Sliding functor and polarization functor for multigraded modules

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.