English
The infimum (greatest lower bound) of a family of subalgebras of a field extension is integrally closed if each member is integrally closed.
Русский
Первообразная (наибольшая нижняя грань) семейства подалгебр расширения поля интегрално замкнута, если каждый элемент - интегрално замкнут.
LaTeX
$$$\\forall i,\\; IsIntegrallyClosed(S_i) \\Rightarrow IsIntegrallyClosed(\\bigwedge_i S_i)$$$
Lean4
theorem exact_of_isLocalized_span
(H : ∀ r : s, Function.Exact (map (.powers r.1) (f r) (g r) F) (map (.powers r.1) (g r) (h r) G)) :
Function.Exact F G := by
simp only [LinearMap.exact_iff] at H ⊢
apply Submodule.eq_of_isLocalized₀_span s spn Nₚ g
intro r
rw [← LinearMap.range_localizedMap_eq_localized₀_range _ (f r) (g r) F]
rw [← LinearMap.ker_localizedMap_eq_localized₀_ker (.powers r.1) (g r) (h r) G]
have := SetLike.ext_iff.mp <| H r
ext x
simp only [mem_range, mem_ker] at this ⊢
exact this x
