English
Localization commutes with quotient constructions and with scalar extension along the localization: the canonical localizedQuotientEquiv is compatible with the scalar extension to Localization p.
Русский
Локализация commuting с квартификацией и линейным расширением по локализации: каноническая локализационная эквивалентность совместима с скаляром по Localization p.
LaTeX
$$$\mathrm{localizedQuotientEquiv}: (\mathrm{LocalizedModule}_p M / M'.localized(p)) \\cong_{\mathrm{Localization}(p)} \mathrm{LocalizedModule}_p (M / M')$$$
Lean4
theorem ker_localizedMap_eq_localized₀_ker (g : M →ₗ[R] P) : ker (map p f f' g) = (ker g).localized₀ p f :=
by
ext x
simp only [Submodule.mem_localized₀, mem_ker]
refine ⟨fun h ↦ ?_, ?_⟩
· obtain ⟨⟨a, b⟩, rfl⟩ := IsLocalizedModule.mk'_surjective p f x
simp only [Function.uncurry_apply_pair, map_mk', mk'_eq_zero, eq_zero_iff p f'] at h
obtain ⟨c, hc⟩ := h
refine ⟨c • a, by simpa, c * b, by simp⟩
· rintro ⟨m, hm, a, ha, rfl⟩
simp [IsLocalizedModule.map_mk', hm]
