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1. CMB 2011 (vol 55 pp. 81)
Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals Let $\mathfrak a$ be an ideal of a commutative Noetherian
ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our
main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized
local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite.
Keywords:cofinite modules, generalized local cohomology modules, local cohomology modules Categories:13D45, 13E05, 13E10 |
2. CMB 2011 (vol 55 pp. 153)
Artinianness of Certain Graded Local Cohomology Modules
We show that if
$R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring
with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and
$M$ a finitely generated graded $R$-module, then
$H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where
$t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely
generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for
each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is
Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is
Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$
with respect to $R_+$. This improves some results of R. Sazeedeh.
Keywords:graded local cohomology, Artinian modules Categories:13D45, 13E10 |
3. CMB 2011 (vol 54 pp. 619)
Artinian and Non-Artinian Local Cohomology Modules Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized Cohen-Macaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords:local cohomology modules, cohomological dimensions, Bass numbers Categories:13D45, 13E10 |
4. CMB 2007 (vol 50 pp. 598)
Artinian Local Cohomology Modules Let $R$ be a commutative Noetherian ring, $\fa$ an ideal
of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a
non-negative integer. It is known that if the local cohomology
module $\H^i_\fa(M)$ is finitely generated for all $i Keywords:local cohomology module, Artinian module, reflexive module Categories:13D45, 13E10, 13C05 |