Non-Commutative Resolutions of Toric Varieties

Abstract

Let \(R\) be the coordinate ring of an affine toric variety. We show that the endomorphism ring \(End_R(\mathbb A),\) where \(\mathbb A\) is the (finite) direct sum of all (isomorphism classes of) conic \(R\)-modules, has finite global dimension. Furthermore, we show that \(End_R(\mathbb A)\) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field \(k\) of prime characteristic, we show that the ring of differential operators \(D_\mathsf{k}(R)\) has finite global dimension.