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Rational symbolic powers of ideals

Souvik Dey, Tai Huy Ha, Dipendranath Mahato

TL;DR

The paper introduces rational symbolic powers $\overline{I^{(u)}}$ for ideals in regular local or graded rings and develops a robust parallel to classical symbolic powers, including a primary-decomposition formula for radical ideals, a concrete membership test, and a rational Zariski–Nagata theorem in characteristic $0$. It then connects rational symbolic powers to saturated powers, proving that binomial expansion formulas extend from ordinary to saturated and subsequently to rational symbolic powers in mixed-sum settings. In the monomial case, a convex-geometric description via the symbolic polyhedron yields asymptotic stability of the rational-symbolic filtration and guarantees the existence of limits for asymptotic invariants like regularity and depth, with analogous rational-length limits under suitable hypotheses. Collectively, these results extend core symbolic-power theory to the rational setting and provide tools for understanding asymptotics and convex-geometric structure, especially for squarefree monomial ideals.

Abstract

We introduce and study rational symbolic powers of an ideal in a regular local or graded ring. We show that important results for symbolic powers, such as the Zariski--Nagata theorem or the ideal containment, extend naturally to rational symbolic powers. We investigate the binomial expansion formula for rational symbolic powers of mixed sums of ideals. For a squarefree monomial ideal, we give a convex-geometric description of its rational symbolic powers. We also show that the filtration of rational symbolic powers is asymptotically stable and, as a consequence, deduce that the asymptotic regularity and asymptotic depth for this filtration exist.

Rational symbolic powers of ideals

TL;DR

The paper introduces rational symbolic powers for ideals in regular local or graded rings and develops a robust parallel to classical symbolic powers, including a primary-decomposition formula for radical ideals, a concrete membership test, and a rational Zariski–Nagata theorem in characteristic . It then connects rational symbolic powers to saturated powers, proving that binomial expansion formulas extend from ordinary to saturated and subsequently to rational symbolic powers in mixed-sum settings. In the monomial case, a convex-geometric description via the symbolic polyhedron yields asymptotic stability of the rational-symbolic filtration and guarantees the existence of limits for asymptotic invariants like regularity and depth, with analogous rational-length limits under suitable hypotheses. Collectively, these results extend core symbolic-power theory to the rational setting and provide tools for understanding asymptotics and convex-geometric structure, especially for squarefree monomial ideals.

Abstract

We introduce and study rational symbolic powers of an ideal in a regular local or graded ring. We show that important results for symbolic powers, such as the Zariski--Nagata theorem or the ideal containment, extend naturally to rational symbolic powers. We investigate the binomial expansion formula for rational symbolic powers of mixed sums of ideals. For a squarefree monomial ideal, we give a convex-geometric description of its rational symbolic powers. We also show that the filtration of rational symbolic powers is asymptotically stable and, as a consequence, deduce that the asymptotic regularity and asymptotic depth for this filtration exist.

Paper Structure

This paper contains 4 sections, 16 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Rational symbolic powers of radical ideals
  3. Rational symbolic powers and saturated powers
  4. Rational symoblic powers of monomial ideals

Key Result

Theorem 2.7

Let $I \subseteq R$ be an ideal with $\mathop{\mathrm{Ass}}\nolimits(R/I)=\mathop{\mathrm{Min}}\nolimits(R/I)=\{{\mathfrak p}_1,\ldots,{\mathfrak p}_n\}$, and let $Q_i$ be the ${\mathfrak p}_i$-primary component of $I$. Then, for all $u\in \mathbb Q_{>0}$, we have $\overline{I^u}R_{{\mathfrak p}_i}\ is the irredundant primary decomposition of $\overline{I^{(u)}}$, and

Theorems & Definitions (54)

  • Definition 2.1: sh
  • Definition 2.2: Rational Symbolic Powers
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 44 more