Related papers: Depth functions of powers of homogeneous ideals

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given…

We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals.

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$.…

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…

We study the limit and initial behavior of the numerical function $f(k)=\depth S/I^k$. General properties of this function together with concrete examples arising from combinatorics are discussed.

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I\subsetneq J \subset S$, we present a new method to compute the Hilbert depth of $J/I$. As an application, we show that if $u\in S$…

Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that,…

In this paper, we prove that if $P$ is a homogeneous prime ideal inside a standard graded polynomial ring $S$ with $\dim(S/P)=d$, and for $s \leq d$, adjoining $s$ general linear forms to the prime ideal changes the $(d-s)$-th Hilbert…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and…

Let S=K[x_1,x_2,...,x_n] be a polynomial ring in n variables over a field K. Stanley's conjecture holds for the modules I and S/I, when I is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $I$ is a squarefree monomial ideal of $S$. For every integer $k\geq 1$, we denote the $k$-th squarefree…

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\…

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of…

The Hilbert depth of a module M is the maximum depth that occurs among all modules with the same Hilbert function as M. In this note we compute the Hilbert depths of the powers of the irrelevant maximal ideal in a standard graded polynomial…

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 $J=J(G)$ is its cover ideal. We…

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\leq 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\geq 1$ and we disprove a motivation question that was appeared in…

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th symbolic…

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds…

We give different bounds for the Stanley depth of a monomial ideal $I$ of a polynomial algebra $S$ over a field $K$. For example we show that the Stanley depth of $I$ is less or equal with the Stanley depth of any prime ideal associated to…

Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of…