Back to Commutative Algebra Seminar: Spring 2023
Commutative Algebra Seminar
Date: Friday, Feb 3, 2023
Time: 2:00PM - 3:00PM
Location: LCB 222
Hanlin Cai
University of Utah
Title |
Perfectoid Signature and Perfectoid Hilbert-Kunz Multiplicity |
Abstract |
In this talk I’ll talk about a (perfectoid) mixed characteristic version of F-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings’ normalized length. These definitions coincide with the classical theory in equal characteristic. We prove that a ring is regular if and only if either its perfectoid signature or perfectoid Hilbert-Kunz multiplicity is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM closure and extended plus closure of ideals. We demonstrate that perfectoid signature detects BCM regularity and transforms similarly to F-signature or normalized volume under quasi-étale maps. As a consequence, we prove that BCM-regular rings have finite local étale fundamental group and torsion part of their divisor class groups. This is joint work with Seungsu Lee, Linquan Ma, Karl Schwede and Kevin Tucker. |
Commutative Algebra