Theorems · Theorem · algebraic geometry

AlgebraicGeometry.HasRingHomProperty.locally_of_iff

∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme}
  {Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop},
  (RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => Q) →
    (RingHom.StableUnderCompositionWithLocalizationAway fun {R S} [CommRing R] [CommRing S] => Q) →
      (∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),
          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))) →
        AlgebraicGeometry.HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] =>
          RingHom.Locally fun {R S} [CommRing R] [CommRing S] => Q

Defined in

Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Cited by

0 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound