Theorems · Theorem · commutative algebra

RingHom.locally_iff_isLocalization

∀ {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 ↔
        ∃ s,
          ∃ (_ : Ideal.span ↑s = ⊤),
            ∀ t ∈ s,
              ∀ (Sₜ : Type u) [inst_2 : CommRing Sₜ] [inst_3 : Algebra S Sₜ] [IsLocalization.Away t Sₜ],
                P ((algebraMap S Sₜ).comp f)

In the definition of Locally we may replace Localization.Away with an arbitrary algebra satisfying IsLocalization.Away.

Defined in

Mathlib.RingTheory.RingHom.Locally

Cited by

1 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRing