| import OAI.Algebra.Finitistic.Bar.DerivedTensorBarSplitting |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex TrivSqZeroExt |
| open scoped DirectSum |
| noncomputable section |
| namespace DerivedBar |
| universe u |
| variable (D : Type u) [Ring D] |
|
|
| /-- Projection from the original chain direct sum to its r-th complex. -/ |
| def directSumComplexProjection (S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ) (r : ℕ) : |
| directSumComplex S ⟶ S r where |
| f n := ModuleCat.ofHom (DirectSum.component D ℕ (fun s => (S s).X n) r) |
| comm' i j hij := by |
| change j+1=i at hij |
| subst i |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro v |
| simp only [directSumComplex, ChainComplex.of_d, ModuleCat.hom_comp, |
| ModuleCat.hom_ofHom] |
| rfl |
|
|
| @[simp] theorem directSumComplex_inclusion_projection |
| (S : ℕ → ChainComplex (ModuleCat.{u} D) ℕ) (r : ℕ) : |
| directSumComplexInclusion D S r ≫ directSumComplexProjection D S r = 𝟙 (S r) := by |
| ext degree value |
| change DirectSum.component D ℕ (fun summand => (S summand).X degree) r |
| (DirectSum.lof D ℕ (fun summand => (S summand).X degree) r value) = value |
| exact DirectSum.component.lof_self D (M := fun summand => (S summand).X degree) r value |
|
|
| end DerivedBar |
| namespace DerivedBar.ParityBridge |
|
|
| private theorem mapped_retract {C E : Type*} [Category C] [Category E] |
| (F : C ⥤ E) {V W : C} {U T : E} (e : U ≅ F.obj V) (f : T ≅ F.obj W) |
| (i : V ⟶ W) (p : W ⟶ V) (h : i ≫ p = 𝟙 V) : |
| (e.hom ≫ F.map i ≫ f.inv) ≫ (f.hom ≫ F.map p ≫ e.inv) = 𝟙 U := by |
| simp only [Category.assoc, Iso.inv_hom_id_assoc] |
| rw [← Functor.map_comp_assoc, h, CategoryTheory.Functor.map_id, Category.id_comp, |
| Iso.hom_inv_id] |
|
|
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance countRetractsScalarModule (V : ModuleCat D) : Module k V := |
| BimoduleTensor.scalarModule k D V |
| local instance countRetractsScalarTower (V : ModuleCat D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| variable (N : ModuleCat D) |
| local notation "A" => TrivSqZeroExt D X |
| local instance countRetractsInflatedModule : Module A N := inflatedModule D X N |
| local instance countRetractsInflatedScalarTower : IsScalarTower k A N := |
| IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
| local instance countRetractsHasDerivedCategory : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
| local notation "S" => SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) |
| local notation "T" => tensorBarComplex k A D N (fstHom k D X) |
|
|
| /-- The original count subcomplex included into the actual balanced tensor bar. -/ |
| def countTensorBarInclusion (r : ℕ) : S r ⟶ T := |
| directSumComplexInclusion D S r ≫ |
| (SquareZero.tensorCountSplitting k D X N (inflated_ideal_zero D X N)).inv |
|
|
| def countTensorBarProjection (r : ℕ) : T ⟶ S r := |
| (SquareZero.tensorCountSplitting k D X N (inflated_ideal_zero D X N)).hom ≫ |
| directSumComplexProjection D S r |
|
|
| @[simp] theorem countTensorBar_retract (r : ℕ) : |
| countTensorBarInclusion k D X N r ≫ countTensorBarProjection k D X N r = 𝟙 (S r) := by |
| simp only [countTensorBarInclusion, countTensorBarProjection, Category.assoc, |
| Iso.inv_hom_id_assoc, directSumComplex_inclusion_projection] |
|
|
| /-- The genuine shifted iterate included into actual left-derived base change. -/ |
| def shiftedIterateInclusion (r : ℕ) : |
| (shiftedIterateDiagram k D X N).obj ⟨r⟩ ⟶ |
| (derivedBaseChange D (fstHom k D X)).obj (ModuleCat.of A N) := |
| (derivedIterateWeightShiftIso k D X N r).hom ≫ |
| (integerQ D).map (countTensorBarInclusion k D X N r) ≫ |
| (derivedBaseChangeBarIso D (fstHom k D X) N).inv |
|
|
| def shiftedIterateProjection (r : ℕ) : |
| (derivedBaseChange D (fstHom k D X)).obj (ModuleCat.of A N) ⟶ |
| (shiftedIterateDiagram k D X N).obj ⟨r⟩ := |
| (derivedBaseChangeBarIso D (fstHom k D X) N).hom ≫ |
| (integerQ D).map (countTensorBarProjection k D X N r) ≫ |
| (derivedIterateWeightShiftIso k D X N r).inv |
|
|
| @[simp] theorem shiftedIterate_retract (r : ℕ) : |
| shiftedIterateInclusion k D X N r ≫ shiftedIterateProjection k D X N r = |
| 𝟙 ((shiftedIterateDiagram k D X N).obj ⟨r⟩) := by |
| exact mapped_retract (integerQ D) (derivedIterateWeightShiftIso k D X N r) |
| (derivedBaseChangeBarIso D (fstHom k D X) N) _ _ (countTensorBar_retract k D X N r) |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
