Related papers: A note on monomial ideals

We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property.

Consider the polynomial ring $R_n = k[x_1,...,x_n]$, where $k$ is a field. Let $m = (x_1,...,x_n)$ and $I$ be an $m$-primary monomial ideal in $R$. We consider the problem of determining whether such ideals are in the Gorenstein liasion…

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial…

When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…

Let I be a monomial ideal of height c in a polynomial ring S over a field k. If I is not generated by a regular sequence, then we show that the sum of the betti numbers of S/I is at least 2^c + 2^{c-1} and characterize when equality holds.…

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…

The depth of squarefree powers of a squarefree monomial ideal is introduced. Let $I$ be a squarefree monomial ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$. The $k$-th squarefree power $I^{[k]}$ of $I$ is the ideal of $S$ generated by…

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…

We study the irreducible representations of simple algebraic groups in which some non-central semisimple element has at most one eigenvalue of multiplicity greater than 1. We bound the multiplicity of this eigenvalue in terms of the rank of…

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose…

We prove a tight lower bound on the Betti numbers of tree and forest ideals and a tight upper bound on certain graded Betti numbers of squarefree monomial ideals.

Let M in k[x,y] be a monomial ideal M=(m_1,m_2,...,m_r), where the m_i are a minimal generating set of M. We construct an explicit free resolution of k over S=k[x,y]/M for all monomial ideals M, and provide recursive formulas for the Betti…

We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is…

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 squarefree monomial ideal is called an $f$-ideal if its Stanley-Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables…

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex…

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve…

In this article we study inequalities of ideal norms. We prove that in a subring $R$ of a number field every ideal can be generated by at most $3$ elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for every pair of…

Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $\nu(I_p)$ of $I$ that are in degree $p$ is bounded…

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…