Commutative Algebra Seminar

I will discuss a Grothendieck topology on the category of quasi-compact quasi-separated schemes called the "arc-topology." Covers in the arc-topology are tested via rank <=1 valuation rings. This topology is motivated by classical questions in algebraic K-theory, and leads to Mayer-Vietoris style sequences. Our main result is that étale cohomology with torsion coefficients satisfies arc-descent. This is joint work with Bhargav Bhatt.

In this talk we will discuss a surprising uniform property concerning the class of Cohen-Macaulay modules over strongly F-regular rings. As an application, we show that the torsion subgroup of the divisor class group of a local strongly F-regular ring is finite.

Strongly F-regular inversion of adjunction is the positive characteristic analog of the klt/plt inversion of adjunction in birational geometry. In characteristic 0, the klt/plt inversion of adjunction statement can made quantitative with the normalized volume. In this talk, we discuss an analogous quantitative refinement of strongly F-regular inversion of adjunction via the F-signature.