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Hilbert-Kunz multiplicity of powers of an ideal III

Người báo cáo: Jugal Verma (Indian Institute of Technology, Mumbai)

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

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

Abstract: Let $(R, m a t h f r a k m)$ be a $d$ -dimensional local ring, $I$ be an $m a t h f r a k m$ -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 $L_{1} (I) = l i m_{q t o i n f t y} e_{1} (I^{[q]}) / q^{d}$ exists. Illya Smirnov proved Trivedi's result for $m a t h f r a k m$ -primary ideals of all local rings. Smirnov asked if $L_{k} (I) = l i m_{q t o i n f t y} e_{k} (I^{[q]}) / q^{d}$ exists for $k = 2, 3, l d o t s, d$ 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.