Related papers: Monomial ideals whose depth function has any given…

In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and…

Let $I\supsetneq J$ be two square free monomial ideals of a polynomial algebra over a field generated in degree $\geq 1$, resp. $\geq 2$ . Almost always when $I$ contains precisely one variable, the other generators having degrees $\geq 2$,…

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of…

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real…

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S=k[x_1,x_2,...,x_n]; this includes the case of…

Given a monomial ideal in a polynomial ring over a field, we define the LCM-dual of the given ideal. We show good properties of LCM-duals. Including the isomorphism between the special fiber of LCM-dual and the special fiber of given…

If $J\subset I$ are two monomials ideals, we give a practical upper bound for the Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$. Also, we compute the quasi-depth of several classes of square free monomial ideals.…

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…

We count the numbers of associated primes of powers of ideals as defined by Bandari, Hibi, and Herzog in 2014. We generalize those ideals to monomial ideals $\operatorname{BHH}(m,r,s)$ for $r \ge 2$, $m$, $s \ge 1$; we establish partially…

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is generated by three monomials of degrees $d$. If the Stanley depth of $I/J$ is…

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.

The Hilbert series of local cohomologies for monomial ideals, which are not necessarily square-free, is established. As applications, we give a sharp lower bound of the non-vanishing degree of local cohomologies and also a sharp lower bound…

Let $I$ be a monomial squarefree ideal of a polynomial ring $S$ over a field $K$ such that the sum of every three different of its minimal prime ideals is the maximal ideal of $S$, or more general a constant ideal. We associate to $I$ a…

We show that $\depth(S/I)=0$ if and only if $\sdepth(S/I)=0$, where $I\subset S=K[x_1,...,x_n]$ is a monomial ideal. We give an algorithm to compute the Stanley depth of $S/I$, where $I\subset S=K[x_1,x_2,x_3]$ is a monomial ideal. Also, we…

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by three monomials of degrees $d$ and a set of monomials of…

Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$ square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not principal) than the number of square free monomials of $I$ of degree $d+1$, then…

Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of…

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.

We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal…

A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by…