Betti posets and the Stanley depth

TL;DR

This paper explores the conjecture that the Betti poset of a monomial ideal determines its Stanley projective dimension, linking it to the Stanley conjecture and providing insights into the bounds of Stanley depth.

Contribution

It introduces a conjecture connecting Betti posets to Stanley projective dimension and demonstrates its implications for the Stanley conjecture and depth bounds.

Findings

The conjecture implies the Stanley conjecture for monomial ideals.

It shows that epth S/I \u2265 epth S/I - 1 if the conjecture holds.

Counterexamples to the Stanley conjecture satisfy the equality epth S/I = epth S/I - 1.

Abstract

Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is best possible.