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