K-Clean Monomial Ideals

arXiv: Commutative Algebra, 2017

We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.

2021

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 ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in squarefree case, an upper bound for the regularity of such ideals. Introduction A simplicial complex ∆ on the vertex set V is a collection of subsetes of V , such that ∪F∈∆F = V and ∆ is closed under the operation of taking subsets. The elements of ∆ are called faces. The maximal faces of ∆, under inclusion, are called the facets of ∆ and a simplicial complex with facets F1, . . . , Fm is often denoted by 〈F1, . . . , Fm〉. A simplicial complex with only one facet is called a simplex. For any face F ∈ ∆ the link and the deletion of F in ∆ are defined as link∆(F ) = {G ∈ ∆: G ∩ F = ∅ and ...

2011

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.

Annals of Combinatorics, 2004

Given a simplicial complex, it is easy to construct a generic deformation of its Stanley-Reisner ideal. The main question under investigation in this paper is how to characterize the simplicial complexes such that their Stanley-Reisner ideals have Cohen-Macaulay generic deformations. Algorithms are presented to construct such deformations for matroid complexes, shifted complexes, and tree complexes.

2010

Monomial ideals are at the intersection between commutative algebra, combinatorics, and algebraic geometry, being in the center of many important problems in polynomial rings. Their study developed extensively especially in the last decades when it became a standard technique to get information about the algebraic and homological invariants of polynomial ideals by passing to initial monomial ideals. By using a standard procedure, one may also use specific combinatorial techniques to study invariants of the so called squarefree monomial ideals. All these new developments led to a spectacular progress in the new branch of commutative algebra, which is usually called combinatorial commutative algebra. In this book, several classes of monomial ideals are studied by using algebraic and combinatorial techniques. Special attention is given to lexsegment ideals whose properties concerning resolutions and invariants are presented in detail, and to constructible ideals, for which, deep connec...

Journal of Pure and Applied Algebra, 2015

The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of n + 1 general hyperplanes in the n-dimensional projective space over an arbitrary field.

Manuscripta Mathematica, 2016

Let ∆ be a simplicial complex. We study the expansions of ∆ mainly to see how the algebraic and combinatorial properties of ∆ and its expansions are related to each other. It is shown that ∆ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or k-decomposable, if and only if an arbitrary expansion of ∆ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of ∆ and those of their expansions are compared.

2021

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I = I ∆ is the Stanley-Reisner ideal of a simplicial complex ∆, then reg(I (n)) δ(n − 1) + b for all n 1, where δ = lim n→∞ reg(I (n))/n, and b = max{reg(I Γ) | Γ is a subcomplex of ∆ with F (Γ) ⊆ F (∆)}. This bound is sharp for any n. When I = I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I (n)) 2n + order-match(G) − 1, where order-match(G) is the ordered matching number of G.

Communications in Algebra, 2012

In this paper we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen-Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen-Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.

2021

Let R = k[x1, . . . , xn] be the polynomial ring in n variables over a field k and let I be a monomial ideal of R. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.