Related papers: Nonincreasing depth functions of monomial ideals

Let $I\supsetneq J$ be two monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . We study when the Stanley Conjecture holds for $I/J$ using the recent result of \cite{IKM} concerning the…

Let $\mathcal{I}$ be an ideal on $\omega$. For $f,g\in\omega^\omega$ we write $f \leq_{\mathcal{I}} g$ if $f(n) \leq g(n)$ for all $n\in\omega\setminus A$ with some $A\in\mathcal{I}$. Moreover, we denote…

$(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide…

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$…

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence…

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $K$ and $I$ be a matroidal ideal of degree $d$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integers $l$ and $k$, for which $\Ass(I^l)$ and $\depth(R/I^k)$…

Let $I\subset S$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$, and let $\text{v}(I)$ be the $\text{v}$-number of $I$. In previous work, we showed that for any graded ideal $I\subset S$…

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector, where $\delta_{\mathcal{F}}(I)$ ($\delta_{\mathcal{N}}(I)$, respectively) is the facet…

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…

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…

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that…

Let $K$ be a field of characteristic zero, let $I \subset S = K[x_1,\dots,x_n]$ be a homogeneous ideal, and let $\partial(I)$ be its gradient ideal. We study the relationship between $\mathrm{reg}\,I$ and $\mathrm{reg}\,\partial(I)$. While…

We extend a result of Minh and Trung to get criteria for $\depth I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial complexes…

In this paper we study squarefree monomial ideals which have constant depth functions. Edge ideals, matroidal ideals and facet ideals of pure simplicial forests connected in codimension one with this property are classified.

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $p$ is the number of its…

Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let ${\rm sgn}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R/I$ is the depth of $R/{\rm sgn}(I)$, and…

Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…