2015/09/30 by Lukas Katthän
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Betti number #Combinatorics #Commutative Algebra and Its Applications #Conjecture #Counterexample #Dimension (graph theory) #Discrete mathematics #Ideal (ethics) #Mathematical analysis #Mathematics #Monomial #Monomial ideal #Partially ordered set #Philosophy #Polynomial #Polynomial and algebraic computation #Polynomial ring #math.AC #math.CO
paper · pdf · doi:10.1007/s40598-016-0039-5
published as Arnold Mathematical Journal 2(2), 267-276 (2016) · 10 pages. Clarified the proof of 3.6. To appear in the Arnold Mathematical Journal
arxiv created 2016/02/05 · openalex publication_date 2016/02/15 · arxiv updated 2016/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let S be a polynomial ring and let I ⊆ 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 sdepthS/I ≥ depthS/I - 1. Recently, Duval et al. (A non-partitionable Cohen–Macaulay simplicial complex, arXiv:1504.04279 , 2015) found a counterexample to the Stanley conjecture, and their counterexample satisfies sdepthS/I = depthS/I - 1 . So if our conjecture is true, then the conclusion is best possible.