Back to Commutative Algebra Seminar: Spring 2022
Commutative Algebra Seminar
Date: Friday, Apr 29, 2022
Time: 2:00PM - 3:00PM
Location: LCB 222
Kriti Goel
University of Utah
Title |
Hilbert-Kunz multiplicity of powers of an ideal |
Abstract |
We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $m$-primary ideal exists in a Noetherian local ring $(R,m)$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal. |
Commutative Algebra