Related papers: Homological shift ideals

A natural candidate for a generating set of the (necessarily prime) defining ideal of an $n$-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of $n$ critical binomials. In a somewhat modified and…

Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of…

We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal…

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the…

Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…

In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the…

Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X, an integer k>=-1 and an upper set in…

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation…

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial…

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…

Theorem 1.2.6 of [ATW20] provides a relatively functorial logarithmic principalization of ideals on relative logarithmic orbifolds $X\to B$ in characteristic 0, relying on a delicate monomialization theorem for Kummer ideals. The paper…

In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the…

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…

We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…

We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in…

Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R $ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and…

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…

In the present paper, motivated by a conjecture of Jahan and Zheng, we prove that componentwise polymatroidal ideals have linear quotients. This solves positively a conjecture of Bandari and Herzog.

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ and $I \subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by…