openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Simplicial /FirstBarComponentsFinite.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Homology.AbelianHomologyFinite | |
| namespace OAI | |
| section | |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory HomologicalComplex SimplicialObject Simplicial Opposite AlgebraicTopology | |
| namespace IntervalBar.Diagram | |
| open FreeChains SimplicialDiagonal | |
| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] | |
| noncomputable abbrev homologyRow (q:ℕ) : ChainComplex A ℕ := | |
| AlternatingFaceMapComplex.obj (simplicial (C:=C) ⋙ nerveFunctor ⋙ SSet.homologyFunctor Z q) | |
| omit [SymmetricCategory C] in | |
| lemma first_row_finite (q:ℕ) | |
| [Module.Finite ℤ ((bar (C:=C)).homology Z (1+q) : A)] | |
| (hf:∀p r,p+r=q+2 → 3≤p → Module.Finite ℤ ((homologyRow (C:=C) r).homology p)) : | |
| Module.Finite ℤ ((homologyRow (C:=C) q).homology 1) := by | |
| let X := simplicial (C:=C) ⋙ nerveFunctor | |
| let K := EilenbergZilber.twoComplex (uncurry X) | |
| have : Module.Finite ℤ ((K.total c).homology (1+q)) := by | |
| let e : ((bar (C:=C)).homology Z (1+q) : A) ≅ (K.total c).homology (1+q) := | |
| EilenbergZilber.homologyIsoN (uncurry X) (1+q) | |
| exact Module.Finite.equiv e.toLinearEquiv | |
| have : Module.Finite ℤ ((TotalFiniteness.row K q).homology 1) := by | |
| apply TotalFiniteness.finite_first K q | |
| intro p r hpr hp | |
| have : Module.Finite ℤ ((AlternatingFaceMapComplex.obj (X ⋙ SSet.homologyFunctor DiagonalResolution.Z r)).homology p) := hf p r hpr hp | |
| exact Module.Finite.equiv (homologyRowHomologyIso X p r).symm.toLinearEquiv | |
| exact Module.Finite.equiv (homologyRowHomologyIso X 1 q).toLinearEquiv | |
| lemma component_homology_finite (n:ℕ) | |
| [Module.Finite ℤ ((bar (C:=C)).homology Z 1 : A)] : | |
| Module.Finite ℤ ((nerve (SingleObj (Skeleton C))).homology Z n : A) := by | |
| have : Module.Finite ℤ ((homologyRow (C:=C) 0).homology 1) := | |
| first_row_finite 0 (by intro p r hpr hp; omega) | |
| have : Module.Finite ℤ ((nerve (SingleObj (Skeleton C))).homology Z 1 : A) := by | |
| let e : (homologyRow (C:=C) 0).homology 1 ≅ (nerve (SingleObj (Skeleton C))).homology Z 1 := | |
| (homologyFunctor A c 1).mapIso (componentRowChainIso (C:=C) ≪≫ (complexIso _).symm) | |
| exact Module.Finite.equiv e.toLinearEquiv | |
| exact TranslationNerve.homology_finite_of_one _ n | |
| lemma first_row_one_finite | |
| [Module.Finite ℤ ((bar (C:=C)).homology Z 1 : A)] | |
| [Module.Finite ℤ ((bar (C:=C)).homology Z 2 : A)] : | |
| Module.Finite ℤ ((homologyRow (C:=C) 1).homology 1) := by | |
| apply first_row_finite 1 | |
| intro p r hpr hp | |
| have hr:r=0:=by omega | |
| subst r | |
| have:=component_homology_finite (C:=C) 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 | |
| end IntervalBar.Diagram | |
| end | |
| end OAI | |