Related papers: The depth of the associated graded ring of ideals …

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two…

Let A be a Cohen-Macaulay local ring of dimension d and I an ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. Let I be a locally complete intersection ideal of analytic deviation one and reduction number at most…

In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and regularity of A and R.

In a local Cohen-Macaulay ring $(A, \mathrm{m})$, we study the Hilbert function of an $\mathrm{m}$-primary ideal $I$ whose reduction number is two. It is a continuous work of the papers of Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With…

Assume that $G$ is a graph with edge ideal $I(G)$. We provide sharp lower bounds for the depth of $I(G)^2$ in terms of the star packing number of $G$.

Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $\rho(I)$ as…

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when…

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for…

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…

Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new…

Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal.…

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at…

A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by…

Let $G$ be a graph on the vertex set $[n]$ and $J_G$ the associated binomial edge ideal in the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate the depth of binomial edge ideals. More precisely, we…

The classical "generalized principal ideal theorems" of Macaulay, Eagon-Northcott, and others give sharp bounds on the heights of determinantal ideals in arbitrary rings. But in regular local rings (or graded polynomial rings) these are far…

We prove that the arithmetic degree of a graded or local ring is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideals $I$ in $A$. In particular, if $Spec (A)$ is equidimensional and has an…

We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with…

We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity…

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…

The aim of this note is to study the class of one dimensional Cohen-Macaulay local rings, $(R, \mathfrak{m})$ say, possessing a canonical ideal $K$ which is a reduction of $\mathfrak{m}$. We call $R$ to have canonical reduction $K$. We show…