Proof of de Smit’s conjecture: a freeness criterion

Sylvain Brochard (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Let A -> 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 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 A-flat -module, then A -> B 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.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Sylvain Brochard. Proof of de Smit’s conjecture: a freeness criterion. Compositio Mathematica, 2017, 153 (11), pp.2310 - 2317. ⟨10.1112/S0010437X17007370⟩. ⟨hal-01819651⟩