Related papers: Shellability and the strong gcd-condition

For a group $G$ (of type $F$) acting properly on a coarse Poincar\'{e} duality space $X$, Kapovich-Kleiner introduced a coarse version of Alexander duality between $G$ and its complement in $X$. More precisely, the cohomology of $G$ with…

Let $\Delta$ be a simplicial complex. We study the expansions of $\Delta$ mainly to see how the algebraic and combinatorial properties of $\Delta$ and its expansions are related to each other. It is shown that $\Delta$ is Cohen-Macaulay,…

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is…

We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…

Suppose a group $G$ acts properly on a simplicial complex $\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\Gamma$ must be a subcomplex of…

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is…

We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal…

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for…

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…

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a…

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…

Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new…

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground…

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no…

We prove G\"ortz's combinatorial conjecture \cite{Go01} on dual shellability of admissible sets in Iwahori-Weyl groups, proving that the augmented admissible set $\widehat{\mathrm{Adm}}(\mu)$ is dual shellable for any dominant coweight…

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of…

We study combinatorial and algebraic properties of $t$-clique-free complexes, a family of simplicial complexes associated with finite simple graphs that generalize the classical independence complex. For a graph $G$ and an integer $t \ge…

We recall a numerical criteria for Cohen--Macaulayness related to system of parameters, and introduce monomial ideals of K\"onig type which include the edge ideals of K\"onig graphs. We show that a monomial ideal is of K\"onig type if and…

Let $\Phi$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $\Delta^m (\Phi)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $\Delta^m (\Phi)$ is…

We describe the simplicial complex $\Delta$ such that the initial ideal of $J_G$ is the Stanley-Reisner ideal of $\Delta$. By $\Delta$ we show that if $J_G$ is $(S_2)$ then $G$ is accessible. We also characterize all accessible blocks with…