{"paper":{"title":"Proof of de Smit's conjecture: a freeness criterion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.NT"],"primary_cat":"math.AC","authors_text":"Sylvain Brochard","submitted_at":"2016-07-07T15:14:22Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02044","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}