Related papers: The face ideal of a simplicial complex
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 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…
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.
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…
Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is…
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…
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…
Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call…
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…
Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure…
We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets)…
The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner…
The free resolution and the Alexander dual of squarefree monomial ideals associated with certain subsets of distributive lattices are studied.
When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…
We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except…
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…
A numerical characterization is given of the so-called h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result characterizes the number of faces of various dimensions and codimensions in such a complex, generalizing the…
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…
A square-free monomial ideal $I$ of $k[x_1,\ldots,x_n]$ is said to be an $f$-ideal if the facet complex and non-face complex associated with $I$ have the same $f$-vector. We show that $I$ is an $f$-ideal if and only if its Newton…
We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and…