Related papers: The projective dimension of sequentially Cohen-Mac…
We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the…
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…
We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen-Macaulay property. Indeed, we study the class of monomial ideals $I$, whose projective dimension is stable under monomial…
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the…
We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially…
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is…
We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…
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…
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also…
We give a structure theorem for Cohen Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the…
Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a…
In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.
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)…
Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex) weighted oriented chordal and simplicial graphs, a more general class of monomial ideals.…
Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal…
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 give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.
We generalize some known results on the relation between the cohomological and projective dimension. Then we examine the set-theoretically Cohen-Macaulay ideals to find some cohomological characterization of these kind of ideals.
We study the class of squarefree principal vector-spread Borel ideals. We compute the minimal primary decomposition of these ideals and thereby we prove that they are sequentially Cohen-Macaulay. As the final conclusion of our results, we…
The principal result is a primary decomposition of ideals generated by the (2x2)-subpermanents of a generic matrix. These permanental ideals almost always have embedded components and their minimal primes are of three distinct heights. Thus…