Hilbert-Kunz multiplicity of powers of an ideal III

Time: 9:00 -- 10:15, December 18, 2024

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: Let $(R, mathfrak m)$ be a $d$-dimensional local ring, $I$ be an $mathfrak m$-primary ideal and let $R$ have prime characteristic $p.$
K.-I. Watanabe and K.-I. Yoshida investigated the Hilbert-Kunz multiplicity of powers of $I$ in terms of Hilbert coefficients of I and its Frobenius powers $I^{[q]}$ where $q=p^n.$ It was proved by V. Trivedi that if $I$ is zero-dimensional graded ideal of a standard graded ring $R$ of dimension $d$ over a field, then $L_1(I)=lim_{qto infty} e_1(I^{[q]})/q^d$ exists. Illya Smirnov proved Trivedi's result for $mathfrak m$-primary ideals of all local rings. Smirnov asked if $L_k(I)=lim_{qto infty} e_k(I^{[q]})/q^d$ exists for $k=2,3,ldots, d$ and whether the HK multiplicity of $I^n$ for all large $n$ is given by the formula [e_{HK}(I^n)=sum_{k=0}^d (-1)^k L_k(I)binom{n+d-k-1}{d-k}.] Smirnov also conjectured that ideals of reduction number one can be characterised in terms of the HK multiplicity. I will report on joint works with {bf Kriti Goel, Arindam Banerjee and Shreedevi Masuti, Marilina Rossi and Alessandro De Stefani, } which provide partial answers to Smirnov's questions.