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Graph rings and ideals: Wolmer Vasconcelos' contributions

Maria Vaz Pinto, Rafael H. Villarreal

TL;DR

This survey assembles Vasconcelos' influential contributions at the intersection of computational methods, Koszul homology, and graph-theoretic monomial ideals. It highlights how edge ideals and graph rings serve as a bridge between combinatorics, polyhedral geometry, and commutative algebra, with key results on normality, symbolic powers, and Rees algebras. The work develops practical tools (LP tests, Normaliz, Macaulay2) and deep structural criteria (Hochster configurations, bowties, MFMC) that connect algebraic properties to graph and clutter combinatorics, enabling precise control of invariants like reg, $a$-invariants, multiplicities, and reductions. The survey emphasizes the unity of algebra and geometry in understanding blowup algebras, normalization, and the containment problem, with broad implications for coding theory, resurgence, and polyhedral methods.

Abstract

This is a survey article featuring some of Wolmer Vasconcelos' contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other areas to the present. We discuss the Vasconcelos' function and the Vasconcelos' number (v-number for short) of graded ideals and their relation to coding theory, and the interplay of Simis and normal monomial ideals with combinatorial optimization problems, blowup algebras, and resurgence theory. The regularity of subrings of normal k-uniform monomial ideals is shown to be a monotone function, and we give a normality criterion for edge ideals of graphs using Ehrhart rings.

Graph rings and ideals: Wolmer Vasconcelos' contributions

TL;DR

This survey assembles Vasconcelos' influential contributions at the intersection of computational methods, Koszul homology, and graph-theoretic monomial ideals. It highlights how edge ideals and graph rings serve as a bridge between combinatorics, polyhedral geometry, and commutative algebra, with key results on normality, symbolic powers, and Rees algebras. The work develops practical tools (LP tests, Normaliz, Macaulay2) and deep structural criteria (Hochster configurations, bowties, MFMC) that connect algebraic properties to graph and clutter combinatorics, enabling precise control of invariants like reg, -invariants, multiplicities, and reductions. The survey emphasizes the unity of algebra and geometry in understanding blowup algebras, normalization, and the containment problem, with broad implications for coding theory, resurgence, and polyhedral methods.

Abstract

This is a survey article featuring some of Wolmer Vasconcelos' contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other areas to the present. We discuss the Vasconcelos' function and the Vasconcelos' number (v-number for short) of graded ideals and their relation to coding theory, and the interplay of Simis and normal monomial ideals with combinatorial optimization problems, blowup algebras, and resurgence theory. The regularity of subrings of normal k-uniform monomial ideals is shown to be a monotone function, and we give a normality criterion for edge ideals of graphs using Ehrhart rings.
Paper Structure (12 sections, 128 theorems, 120 equations, 2 figures)

This paper contains 12 sections, 128 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computational methods
  3. The first Koszul homology module and the conormal module
  4. Origins of graph ideals: Syzygetic and linear type ideals
  5. Edge ideals of graphs and clutters
  6. Rees algebras, symbolic powers and normalizations of graph ideals
  7. Normality of monomial ideals
  8. Normality and Ehrhart rings: $a$-invariant and regularity
  9. Normalization index, reduction numbers, and Noether normalizations
  10. Multiplicities, Hilbert functions and $\mathfrak{m}$-fullness of monomial ideals
  11. Blowup algebras of edge ideals of clutters
  12. The Vasconcelos formula for the generalized Hamming weights

Key Result

Theorem 3.2

(Huneke, Ulrich, Vasconcelos HUV) Let $(S,\mathfrak{m})$ be a regular local ring with infinite residue class field, and let $I\neq\mathfrak{m}$ be a reduced strongly Cohen--Macaulay $S$-ideal such that $\mu(I)= \dim(S)$ and $\mu(I_\mathfrak{p})\leq\max\{{\rm ht}(I),\dim(S_\mathfrak{p})-1\}$ for all

Figures (2)

  • Figure 1: A $5$-cycle and a $3$-cycle joined by a path of length $2$.
  • Figure 2: Region defined by $I=(t_1^6,\,t_2^5,\,t_1^2t_2^2,\,t_1^3t_2)$.

Theorems & Definitions (173)

  • Conjecture 3.1
  • Theorem 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 5.1
  • Definition 5.2
  • Theorem 5.3
  • Proposition 5.4
  • ...and 163 more