A Volume = Multiplicity formula for 𝑝-families of ideals

Abstract

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