etale_iff_formallyUnramified_and_smooth

English

Equivalence of Etale to being formally unramified and smooth holds uniformly across base changes; the formal equivalence is expressed by equality of Etale with And(FormallyUnramified, Smooth).

Русский

Эквивалентность Etale к состоянию формально нерасслоения и гладкости сохраняется под основанием; формальная эквивалентность выражается через равенство Etale и And(FormallyUnramified, Smooth).

LaTeX

$$$f.Etale = (f.FormallyUnramified \\land f.Smooth)$$$

Lean4

theorem etale_iff_formallyUnramified_and_smooth : f.Etale ↔ f.FormallyUnramified ∧ f.Smooth :=
  by
  algebraize [f]
  simp only [Etale, Smooth, FormallyUnramified]
  refine ⟨fun h ↦ ⟨inferInstance, ?_⟩, fun ⟨h1, h2⟩ ↦ ⟨?_, inferInstance⟩⟩
  · constructor <;> infer_instance
  · rw [Algebra.FormallyEtale.iff_unramified_and_smooth]
    constructor <;> infer_instance

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