A Volume = Multiplicity formula for $p$-families of ideals

TL;DR

This paper introduces a volume-multiplicity formula for p-families of ideals in rings of prime characteristic, linking geometric objects called p-bodies to algebraic invariants like Hilbert-Kunz multiplicity.

Contribution

It establishes a novel volume equals multiplicity formula connecting p-bodies and Hilbert-Kunz multiplicity for p-families of ideals in Noetherian local rings.

Findings

Defined p-bodies as Euclidean objects associated with p-families.

Proved a volume = multiplicity formula relating p-bodies to Hilbert-Kunz multiplicity.

Extended the theory of Newton Okounkov bodies to p-families of ideals.

Abstract

In this paper, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach a Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish in this paper a Volume = Multiplicity formula for $p$-families of $\mathfrak{m}_{R}$-primary ideals in a Noetherian local ring $R$.