Description
Date and Time: 4 December 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EDT
(joining time: 6:15 pm IST - 6:30 pm IST)
Speaker: Ian Aberbach, University of Missouri
Google meet link: meet.google.com/aum-zrru-xtg
Title: On the equivalence of weak and strong F-regularity
Abstract: Let $(R, m, k)$ be a (Noetherian) local ring of positive prime
characteristic $p.$ Assume also, for simplicity, that $R$ is complete (or,
more generally, excellent). In such rings we have the notion of tight
closure of an ideal, defined by Hochster and Huneke, using the Frobenius
endomorphism. The tight closure of an ideal sits between the ideal itself
and its integral closure. When the tight closure of an ideal $I$ is $I$
itself we call $I$ tightly closed. For particularly nice rings, e.g.,
regular rings, every ideal is tightly closed. We call such rings weakly
$F$-regular. Unfortunately, tight closure is known not to commute with
localization, and hence this property of being weakly $F$-regular is not
known to localize. To deal with this problem, Hochster and Huneke defined
the notion of strongly $F$-regular (assuming $R$ is $F$-finite), which
does localize, and implies that $R$ is weakly $F$-regular. It is still an
open question whether or not the two notions are equivalent, although it
has been shown in some classes of rings. Not much progress has been made
in the last 15-20 years. I will discuss the problem itself, the cases that
are known, and also outline recent progress made by myself and Thomas
Polstra.