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Generalized Hilbert-Kunz Multiplicity for Families of Ideals

Stephen Landsittel, Sudipta Das

TL;DR

This work develops a comprehensive framework for generalized Hilbert-Kunz multiplicities of $p$-families of ideals in Noetherian local rings of positive characteristic. It introduces the Amao-type multiplicity $a_F(I,J)$ and proves that, under linear growth (and LC) conditions, the generalized HK multiplicity $e_{gHK}(I_{\bullet})$ equals the asymptotic limit of Amao-type values, $e_{gHK}(I_{\bullet}) = \lim_{q'\to\infty} a_F(I_{q'}, (I_{q'})^{sat})/(q')^d$. The core technical advance is a general volume formula linking limits of lengths to volumes of truncated $p$-bodies derived from OK-domain valuations and cone geometry; this yields a volume-equals-multiplicity correspondence and enables applications to BBL weakly $p$-families. The results unify and extend classical HK theory, provide existence criteria, and connect asymptotic invariants to geometric data via a robust valuation-driven framework.

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.

Generalized Hilbert-Kunz Multiplicity for Families of Ideals

TL;DR

This work develops a comprehensive framework for generalized Hilbert-Kunz multiplicities of -families of ideals in Noetherian local rings of positive characteristic. It introduces the Amao-type multiplicity and proves that, under linear growth (and LC) conditions, the generalized HK multiplicity equals the asymptotic limit of Amao-type values, . The core technical advance is a general volume formula linking limits of lengths to volumes of truncated -bodies derived from OK-domain valuations and cone geometry; this yields a volume-equals-multiplicity correspondence and enables applications to BBL weakly -families. The results unify and extend classical HK theory, provide existence criteria, and connect asymptotic invariants to geometric data via a robust valuation-driven framework.

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.

Paper Structure

This paper contains 9 sections, 15 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cones in $\mathbb{R}^d$
  4. $p$-systems and $p$-families
  5. $OK$-Domain
  6. Generalized Hilbert-Kunz multiplicities and existence conditions
  7. A general volume formula
  8. Applications to generalized Hilbert-Kunz multiplicities
  9. Application to BBL weakly $p$-families

Key Result

Theorem A

(thm6-1, thm6-2) Let $(R,m)$ be a $d$-dimensional regular local ring of prime characteristic $p>0$. Let $I_{\bullet} = \{I_q\}$ be a $p$-family in $R$ for which there is $c>0$ such that $I_q\cap m^{cq} = I_q^{{\normalfont\text{sat}}}\cap m^{cq}$ for all $q$. Suppose that for $P\in\text{Assh}(R)$ suc Moreover, let $(R, \mathfrak{m})$ is a $d$-dimensional analytically unramified local ring and $I \s

Theorems & Definitions (52)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Example 1
  • ...and 42 more