相关论文: Face numbers of sequentially Cohen-Macaulay comple…

This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we…

A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Bjorner-Frankl-Stanley (sufficiency) that characterizes…

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a…

The starting point is the class of the following simplicial complexes $\Delta$ with 2-linear resolutions. The facets of $\Delta$ are $F_1,\ldots,F_n$, and we demand that for each $i$ $F_i\cap (F_1\cup \cdots\cup F_{i-1}\cup…

For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $\Delta$ such that ${\rm dim}(\Delta)={\rm dim}(\Delta^c)$ and $\mathcal{F}(\Delta)\cap…

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and…

The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face…

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is…

We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear…

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and…

Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the…

A triangulation of a simplicial complex $\Delta$ is called uniform if the $f$-vector of its restriction to a face of $\Delta$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform…

Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$,…

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is `nonsingular', i.e., has the homology of a wedge of spheres of the…

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The…

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at…

Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting…

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $\Delta$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in…

We show that if a $d$-dimensional Cohen-Macaulay complex is, in a certain sense, sufficiently "close" to being balanced, then there is a $d$-dimensional balanced Cohen-Macaulay complex having the same $f$-vector. This in turn provides some…