isSeparable — Mathlib

English

If K ⟶ L is purely inseparable and L is FormallyUnramified over K and EssFiniteType, then algebraMap K L has range equal to the top subring.

Русский

Если K ⟶ L чисто инсекприруемо и L формально неразветвлено над K и EssFiniteType, тогда отображение алгебра-карты имеет полный диапазон.

LaTeX

$$$[IsPurelyInseparable K L] \Rightarrow (algebraMap\ K L).range = ⊤$$$

Lean4

theorem isSeparable : Algebra.IsSeparable K L :=
  by
  have := finite_of_free (R := K) (S := L)
  rw [← separableClosure.eq_top_iff]
  have := of_comp K (separableClosure K L) L
  have := EssFiniteType.of_comp K (separableClosure K L) L
  ext
  change _ ↔ _ ∈ (⊤ : Subring _)
  rw [← range_eq_top_of_isPurelyInseparable (separableClosure K L) L]
  simp

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