StableUnderCompositionWithLocalizationAwaySource

English

A property P of ring homs is stable under composition with localization away on both source and target if P is preserved under pre- and post-composition with localization maps.

Русский

Свойство P гомоморфизмов сохраняется при композиции с локализацией вдали по источнику и цели, если сохраняется при предподстановке и пост-подстановке локализаций.

LaTeX

$$$P(f) \\Rightarrow P(\\text{comp with localization away on source and target}).$$$

Lean4

/-- A property `P` of ring homs satisfies `RingHom.StableUnderCompositionWithLocalizationAwaySource`
if whenever `P` holds for `f` it also holds for the composition with
localization maps on the source. -/
def StableUnderCompositionWithLocalizationAwaySource : Prop :=
  ∀ ⦃R : Type u⦄ (S : Type u) ⦃T : Type u⦄ [CommRing R] [CommRing S] [CommRing T] [Algebra R S] (r : R)
    [IsLocalization.Away r S] (f : S →+* T), P f → P (f.comp (algebraMap R S))

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