The Stanley Depth in the Upper Half of the Koszul Complex

Abstract

Let R = K[X1, ⋅c, Xn] be a polynomial ring over some field K. In this article, we prove that the kth syzygy module of the residue class field K of R has Stanley depth n − 1 for ⌊n/2⌋ ≤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.