Theorems · Theorem · commutative algebra

RingHom.IsStableUnderBaseChange.isLocalization_map

∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop},
  RingHom.IsStableUnderBaseChange P →
    ∀ {R S Rᵣ Sᵣ : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing Rᵣ] [inst_3 : CommRing Sᵣ]
      [inst_4 : Algebra R Rᵣ] [inst_5 : Algebra S Sᵣ] (M : Submonoid R) [inst_6 : IsLocalization M Rᵣ] (f : R →+* S)
      [inst_7 : IsLocalization (Submonoid.map f M) Sᵣ], P f → P (IsLocalization.map Sᵣ f ⋯)

If P is stable under base change and holds for f, then P holds for f localized at any submonoid M of R.

Defined in

Mathlib.RingTheory.LocalProperties.Basic

Cited by

1 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingCommRingAlgebraAlgebraIsLocalizationIsLocalization