  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.
