| import OAI.Algebra.Finitistic.Iteration.BinomialVanishing |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits |
| noncomputable section |
| namespace ExternalTensor.Binomial |
| universe v u w |
| variable {C : Type u} [Category.{v} C] [Preadditive C] |
| [HasFiniteBiproducts C] [HasBinaryBiproducts C] |
| [HasShift C ℤ] [∀ n : ℤ, (shiftFunctor C n).Additive] |
| variable {S : Type w} (F : C ⥤ C) [F.Additive] [F.CommShift ℤ] |
| (M : S → C) (H : S → S) (b : ℤ) |
|
|
| / |
| def evenIteration |
| (split : ∀ Y, F.obj (F.obj (M Y)) ≅ (M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+3⟧) |
| (j : ℕ) (Y : S) : |
| (F.obj^[2*j]) (M Y) ≅ |
| ⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) => |
| (M ((H^[j]) Y))⟦(j : ℤ)*b + 3*(q.val : ℤ)⟧ := by |
| have he : (F ⋙ F).obj = F.obj ∘ F.obj := rfl |
| simpa only [he, binomialObject, Function.iterate_mul, Function.iterate_succ, |
| Function.iterate_zero, Function.id_comp, Functor.comp_obj] using |
| binomialIteration (F ⋙ F) M H b 3 split j Y |
|
|
| / |
| theorem evenIteration_isZero_iff |
| (split : ∀ Y, F.obj (F.obj (M Y)) ≅ (M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+3⟧) |
| (j : ℕ) (Y : S) : |
| IsZero ((F.obj^[2*j]) (M Y)) ↔ IsZero (M ((H^[j]) Y)) := |
| (evenIteration F M H b split j Y).isZero_iff.trans |
| (binomialObject_isZero_iff b 3 j _) |
|
|
| / |
| theorem even_extinction_pair |
| (split : ∀ Y, F.obj (F.obj (M Y)) ≅ (M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+3⟧) |
| (m : ℕ) (Y : S) |
| (vanishes : IsZero (M ((H^[m+1]) Y))) |
| (survives : ¬ IsZero (M ((H^[m]) Y))) : |
| IsZero ((F.obj^[2*(m+1)]) (M Y)) ∧ ¬ IsZero ((F.obj^[2*m]) (M Y)) := |
| ⟨(evenIteration_isZero_iff F M H b split (m+1) Y).mpr vanishes, |
| fun h => survives ((evenIteration_isZero_iff F M H b split m Y).mp h)⟩ |
| end ExternalTensor.Binomial |
|
|
| end |
|
|
| end OAI |
|
|
