| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleTotalMaps | |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryRestrictionHomology | |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryTensorTotalQuasiIso | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.BimoduleComplexTensor | |
| universe u | |
| variable (k S A : Type u) [Field k] [Ring S] [Ring A] [Algebra k S] [Algebra k A] | |
| variable (Q : CochainComplex (ModuleCat.{u} (S ⊗[k] Aᵐᵒᵖ)) ℤ) | |
| variable {l : ℕ} (p : Fin l → ℤ) (hp : StrictMono p) | |
| /-- The actual tensor total is a functor retaining its outside action on every complex. -/ | |
| def totalFunctor : CochainComplex (ModuleCat.{u} A) ℤ ⥤ CochainComplex (ModuleCat.{u} S) ℤ where | |
| obj Z := total k S A Q Z p hp | |
| map f := totalMap k S A Q f p hp | |
| map_id Z := by | |
| apply HomologicalComplex.Hom.ext | |
| funext n | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro z | |
| funext i | |
| change termInputMap k S A Q (𝟙 Z) (p i) (n + -(p i)) (z i) = z i | |
| rw [termInputMap_id] | |
| rfl | |
| map_comp f g := by | |
| apply HomologicalComplex.Hom.ext | |
| funext n | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro z | |
| funext i | |
| change termInputMap k S A Q (f ≫ g) (p i) (n + -(p i)) (z i) = | |
| termInputMap k S A Q g (p i) (n + -(p i)) (termInputMap k S A Q f (p i) (n + -(p i)) (z i)) | |
| rw [termInputMap_comp] | |
| rfl | |
| /-- The forgetful chain comparison is natural in all input complexes. -/ | |
| def forgetTotalFunctorIso : | |
| totalFunctor k S A Q p hp ⋙ (ModuleCat.restrictScalars (algebraMap k S)).mapHomologicalComplex (.up ℤ) ≅ | |
| ComplexTensor.finiteTotalFunctor k A (rightComplex k S A Q) p hp := | |
| NatIso.ofComponents (fun Z => forgetTotalIso k S A Q Z p hp) | |
| (fun f => forgetTotalIso_natural k S A Q f p hp) | |
| /-- The genuine outside-action tensor preserves arbitrary quasi-isomorphisms when its right terms | |
| are projective. There are no restrictions on either input complex. -/ | |
| theorem totalFunctor_quasiIso | |
| [∀ i, Module.Projective Aᵐᵒᵖ (ComplexTensor.rightTerm k A (rightComplex k S A Q) (p i))] | |
| {Z Z' : CochainComplex (ModuleCat.{u} A) ℤ} (f : Z ⟶ Z') [QuasiIso f] : | |
| QuasiIso ((totalFunctor k S A Q p hp).map f) := by | |
| apply (RestrictionHomology.quasiIso_iff (algebraMap k S) _).mp | |
| let g := ((ModuleCat.restrictScalars (algebraMap k S)).mapHomologicalComplex (.up ℤ)).map | |
| (totalMap k S A Q f p hp) | |
| have := ComplexTensor.finiteTotalMap_quasiIso k A (rightComplex k S A Q) f p hp | |
| have hg : QuasiIso (g ≫ (forgetTotalIso k S A Q Z' p hp).hom) := by | |
| dsimp only [g] | |
| rw [forgetTotalIso_natural] | |
| infer_instance | |
| exact (quasiIso_iff_comp_right g (forgetTotalIso k S A Q Z' p hp).hom).mp hg | |
| end LittleFinitistic.BimoduleComplexTensor | |
| end | |
| end OAI | |