English
If B is a faithfully flat A-algebra, the induced map on prime spectra Spec(B) → Spec(A) is surjective.
Русский
Если B напоминает плоское над A расширение, отображение между спектрами простых идеалов Spec(B) → Spec(A) нацелено на сюрьективность.
LaTeX
$$$Spec\_A(B)\;\text{is surjective, i.e. } Spec(B) \to Spec(A)\text{ is onto}$$$
Lean4
/-- If `A →+* B` is flat and surjective on prime spectra, `B` is a faithfully flat `A`-algebra. -/
theorem of_specComap_surjective [Flat A B] (h : Function.Surjective ((algebraMap A B).specComap)) :
Module.FaithfullyFlat A B := by
refine ⟨fun m hm ↦ ?_⟩
obtain ⟨m', hm'⟩ := h ⟨m, hm.isPrime⟩
have : m = Ideal.comap (algebraMap A B) m'.asIdeal := by
rw [← PrimeSpectrum.specComap_asIdeal (algebraMap A B) m', hm']
rw [Ideal.smul_top_eq_map, this]
exact
(Submodule.restrictScalars_eq_top_iff _ _ _).ne.mpr fun top ↦
m'.isPrime.ne_top <| top_le_iff.mp <| top ▸ Ideal.map_comap_le
