Proof of de Smit's conjecture: A freeness criterion

Abstract

Let be a morphism of Artin local rings with the same embedding dimension. We prove that any -flat -module is -flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond's criterion [The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379-391, Theorem 2.1]. We also prove that if there is a nonzero -flat -module, then is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles's proof of Fermat's last theorem: we do not need the so-called 'Taylor-Wiles systems' any more.