| import OAI.Algebra.Finitistic.Ordinary.OrdinaryTensorBimoduleResolution |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryRestrictionHomology |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory Limits |
| open scoped TensorProduct |
| namespace LittleFinitistic.OrdinarySimulation.TensorResolution |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] (l : ℕ) |
|
|
| abbrev bimoduleTerminal := RetainedBimodule k B (B ⊗[k] ChainAlgebra k l) (TerminalTensor B) |
| abbrev bimoduleTerminalObject : ModuleCat.{u} (B ⊗[k] (B ⊗[k] ChainAlgebra k l)ᵐᵒᵖ) := |
| ModuleCat.of _ (bimoduleTerminal k B l) |
|
|
| def bimoduleAugmentationHom : (bimoduleComplex k B l).X l ⟶ bimoduleTerminalObject k B l := |
| ModuleCat.ofHom (RetainedBimodule.map (k := k) (S := B) (A := B ⊗[k] ChainAlgebra k l) |
| (augmentation k B l) (augmentation_left k B l)) |
|
|
| /-- The actual augmentation of the genuine finite bimodule resolution. -/ |
| def bimoduleAugmentationMap : bimoduleComplex k B l ⟶ |
| (HomologicalComplex.single _ (.up ℕ) l).obj (bimoduleTerminalObject k B l) := |
| HomologicalComplex.mkHomToSingle (bimoduleAugmentationHom k B l) (by |
| intro i hi |
| change i+1=l at hi |
| subst l |
| change CochainComplex.of.d _ _ i (i+1) ≫ bimoduleAugmentationHom k B (i+1) = 0 |
| erw [CochainComplex.of_d] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| exact congrArg (fun f => f x) (augmentation_comp k B i)) |
|
|
| def forgetBimoduleTerminal : (forgetBimodule k B l).obj (bimoduleTerminalObject k B l) ≅ |
| ModuleCat.of (B ⊗[k] ChainAlgebra k l)ᵐᵒᵖ (TerminalTensor B) := by |
| let e : (forgetBimodule k B l).obj (bimoduleTerminalObject k B l) ≃ₗ[(B ⊗[k] ChainAlgebra k l)ᵐᵒᵖ] |
| TerminalTensor B := |
| { toFun := fun x => x |
| invFun := fun x => x |
| left_inv := fun _ => rfl |
| right_inv := fun _ => rfl |
| map_add' := fun _ _ => rfl |
| map_smul' := fun r x => by |
| change (1:B) * (TerminalTensor.value B x * terminalAugmentation k B l r.unop) = |
| TerminalTensor.value B x * terminalAugmentation k B l r.unop |
| exact one_mul _ } |
| exact e.toModuleIso |
|
|
| def forgetAugmentationTargetIso : |
| ((forgetBimodule k B l).mapHomologicalComplex (.up ℕ)).obj |
| ((HomologicalComplex.single _ (.up ℕ) l).obj (bimoduleTerminalObject k B l)) ≅ |
| (HomologicalComplex.single _ (.up ℕ) l).obj |
| (ModuleCat.of (B ⊗[k] ChainAlgebra k l)ᵐᵒᵖ (TerminalTensor B)) := |
| (HomologicalComplex.singleMapHomologicalComplex (forgetBimodule k B l) (.up ℕ) l).app |
| (bimoduleTerminalObject k B l) ≪≫ |
| (HomologicalComplex.single _ (.up ℕ) l).mapIso (forgetBimoduleTerminal k B l) |
| end LittleFinitistic.OrdinarySimulation.TensorResolution |
|
|
| end |
|
|
| end OAI |
|
|
