| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.FirstBarDegreeTwoFinite |
|
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| namespace OAI |
|
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| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite HomologicalComplex |
| namespace MarkedH1 |
| open FreeChains ComponentTranslation IntervalBar.Diagram |
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| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] |
| noncomputable def lowOneHomologyIso : |
| ((RegularCoefficient.double (homologyDiagram (C:=C) 1)).total c).homology 1 ≅ |
| (homologyRow (C:=C) 1).homology 2 := by |
| let K := (RegularCoefficient.double (homologyDiagram (C:=C) 1)).total c |
| let L := homologyRow (C:=C) 1 |
| let e : K.sc' 2 1 0 ≅ L.sc' 3 2 1 := |
| ShortComplex.isoMk (gradedIso 2) (gradedIso 1) (gradedIso 0) |
| (differential2 (C:=C)).symm (differential1 (C:=C)).symm |
| exact (ShortComplex.homologyFunctor A).mapIso |
| (K.isoSc' 2 1 0 (c.prev_eq' rfl) (c.next_eq' rfl) ≪≫ e ≪≫ |
| (L.isoSc' 3 2 1 (c.prev_eq' rfl) (c.next_eq' rfl)).symm) |
| lemma row_one_two_finite |
| [Module.Finite ℤ (ComponentStable.object (C:=C) 1:A)] |
| (hf:∀i,i≤1 → Module.Finite ℤ ((nerve (SingleObj (Skeleton C))).homology Z i:A)) : |
| Module.Finite ℤ ((homologyRow (C:=C) 1).homology 2) := by |
| have : Module.Finite ℤ (colimit (homologyDiagram (C:=C) 1):A) := inferInstanceAs |
| (Module.Finite ℤ (ComponentStable.object (C:=C) 1:A)) |
| have := RegularCoefficient.total_finite (homologyDiagram (C:=C) 1) 1 (by |
| intro i hi |
| have := hf i hi |
| exact Module.Finite.equiv (CommMonoidOpposite.homologyIso (Skeleton C) i).symm.toLinearEquiv) |
| exact Module.Finite.equiv (lowOneHomologyIso (C:=C)).toLinearEquiv |
| end MarkedH1 |
| namespace IntervalBar.Diagram |
| open FreeChains ComponentTranslation SimplicialDiagonal |
|
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| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] |
| lemma bar_low_finite |
| [Module.Finite ℤ (ComponentStable.object (C:=C) 1:A)] |
| [Module.Finite ℤ (ComponentStable.object (C:=C) 2:A)] |
| (hf:∀i,Module.Finite ℤ ((nerve (SingleObj (Skeleton C))).homology Z i:A)) |
| (n:ℕ) (hn:n≤3) : Module.Finite ℤ ((bar (C:=C)).homology Z n:A) := by |
| let X:=simplicial (C:=C) ⋙ nerveFunctor |
| apply EilenbergZilber.finite_diagonal (uncurry X) n |
| intro p q hpq |
| have hr : Module.Finite ℤ ((homologyRow (C:=C) q).homology p) := by |
| by_cases hq:q=0 |
| · subst q |
| have := hf p |
| let e : ((nerve (SingleObj (Skeleton C))).homology Z p:A) ≅ (homologyRow (C:=C) 0).homology p := |
| (homologyFunctor A c p).mapIso ((complexIso _) ≪≫ (componentRowChainIso (C:=C)).symm) |
| exact Module.Finite.equiv e.toLinearEquiv |
| · by_cases hp:p=0 |
| · subst p |
| have : Subsingleton ((homologyRow (C:=C) q).X 0) := ModuleCat.isZero_iff_subsingleton.mp |
| (nerve.positive_zero (fun U V:IntervalBar.Diagram C (Fin 1)=>⟨zeroHom C U V⟩) q hq) |
| have : Module.Finite ℤ ((homologyRow (C:=C) q).X 0) := inferInstance |
| exact BarHomology.finite_homology _ _ |
| · have hq2:q≤2:=by omega |
| interval_cases q |
| · contradiction |
