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Proof of de Smit’s conjecture: a freeness criterion

Part of: Algebraic geometry: Foundations Discontinuous groups and automorphic forms Theory of modules and ideals

Published online by Cambridge University Press:  14 August 2017

Sylvain Brochard
Sylvain Brochard*
Affiliation:
Institut Montpelliérain Alexander Grothendieck, CNRS, Université de Montpellier, Place Eugène Bataillon CC051, 34 095 Montpellier CEDEX 5, France email sylvain.brochard@umontpellier.fr
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Abstract

Let $A\rightarrow B$ 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 $B$ -module, then $A\rightarrow B$ 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.

Keywords

flatnessArtin local ringsembedding dimensionpatching

MSC classification

Primary: 13C10: Projective and free modules and ideals 11F80: Galois representations 14A05: Relevant commutative algebra
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 11 , November 2017 , pp. 2310 - 2317
Copyright
© The Author 2017 

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