On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley--Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley--Reisner- and affine monoid algebras. We consider (non-pure) shellable fan's and the Cohen--Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.Comment: 22 pages; Revised version of the pape

The behavior of Stanley depth under polarization

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\ {I^p/J^p}$, where$\mathrm{sdepth}\ I/J$denotes the Stanley depth and$I^p$denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form$I/J$) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form$R/I$ and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.Comment: Version 2: several proofs were clarified and a minor result was added. Version 3: further improvements based on several readers feedbac

Lcm-lattices and Stanley depth: a first computational approach

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that $\mathrm{depth}_S S/I = \mathrm{sdepth}_S S/I < \mathrm{sdepth}_S I$. This shows that the Stanley conjecture is true for $S/I$ and $I$, if $I$ can be generated by at most 5 monomials. The result also brings additional computational evidence for a conjecture made by Herzog.Comment: To appear in Experimental Math. ArXiv admin note: text overlap with arXiv:1405.360

Stanley depth and the lcm-lattice

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in the reference [IKMF14]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.Comment: V2: Updated version of V1 named "The behavior of depth and Stanley depth under maps of the lcm-lattice". V3: Sect. 3 rewritten; results reformulated in terms of lcm-lattices, instead of semilattices; new formulation of main results 3.4, 4.5, 4.9 is equivalent to former versions; examples added, references updated. V4: Thm 4.9. contained a typo: we wrote spdim instead of pdim; references update