Module: Mathlib.RingTheory.LocalRing.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R M
inst✝¹ : Module.Finite R N
inst✝ : Free R N
l : M →ₗ[R] N
⊢ (∃ l', l' ∘ₗ l = LinearMap.id) ↔ Function.Injective ⇑(LinearMap.lTensor k l)

Tactic:
by
  constructor
  · intro ⟨l', hl⟩
    have : l'.lTensor (ResidueField R) ∘ₗ l.lTensor (ResidueField R) = .id := by
      rw [← LinearMap.lTensor_comp, hl, LinearMap.lTensor_id]
    exact Function.HasLeftInverse.injective ⟨_, LinearMap.congr_fun this⟩
  · intro h
    -- By `Module.free_of_lTensor_residueField_injective`, `k ⊗ l` injective => `N ⧸ l(M)` free.
    have := Module.free_of_lTensor_residueField_injective l (LinearMap.range l).mkQ
      (Submodule.mkQ_surjective _) l.exact_map_mkQ_range h
    -- Hence `l(M)` is projective because `0 → l(M) → N → N ⧸ l(M) → 0` splits.
    have : Module.Projective R (LinearMap.range l) := by
      have := (Exact.split_tfae (LinearMap.exact_subtype_mkQ (LinearMap.range l))
        Subtype.val_injective (Submodule.mkQ_surjective _)).out 0 1
      obtain ⟨l', hl'⟩ := this.mp
         (Module.projective_lifting_property _ _ (Submodule.mkQ_surjective _))
      exact Module.Projective.of_split _ _ hl'
    -- Then `0 → ker l → M → l(M) → 0` splits.
    obtain ⟨l', hl'⟩ : ∃ l', l' ∘ₗ (LinearMap.ker l).subtype = LinearMap.id := by
      have : Function.Exact (LinearMap.ker l).subtype
          (l.codRestrict (LinearMap.range l) (LinearMap.mem_range_self l)) := by
        rw [LinearMap.exact_iff, LinearMap.ker_rangeRestrict, Submodule.range_subtype]
      have := (Exact.split_tfae this
        Subtype.val_injective (fun ⟨x, y, e⟩ ↦ ⟨y, Subtype.ext e⟩)).out 0 1
      exact this.mp (Module.projective_lifting_property _ _ (fun ⟨x, y, e⟩ ↦ ⟨y, Subtype.ext e⟩))
    have : Module.Finite R (LinearMap.ker l) := by
      refine Module.Finite.of_surjective l' ?_
      exact Function.HasRightInverse.surjective ⟨_, DFunLike.congr_fun hl'⟩
    -- And tensoring with `k` preserves the injectivity of the first arrow.
    -- That is, `k ⊗ ker l → k ⊗ M` is also injective.
    have H : Function.Injective ((LinearMap.ker l).subtype.lTensor k) := by
      apply_fun (LinearMap.lTensor k) at hl'
      rw [LinearMap.lTensor_comp, LinearMap.lTensor_id] at hl'
      exact Function.HasLeftInverse.injective ⟨l'.lTensor k, DFunLike.congr_fun hl'⟩
    -- But by assumption `k ⊗ M → k ⊗ l(M)` is already injective, so `k ⊗ ker l = 0`.
    have : Subsingleton (k ⊗[R] LinearMap.ker l) := by
      refine (subsingleton_iff_forall_eq 0).mpr fun y ↦ H (h ?_)
      rw [map_zero, map_zero, ← LinearMap.comp_apply, ← LinearMap.lTensor_comp,
        l.exact_subtype_ker_map.linearMap_comp_eq_zero, LinearMap.lTensor_zero,
        LinearMap.zero_apply]
    -- By Nakayama's lemma, `l` is injective.
    have : Function.Injective l := by
      rwa [← LinearMap.ker_eq_bot, ← Submodule.subsingleton_iff_eq_bot,
        ← IsLocalRing.subsingleton_tensorProduct (R := R)]
    -- Whence `M ≃ l(M)` is projective and the result follows.
    have := (Exact.split_tfae l.exact_map_mkQ_range this (Submodule.mkQ_surjective _)).out 0 1
    rw [← this]
    exact Module.projective_lifting_property _ _ (Submodule.mkQ_surjective _)

Goal state after: