Related papers: Cofiniteness and associated primes of local cohomo…

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$…

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study…

The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete…

This paper centers around Artinianness of the local cohomology of $ZD$-modules. Let $\fa$ be an ideal of a commutative Noetherian ring $R$. The notion of $\fa$-relative Goldie dimension of an $R$-module $M$, as a generalization of that of…

Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak…

Let R be a local complete ring. For an R-module M the canonical ring map R\to End_R(M) is in general neither injective nor surjective; we show that it is bijective for every local cohomology module M := H^h_I(R) if H^l_I(R) = 0 for every…

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)]…

In this paper, at first, we show that for a ramified regular local ring $S$, which is an Eisenstein extension of an unramified regular local ring $R$, when an ideal $I$ of $S$ is extended from an ideal $J$ of $R$, the punctured spectrum of…

We consider the (graded) Matlis dual $\DD(M)$ of a graded $\D$-module $M$ over the polynomial ring $R = k[x_1, \ldots, x_n]$ ($k$ is a field of characteristic zero), and show that it can be given a structure of $\D$-module in such a way…

Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is…

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $n$ be a non-negative integer. In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\frak a}^i(M)$ such that $\dim{\rm Supp…

Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that 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 $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning…

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…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c =…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global…

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$.…

Let $M$ be a finitely generated bigraded module over the standard bigraded polynomial ring $S=K[x_1,...,x_m, y_1,...,y_n]$, and let $Q=(y_1,...,y_n)$. The local cohomology modules $H^k_Q(M)$ are naturally bigraded, and the components…