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 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 

3. 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 
