Local Cohomology of Module of Differentials of Integral Extensions II

Abstract

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.