Sliding functor and polarization functor for multigraded modules

TL;DR

This paper introduces sliding and polarization functors for multigraded modules, preserving key invariants like depth and Stanley depth, and applies these concepts to study simplicial spheres of Bier-Murai type.

Contribution

It defines new exact endofunctors for multigraded modules and offers a novel approach to polarization, extending previous work by Bruns-Herzog and Sbarra.

Findings

Sliding functors preserve depth and Stanley depth.

Polarization functor is defined with a new approach.

Application to simplicial spheres of Bier-Murai type.

Abstract

We define "sliding functors", which are exact endofunctors of the category of multi-graded modules over a polynomial ring. They preserve several invariants of modules, especially the (usual) depth and Stanley depth. In a similar way, we can also define the "polarization functor". While this idea has appeared in papers of Bruns-Herzog and Sbarra, we give slightly different approach. Keeping these functors in mind, we treat simplicial spheres of Bier-Murai type.