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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 51 52 53 54 | import OAI.Algebra.Finitistic.Selection.FiniteSelectionFamily
import OAI.Algebra.Finitistic.Iteration.ObjectwiseSplitting
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
attribute [local instance] HasDerivedCategory.standard HasExt.standard
namespace LittleFinitistic.SelectedIteration
open SelectedFixedDiagram QuadraticEncoding FiniteSelectionFamily
variable (Phi : DerivedCategory (ModuleCat (B P)) ⥤ DerivedCategory (ModuleCat (B P)))
[Phi.Additive] [Phi.CommShift ℤ]
/- This is solely the remaining derived-tensor/stalk comparison. The realization
and exact reference cohomology have already been proved independently. -/
variable (comparison : ∀ Y : S, Phi.obj (derived Y) ≅
SelectedTensorSplitting.object (representation Y).Evaluation)
include comparison
def split (Y : S) : Phi.obj (derived Y) ≅ derived (H Y) ⊞ (derived (H Y))⟦(3 : ℤ)⟧ :=
comparison Y ≪≫ FiniteSelectionFamily.realization Y
/-- The exact binomial formula, instantiated with the actual finite module family. -/
def iteration (j : ℕ) (Y : S) :
(Phi.obj^[j]) (derived Y) ≅
⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) =>
(derived ((H^[j]) Y))⟦(3 : ℤ)*(q.val : ℤ)⟧ :=
ExternalTensor.Binomial.section4Iteration Phi derived H (split Phi comparison) j Y
private def shiftedSplit (Y : S) :
Phi.obj (derived Y) ≅ (derived (H Y))⟦(0 : ℤ)⟧ ⊞ (derived (H Y))⟦(0+3 : ℤ)⟧ :=
split Phi comparison Y ≪≫ biprod.mapIso (shiftZero ℤ (derived (H Y))).symm (Iso.refl _)
theorem isZero_iff (j : ℕ) (Y : S) :
IsZero ((Phi.obj^[j]) (derived Y)) ↔ Subsingleton (selectionIterate Y.val j) := by
rw [ExternalTensor.Binomial.iteration_isZero_iff Phi derived H 0 3 (shiftedSplit Phi comparison)]
rw [derived_zero, iterate_val]
/-- Actual Section 2 tests give arbitrarily long finite extinction for this tensor
functor, once its stated stalk comparison is instantiated. -/
theorem extinction (m : ℕ) [NeZero m] :
(∀ j < m, ¬ IsZero ((Phi.obj^[j]) (derived (test m)))) ∧
IsZero ((Phi.obj^[m]) (derived (test m))) := by
constructor
· intro j hj hz
have hs := (isZero_iff Phi comparison j (test m)).mp hz
have hn := selectionIterate_test_nontrivial m j hj
exact @not_subsingleton _ hn hs
· exact (isZero_iff Phi comparison m (test m)).mpr (selectionIterate_test_subsingleton m)
end LittleFinitistic.SelectedIteration
end
end OAI
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