THE STANLEY DEPTH IN THE UPPER HALF OF THE KOSZUL COMPLEX

Abstract

Let R=K[X-1, ..., X-n] 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 left perpendicularn/2right perpendicular <= 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.