Graph rings and ideals: Wolmer Vasconcelos' contributions

TL;DR

This survey highlights Wolmer Vasconcelos' influential work in commutative algebra, focusing on concepts like the Vasconcelos' function and number, and their applications to coding theory, combinatorics, and algebraic geometry.

Contribution

It synthesizes Vasconcelos' key contributions and explores their impact on the development of commutative algebra and related fields.

Findings

Vasconcelos' function and v-number relate to coding theory.

Normality of subrings is a monotone function.

A normality criterion for edge ideals of graphs is provided.

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.