On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

TL;DR

This paper investigates the asymptotic behavior of Hilbert-Samuel coefficients of Frobenius powers of ideals in prime characteristic rings, providing conditions for limits, explicit formulas, and counterexamples to existing conjectures.

Contribution

It establishes conditions for the existence of limits of Hilbert-Samuel coefficients, relates Hilbert-Kunz multiplicity to these limits, and analyzes polynomial behavior of generalized Hilbert-Kunz functions in face rings.

Findings

Limit of Hilbert-Samuel coefficients exists under certain conditions.

Hilbert-Kunz multiplicity can be expressed via Hilbert-Samuel multiplicity.

Generalized Hilbert-Kunz functions are polynomial in face rings.

Abstract

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ 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.