English
There is a canonical linear isomorphism between the localization of a quotient and the quotient of the localized module: localizedQuotientEquiv.
Русский
Существует каноническое линейное изоморфизм между локализацией квадрата и квадратом локализованного модуля: localizedQuotientEquiv.
LaTeX
$$$(\mathrm{LocalizedModule}_p M / M'.localized(p)) \cong_{\mathrm{Localization}(p)} \mathrm{LocalizedModule}_p (M / M')$$$
Lean4
/-- The canonical isomorphism between the localization of a quotient module and its realization
as a quotient of the localized module. -/
noncomputable def localizedQuotientEquiv :
(LocalizedModule p M ⧸ M'.localized p) ≃ₗ[Localization p] LocalizedModule p (M ⧸ M') :=
have := IsLocalization.linearMap_compatibleSMul p
IsLocalizedModule.linearEquiv p (M'.toLocalizedQuotient p) (LocalizedModule.mkLinearMap _ _) |>.restrictScalars _
