Theorems · Theorem · commutative algebra

RingHom.OfLocalizationSpan.ofIsLocalization

∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop},
  (RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => P) →
    (RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P) →
      ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S) (s : Set R),
        Ideal.span s = ⊤ →
          (∀ (r : ↑s),
              ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
                ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
                  ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ) →
            P f

Defined in

Mathlib.RingTheory.LocalProperties.Basic

Cited by

1 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRing