Alexander duality and Stanley depth of multigraded modules

Abstract

We apply Miller's theory on multigraded modules over a polynomial ring to the study of the Stanley depth of these modules. Several tools for Stanley's conjecture are developed, and a few partial answers are given. For example, we show that taking the Alexander duality twice (but with different "centers") is useful for this subject. Generalizing a result of Apel, we prove that Stanley's conjecture holds for the quotient by a cogeneric monomial ideal.

Stanley depth and the lcm-lattice

“…Let K be a field, S a N n -graded K-algebra and M a finitely generated Z n -graded Smodule. The Stanley depth of M, denoted sdepth S M, is a combinatorial invariant of M which was introduced by Apel in [Ape03a] and has attracted the attention of many researchers [HP06, HVZ09, BHK + 10, OY11,DGKM16]. We refer the reader to the survey of Herzog [Her13] for an introduction to this subject.…”

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]).…”

“…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]). However, our proofs of Lemma 2 and Corollary 1 are completely different from those appeared in the quoted papers and we keep them for the sake of our completeness.…”

“…The depth and the Stanley depth of such a factor is invariant under taking the canonical form. This can be seen using a result of Okazaki and Yanagawa [7]. In the case of depth we present in this paper a different proof.…”

Monomial Ideals, Computations and Applications

“…Then sreg.I / C sdepth.KOE _ / D n and sdepth.I / C sreg.KOE _ / D n:A remarkable generalization of this result due to Okazaki and Yanagawa can be found in[155, Theorem 3.13]. Then sreg.I / C sdepth.KOE _ / D n and sdepth.I / C sreg.KOE _ / D n:A remarkable generalization of this result due to Okazaki and Yanagawa can be found in[155, Theorem 3.13].…”