Stanley depth and the lcm-lattice

Abstract

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in the reference [IKMF14]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.Comment: V2: Updated version of V1 named "The behavior of depth and Stanley depth under maps of the lcm-lattice". V3: Sect. 3 rewritten; results reformulated in terms of lcm-lattices, instead of semilattices; new formulation of main results 3.4, 4.5, 4.9 is equivalent to former versions; examples added, references updated. V4: Thm 4.9. contained a typo: we wrote spdim instead of pdim; references update