Theorems · Theorem · algebraic geometry

AlgebraicGeometry.HasAffineProperty.affineAnd_iff

∀ {Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop}
  (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme),
  (RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q) →
    (RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => Q) →
      (RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => Q) →
        (AlgebraicGeometry.HasAffineProperty P (AlgebraicGeometry.affineAnd fun {R S} [CommRing R] [CommRing S] => Q) ↔
          ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),
            P f ↔
              AlgebraicGeometry.IsAffineHom f ∧
                ∀ (U : Y.Opens),
                  AlgebraicGeometry.IsAffineOpen U → Q (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.app f U)))

A convenience constructor for HasAffineProperty P (affineAnd Q). The IsAffineHom is bundled, since this goes well with defining morphism properties via extends IsAffineHom.

Defined in

Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Cited by

0 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound