Non-Commutative Resolutions of Toric Varieties

TL;DR

This paper demonstrates that for affine toric varieties, the endomorphism ring of a specific module has finite global dimension and provides conditions under which it is a non-commutative crepant resolution, also analyzing differential operators in prime characteristic.

Contribution

It establishes finite global dimension of endomorphism rings for conic modules and characterizes when these are non-commutative crepant resolutions for toric varieties.

Findings

Endomorphism ring has finite global dimension for affine toric varieties.

Non-commutative crepant resolution occurs if and only if the variety is simplicial.

Ring of differential operators has finite global dimension in prime characteristic.

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.