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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 | import OAI.Algebra.Finitistic.Selected.SelectedRectification
import OAI.Algebra.Finitistic.Localization.TensorCohomologyDiagram
import OAI.Algebra.Finitistic.Rectification.FullTensorProperties
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
namespace LittleFinitistic.SelectedTensor
open SelectedFixedDiagram SelectedRectification QuadraticEncoding
variable (N : Type) [AddCommGroup N] [Module ℂ N] [Module (B P) N]
/-- Evaluate the one fixed selecting bimodule, retaining its full left action. -/
abbrev complex := FullTensorRectification.tensor P D0 D1 D2 f g relations N
abbrev source (v : Fin 3) := FullTensorRectification.source P D0 D1 D2 N v
abbrev comparison (v : Fin 3) (n : ℤ) :=
TensorCohomologyDiagram.comparison P D0 D1 D2 f g relations N v n
/-- Every cohomology comparison is invertible for the fixed selected representatives. -/
instance comparison_isIso (v : Fin 3) (n : ℤ) : IsIso (comparison N v n) := by
obtain ⟨l₀,h₀,b₀⟩ := UnderlyingRepresentatives.bounded P fixed.D0
obtain ⟨l₁,h₁,b₁⟩ := UnderlyingRepresentatives.bounded P fixed.D1
obtain ⟨l₂,h₂,b₂⟩ := UnderlyingRepresentatives.bounded P fixed.D2
let lo := min l₀ (min l₁ l₂)
let hi := max h₀ (max h₁ h₂)
refine TensorCohomologyDiagram.comparison_isIso P D0 D1 D2 f g relations N v n lo hi
(fun n hn => b₀ n ?_) (fun n hn => b₁ n ?_) (fun n hn => b₂ n ?_)
(UnderlyingRepresentatives.projective P fixed.D0)
(UnderlyingRepresentatives.projective P fixed.D1)
(UnderlyingRepresentatives.projective P fixed.D2)
all_goals dsimp [lo,hi] at hn; omega
def comparisonIso (v : Fin 3) (n : ℤ) := asIso (comparison N v n)
/-- The actual tensor complex has bounded support. -/
theorem bounded : ∃ lo hi : ℤ, ∀ n, n < lo ∨ hi < n → IsZero ((complex N).X n) := by
obtain ⟨lo,hi,h⟩ := SelectedRectification.verified
exact ⟨lo-2,hi,FullTensorProperties.bounded ℂ (B P) N bimodule (lo-2) hi h.bounded⟩
/-- Every actual evaluated term is finite-dimensional. -/
theorem finite [FiniteDimensional ℂ N] (n : ℤ) :
FiniteDimensional ℂ ((ModuleCat.restrictScalars (algebraMap ℂ (B P))).obj ((complex N).X n)) := by
obtain ⟨lo,hi,h⟩ := SelectedRectification.verified
have : FiniteDimensional ℂ ((ExternalTensor.Section4.forgetK ℂ (B P)).obj (bimodule.X n)) := h.finite n
exact FullTensorProperties.finite ℂ (B P) N (bimodule.X n)
end LittleFinitistic.SelectedTensor
end
end OAI
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