Related papers: Single spot ideals of codimension 3 and long Bourb…

This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and…

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$…

Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes…

Cofiniteness of the generalized local cohomology modules Hai(M, N) of two R-modules M and N with respect to an ideal a is studied for some i,s with a specified property. Furthermore, Artinianness of Hbj 0(Hai(M, N)) is investigated by using…

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…

We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $\Theta$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the…

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…

Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim…

Let $(A,\mathfrak{m})$ be an excellent equi-charateristic Gorenstein isolated singularity of dimension $d \geq 2$. Assume the residue field of $A$ is perfect. Let $I$ be any $\mathfrak{m}$-primary ideal. Let $G_I(A) = \bigoplus_{n \geq…

Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$ and let $I$ be an ideal in $R$. Let $A_n(K) = K<X_1,...,X_n, \partial_1,..., \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. By a result due to Lyubeznik the local…

For an ideal $I$ in a regular local ring or a graded ideal $I$ in the polynomial ring we study the limiting behavior of the Betti numbers of S/I^k as k goes to infinity. By Kodiyalam's result it is known that in each homological degree the…

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal…

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We…

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our…

An ideal $I$ of a local Gorenstein ring $(R, \mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \not= \height I.$ Here $H^i_I(R), i \in \mathbb Z,$ denotes the local cohomology of $R$ with…

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I.

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of…

Let $K$ be a field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$ and let $M = H^i_I(R)$ be the $i^{th}$-local cohomology module of $R$ with respect to $I$. Let $c = \ injdim \ M$. We prove that if $P$ is…

Let $R$ be a commutative noetherian ring and $I$ an ideal of $R$. Assume that for all integers $i$ the local cohomology module $H_I^i(R)$ is $I$-cofinite. Suppose that $R_\mathfrak{p}$ is a regular local ring for all prime ideals…

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,…