# F-rational singularities mod p implies rational singularities in mixed characteristic

### F-rational singularities mod p implies rational singularities in mixed characteristic

**Series:**Algebraic Geometry Seminar

**Location:**ENR 2 S395

**Presenter:**Karl Schwede, University of Utah

In characteristic zero, Elkik proved that if R is a

local domain and that f in R is nonzero such that R/fR has rational

singularities, then R has rational singularities as well. In

characteristic p, there is an analog of rational singularities called

F(robenius)-rational singularities and the analogous result to Elkik's

also holds in that context. Additionally K. Smith showed that if R in

characteristic zero has F-rational singularities after reduction to p >> 0

then R has rational singularities in characteristic zero.

Suppose now that R is a local ring of mixed characteristic (0, p) and

that R/pR has F-rational singularities. We show that this implies that R

has a form of rational singularities which we call BCM-rational. This

implies that R has rational singularities in the usual sense and hence the

localization to characteristic zero also has rational singularities. In

particular, if R in characteristic zero has rational singularities after

reduction to a single characteristic p (even a small one), then R has

rational singularities in characteristic zero. This also gives a way for

a computer to show that a ring in characteristic zero has rational

singularities.

If time permits, I will also discuss generalizations to other ways to

measure singularities such as multiplier ideals. This is all joint work

with Linquan Ma.