Betti posets and the Stanley depth

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 $\mathrm{sdepth}S/I=\mathrm{depth}S/I-1$. So if our conjecture is true, then the conclusion is best possible.

10 pages. Clarified the proof of 3.6. To appear in the Arnold Mathematical Journal

References in corpus (4)

• A non-partitionable Cohen-Macaulay simplicial complex
• Stanley depth and the lcm-lattice
• Modules over categories and Betti posets of monomial ideals
• The Betti poset in monomial resolutions

Cited by in corpus (2)

• A non-partitionable Cohen-Macaulay simplicial complex
• On the Stanley depth of powers of some classes of monomial ideals