File size: 4,203 Bytes
ae00908 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Algebra.Finitistic.Selected.SelectedReferenceCohomology
import OAI.Algebra.Finitistic.Selected.SelectedTensorSplitting
import OAI.Algebra.Finitistic.Selected.SelectedChainAction
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
attribute [local instance] HasDerivedCategory.standard HasExt.standard
namespace LittleFinitistic.SelectedRealization
open SelectedFixedDiagram QuadraticEncoding SelectedNativeEvaluation
variable {Y : Type} [AddCommGroup Y] [Module ℂ Y] [FiniteDimensional ℂ Y]
[Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y]
local instance selectedModule : Module SelectionRing (H (Y := Y)) :=
inferInstanceAs (Module SelectionRing (selectionStep (ModuleCat.of SelectionRing Y)))
local instance selectedTower : IsScalarTower ℂ SelectionRing (H (Y := Y)) :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
abbrev input := (SelectedChainHomology.representation (Y := Y)).Evaluation
abbrev selectedRepresentation := SourceActionEvaluation.representation (Y := H (Y := Y)) encoding
abbrev reference := EvaluationVertexDiagram.module (selectedRepresentation (Y := Y))
omit [FiniteDimensional ℂ Y] in
lemma selected_action (a : Fin (encoding.generators+1)) :
EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a =
HAction (Y := Y) (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
rw [HAction_apply]
refine Fin.cases ?_ (fun j => ?_) a
· change x = (1 : SelectionRing) • x
rw [one_smul]
· rfl
def coordinates (n : ℤ) (hn : n = 0 ∨ n = -3) :
∀ v : Fin 3, (SelectedTensor.source (input (Y := Y)) v).homology n ≅ H (Y := Y) :=
Fin.cases (SelectedChainHomology.D0Iso n hn)
(Fin.cases (SelectedChainHomology.D1Iso n hn)
(fun _ => SelectedChainHomology.D2Iso n hn))
lemma coordinates01 (a : Fin (encoding.generators+1)) (n : ℤ) (hn : n = 0 ∨ n = -3) :
homologyMap (((BalancedTensor.tensorFunctor ℂ (B P) (input (Y := Y))).mapHomologicalComplex (.up ℤ)).map
(SelectedRectification.f a)) n ≫ (coordinates (Y := Y) n hn 1).hom =
(coordinates (Y := Y) n hn 0).hom ≫ EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a := by
erw [selected_action (Y := Y)]
exact SelectedChainHomology.fixed_arrow01 a n hn
lemma coordinates12 (a : Fin (encoding.generators+1)) (n : ℤ) (hn : n = 0 ∨ n = -3) :
homologyMap (((BalancedTensor.tensorFunctor ℂ (B P) (input (Y := Y))).mapHomologicalComplex (.up ℤ)).map
(SelectedRectification.g a)) n ≫ (coordinates (Y := Y) n hn 2).hom =
(coordinates (Y := Y) n hn 1).hom ≫ EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a := by
erw [selected_action (Y := Y)]
exact SelectedChainHomology.fixed_arrow12 a n hn
/-- The entire cohomology B-module is the manuscript's M(H(Y)), in both degrees. -/
def cohomology (n : ℤ) (hn : n = 0 ∨ n = -3) :
(SelectedTensor.complex (input (Y := Y))).homology n ≅ reference (Y := Y) :=
SelectedReferenceCohomology.iso _ (selectedRepresentation (Y := Y)) n (coordinates (Y := Y) n hn)
(fun a => coordinates01 a n hn) (fun a => coordinates12 a n hn)
lemma source_zero (n : ℤ) (hn0 : n ≠ 0) (hn3 : n ≠ -3) (v : Fin 3) :
IsZero ((SelectedTensor.source (input (Y := Y)) v).homology n) := by
fin_cases v
· exact SelectedChainHomology.D0_zero n hn0 hn3
· exact SelectedChainHomology.D1_zero n hn0 hn3
· exact SelectedChainHomology.D2_zero n hn0 hn3
/-- The one fixed selecting bimodule realizes H on every finite-dimensional Y.
This is an isomorphism in the full B-module derived category. -/
def realization :
SelectedTensorSplitting.object (input (Y := Y)) ≅
(DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj (reference (Y := Y)) ⊞
((DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj (reference (Y := Y)))⟦(3 : ℤ)⟧ :=
SelectedTensorSplitting.split _ _ (cohomology 0 (Or.inl rfl))
(cohomology (-3) (Or.inr rfl)) source_zero
end LittleFinitistic.SelectedRealization
end
end OAI
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