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 non-commutative 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.