English
For any f: N → N', injectivity of f.lTensor M is equivalent to injectivity of f, provided M is faithfully flat.
Русский
При верноподобной M неравенство иньекции f равно иньекции f.lTensor M.
LaTeX
$$[simp] forall f: N → N' [Module.FaithfullyFlat R M] : (Function.Injective (f.lTensor M) ) \iff (Function.Injective f)$$
Lean4
/-- If `A` is a faithfully flat `R`-algebra, and `m` is a term of an `R`-module `M`,
then `1 ⊗ₜ[R] m = 0` if and only if `m = 0`. -/
@[simp]
theorem one_tmul_eq_zero_iff {A : Type*} [Ring A] [Algebra R A] [FaithfullyFlat R A] (m : M) :
(1 : A) ⊗ₜ[R] m = 0 ↔ m = 0 := by
constructor; swap
· rintro rfl; rw [tmul_zero]
intro h
let f : R →ₗ[R] M := (LinearMap.lsmul R M).flip m
suffices f = 0 by simpa [f] using DFunLike.congr_fun this 1
rw [Module.FaithfullyFlat.zero_iff_lTensor_zero R A]
ext a
apply_fun (a • ·) at h
rw [smul_zero, smul_tmul', smul_eq_mul, mul_one] at h
simpa [f]
