tensor_map_id_left_injective_of_injective

English

For finite M, AdicCompletion.ofTensorProductEquivOfFiniteNoetherian induces an isomorphism that identifies the finite-Noetherian maps with the canonical tensor maps.

Русский

Для конечного M эквиваленция AdicCompletion.ofTensorProductEquivOfFiniteNoetherian порождает изоморфизм и согласует конечную Ноетерову карту с каноническим тензорным отображением.

LaTeX

$$$\mathrm{ofTensorProductEquivOfFiniteNoetherian}\ I\ M \text{ is bijective and aligns with } \mathrm{ofTensorProduct} I M$$$

Lean4

theorem tensor_map_id_left_injective_of_injective (hf : Function.Injective f) :
    Function.Injective
      (AlgebraTensorModule.map LinearMap.id f :
        AdicCompletion I R ⊗[R] M →ₗ[AdicCompletion I R] AdicCompletion I R ⊗[R] N) :=
  by
  rw [tensor_map_id_left_eq_map I f]
  simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, EmbeddingLike.comp_injective, EquivLike.injective_comp]
  exact map_injective I hf

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