descendsAlong — Mathlib

English

Under hypotheses that P' codescends along Q', every P-descends along quasi-compact morphisms.

Русский

При предположениях, что P' кодесцендирует вдоль Q', любое descent по P-descends вдоль квазик compacto отображений.

LaTeX

$$$[P'.CodescendsAlong Q Q'] \\Rightarrow P.DescendsAlong (P' \\wedge QuasiCompact)$$$

Lean4

/-- Let `P` be the morphism property associated to the ring hom property `Q`. Suppose

- `P'` implies `Q'` on global sections for affine schemes,
- `P'` is satisfied for all surjective, local isomorphisms, and
- `Q` codescend along `Q'`.

Then `P` descends along quasi-compact morphisms satisfying `P'`.

Note: The second condition is in particular satisfied for faithfully flat morphisms.
-/
nonrec theorem descendsAlong [HasRingHomProperty P Q] (hQQ' : RingHom.CodescendsAlong Q Q') :
    P.DescendsAlong (P' ⊓ @QuasiCompact) :=
  by
  apply IsZariskiLocalAtTarget.descendsAlong_inf_quasiCompact _ _ H₁
  introv h hf
  wlog hY : ∃ S, Y = Spec S generalizing Y
  · rw [IsZariskiLocalAtSource.iff_of_openCover (P := P) Y.affineCover]
    intro i
    have heq :
      pullback.fst (Spec.map φ) (Y.affineCover.f i ≫ g) =
        pullback.map _ _ _ _ (𝟙 _) (Y.affineCover.f i) (𝟙 _) (by simp) (by simp) ≫ pullback.fst (Spec.map φ) g :=
      (pullback.lift_fst _ _ _).symm
    exact this _ (heq ▸ AlgebraicGeometry.IsZariskiLocalAtSource.comp hf _) ⟨_, rfl⟩
  obtain ⟨S, rfl⟩ := hY
  apply of_pullback_fst_Spec_of_codescendsAlong _ _ hQQ' H₂ _ h hf
  simp [HasRingHomProperty.Spec_iff (P := P)]

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