iff_characterModule_baer — Mathlib

English

Flatness over R equals Baer property of the character module.

Русский

Плоскость над R равна свойству Баера характерного модуля.

LaTeX

$$$\\text{Flat}(R,M) \\iff \\text{Baer}(R, \\text{CharacterModule}(M))$$$

Lean4

/-- `CharacterModule M` is Baer iff `M` is flat. -/
theorem iff_characterModule_baer : Flat R M ↔ Baer R (CharacterModule M) := by
  rw [equiv_iff (N := ULift.{u} M) ULift.moduleEquiv.symm, iff_characterModule_injective, ← Baer.iff_injective,
    Baer.congr (CharacterModule.congr ULift.moduleEquiv)]

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