BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ian Aberbach (University of Missouri\, Columbia\, MO) DTSTART:20201204T130000Z DTEND:20201204T143000Z DTSTAMP:20260423T020952Z UID:VCAS/45 DESCRIPTION:Title: On the equivalence of weak and strong F-regularity\nby Ian Aberbach (Uni versity of Missouri\, Columbia\, MO) as part of IIT Bombay Virtual Commuta tive Algebra Seminar\n\n\nAbstract\nLet $(R\, \\mathfrak m\, k)$ be a (Noe therian) local ring of positive prime characteristic $p.$ Assume also\, f or 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 closur e of an ideal sits between the ideal itself and its integral closure. Whe n the tight closure of an ideal $I$ is $I$ itself we call $I$ tightly clos ed. For particularly nice rings\, e.g.\, regular rings\, every ideal is ti ghtly closed. We call such rings weakly $F$-regular. Unfortunately\, tig ht closure is known not to commute with localization\, and hence this prop erty of being weakly $F$-regular is not known to localize. To deal with t his problem\, Hochster and Huneke defined the notion of strongly $F$-regul ar (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 o f 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.\n LOCATION:https://researchseminars.org/talk/VCAS/45/ END:VEVENT END:VCALENDAR