Asymptotic colengths for families of ideals: an analytic approach

TL;DR

This paper develops an analytic framework to establish the existence of asymptotic colengths for various families of ideals in Noetherian local rings, extending previous results and deriving inequalities and volume formulas.

Contribution

It introduces a unified analytic approach to prove the existence of limits for asymptotic colengths and extends the theory to weakly graded and p-families of ideals.

Findings

Proved existence of limits for asymptotic colengths.

Established Brunn-Minkowski type inequalities.

Derived volume equals multiplicity formulas.

Abstract

This article focuses on the existence of asymptotic colengths for families of $\fm_{R}$-primary ideals in a Noetherian local ring $(R,\fm)$. In any characteristic, we generalize graded families to weakly graded families of ideals, and in prime characteristic, we explore various families such as weakly $p$-families and weakly inverse $p$-families. The main contribution of this paper is providing a unified analytic method to prove the existence of limits. Additionally, we establish Brunn-Minkowski type inequalities, positivity results, and volume = multiplicity formulas for these families of ideals.