Stanley decompositions in localized polynomial rings

Abstract

We introduce the concept of Stanley decompositions in the localized polynomial ring S f where f is a product of variables, and we show that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial ideals \({J \subset I \subset S_f}\) the number of maximal Stanley spaces in a Stanley decomposition of I/J is an invariant of I/J. For the proof of this result we introduce Hilbert series for \({\mathbb{Z}^n}\)-graded K-vector spaces.