Generalized Hilbert-Kunz Multiplicity for Families of Ideals

TL;DR

This paper introduces a new asymptotic invariant called Amao-type multiplicity and demonstrates its relationship with the generalized Hilbert-Kunz multiplicity for families of ideals in Noetherian local rings of positive characteristic.

Contribution

It systematically studies the generalized Hilbert-Kunz multiplicity and introduces the Amao-type multiplicity, establishing their connection for p-families of ideals.

Findings

Generalized Hilbert-Kunz multiplicity is the limit of Amao-type multiplicities.

Introduces a new asymptotic invariant for families of ideals.

Provides foundational results for future research in multiplicity theory.

Abstract

In this paper, we initiate a systematic study of the generalized Hilbert-Kunz multiplicity for families of ideals in a Noetherian local ring (R,m) of positive characteristic, and introduce a new asymptotic invariant called the Amao-type multiplicity. We establish that, for a p-family of ideals, the generalized Hilbert-Kunz multiplicity arises as the limit of Amao-type multiplicities.