1. CMB Online first
 Shaul, Liran

Homological dimensions of local (co)homology over commutative DGrings
Let $A$ be a commutative noetherian ring,
let $\mathfrak{a}\subseteq A$ be an ideal,
and let $I$ be an injective $A$module.
A basic result in the structure theory of injective modules states
that
the $A$module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$torsion elements
is also an injective $A$module.
Recently, de Jong proved a dual result: If $F$ is a flat $A$module,
then the $\mathfrak{a}$adic completion of $F$ is also a flat $A$module.
In this paper we generalize these facts to commutative noetherian
DGrings:
let $A$ be a commutative nonpositive DGring such that $\mathrm{H}^0(A)$
is a noetherian ring,
and for each $i\lt 0$, the $\mathrm{H}^0(A)$module $\mathrm{H}^i(A)$
is finitely generated.
Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$,
we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$
associated to $\bar{\mathfrak{a}}$ does not increase injective dimension.
Dually, the derived $\bar{\mathfrak{a}}$adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$
does not increase flat dimension.
Keywords:local cohomology, derived completion, homological dimension, commutative DGring Categories:13B35, 13D05, 13D45, 16E45 

2. CMB 2017 (vol 60 pp. 225)
 Bahmanpour, Kamal; Naghipour, Reza

Faltings' Finiteness Dimension of Local Cohomology Modules Over Local CohenMacaulay Rings
Let $(R, \frak m)$ denote a local CohenMacaulay ring and $I$
a nonnilpotent ideal of $R$. The purpose of this article is
to investigate Faltings' finiteness
dimension $f_I(R)$ and equidimensionalness of certain homomorphic
image of $R$. As a consequence
we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$
and if $\operatorname{mAss}_R(R/I)$
is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq
1}(0:_RI^n)$ is equidimensional of dimension $\dim R1$.
Moreover, we will obtain a lower bound for injective dimension
of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case
$(R, \frak m)$ is a complete equidimensional local ring.
Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology Categories:13D45, 14B15 

3. CMB 2016 (vol 59 pp. 271)
 DehghaniZadeh, Fatemeh

Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with
local base ring $(R_{0}, \mathfrak{m}_{0})$ and let
$R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated
graded $R$modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We
show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M,
N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and
$j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal
of
$R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$primary ideal.
Keywords:generalized local cohomology, Artinian, graded module Categories:13D45, 13E10, 16W50 

4. CMB 2016 (vol 59 pp. 403)
 Zargar, Majid Rahro; Zakeri, Hossein

On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules
Let $\mathfrak{a}$ be an ideal of a Noetherian local
ring $R$ and let $C$ be a semidualizing $R$module. For an $R$module
$X$, we denote any of the quantities $\mathfrak{d}_R X$,
$\operatorname{\mathsf{Gfd}}_R X$ and
$\operatorname{\mathsf{G_Cfd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$module such that
$\operatorname{H}_{\mathfrak{a}}^i(M)=0$
for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then
$\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever
$M$ is finitely generated. With the aid of these results, among
other things, we characterize CohenMacaulay modules, dualizing
modules and Gorenstein rings.
Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative CohenMacaulay module, semidualizing module Categories:13D05, 13D45, 18G20 

5. CMB 2015 (vol 58 pp. 664)
 Vahidi, Alireza

Betti Numbers and Flat Dimensions of Local Cohomology Modules
Assume that $R$ is a commutative Noetherian ring with nonzero
identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$module.
In this paper, we first study the finiteness of Betti numbers
of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some
inequalities between the Betti numbers of $X$ and those of its
local cohomology modules. Finally, we present many upper bounds
for the flat dimension of $X$ in terms of the flat dimensions
of its local cohomology modules and an upper bound for the flat
dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of
the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.
Keywords:Betti numbers, flat dimensions, local cohomology modules Categories:13D45, 13D05 

6. CMB 2014 (vol 57 pp. 477)
 Eghbali, Majid

On Set Theoretically and Cohomologically Complete Intersection Ideals
Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities
\[
\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq
\operatorname{ara}(\mathfrak a) \leq
l(\mathfrak a) \leq \mu(\mathfrak a)
\]
are known. It is an interesting and longstanding problem to find
out the cases giving equality. Thanks to the formal grade we give
conditions in which the above inequalities become
equalities.
Keywords:settheoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of ideals Categories:13D45, 13C14 

7. CMB 2012 (vol 56 pp. 491)
 Bahmanpour, Kamal

A Note on Homological Dimensions of Artinian Local Cohomology Modules
Let $(R,{\frak m})$ be a nonzero commutative Noetherian local ring
(with identity), $M$ be a nonzero finitely generated $R$module. In
this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that
$
\operatorname{{\rm injdim_{_{R_{\frak p}}}}}
H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and
${\rm fd}_{R_{\p}} H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak
p}}(M_{\frak p})$ are bounded from above by $
\operatorname{{\rm injdim_{_{R}}}}
H^i_{\frak
m}(M)$ and
$ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.
Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology Category:13D45 

8. CMB 2011 (vol 55 pp. 315)
 Hellus, M.

A Note on the Vanishing of Certain Local Cohomology Modules
For a finite module $M$ over a local, equicharacteristic ring $(R,m)$,
we show that the wellknown formula $\textrm{cd}(m,M)=\dim M$ becomes trivial
if ones uses Matlis duals of local cohomology modules together with spectral sequences.
We also prove a new ringtheoretic vanishing criterion for local cohomology modules.
Category:13D45 

9. CMB 2011 (vol 55 pp. 81)
10. CMB 2011 (vol 55 pp. 153)
 Mafi, Amir; Saremi, Hero

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 

11. CMB 2011 (vol 54 pp. 619)
 Dibaei, Mohammad T.; Vahidi, Alireza

Artinian and NonArtinian 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 CohenMacaulay 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
nonempty 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 

12. CMB 2010 (vol 53 pp. 667)
 Khashyarmanesh, Kazem

On the Endomorphism Rings of Local Cohomology Modules
Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence Categories:13D45, 13D07, 13D25 

13. CMB 2010 (vol 53 pp. 577)
14. CMB 2007 (vol 50 pp. 598)
15. CMB 2002 (vol 45 pp. 119)