  In this note ($R, m$) denotes a complete regular local ring and $B$ mostly
denotes its absolute integral closure. The four objectives of this paper are
the following: i) to determine the highest non-vanishing local cohomology of
$\Omega_{B/R}$ in equicharacteristic $0$, ii) to establish a connection between
each of $\Omega_{B/R}$ and $\Omega_{B/V}$ and pull-back of $\Omega_{A/V}$ via a
short exact sequence together with new observations on corresponding local
cohomologies in mixed characteristic where $V$ is the coefficient ring of $R$
and $A$ is its absolute integral closure, iii) to demonstrate that
$\Omega_{B/R}$ can be mapped onto a cohomologically Cohen-Macaulay module and
iv) to study torsion-free property for $\Omega_{C/V}$ and $\Omega_{C/k}$ along
with their respective completions where $C$ is an integral domain and a module
finite extension of $R$. In this connection an extension of Suzuki's theorem on
normality of complete intersections to the formal set-up in all characteristics
is accomplished.
