smooth algebra [M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, math.AG/9812158, 1998]) the one-quiver Q(1)(A) and dimension vector alpha(1)(A). This pair contains enough information to reconstruct for all n is an element of N the GL(n)-etale local structure of the representation scheme rep, A. In an appendix we indicate how one might extend this to qurves over nonalgebraically closed fields. Further, we classify all finitely generated groups G such that the group algebra kG is a k-qurve. If char(k) = 0 these are exactly the virtually free groups. We determine the one-quiver setting in this case and indicate how it can be used to study the finite-dimensional representations of virtually free groups. As this approach also applies to fundamental algebras of graphs of separable k-algebras, we state the results in this more general setting. (c) 2005 Elsevier Inc. All rights reserved.