Proof of de Smit's conjecture: a freeness criterion

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.