Non-commutative resolutions and Grothendieck groups

Abstract

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a noncommutative desingularization of Spec($R$). We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over $R$. In some cases those conditions are enough to imply that Spec($R$) has only rational singularities.