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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 | import OAI.Algebra.Finitistic.Bimodule.BimoduleBarResolution
import OAI.Algebra.Finitistic.Tensor.BalancedProjectiveLeft
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.BimoduleTotal
open ExternalTensor.Section4
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
abbrev bimoduleStalk (X : ModuleCat.{u} (Enveloping k B)) : BiComplex k B :=
(HomologicalComplex.single (ModuleCat (Enveloping k B)) (.up ℤ) 0).obj X
/-- Actual bounded right-projective resolution data, including the bimodule
augmentation and its quasi-isomorphism; no tensor comparison is assumed. -/
structure AugmentedProjective (P : BiComplex k B) (X : ModuleCat.{u} (Enveloping k B)) where
data : BoundedProjective k B P
augmentation : P ⟶ bimoduleStalk k B X
quasiIso : QuasiIso augmentation
namespace AugmentedProjective
variable {k B} {P : BiComplex k B} {X : ModuleCat.{u} (Enveloping k B)}
variable (res : AugmentedProjective k B P X)
/-- The augmentation induces the concrete finite-total transformation. -/
def tensorAugmentation : functor k B P res.data.degrees
(intervalDegrees_strictMono res.data.lower res.data.upper) ⟶
functor k B (bimoduleStalk k B X) res.data.degrees
(intervalDegrees_strictMono res.data.lower res.data.upper) :=
leftTensorNat res.data.degrees (intervalDegrees_strictMono res.data.lower res.data.upper)
res.augmentation
/-- Each horizontal augmentation row is a B-linear quasi-isomorphism on a
projective left-module input, including every term of the total bar. -/
theorem projectiveRow_quasiIso (N : ModuleCat.{u} B) [Projective N] :
QuasiIso (((BimoduleBalancedTensor.functor k B
(BalancedTensor.ScalarLeftModule k B N)).mapHomologicalComplex (.up ℤ)).map
res.augmentation) := by
let : Module.Projective B (BalancedTensor.ScalarLeftModule k B N) :=
inferInstanceAs (Module.Projective B N)
let := res.quasiIso
exact BimoduleBalancedTensor.projectiveInput_quasiIso k B _ res.augmentation
/-- In particular the actual resolution supplied by DerivedBar has all the
projective rows needed for the remaining finite-total quasi-isomorphism proof. -/
theorem barRow_quasiIso (Z : ChainComplex (ModuleCat.{u} B) ℕ) (n : ℕ) :
QuasiIso (((BimoduleBalancedTensor.functor k B
(BalancedTensor.ScalarLeftModule k B ((DerivedBar.barTotal k B Z).X n))).mapHomologicalComplex
(.up ℤ)).map res.augmentation) := by
let : Projective ((DerivedBar.barTotal k B Z).X n) := DerivedBar.barTotal_projective k B Z n
exact projectiveRow_quasiIso res _
end AugmentedProjective
end LittleFinitistic.BimoduleTotal
end
end OAI
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