| · rcases (show p=1 ∨ p=2 by omega) with rfl|rfl |
| · exact StableBarOne.row_one_finite 1 (by decide) |
| · exact MarkedH1.row_one_two_finite (fun i _=>hf i) |
| · have hp1:p=1:=by omega |
| subst p |
| exact StableBarOne.row_one_finite 2 (by decide) |
| have : Module.Finite ℤ ((AlgebraicTopology.AlternatingFaceMapComplex.obj (X ⋙ SSet.homologyFunctor DiagonalResolution.Z q)).homology p) := hr |
| exact Module.Finite.equiv (homologyRowHomologyIso X p q).symm.toLinearEquiv |
| end IntervalBar.Diagram |
|
|
| end |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite |
| namespace SimplicialDiagonal |
| open FreeChains |
|
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| variable (X:I ⥤ SSet) |
| lemma finite_forward (r:ℕ) |
| (hc:∀p,(X.obj p).IsConnected) |
| (hz:∀q,0<q → IsZero ((X.obj (op ⦋0⦌)).homology Z q)) |
| (hf:∀p q,q≤r → Module.Finite ℤ ((X.obj (op ⦋p⦌)).homology Z q : A)) : |
| ∀n,n≤r+1 → Module.Finite ℤ ((diagonal.obj X).homology Z n:A) := by |
| intro n hn |
| apply EilenbergZilber.finite_diagonal (uncurry X) n |
| intro p q hpq |
| by_cases hq:q=0 |
| · subst q |
| exact BarHomology.zeroRow_finite _ (rows_connected X hc) p |
| · have : Module.Finite ℤ (SSet.homology ((EilenbergZilber.bisimplicial (uncurry X)).obj (op ⦋p⦌)) Z q:A) := by |
| by_cases hp:p=0 |
| · subst p |
| have hz' := (hz q (by omega)).of_iso ((SSet.homologyFunctor Z q).mapIso (rowIso X (op ⦋0⦌))) |
| have : Subsingleton (SSet.homology ((EilenbergZilber.bisimplicial (uncurry X)).obj (op ⦋0⦌)) Z q:A) := |
| ModuleCat.isZero_iff_subsingleton.mp hz' |
| infer_instance |
| · have : Module.Finite ℤ ((SSet.homologyFunctor Z q).obj (X.obj (op ⦋p⦌)) : A) := hf p q (by omega) |
| exact Module.Finite.equiv ((SSet.homologyFunctor Z q).mapIso (rowIso X (op ⦋p⦌))).symm.toLinearEquiv |
| exact BarHomology.E2_finite _ p q |
| end SimplicialDiagonal |
| namespace IntervalBar.Diagram |
| open FreeChains BarFinitePower SimplicialDiagonal |
|
|
| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] |
| lemma bar₂_finite_forward (r:ℕ) |
| (hf:∀n,n≤r → Module.Finite ℤ ((bar (C:=C)).homology Z n : A)) : |
| ∀q,q≤r+1 → Module.Finite ℤ ((bar₂ (C:=C)).homology Z q : A) := by |
| apply finite_forward (rows₂ (C:=C)) r (fun _=>bar_connected _) zero_bar_acyclic |
| intro p q hq |
| apply singleRow_finite |
| exact fun i hi=>hf i (by omega) |
| lemma bar₃_finite_forward (r:ℕ) |
| (hf:∀n,n≤r → Module.Finite ℤ ((bar₂ (C:=C)).homology Z n : A)) : |
| ∀q,q≤r+1 → Module.Finite ℤ ((bar₃ (C:=C)).homology Z q : A) := by |
| apply finite_forward (rows₃ (C:=C)) r (fun _=>bar₂_connected) |
| (fun q hq=>by |
| change IsZero (SSet.homology (C:=A) (bar₂ (C:=Diagram C (Fin 1))) Z q) |
| exact bar₂_acyclic (fun _ _=>inferInstance) q hq) |
| intro p q hq |
| apply doubleRow_finite |
| exact fun i hi=>hf i (by omega) |
| lemma triple_finite_of_bar |
| (hf:∀n,n≤3 → Module.Finite ℤ ((bar (C:=C)).homology Z n : A)) : |
| ∀q,q≤5 → Module.Finite ℤ ((bar₃ (C:=C)).homology Z q : A) := |
| bar₃_finite_forward 4 (bar₂_finite_forward 3 hf) |
| end IntervalBar.Diagram |
|
|
| end |
|
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| end OAI |
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