File size: 2,777 Bytes
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 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Algebra.Finitistic.Rectification.FiltrationExactness
import OAI.Algebra.Finitistic.Bimodule.BimoduleVertex
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
namespace LittleFinitistic.ModuleComplexExact
universe u
variable {S : Type u} [Ring S]
variable {X Y Z : CochainComplex (ModuleCat.{u} S) ℤ} (f : X ⟶ Y) (g : Y ⟶ Z)
theorem comp_zero (h : ∀ n, Function.Exact (f.f n) (g.f n)) : f ≫ g = 0 := by
ext n x
exact (h n).apply_apply_eq_zero x
def sequence (h : ∀ n, Function.Exact (f.f n) (g.f n)) :
ShortComplex (CochainComplex (ModuleCat.{u} S) ℤ) := .mk f g (comp_zero f g h)
theorem shortExact (h : ∀ n, Function.Injective (f.f n) ∧
Function.Exact (f.f n) (g.f n) ∧ Function.Surjective (g.f n)) :
(sequence f g (fun n => (h n).2.1)).ShortExact := by
apply HomologicalComplex.shortExact_of_degreewise_shortExact
intro n
exact ModuleCat.shortComplex_shortExact _ (h n).2.1 (h n).1 (h n).2.2
end LittleFinitistic.ModuleComplexExact
namespace ExternalTensor.Section4.SourceFiltration.Complex
universe u
variable {k B : Type u} [Field k] [Ring B] [Algebra k B]
variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R]
variable (s : Quadratic k B (A := A) (J := J) (R := R))
variable (D₀ D₁ D₂ : RightComplex B)
variable (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂))
variable (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0)
open Exactness LittleFinitistic
def source0 := ModuleComplexExact.sequence (i1 s D₀ D₁ D₂ f g H)
(pi0 s D₀ D₁ D₂ f g H)
(fun n => (exact_source_filtration s D₀ D₁ D₂ f g H n).1.2.1)
def source1 := ModuleComplexExact.sequence (i2 s D₀ D₁ D₂ f g)
(pi1 s D₀ D₁ D₂ f g)
(fun n => (v2_shortExact (data s D₀ D₁ D₂ f g) n).2.1)
theorem source0_shortExact : (source0 s D₀ D₁ D₂ f g H).ShortExact :=
ModuleComplexExact.shortExact _ _
(fun n => (exact_source_filtration s D₀ D₁ D₂ f g H n).1)
omit [Fintype R] in
theorem source1_shortExact : (source1 s D₀ D₁ D₂ f g).ShortExact :=
ModuleComplexExact.shortExact _ _ (v2_shortExact (data s D₀ D₁ D₂ f g))
/-- Every original source filtration remains short exact after taking a left vertex. -/
theorem vertex_shortExact (e : B) (he : e*e=e)
(S : ShortComplex (BiComplex k B)) (hS : S.ShortExact) :
(S.map ((BimoduleVertex.functor k B e).mapHomologicalComplex (.up ℤ))).ShortExact := by
apply HomologicalComplex.shortExact_of_degreewise_shortExact
intro n
exact BimoduleVertex.map_shortExact k B e he
(S.map (HomologicalComplex.eval _ _ n))
((HomologicalComplex.shortExact_iff_degreewise_shortExact S).1 hS n)
end ExternalTensor.Section4.SourceFiltration.Complex
end
end OAI
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