A tight closure approach to a result of G. Faltings

TL;DR

This paper introduces a new criterion for complete intersection rings in characteristic p>0 using tight closure theory, providing a novel proof of a key algebraic result by Faltings that aided the proof of the Taylor-Wiles theorem.

Contribution

It presents a new tight closure-based criterion for complete intersection rings and offers a different proof of Faltings' algebraic result relevant to deformation theory.

Findings

Established a criterion for complete intersection rings in characteristic p>0.

Provided a new proof of Faltings' algebraic result used in deformation problems.

Simplified the proof of the minimal deformation problem in number theory.

Abstract

Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G. Faltings that was used by Taylor and Wiles in \cite{TW} for a simplification of the proof of the minimal deformation problem.