Theorems · Theorem · commutative algebra

RingHom.locally_propertyIsLocal

∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop},
  (RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P) →
    (RingHom.StableUnderCompositionWithLocalizationAway fun {R S} [CommRing R] [CommRing S] => P) →
      RingHom.PropertyIsLocal fun {R S} [CommRing R] [CommRing S] =>
        RingHom.Locally fun {R S} [CommRing R] [CommRing S] => P

If P is preserved by localizations and stable under composition with localization away maps, then Locally P is a local property of ring homomorphisms.

Defined in

Mathlib.RingTheory.RingHom.Locally

Cited by

1 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound