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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 | import OAI.Algebra.Finitistic.Selected.SelectedCohomologyVanishing
import OAI.Algebra.Finitistic.Localization.TwoDegreeFormalityNaturality
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
attribute [local instance] HasDerivedCategory.standard HasExt.standard
namespace LittleFinitistic.SelectedTensorSplitting
open SelectedFixedDiagram QuadraticEncoding
variable (N : Type) [AddCommGroup N] [Module ℂ N] [Module (B P) N]
/-- The actual derived object represented by the full B-valued fixed tensor complex. -/
abbrev object : DerivedCategory (ModuleCat (B P)) := DerivedCategory.Q.obj (SelectedTensor.complex N)
def homologyIso (n : ℤ) :
(DerivedCategory.homologyFunctor (ModuleCat (B P)) n).obj (object N) ≅
(SelectedTensor.complex N).homology n :=
(DerivedCategory.homologyFunctorFactors (ModuleCat (B P)) n).app (SelectedTensor.complex N)
/-- Support of the evaluated representative diagram implies actual derived
cohomological support of the full tensor object. -/
theorem twoDegrees
(h : ∀ n : ℤ, n ≠ 0 → n ≠ -3 → ∀ v : Fin 3,
IsZero ((SelectedTensor.source N v).homology n)) :
DerivedTruncation.TwoDegrees (object N) := by
intro n hn0 hn3
exact (homologyIso N n).isZero_iff.mpr
(SelectedCohomologyVanishing.cohomology_isZero N n (h n hn0 hn3))
/-- Apply the proved B-linear truncation splitting to the actual fixed tensor,
using full module cohomology models and source-diagram support. -/
def split (R : ModuleCat (B P))
(h0 : (SelectedTensor.complex N).homology 0 ≅ R)
(h3 : (SelectedTensor.complex N).homology (-3) ≅ R)
(hz : ∀ n : ℤ, n ≠ 0 → n ≠ -3 → ∀ v : Fin 3,
IsZero ((SelectedTensor.source N v).homology n)) :
object N ≅ (DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj R ⊞
((DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj R)⟦(3 : ℤ)⟧ :=
TwoDegreeFormality.splitIsoWithModels P (object N) (twoDegrees N hz) R R
(homologyIso N 0 ≪≫ h0) (homologyIso N (-3) ≪≫ h3)
end LittleFinitistic.SelectedTensorSplitting
end
end OAI
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