Local Cohomology of Module of Differentials of Integral Extensions II

TL;DR

This paper investigates the local cohomology of modules of differentials in integral extensions, establishing new results in both equicharacteristic zero and mixed characteristic, with implications for Cohen-Macaulay modules and normality criteria.

Contribution

It determines the highest non-vanishing local cohomology of differential modules and extends Suzuki's theorem to formal settings across all characteristics.

Findings

Identified the highest non-vanishing local cohomology of _{B/R} in equicharacteristic zero.

Established connections between _{B/R} and _{B/V} with pull-back sequences.

Extended Suzuki's theorem on normality of complete intersections to formal schemes.

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.