How to compute the Stanley depth of a module

Abstract

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module M over the polynomial ring K[X1, . . . , Xn]. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog in [Her13]. Moreover, we show that the question whether M has Stanley depth at least r can be reduced to the question whether a certain combinatorially defined polytope P contains a Z-lattice point.