Fri, March 12, 2021
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6:00pm [6:30pm] Thomas Polstra, University of Virginia, Charlottesville, VA, USA
Description:
Speaker: Thomas Polstra, University of Virginia, Charlottesville, VA, USA Date/Time: 12 March 2021, 6:30pm IST/ 1:00pm GMT / 8:00am EST (joining time 6:15pm IST). Google meet link: https://meet.google.com/xze-mbdb-qdb Title: Strongly $F$-regular rings, maximal Cohen-Macaulay modules, and the $F$-signature Abstract: The singularities of a local prime characteristic ring are best understood through the behavior of the Frobenius endomorphism. A singularity class of central focus is the class of strongly $F$-regular rings. Examples of strongly $F$-regular rings include normal affine toric rings, direct summands of regular rings, and determinantal rings. Every strongly $F$-regular ring enjoys the property of being a normal Cohen-Macaulay domain. In particular, the study of finitely generated maximal Cohen-Macaulay modules over such rings is a warranted venture. We will demonstrate a surprising uniform behavior enjoyed by the category of maximal Cohen-Macaulay modules over a strongly $F$-regular local ring. Consequently, we can redrive Aberbach and Leuschke's theorem that the $F$-signature of a strongly $F$-regular ring is positive in a novel and elementary manner. Time permitting, we will present applications on the structure of the divisor class group of a local strongly $F$-regular ring. For more information and links to previous seminars, visit the website of VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar