Skeletons of monomial ideals

TL;DR

This paper introduces the concept of skeletons for monomial ideals, enabling depth computation and providing new insights into Stanley's conjecture and related conjectures in commutative algebra.

Contribution

It defines skeletons of monomial ideals, relates them to depth, and connects Stanley's conjecture to Cohen--Macaulay cases and Soleyman-Jahan's conjecture.

Findings

Skeletons of monomial ideals can be used to compute depth.

Stanley's conjecture holds if true for Cohen--Macaulay cases.

Proving Soleyman-Jahan's conjecture reduces to ideals with linear resolution.

Abstract

In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of $S/I$ holds provided it holds whenever $S/I$ is Cohen--Macaulay. We also discuss a conjecture of Soleyman-Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution.