Proof of de Smit's conjecture: a freeness criterion

TL;DR

This paper proves de Smit's conjecture on a freeness criterion for modules over Artin local rings, showing that flat modules are free under certain conditions, and simplifies parts of Wiles's proof of Fermat's Last Theorem.

Contribution

It proves de Smit's conjecture, establishing a new freeness criterion for modules over Artin local rings and simplifying key aspects of Wiles's proof of Fermat's Last Theorem.

Findings

Any A-flat B-module is B-flat when A→B are Artin local rings with same embedding dimension.

If a nonzero A-flat B-module exists, then A→B is flat and a complete intersection.

Simplifies Wiles's proof by removing the need for Taylor-Wiles systems.

Abstract

Let $A\to B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond's Theorem 2.1 from his 1997 paper "The Taylor-Wiles construction and multiplicity one". We also prove that if there is a nonzero $A$-flat $B$-module, then $A\to B$ is flat and is a relative complete intersection (i.e. $B/\mathfrak{m}_AB$ is a complete intersection). Then we explain how this result allows to simplify Wiles's proof of Fermat's Last Theorem: we do not need the so-called "Taylor-Wiles systems" anymore.