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Asymptotic colengths for families of ideals: an analytic approach

Sudipta Das, Cheng Meng

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.

Asymptotic colengths for families of ideals: an analytic approach

Abstract

This article focuses on the existence of asymptotic colengths for families of -primary ideals in a Noetherian local ring . In any characteristic, we generalize graded families to weakly graded families of ideals, and in prime characteristic, we explore various families such as weakly -families and weakly inverse -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.

Paper Structure

This paper contains 26 sections, 105 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Results in present work
  3. Applications
  4. Conventions and basic notions
  5. Notations in Euclidean space
  6. Pointed cones and upward cones
  7. Semigroups, semigroup ideals, and cones
  8. OK valuation
  9. Lattice points and cubes in Lg
  10. Some theorems in real analysis
  11. Pointed cone, upward cone, and height function
  12. Families of ideals
  13. Approximation of sets of valuations and bounds for the valuation of BBL families
  14. Proof of the convergence for weakly graded families
  15. Convergence of $F_{1/n\nu^c(I_n),1/n}(\mathbf x)$ for BBL families
  16. ...and 11 more sections

Key Result

Theorem A

Let $R$ be a Noetherian local ring with $\dim \left(\mathop{\mathrm{nil}}\nolimits(\hat{R})\right)<\dim R$.

Theorems & Definitions (227)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem A
  • Remark 2
  • Definition 3
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 4
  • ...and 217 more