| import OAI.Algebra.Finitistic.Bimodule.BimoduleBarStalk |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.BimoduleTotal |
| open ExternalTensor.Section4 |
| universe u |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] |
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} B) := |
| HasDerivedCategory.standard (ModuleCat.{u} B) |
| variable (P : BiComplex k B) (X : ModuleCat.{u} (Enveloping k B)) (length : ℕ) |
| variable (ε : P ⟶ bimoduleStalk k B X) (hε : QuasiIso ε) |
| variable (hproj : ∀ n, Module.Projective Bᵐᵒᵖ (ComplexTensor.rightTerm k B (right k B P) n)) |
| variable (hbound : ∀ n, n < -(length : ℤ) ∨ 0 < n → IsZero (P.X n)) |
|
|
| / |
| API. Augmentation, quasi-isomorphism, projectivity and support are all explicit. -/ |
| def ofNonpositiveResolution : AugmentedProjective k B P X where |
| data := { lower := -(length : ℤ), upper := 0, bounded := hbound, projective := hproj } |
| augmentation := ε |
| quasiIso := hε |
|
|
| local notation "R" => ofNonpositiveResolution P X length ε hε hproj hbound |
|
|
| / |
| with the genuine bar-derived functor after chainToIntegerDerived. -/ |
| def nonpositiveResolutionBarIso : |
| DerivedBar.chainToIntegerDerived B ⋙ (R).data.derived ≅ |
| DerivedBar.derivedTensor k B (barModule (X := X)) ⋙ DerivedBar.chainToIntegerDerived B := |
| nativeBarIso R (zeroDegreeIndex R (by dsimp [ofNonpositiveResolution]; omega) (by rfl)) |
| (zeroDegreeIndex_eq R (by dsimp [ofNonpositiveResolution]; omega) (by rfl)) |
|
|
| / |
| by the genuine Section 6 indexed multibar. All comparison data is constructed. -/ |
| def nonpositiveResolutionPowerIso (N : ModuleCat.{u} B) (r : ℕ) : |
| ((R).data.derived.obj^[r]) (DerivedCategory.Q.obj (singleInput B N)) ≅ |
| DerivedCategory.Q.obj ((DerivedBar.integerExtension B).obj |
| (DerivedBar.RecursiveTotal.indexedMultibar k B (barModule (X := X)) N r)) := |
| (eqToIso (nativeIterate_obj R r _)).symm ≪≫ |
| nativeStalkIndexedIso R (by dsimp [ofNonpositiveResolution]; omega) (by rfl) N r |
| end LittleFinitistic.BimoduleTotal |
|
|
| end |
|
|
| end OAI |
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|
