localRingHom — Mathlib

English

Localization of a flat ring hom yields a flat ring hom.

Русский

Локализация слабого коленного отображения даёт плоское отображение.

LaTeX

$$$\\text{Flat}(\\mathrm{Localization.localRingHom} Q P f hQP)$$$

Lean4

theorem localRingHom {f : R →+* S} (hf : f.Flat) (P : Ideal S) [P.IsPrime] (Q : Ideal R) [Q.IsPrime]
    (hQP : Q = Ideal.comap f P) : (Localization.localRingHom Q P f hQP).Flat :=
  by
  subst hQP
  algebraize [f, Localization.localRingHom (Ideal.comap f P) P f rfl]
  have : IsScalarTower R (Localization.AtPrime (Ideal.comap f P)) (Localization.AtPrime P) :=
    .of_algebraMap_eq fun x ↦ (Localization.localRingHom_to_map _ _ _ rfl x).symm
  rw [RingHom.Flat,
    Module.flat_iff_of_isLocalization (S := (Localization.AtPrime (Ideal.comap f P))) (p :=
      (Ideal.comap f P).primeCompl)]
  exact Module.Flat.trans R S (Localization.AtPrime P)

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