of_flat_of_isBaseChange — Mathlib

English

Under flatness on the left, the map hf.map hg LinearMap.id is injective when i is injective, for IsTensorProduct f and g.

Русский

При плоскости слева инъективно отображение hf.map hg LinearMap.id если i инъективно, для IsTensorProduct f и g.

LaTeX

$$$\\forall i,\\; \\text{Injective}(i) \\Rightarrow [\\text{Module.Flat } R M_1] \\Rightarrow \\text{Injective}(hf.map hg LinearMap.id)$$$

Lean4

theorem of_flat_of_isBaseChange {f : M →ₗ[R] N} (hf : IsBaseChange S f) {x : R} (reg : IsSMulRegular M x) :
    IsSMulRegular N (algebraMap R S x) :=
  by
  have h := hf.map_id_injective_of_flat_left hf (LinearMap.lsmul R M x) reg
  rwa [hf.map_id_lsmul_eq_lsmul_algebraMap] at h

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