free_of_finite_type_torsion_free — Mathlib

English

Under suitable finite conditions, one can extract a basis for a torsion-free module in a similar fashion, with apostrophe to denote variant.

Русский

При подходящих условиях можно выделить базис для безtorsion модуля, обозначая вариант апострофом.

LaTeX

$$$\\exists n, Basis (Fin n) R M$ (variant)$$

Lean4

theorem free_of_finite_type_torsion_free [_root_.Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
    [NoZeroSMulDivisors R M] : Module.Free R M :=
  by
  cases nonempty_fintype ι
  obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs
  exact Module.Free.of_basis b

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