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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.

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