Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

TL;DR

This paper investigates the algebraic properties of bipartite graph edge ideals, establishing formulas for regularity and depth via associated directed graphs, and exploring conditions where arithmetic rank equals projective dimension.

Contribution

It introduces a novel approach linking directed graph invariants to algebraic properties of bipartite edge ideals and identifies classes where arithmetic rank equals projective dimension.

Findings

Derived formulas for regularity and depth using directed graph invariants.

Established conditions where arithmetic rank equals projective dimension.

Analyzed minimal free resolutions of bipartite edge ideals.

Abstract

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.