mk' — Mathlib

English

`IsZariskiLocalAtSource` is constructed by providing a restriction map and a gluing rule over zero-hypercovers.

Русский

IsZariskiLocalAtSource строится из отображения ограничений и правила склейки по нулевым гиперпокрытиям.

LaTeX

$$$IsZariskiLocalAtSource(P)$ строится из restrictions и of_zeroHypercover rules.$$

Lean4

/-- `P` is local at the source if
1. `P` respects isomorphisms.
2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `U.ι ≫ f` for any `U`.
3. If `P` holds for `U.ι ≫ f` for an open cover `U` of `X`, then `P` holds for `f`.
-/
protected theorem mk' {P : MorphismProperty Scheme} [P.RespectsIso]
    (restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens), P f → P (U.ι ≫ f))
    (of_sSup_eq_top :
      ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens), iSup U = ⊤ → (∀ i, P ((U i).ι ≫ f)) → P f) :
    IsZariskiLocalAtSource P
    where
  comp 𝒰 i
    H :=
    by
    rw [←
      IsOpenImmersion.isoOfRangeEq_hom_fac (𝒰.f i) (Scheme.Opens.ι _)
        (congr_arg Opens.carrier (𝒰.f i).opensRange.opensRange_ι.symm),
      Category.assoc, P.cancel_left_of_respectsIso]
    exact restrict _ _ H
  of_zeroHypercover {X Y} f 𝒰
    h := by
    refine of_sSup_eq_top f _ (Scheme.OpenCover.iSup_opensRange 𝒰) fun i ↦ ?_
    rw [←
      IsOpenImmersion.isoOfRangeEq_inv_fac (𝒰.f i) (Scheme.Opens.ι _)
        (congr_arg Opens.carrier (𝒰.f i).opensRange.opensRange_ι.symm),
      Category.assoc, P.cancel_left_of_respectsIso]
    exact h _

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