Related papers: Homological shift ideals

Principal symmetric ideals were recently introduced by Harada, Seceleanu, and Sega, with a focus on their homological properties. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial…

The class of equidimensional polymatroidal ideals are studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in…

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal…

Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for…

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been…

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. As a key ingredient of our work, we introduce the class of cointerval…

Let I be a conjugation-invariant ideal in the complex polynomial ring with variables z_1,...,z_n and their conjugates. The ideal I has the Quillen property if every real valued, strictly positive polynomial on the real zero set of I in C^n…

It gives a class of $p$-Borel principal ideals of a polynomial algebra over a field $K$ for which the graded Betti numbers do not depend on the characteristic of $K$ and the Koszul homology modules have monomial cyclic basis. Also it shows…

For a non-negative integer $k$, let $\mathrm{HS}_{k}(J(G))$ denote the $k^{\text{th}}$ homological shift ideal of the vertex cover ideal $J(G)$ of a graph $G$. For each $k\geq 2$, we construct a Cohen-Macaulay very well-covered graph $G_k$…

In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all…

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial…

For a monomial ideal $I$ of a polynomial ring $S$, a "polarization" of $I$ is a \textit{squarefree} monomial ideal $J$ of a larger polynomial ring $S'$ such that $S/I$ is a quotient of $S'/J$ by a regular sequence (consisting of degree 1…

Inspired by the notion of K\"onig graphs we introduce graded ideals of K\"onig type with respect to a monomial order $<$. It is shown that if $I$ is of K\"onig type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the…

We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a…

An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional…

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and…

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search…

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S…

In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results…