isOpenImmersion_SpecMap_lmul — Mathlib

English

A flat surjective morphism is an epimorphism in the category of schemes.

Русский

Плоский сюръективный морфизм является эпиморфизмом в категории схем.

LaTeX

$$Epi(f) if Flat(f) and Surjective(f)$$

Lean4

/-- If `S` is a formally unramified `R`-algebra, essentially of finite type, the diagonal is an
open immersion. -/
instance isOpenImmersion_SpecMap_lmul {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
    [Algebra.FormallyUnramified R S] [Algebra.EssFiniteType R S] :
    IsOpenImmersion (Spec.map (CommRingCat.ofHom (TensorProduct.lmul' R (S := S)).toRingHom)) :=
  by
  rw [isOpenImmersion_SpecMap_iff_of_surjective _ (fun x ↦ ⟨1 ⊗ₜ x, by simp⟩)]
  apply (Ideal.isIdempotentElem_iff_of_fg _ (KaehlerDifferential.ideal_fg R S)).mp
  apply (Ideal.cotangent_subsingleton_iff _).mp
  exact inferInstanceAs <| Subsingleton Ω[S⁄R]

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