Hilbert-Kunz multiplicity of the powers of an ideal

TL;DR

This paper investigates the Hilbert-Kunz multiplicity of ideal powers, establishing the existence of the second coefficient in generality, and relates it to Hilbert coefficients of Frobenius powers, with properties like additivity and inequalities.

Contribution

It extends recent results by proving the existence of the second coefficient in a broad setting and characterizes it via limits of Hilbert coefficients, also showing its additive nature.

Findings

Second coefficient exists in general settings.

Second coefficient equals the limit of Hilbert coefficients of Frobenius powers.

The second coefficient is additive in short exact sequences.

Abstract

We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.