mem_stabilizer_mk_iff — Mathlib

English

If w is unramified over k, then it is unramified after comapping along algebra maps K→F.

Русский

Если w неразморожено над k, то при переносе по алгебраическим картам K→F сохраняется неразм.

LaTeX

$$$IsUnramified\\; k\\; w \\Rightarrow IsUnramified\\; k\\; (w.comap(\\text{algebraMap } K F))$$$

Lean4

theorem mem_stabilizer_mk_iff (φ : K →+* ℂ) (σ : K ≃ₐ[k] K) : σ ∈ Stab (mk φ) ↔ σ = 1 ∨ ComplexEmbedding.IsConj φ σ :=
  by
  simp only [MulAction.mem_stabilizer_iff, smul_mk, mk_eq_iff]
  rw [← ComplexEmbedding.isConj_symm, ComplexEmbedding.conjugate, star_eq_iff_star_eq]
  refine or_congr ⟨fun H ↦ ?_, fun H ↦ H ▸ rfl⟩ Iff.rfl
  exact congr_arg AlgEquiv.symm (AlgEquiv.ext (g := AlgEquiv.refl) fun x ↦ φ.injective (RingHom.congr_fun H x))

Похожие утверждения