isField_of_universallyClosed — Mathlib

English

If f is universally closed and Y is affine, then the induced map on global sections is integral.

Русский

Если f универсально замкнут и Y аффинный, то отображение на глобальные секции иррационально интегрировано.

LaTeX

$$$[UniversallyClosed(f)] \land [IsAffine Y] \Rightarrow f.appTop.hom.IsIntegral$$$

Lean4

/-- If `X` is an integral scheme that is universally closed over `Spec K`,
then `Γ(X, ⊤)` is a field. -/
theorem isField_of_universallyClosed (f : X ⟶ (Spec <| .of K)) [IsIntegral X] [UniversallyClosed f] : IsField Γ(X, ⊤) :=
  by
  let F := (Scheme.ΓSpecIso _).inv ≫ f.appTop
  have : F.hom.IsIntegral :=
    by
    apply RingHom.isIntegral_respectsIso.2 (e := (Scheme.ΓSpecIso _).symm.commRingCatIsoToRingEquiv)
    exact isIntegral_appTop_of_universallyClosed f
  algebraize [F.hom]
  exact isField_of_isIntegral_of_isField' (Field.toIsField K)

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