Sliding Functor and Polarization Functor for Multigraded Modules

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.

Depth and Stanley Depth of the Canonical Form of a factor of monomial ideals

“…We owe thanks to Y.-H. Shen who noticed our results in a previous arXiv version and showed us the papers of Okazaki and Yanagawa [7] and [13], because they are strongly connected with our topic. Indeed Proposition 1 and Corollary 1 follow from [7, Theorem 5.2] (see also [7,Section 2,3]).…”

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

“…Yanagawa [41] has constructed the polarization functor pol a from the category of positively a-determined modules to the category of squarefree modules. It is well-known that (see [41,Section 4]) if I is a monomial ideal of R, then The following statement is a corollary to the above proposition.…”

Properties of Lyubeznik numbers under localization and polarization

“…For more detail about local cohomology with support in monomial ideals, we refer to [ÀMGLZA03, ÀM04, ÀMV14, Mus00, Ter99, Yan00, Yan01]. We also refer to [Pee11,Far06,Yan12] for details about polarization.…”

“…An essentially same functor had been introduced by Sbarra [Sba01], but we use the convention of[Yan12] here.…”