surjectiveOnStalks_of_isLocalization

English

If M is a localization data for R, and S is a localization, then the base map induces surjectivity on stalks: f.SurjectiveOnStalks.

Русский

Если имеется локализация M и S, то базовый отображение порождает сюръективность на стэках.

LaTeX

$$$\\text{IsLocalization } M S \\Rightarrow f.SurjectiveOnStalks$$$

Lean4

theorem surjectiveOnStalks_of_isLocalization [Algebra R S] [IsLocalization M S] : SurjectiveOnStalks (algebraMap R S) :=
  by
  refine surjectiveOnStalks_of_exists_div fun s ↦ ?_
  obtain ⟨x, s, rfl⟩ := IsLocalization.mk'_surjective M s
  exact ⟨x, s, IsLocalization.map_units S s, IsLocalization.mk'_spec' S x s⟩

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