The combinatorial calculation of algebraic invariants of a monomial ideal

TL;DR

This paper introduces a combinatorial approach to analyze algebraic invariants of monomial ideals, providing new methods to determine Lyubeznik properties and resolutions.

Contribution

It develops a combinatorial Lyubeznik resolution, characterizes its minimality, and classifies Lyubeznik symbols using combinatorial criteria.

Findings

A combinatorial method to identify Lyubeznik ideals.

A classification of Lyubeznik symbols via combinatorial criteria.

A combinatorial formula for projective dimension, Lyubeznik length, and arithmetical rank.

Abstract

We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a Lyubeznik ideal. Furthermore, the minimality of the Lyubeznik resolution is characterized and we classify all the Lyubeznik symbols using combinatorial criteria. We get a combinatorial expression for the projective dimension, the length of Lyubeznik, and the arithmetical rank of a monomial ideal. We define the Lyubeznik totally ideals as those ideals that yield a minimal free resolution under any total order. Finally, we present that for a family of graphics, that their edge ideals are Lyubeznik totally ideals.