Stanley decompositions in localized polynomial rings

TL;DR

This paper extends the concept of Stanley decompositions to localized polynomial rings, demonstrating invariance of Stanley depth under localization and introducing Hilbert series for graded vector spaces.

Contribution

It introduces Stanley decompositions in localized polynomial rings and proves the invariance of Stanley depth and the number of Stanley spaces as an invariant of quotient modules.

Findings

Stanley depth does not decrease upon localization.

Number of Stanley spaces in a decomposition is an invariant of the quotient.

Hilbert series for $bZ^n$-graded vector spaces are introduced.

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 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 $\ZZ^n$-graded $K$-vector spaces.