Theorems · Theorem · commutative algebra

IsLocalization.map_injective_of_injective

∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) (S : Type u_2) [inst_1 : CommSemiring S]
  [inst_2 : Algebra R S] {P : Type u_3} [inst_3 : CommSemiring P] [inst_4 : IsLocalization M S] {g : R →+* P}
  (Q : Type u_4) [inst_5 : CommSemiring Q] [inst_6 : Algebra P Q],
  Function.Injective ⇑g →
    ∀ [inst_7 : IsLocalization (Submonoid.map g M) Q], Function.Injective ⇑(IsLocalization.map Q g ⋯)

Injectivity of a map descends to the map induced on localizations.

Defined in

Mathlib.RingTheory.Localization.Defs

Cited by

6 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringIsLocalizationCommSemiringAlgebraIsLocalization