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Symbolic powers and integral closures via extremal ideals

Trung Chau, Art Duval, Sara Faridi, Thiago Holleben, Susan Morey, Liana Şega

TL;DR

The paper develops a universal framework to study powers and symbolic powers of square-free monomial ideals by embedding them into the powers and symbolic powers of a single highly symmetric extremal ideal $\\mathcal{E}_q$ via a ring homomorphism $\\psi_I$. It proves that $\\overline{I^r} =\\psi_I(\\overline{\\mathcal{E}_q^r})R$ and $I^{(r)} =\\psi_I(\\mathcal{E_q^{(r)}})R$, providing test elements $g(r,q)$ and $h(r,q)$ to decide equality with ordinary powers and to detect non-equality in many cases. The framework yields explicit descriptions and generators for $I^{(2)}$, sharp bounds for integral defects, symbolic defects, and resurgences, and translates many computations into linear programming problems on $\\mathcal{E}_q$, while preserving LCM structures to transfer Betti-number information from the extremal ideal to any $I$. This reduces infinitely many algebraic questions to a finite, symmetric model, enabling concrete algorithmic and geometric approaches to longstanding containment and homological problems.

Abstract

This paper demonstrates that extremal ideals can be used to great effect to compute integral closures of powers and symbolic powers of square-free monomial ideals. We show that the generators of these powers are images of the generators of the corresponding powers of extremal ideals under a specific ring homomorphism. Extremal ideals provide sharp bounds for a variety of invariants widely studied in the literature, including resurgence, asymptotic resurgence, and symbolic defect, as well as Betti numbers of symbolic powers and of integral closures of powers of square-free monomial ideals. When restricted to the class of extremal ideals, algebraic computations are reduced to problems of discrete geometry and linear programming, allowing the use of a wide variety of techniques. As a result, in situations where computations are feasible for extremal ideals, we provide concrete sharp bounds for many of these invariants. Our methods reduce finding homological invariants and algebraic constructions for infinitely many ideals to computations for a single highly symmetric ideal, based solely on the number of generators.

Symbolic powers and integral closures via extremal ideals

TL;DR

The paper develops a universal framework to study powers and symbolic powers of square-free monomial ideals by embedding them into the powers and symbolic powers of a single highly symmetric extremal ideal via a ring homomorphism . It proves that and , providing test elements and to decide equality with ordinary powers and to detect non-equality in many cases. The framework yields explicit descriptions and generators for , sharp bounds for integral defects, symbolic defects, and resurgences, and translates many computations into linear programming problems on , while preserving LCM structures to transfer Betti-number information from the extremal ideal to any . This reduces infinitely many algebraic questions to a finite, symmetric model, enabling concrete algorithmic and geometric approaches to longstanding containment and homological problems.

Abstract

This paper demonstrates that extremal ideals can be used to great effect to compute integral closures of powers and symbolic powers of square-free monomial ideals. We show that the generators of these powers are images of the generators of the corresponding powers of extremal ideals under a specific ring homomorphism. Extremal ideals provide sharp bounds for a variety of invariants widely studied in the literature, including resurgence, asymptotic resurgence, and symbolic defect, as well as Betti numbers of symbolic powers and of integral closures of powers of square-free monomial ideals. When restricted to the class of extremal ideals, algebraic computations are reduced to problems of discrete geometry and linear programming, allowing the use of a wide variety of techniques. As a result, in situations where computations are feasible for extremal ideals, we provide concrete sharp bounds for many of these invariants. Our methods reduce finding homological invariants and algebraic constructions for infinitely many ideals to computations for a single highly symmetric ideal, based solely on the number of generators.
Paper Structure (6 sections, 34 theorems, 152 equations, 1 figure)

This paper contains 6 sections, 34 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The map $\psi_I$ and extremal ideals
  3. Integral Closures via $\psi_I$
  4. Symbolic Powers and $\psi_I$
  5. The Second Symbolic Defect
  6. Free resolutions and Betti numbers via $\psi_I$

Key Result

Theorem 1

[theorem]t:intro If $I$ is an ideal minimally generated by $q \ge 1$ square-free monomials in a polynomial ring $R$, and $r$ is a positive integer, then

Figures (1)

  • Figure 1: Dots and points above the solid line represent pairs $(x, y)$ such that $I^{(x)} \subseteq \overline{I^y}$ for every square-free monomial ideal generated by $q$ elements, and an $\times$ or point below the dotted line means ${\mathcal{E}_q}^{(x)} \not\subseteq \overline{{\mathcal{E}_q}^y}$.

Theorems & Definitions (74)

  • Theorem 1: \ref{['thm:integral-closure-extremal', 'thm:symbolic-extremal']}
  • Theorem 2: \ref{['c:integralclosurepsi', 't:general-I-ic', 'thm:integral-defect']}
  • Theorem 3: \ref{['c:symbolic=ordinary', 'c:E2-symbolic', 't:general-I-symbolic']}
  • Theorem 4: \ref{['t:upperbound', 'thm:integral-defect', 'c:sdefectbound', 'c:rho', 't:general-I-symbolic', 'p:resurgence-cases']}
  • Definition 2.1: Extremal ideals
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5: The ring homomorphism $\psi_I$
  • Example 2.6
  • ...and 64 more