The Stanley Depth in the Upper Half of the Koszul Complex

2014/08/25 by Lukas Katthän, Katthän, Lukas, Richard Sieg +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary: 05E40 #Secondary: 13F20

paper · pdf · doi:10.48550/arxiv.1408.5727

openalex publication_date 2014/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R = K[X₁, ..., X_n] be a polynomial ring over some field K. In this paper, we prove that the k-th syzygy module of the residue class field K of R has Stanley depth n-1 for \lfloor n/2 \rfloor ≤ k < n, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than 1. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.

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