range_eq_top_of_isPurelyInseparable

English

If K,L are fields and L is FormallyUnramified over K with EssFiniteType, then L is a reduced ring.

Русский

Если K,L — поля, L формально неразветвлено над K и имеет EssFiniteType, то L — редкое кольцо.

LaTeX

$$$[\text{IsReduced } L]$$$

Lean4

theorem range_eq_top_of_isPurelyInseparable [IsPurelyInseparable K L] : (algebraMap K L).range = ⊤ := by
  classical
  have : Nontrivial (L ⊗[K] L) :=
    by
    rw [← not_subsingleton_iff_nontrivial, ← rank_zero_iff (R := K), rank_tensorProduct', mul_eq_zero, or_self,
      rank_zero_iff, not_subsingleton_iff_nontrivial]
    infer_instance
  rw [← top_le_iff]
  intro x _
  obtain ⟨n, hn⟩ := IsPurelyInseparable.pow_mem K (ringExpChar K) x
  have : ExpChar (L ⊗[K] L) (ringExpChar K) := by
    refine expChar_of_injective_ringHom (algebraMap K _).injective (ringExpChar K)
  have : (1 ⊗ₜ x - x ⊗ₜ 1 : L ⊗[K] L) ^ (ringExpChar K) ^ n = 0 :=
    by
    rw [sub_pow_expChar_pow, TensorProduct.tmul_pow, one_pow, TensorProduct.tmul_pow, one_pow]
    obtain ⟨r, hr⟩ := hn
    rw [← hr, algebraMap_eq_smul_one, TensorProduct.smul_tmul, sub_self]
  have H : (1 ⊗ₜ x : L ⊗[K] L) = x ⊗ₜ 1 :=
    by
    have inst : IsReduced (L ⊗[K] L) := isReduced_of_field L _
    exact sub_eq_zero.mp (IsNilpotent.eq_zero ⟨_, this⟩)
  by_cases h' : LinearIndependent K ![1, x]
  · have h := h'.linearIndepOn_id
    let S := h.extend (Set.subset_univ _)
    let a : S := ⟨1, h.subset_extend _ (by simp)⟩
    have ha : Basis.extend h a = 1 := by simp [a]
    let b : S := ⟨x, h.subset_extend _ (by simp)⟩
    have hb : Basis.extend h b = x := by simp [b]
    by_cases e : a = b
    · obtain rfl : 1 = x := congr_arg Subtype.val e
      exact ⟨1, map_one _⟩
    have := DFunLike.congr_fun (DFunLike.congr_arg ((Basis.extend h).tensorProduct (Basis.extend h)).repr H) (a, b)
    simp only [Basis.tensorProduct_repr_tmul_apply, ← ha, ← hb, Basis.repr_self, smul_eq_mul, Finsupp.single_apply, e,
      Ne.symm e, ↓reduceIte, mul_one, mul_zero, one_ne_zero] at this
  · rw [LinearIndependent.pair_iff] at h'
    simp only [not_forall, not_and, exists_prop] at h'
    obtain ⟨a, b, e, hab⟩ := h'
    have : IsUnit b := by
      rw [isUnit_iff_ne_zero]
      rintro rfl
      rw [zero_smul, ← algebraMap_eq_smul_one, add_zero,
        (injective_iff_map_eq_zero' _).mp (algebraMap K L).injective] at e
      cases hab e rfl
    use (-this.unit⁻¹ * a)
    rw [map_mul, ← Algebra.smul_def, algebraMap_eq_smul_one, eq_neg_iff_add_eq_zero.mpr e, smul_neg, neg_smul, neg_neg,
      smul_smul, this.val_inv_mul, one_smul]

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