Proof of de Smit’s conjecture: a freeness criterion

2016/07/31 by Sylvain Brochard
Mathematics · #Algebraic Geometry and Number Theory #Combinatorics #Conjecture #Dimension (graph theory) #Discrete mathematics #Fermat's Last Theorem #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematics #Morphism #Pure mathematics #math.AC #math.AG #math.NT #msc:11F80 #msc:13C10 #msc:14A05

paper · pdf · doi:10.1112/s0010437x17007370

published as Compositio Math. 153 (2017) 2310-2317 · final version, to appear in Compositio Mathematica

arxiv created 2017/06/16 · openalex publication_date 2017/08/14 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 B -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.

Citations

• Complete intersections of dimension zero: variations on a theme of Wiebe
• Modularity of certain potentially Barsotti-Tate Galois representations
• Serre’s modularity conjecture (II)
• Automorphy for some l -adic lifts of automorphic mod l Galois representations. II
• Automorphy for some l -adic lifts of automorphic mod l Galois representations