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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.SelectedCountCohomology
import OAI.Algebra.Finitistic.Iteration.IntegerIterateExtinction
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.SelectedCountExtinction
open SelectedOrdinaryAlgebra SelectedBaseIterates SelectedCountCohomology
local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) :=
HasDerivedCategory.standard (ModuleCat.{0} D)
/-- After actual tensor extinction, all later original count summands vanish. -/
theorem tail_zero (Y : FiniteSelectionFamily.S.{0}) (t r : ℕ) (htr : t ≤ r)
(h : IsZero (object Y t)) : IsZero (count Y r) :=
(SelectedCountCohomology.isZero_iff Y r).mpr
(DerivedBar.IterateSupport.zero_of_le ℂ D X (input Y) t r htr h)
/-- A zero iterate gives one finite cohomological window for the entire original
count family, including the infinitely many counts that have already vanished. -/
theorem uniform_bounded (Y : FiniteSelectionFamily.S.{0}) (t : ℕ)
(h : IsZero (object Y t)) (r : ℕ) :
(count Y r).IsGE (-((t * (3 * length + 5) : ℕ) : ℤ)) ∧ (count Y r).IsLE 0 := by
have := (SelectedCountCohomology.bounded Y r).1
have := (SelectedCountCohomology.bounded Y r).2
constructor
· by_cases htr : t ≤ r
· apply (DerivedCategory.isGE_iff _ _).mpr
intro i _
exact (DerivedCategory.homologyFunctor (ModuleCat D) i).map_isZero (tail_zero Y t r htr h)
· apply DerivedCategory.TStructure.t.isGE_of_ge (count Y r)
(-((t * (3 * length + 5) : ℕ) : ℤ)) (-((r * (3 * length + 5) : ℕ) : ℤ)) ?_
have hm := Nat.mul_le_mul_right (3 * length + 5) (show r ≤ t by omega)
exact neg_le_neg (by exact_mod_cast hm)
· exact DerivedCategory.TStructure.t.isLE_of_le (count Y r) (-(r : ℤ)) 0 (by omega)
/-- Every original count has zero cohomology outside that single finite window. -/
theorem uniform_homology_zero (Y : FiniteSelectionFamily.S.{0}) (t : ℕ)
(h : IsZero (object Y t)) (r : ℕ) (i : ℤ)
(hi : i < -((t * (3 * length + 5) : ℕ) : ℤ) ∨ 0 < i) :
IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) i).obj (count Y r)) := by
have := (uniform_bounded Y t h r).1
have := (uniform_bounded Y t h r).2
exact hi.elim (fun h => DerivedCategory.isZero_of_isGE _ _ i h)
(fun h => DerivedCategory.isZero_of_isLE _ 0 i h)
end LittleFinitistic.SelectedCountExtinction
end
end OAI
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