Related papers: On some local cohomology modules

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

Let $\fa$ be an ideal of a $d$-dimensional commutative Noetherian ring $R$. In this paper we give some information on some last non-zero local cohomology modules known as top local cohomology modules in particular, $H^{d-1}_{\fa}(R)$.

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It…

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…

Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the 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 $R$ 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 $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal $\fm$, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl…

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology…

We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$ such that $\dim R/\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\lc^{i}_{\fa}(M)$. For an arbitrary ideal $\fa$…

Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in…

We prove an equivalent condition for the existence of a link between prime ideals in terms of the structure of a certain cohomology module. We use this formulation to answer an open question regarding the nature of module extensions over…

Let $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $…

For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local…

Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules…

Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local…

For a finite module $M$ over a local, equicharacteristic ring $(R,m)$, we show that the well-known formula $\cohdim(m,M)=\dim M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also…