Theorems · Theorem · commutative algebra

RingHom.locally_iff_exists

∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} {R S : Type u}
  [inst : CommRing R] [inst_1 : CommRing S],
  (RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P) →
    ∀ (f : R →+* S),
      RingHom.Locally (fun {R S} [CommRing R] [CommRing S] => P) f ↔
        ∃ ι s,
          ∃ (_ : Ideal.span (Set.range s) = ⊤),
            ∃ Sₜ x x_1, ∃ (_ : ∀ (i : ι), IsLocalization.Away (s i) (Sₜ i)), ∀ (i : ι), P ((algebraMap S (Sₜ i)).comp f)

Equivalence variant of locally_of_exists. This is sometimes easier to use, if the IsLocalization.Away instance can't be automatically inferred.

Defined in

Mathlib.RingTheory.RingHom.Locally

Cited by

4 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRing