Eleonore Faber

ABSTRACT

This talk is about some results on noncommutative resolutions of toric varieties. Let $R$ be the coordinate ring of a normal affine toric variety over a field of arbitrary characteristic. Then the module $R^{1/q}$ of $q$-th roots of $R$ is a direct sum of so-called conic modules. Furhter, $\text{End}_R(R^{1/q})$ provides a noncommutative resolution of singularities of $R$ for $q \gg 0$, in particular, it is a ring of finite global dimension. This has first been shown by Špela Špenko and Michel Van den Bergh, and in joint work with Greg Muller and Karen Smith we gave an explicit description of the combinatorial structure of these endomorphism rings. I will discuss these results and report on joint work in progress with Milena Hering and Kevin Tucker on noncommutative resolutions for seminormal monoid algebras and their connections to rings of differential operators.