Theorems · Theorem · algebraic geometry

AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally

∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme}
  {Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} {X Y : AlgebraicGeometry.Scheme}
  {f : X ⟶ Y},
  (RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] => Q) →
    (RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q) →
      ∀
        [AlgebraicGeometry.HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] =>
            RingHom.Locally fun {R S} [CommRing R] [CommRing S] => Q],
        P f ↔
          ∀ (x : ↥X),
            ∃ U V,
              ∃ (_ : x ∈ ↑V) (e : ↑V ≤ (TopologicalSpace.Opens.map f.base).obj ↑U),
                Q (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appLE f (↑U) (↑V) e))

If P is induced by Locally Q, it suffices to check Q on affine open sets locally around points of the source.

Defined in

Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Cited by

3 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AlgebraicGeometry.HasRingHomProperty