Fri, December 4, 2020
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6:00pm [6:30pm] Ian Aberbach, University of Missouri
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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.