File size: 2,895 Bytes
0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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
|