Let $R$ be a commutative Noetherian ring and $𝔞$ an ideal of $R$. We introduce the concept of $𝔞$-weakly Laskerian $R$-modules, and we show that if $M$ is an $𝔞$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\mathrm{Ext}}_R^j(R/𝔞,H_{𝔞}^i\left(M\right))$ is $𝔞$-weakly Laskerian for all $i<s$ and all $j$, then for any $𝔞$-weakly Laskerian submodule $X$ of $H_{𝔞}^s\left(M\right)$, the $R$-module ${\mathrm{Hom}}_R(R/𝔞,H_{𝔞}^s\left(M\right)/X)$ is $𝔞$-weakly Laskerian. In particular, the set of associated primes of $H_{𝔞}^s\left(M\right)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an $𝔞$-weakly...

Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of $f_{}^{}\left(N\right)$ relative to in the context of generalized local cohomology modules as $f_{}^{}(M,N):=infi\ge 0\left|\subseteq \surd \right(0{:}_R{H}_{}^i(M,N))$, where M is an R-module. We also show that $f_{}^{}\left(N\right)\le f_{}^{}(M,N)$ for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.

Let $(R,𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\mathrm{hd}(𝔞,M)=n$. We determine the annihilator of the top local homology module ${\mathrm{H}}_n^{𝔞}\left(M\right)$. In fact, we prove that $${\mathrm{Ann}}_R\left({\mathrm{H}}_n^{𝔞}\left(M\right)\right)={\mathrm{Ann}}_R\left(N(𝔞,M)\right),$$ where $N(𝔞,M)$ denotes the smallest submodule of $M$ such that $\mathrm{hd}(𝔞,M/N(𝔞,M\left)\right)<n$. As a consequence, it follows that for a complete local ring $(R,𝔪)$ all associated primes of ${\mathrm{H}}_n^{𝔞}\left(M\right)$ are minimal.

Let $(R,𝔪)$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $𝔭$ is a prime ideal of $R$ such that $dimR/𝔭=1$ and $\left(0{:}_M𝔭\right)$ is not finitely generated and for each $i\ge 2$ the $R$-module ${\mathrm{Ext}}_R^i(M,R/𝔭)$ is of finite length, then the $R$-module ${\mathrm{Ext}}_R^1(M,R/𝔭)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and for all integers $i\ge 0$, the $R$-modules ${\mathrm{Ext}}_R^i(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and...

Let $𝔞$ be an ideal of Noetherian local ring $(R,𝔪)$ and $M$ a finitely generated $R$-module of dimension $d$. In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to $𝔪$. Also we prove that for an arbitrary local ring $(R,𝔪)$ (not necessarily complete), we have ${\mathrm{Att}}_R\left({𝔉}_{𝔞}^d\left(M\right)\right)=\mathrm{Min}\mathrm{V}\left({\mathrm{Ann}}_R{𝔉}_{𝔞}^d\left(M\right)\right).$

Let $ℐ$ be a set of ideals of a commutative Noetherian ring $R$. We use the notion of $ℐ$-closure operation which is a semiprime closure operation on submodules of modules to introduce the class of $ℐ$-Laskerian modules. This enables us to investigate the set of associated prime ideals of certain $ℐ$-closed submodules of local cohomology modules.

The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic invariants such as dimension, codimension and degree. This paper studies a bound on the regularity conjectured by Hoa, and shows this bound and extremal examples in the case of divisors on rational normal scrolls.

Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module $H_{}^i(M,N)$ is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $To{r}_i^R(N,M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $To{r}_i^R(N,H_I^j\left(M\right))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $To{r}_i^R(N,M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...

Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $={M_i}_{i=0}^c$, where c = cd(,M) and $M_i$ denotes the largest submodule of M such that $cd(,M_i)\le i$. Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^c\left(M\right)$, namely $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/M_{c-1})$. As a consequence, there exists an ideal of R such that $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/H{⁰}_{}\left(M\right))$. This generalizes the main results...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\mathrm{Ext}}_R^i(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H_I^i\left(M\right)$ are $I$-cominimax for all $i<t$ and ${\mathrm{Ext}}_R^i(R/I,H_I^t\left(M\right))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\mathrm{Ext}}_R^j(N,H_I^i\left(M\right))$ and ${\mathrm{Tor}}_j^R(N,H_I^i\left(M\right))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i$.