# Commentarii Mathematici Helvetici

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**Volume 74, Issue 4, 1999, pp. 548–574**

**DOI: 10.1007/s000140050105**

Published online: 1999-12-31

Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities

William Crawley-Boevey^{[1]}(1) University of Leeds, UK

For any associative algebra A over a field K we define a family of algebras $ \Pi^\lambda(A) $ for $ \lambda \in K \,\otimes_{\Bbb Z}\,{\rm K_0}(A) $. In case A is the path algebra of a quiver, one recovers the deformed preprojective algebra introduced by M. P. Holland and the author. In case A is the coordinate ring of a smooth curve, the family includes the ring of differential operators for A and the coordinate ring of the cotangent bundle for Spec A. In case A is quasi-free and $ \Omega^1 $ A is a finitely generated A-A-bimodule we prove that $ \Pi^\lambda(A) $ is well-behaved under localization. We use this to prove a Conze embedding for deformations of Kleinian singularities.

*Keywords: *Preprojective algebra, quiver, quasi-free algebra, pseudoflat epimorphism, differential operator, Kleinian singularity, Conze embedding

Crawley-Boevey William: Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities. *Comment. Math. Helv.* 74 (1999), 548-574. doi: 10.1007/s000140050105