English
If P is stable under base change, then localizationPreserves P holds.
Русский
Если P устойчиво к базовым изменениям, то holds для локализации сохраняется.
LaTeX
$$$\\text{IsStableUnderBaseChange }P \\Rightarrow \\text{LocalizationPreserves }P$$$
Lean4
/-- If `P` is stable under base change and holds for `f`, then `P` holds for `f` localized
at any submonoid `M` of `R`. -/
theorem isLocalization_map (M : Submonoid R) [IsLocalization M Rᵣ] (f : R →+* S) [IsLocalization (M.map f) Sᵣ]
(hf : P f) : P (IsLocalization.map Sᵣ f M.le_comap_map : Rᵣ →+* Sᵣ) :=
by
algebraize [f, IsLocalization.map (S := Rᵣ) Sᵣ f M.le_comap_map,
(IsLocalization.map (S := Rᵣ) Sᵣ f M.le_comap_map).comp (algebraMap R Rᵣ)]
haveI : IsScalarTower R S Sᵣ := IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map)
haveI : IsLocalization (Algebra.algebraMapSubmonoid S M) Sᵣ := inferInstanceAs <| IsLocalization (M.map f) Sᵣ
apply hP.of_isLocalization M hf
