相关论文: Alexander Duality for Monomial Ideals and Their Re…

Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice the Alexander dual is computed.

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…

A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of…

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary…

The free resolution and the Alexander dual of squarefree monomial ideals associated with certain subsets of distributive lattices are studied.

In this note, we study Serre's property $(S_i)$, and its relation to Alexander duality for monomial ideals in a polynomial ring over a field. We describe ideals that define the non-Cohen-Macaulay- and the non-$(S_i)$-loci of finitely…

Given a simplicial complex we associate to it a squarefree monomial ideal which we call the face ideal of the simplicial complex, and show that it has linear quotients. It turns out that its Alexander dual is a whisker complex. We apply…

For a pair $(P,Q)$ of finite posets the generators of the ideal $L(P,Q)$ correspond bijectively to the isotone maps from $P$ to $Q$. In this note we determine all pairs $(P,Q)$ for which the Alexander dual of $L(P,Q)$ coincides with…

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in…

We provide a simple method to compute the Betti numbers if the Stanley-Reisner ideal of a simplicial tree and its Alexander dual.

We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to…

The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.

Linear resolutions and the stronger notion of linear quotients are important properties of monomial ideals. In this paper, we fully characterize linear quotients in terms of the lcm-lattice of monomial ideals. We also formulate an analogous…

We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial…

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of…

We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…

In this paper, basic properties of monomial difference ideals are studied. We prove the finitely generated property of well-mixed difference ideals generated by monomials. Furthermore, a finite prime decomposition of radical well-mixed…

Let $\Delta$ be a triangulated homology ball whose boundary complex is $\partial\Delta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $\Delta$, $\mathbb F[\Delta]$, is isomorphic to the…

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of…