File size: 3,438 Bytes
216fb31 | 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.Algebra.Finitistic.Bar.SplitProjectiveTail
import OAI.Algebra.Finitistic.Iteration.FiniteIntegerIterate
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
namespace DerivedBar.IterateSupport
universe u
variable (k D X : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[IsScalarTower k D X]
local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
variable (g : ℕ) (hg : ∀ M : ModuleCat.{u} D, HasProjectiveDimensionLE M g)
variable (N : ModuleCat.{u} D)
abbrev resolvedModel (r : ℕ) := (tensorTotalIterate k D X r).obj
((ChainComplex.single₀ (ModuleCat D)).obj N)
include hg in
/-- A bounded input remains cohomologically bounded after actual derived tensor.
The input is resolved by its genuine projective bar total. -/
theorem tensor_exact_above (Z : ChainComplex (ModuleCat.{u} D) ℕ) (m : ℕ)
(hZ : ∀ i, m < i → Z.ExactAt i) (n : ℕ) (hn : m+g+2 < n) :
((tensorTotalFunctor k D X).obj Z).ExactAt n := by
apply ProjectiveTail.map_exact_tail (barTotal k D Z) (BimoduleTensor.tensorFunctor k D X)
(m+1) g
· intro i hi
exact (exactAt_iff_of_quasiIsoAt (barTotalAugmentation k D Z) i).mpr (hZ i (by omega))
· exact barTotal_projective k D Z
· exact hg
· omega
include hg in
/-- The actual finite iteration has an explicit finite homological support bound.
This bound has a two-degree allowance per step; no optimal upper bound is claimed. -/
theorem resolvedModel_exact_above (r n : ℕ) (hn : r*(g+2) < n) :
(resolvedModel k D X N r).ExactAt n := by
induction r generalizing n with
| zero =>
cases n with
| zero => simp at hn
| succ n => exact ChainComplex.exactAt_succ_single_obj _ _
| succ r ih =>
apply tensor_exact_above k D X g hg (resolvedModel k D X N r) (r*(g+2)) ih n
simpa only [Nat.add_mul, Nat.one_mul, Nat.add_assoc] using hn
def resolvedModelIso (r : ℕ) : object k D X N r ≅
DerivedCategory.Q.obj ((integerExtension D).obj (resolvedModel k D X N r)) :=
(chainToIntegerDerived D).mapIso
((derivedTensorIterateIso k D X r).app ((ChainComplex.single₀ (ModuleCat D)).obj N)) ≪≫
(chainToIntegerDerivedIso D).app (resolvedModel k D X N r)
include hg in
/-- Actual derived tensor iterates have finite lower cohomological support over
any ring with the supplied genuine global projective-dimension bound. -/
theorem isGE_bound (r : ℕ) : (object k D X N r).IsGE (-((r*(g+2) : ℕ) : ℤ)) := by
apply (DerivedCategory.isGE_iff _ _).mpr
intro i hi
obtain ⟨n,rfl⟩ := Int.exists_eq_neg_ofNat (show i ≤ 0 by omega)
let e := (DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).mapIso
(resolvedModelIso k D X N r) ≪≫
(DerivedCategory.homologyFunctorFactors (ModuleCat D) (-(n : ℤ))).app
((integerExtension D).obj (resolvedModel k D X N r)) ≪≫
integerExtensionHomologyIso D (resolvedModel k D X N r) n
exact e.isZero_iff.mpr ((resolvedModel_exact_above k D X g hg N r n (by omega)).isZero_homology)
variable [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k Dᵐᵒᵖ X]
include hg in
theorem bounded_amplitude (r : ℕ) :
(object k D X N r).IsGE (-((r*(g+2) : ℕ) : ℤ)) ∧ (object k D X N r).IsLE 0 :=
⟨isGE_bound k D X g hg N r, inferInstance⟩
end DerivedBar.IterateSupport
end
end OAI
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