| 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 |
|
|
