Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
⊢ (H1CoresCoinf A S).Exact

Tactic:
rw [ShortComplex.moduleCat_exact_iff_ker_sub_range]
  intro x hx
  induction x using H1_induction_on with | @h x =>
  simp only [H1CoresCoinf_X₂, H1CoresCoinf_X₃, LinearMap.mem_ker, H1CoresCoinf_g,
    H1π_comp_map_apply (QuotientGroup.mk' S)] at hx
/- Let `x : Z₁(G, A)` map to 0 in `H₁(G, ⧸ S, A_S)`. Pick `y : C₂(G ⧸ S, A_S)` such that `d(y)`
equals `Z₁(π, π)(x) : Z₁(G ⧸ S, A_S)`. -/
  rcases (H1π_eq_zero_iff _).1 hx with ⟨y, hy⟩
/- Then `Z₁(π, Id)(x) : Z₁(G, A_S)` maps to 0 in `H₁(G ⧸ S, A_S)`. We know
`H₁(S, A_S) ⟶ H₁(G, A_S) ⟶ H₁(G ⧸ S, A_S)` is exact by `H1CoresCoinfOfTrivial_exact`, since
`S` acts trivially on `A_S`. So we can choose `z : Z₁(S, A_S)` with the same homology class as
`Z₁(π, Id)(π)` in `H₁(G, A_S)`. -/
  rcases @(ShortComplex.moduleCat_exact_iff_ker_sub_range _).1
    (H1CoresCoinfOfTrivial_exact (toCoinvariants A S) S)
    (H1π _ <| mapCycles₁ (MonoidHom.id G) (Rep.toCoinvariantsMkQ A S) x) (by
    simpa only [H1CoresCoinfOfTrivial_X₂, H1CoresCoinfOfTrivial_X₃, H1CoresCoinfOfTrivial_g,
      Iso.refl_inv, LinearMap.mem_ker, H1π_comp_map_apply (QuotientGroup.mk' S),
      ← mapCycles₁_comp_apply (x := x)] using! hx) with ⟨z, hz⟩
  induction z using H1_induction_on with | @h z =>
  simp only [H1CoresCoinfOfTrivial_X₂, H1CoresCoinfOfTrivial_X₁, H1CoresCoinfOfTrivial_f] at hz
  rw [H1π_comp_map_apply] at hz
/- Choose `w : C₂(G, A_S)` such that `d(w) = Z₁(i, Id)(z) - Z₁(Id, π)(x)`. -/
  rcases (H1π_eq_iff _ _).1 hz with ⟨w, hzw⟩
/- Choose `Z : C₁(S, A)` mapping to `z : C₁(S, A_S)`, and `W : C₂(G, A)` mapping to
`w : C₂(G, A_S)`. -/
  rcases mapRange_surjective (Coinvariants.mk _) (map_zero _)
    (Coinvariants.mk_surjective _) z.1 with ⟨Z, hZ⟩
  rcases mapRange_surjective (Coinvariants.mk _) (map_zero _)
    (Coinvariants.mk_surjective _) w with ⟨W, hW⟩
/- Let `b : C₁(G, A)` denote `x + d(W) - C₁(i, Id)(z)`. -/
  let b : G →₀ A := (x.1 : G →₀ A) + d₂₁ A W - lmapDomain _ k S.subtype Z
/- Then `b` has coefficients in `I(S)A := ⟨{ρ(s)(a) - a | s ∈ S, a ∈ A}⟩`, since
`C₁(G, I(S)(A)) ⟶ C₁(G, A) ⟶ C₁(G, A_S)` is exact, and `b` is in the kernel of the second map. -/
  have hb : ∀ g, b g ∈ Coinvariants.ker (A.ρ.comp S.subtype) :=
    fun g => (Coinvariants.mk_eq_iff _).1 <| by
      have := Finsupp.ext_iff.1 (congr($((mapShortComplexH1 (B := toCoinvariants A S)
        (MonoidHom.id G) (toCoinvariantsMkQ A S)).comm₁₂.symm) W)) g
      simp only [shortComplexH1, mapShortComplexH1_τ₂, ModuleCat.ofHom_comp, MonoidHom.coe_id,
        lmapDomain_id, ModuleCat.ofHom_id, res_obj_ρ, hom_ofHom, Category.id_comp,
        mapShortComplexH1_τ₁, Prod.map_id, ModuleCat.hom_comp, ModuleCat.hom_ofHom,
        LinearMap.coe_comp, Function.comp_apply, mapRange.linearMap_apply, mapRange_apply] at this
      simp only [← mapRange_apply (f := Coinvariants.mk <| A.ρ.comp S.subtype)
        (hf := map_zero _) (a := g), ← mapRange.linearMap_apply (R := k)]
      simp [← mapDomain_mapRange, hZ, this, hW, hzw, coe_mapCycles₁ S.subtype,
        coe_mapCycles₁ (MonoidHom.id G)]
/- Let `β` be `b` considered as an element of `C₁(G, I(S)(A))`, so that `C₁(Id, i)(β) = b`. -/
  let β : G →₀ Coinvariants.ker (A.ρ.comp S.subtype) :=
    mapRange (Function.invFun <| (Coinvariants.ker (A.ρ.comp S.subtype)).subtype)
    (Function.leftInverse_invFun Subtype.val_injective (0 : Coinvariants.ker _)) b
  have hβb : mapRange Subtype.val rfl β = b := Finsupp.ext fun g => Subtype.ext_iff.1 <|
    Function.leftInverse_invFun Subtype.val_injective ⟨b g, hb g⟩
/- Then, since the image of `C₁(G, I(S)A)` in `C₁(G, A)⧸B₁(G, A)` is contained in the image of
`C₁(S, A)` by `comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top`, we can choose
`α : C₁(S, A)`, `δ : C₂(G, A)` such that `d(δ) = Z₁(i, Id)(α) - Z₁(Id, i)(β)`. -/
  rcases eq_top_iff.1 (comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top A S)
    (by trivial : β ∈ ⊤) with ⟨(α : S →₀ A), hα⟩
  dsimp only [ModuleCat.hom_comp] at hα
  rcases (moduleCat_pOpcycles_eq_iff _ _ _).1 hα with ⟨(δ : G × G →₀ A), hβ⟩
/- Then, by assumption, `d(W + δ) = C₁(i, Id)(α + Z) - x`. -/
  have hαZ : d₂₁ A (W + δ) = mapDomain Subtype.val (α + Z) - x := by
    simp_all [shortComplexH1, coinvariantsShortComplex_X₁,
      ← sub_add, ← sub_sub, sub_add_eq_add_sub, mapDomain_add, b]
/- So we claim that `α + Z` is an element of `Z₁(S, A)` which differs from `x` by a boundary in
`Z₁(G, A)`. -/
  use H1π _ ⟨α + Z, ?_⟩
/- Indeed, by `hαZ`, `d(W + δ)` is the desired boundary: -/
  · simp only [H1CoresCoinf_X₂, H1CoresCoinf_X₁, H1CoresCoinf_f, H1π_comp_map_apply]
    refine (H1π_eq_iff _ _).2 ⟨W + δ, ?_⟩
    simp [hαZ, mapCycles₁_hom]
/- And `α + Z` is a cycle, since `d(W + δ) + x` is. -/
  · rw [mem_cycles₁_iff]
    have : x + d₂₁ A (W + δ) ∈ cycles₁ A := Submodule.add_mem _ x.2 (d₂₁_apply_mem_cycles₁ _)
    rwa [eq_sub_iff_add_eq'.1 hαZ, mem_cycles₁_iff, sum_mapDomain_index_inj
      Subtype.val_injective, sum_mapDomain_index_inj Subtype.val_injective] at this

Goal state after: