| import OAI.Algebra.Finitistic.Bar.Bar |
| import Mathlib.Algebra.Homology.TotalComplex |
| import Mathlib.Algebra.Category.ModuleCat.Colimits |
|
|
| namespace OAI |
| open scoped TensorProduct |
| open TensorProduct CategoryTheory |
| noncomputable section |
|
|
| namespace DerivedBar |
| universe u |
| variable (k A : Type u) [Field k] [Ring A] [Algebra k A] |
| variable (V W U : Type u) [AddCommGroup V] [Module k V] [Module A V] [IsScalarTower k A V] |
| [AddCommGroup W] [Module k W] [Module A W] [IsScalarTower k A W] |
| [AddCommGroup U] [Module k U] [Module A U] [IsScalarTower k A U] |
|
|
| /-- Functorial map on every augmented bar term. -/ |
| def termMap (h : V →ₗ[A] W) : (n : ℕ) → term k A V n →ₗ[A] term k A W n |
| | 0 => h |
| | n+1 => AlgebraTensorModule.lTensor A A ((termMap h n).restrictScalars k) |
|
|
| @[simp] theorem termMap_tmul (h : V →ₗ[A] W) (n : ℕ) (a : A) (v : term k A V n) : |
| termMap k A V W h (n+1) (a ⊗ₜ[k] v) = a ⊗ₜ[k] termMap k A V W h n v := rfl |
|
|
| theorem termMap_comp (h : V →ₗ[A] W) (g : W →ₗ[A] U) (n : ℕ) : |
| termMap k A V U (g.comp h) n = (termMap k A W U g n).comp (termMap k A V W h n) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| apply AlgebraTensorModule.ext |
| intro a v |
| change a ⊗ₜ[k] termMap k A V U (g.comp h) n v = |
| a ⊗ₜ[k] termMap k A W U g n (termMap k A V W h n v) |
| rw [ih] |
| rfl |
|
|
| theorem termMap_id (n : ℕ) : termMap k A V V LinearMap.id n = LinearMap.id := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| apply AlgebraTensorModule.ext |
| intro a v |
| change a ⊗ₜ[k] termMap k A V V LinearMap.id n v = a ⊗ₜ[k] v |
| rw [ih] |
| rfl |
|
|
| theorem termMap_zero (n : ℕ) : termMap k A V W 0 n = 0 := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| apply AlgebraTensorModule.ext |
| intro a v |
| change a ⊗ₜ[k] termMap k A V W 0 n v = 0 |
| rw [ih, LinearMap.zero_apply, TensorProduct.tmul_zero] |
|
|
| /-- The bar differential is natural for all module maps. -/ |
| theorem differential_natural (h : V →ₗ[A] W) (n : ℕ) : |
| (differential k A W n).comp (termMap k A V W h (n+1)) = |
| (termMap k A V W h n).comp (differential k A V n) := by |
| induction n with |
| | zero => |
| apply AlgebraTensorModule.ext |
| intro a v |
| exact (h.map_smul a v).symm |
| | succ n ih => |
| apply AlgebraTensorModule.ext |
| intro a v |
| have hn : differential k A W n (termMap k A V W h (n+1) v) = |
| termMap k A V W h n (differential k A V n v) := LinearMap.congr_fun ih v |
| change a • termMap k A V W h (n+1) v - |
| a ⊗ₜ[k] differential k A W n (termMap k A V W h (n+1) v) = |
| termMap k A V W h (n+1) (a • v - a ⊗ₜ[k] differential k A V n v) |
| rw [map_sub, map_smul] |
| change a • termMap k A V W h (n+1) v - |
| a ⊗ₜ[k] differential k A W n (termMap k A V W h (n+1) v) = |
| a • termMap k A V W h (n+1) v - |
| a ⊗ₜ[k] termMap k A V W h n (differential k A V n v) |
| rw [hn] |
|
|
| /-- Unit insertion is natural over k, the key homotopy for the total bar augmentation. -/ |
| theorem contraction_natural (h : V →ₗ[A] W) (n : ℕ) (v : term k A V n) : |
| termMap k A V W h (n+1) (contraction k A V n v) = |
| contraction k A W n (termMap k A V W h n v) := rfl |
|
|
| /-- An actual chain map between unaugmented free bar resolutions. -/ |
| def complexMap (h : V →ₗ[A] W) : complex k A V ⟶ complex k A W where |
| f n := ModuleCat.ofHom (termMap k A V W h (n+1)) |
| comm' i j hij := by |
| change j+1 = i at hij |
| subst i |
| apply ModuleCat.hom_ext |
| simpa only [complex, ChainComplex.of_d, ModuleCat.hom_comp, ModuleCat.hom_ofHom] using! |
| differential_natural k A V W h (j+1) |
|
|
| local instance (N : ModuleCat.{u} A) : Module k N := Module.compHom N (algebraMap k A) |
| local instance (N : ModuleCat.{u} A) : IsScalarTower k A N := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
|
|
| /-- The explicit free bar resolution is functorial, not just an objectwise choice. -/ |
| def barFunctor : ModuleCat.{u} A ⥤ ChainComplex (ModuleCat.{u} A) ℕ where |
| obj N := complex k A N |
| map h := complexMap k A _ _ h.hom |
| map_id N := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply ModuleCat.hom_ext |
| exact termMap_id k A N (n+1) |
| map_comp h g := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply ModuleCat.hom_ext |
| exact termMap_comp k A _ _ _ h.hom g.hom (n+1) |
|
|
| instance barFunctor_preservesZero : (barFunctor k A).PreservesZeroMorphisms where |
| map_zero V W := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply ModuleCat.hom_ext |
| exact termMap_zero k A V W (n+1) |
|
|
| /-- The bicomplex for the left-module bar resolution of a nonnegative chain complex. |
| Outer degree is the bar length; inner degree is the input complex degree. |
| The two differentials commute by the proved naturality of the absolute bar differential. -/ |
| def barBicomplex (Z : ChainComplex (ModuleCat.{u} A) ℕ) : |
| HomologicalComplex₂ (ModuleCat.{u} A) (ComplexShape.down ℕ) (ComplexShape.down ℕ) := |
| HomologicalComplex₂.flip (((barFunctor k A).mapHomologicalComplex (ComplexShape.down ℕ)).obj Z) |
|
|
| /-- Each entry in the bar bicomplex is genuinely projective over A. -/ |
| instance barBicomplex_projective (Z : ChainComplex (ModuleCat.{u} A) ℕ) (i q : ℕ) : |
| Projective (((barBicomplex k A Z).X i).X q) := by |
| change Projective ((complex k A (Z.X q)).X i) |
| exact complex_projective k A (Z.X q) i |
|
|
| /-- The ordinary direct-sum total bar complex, with standard totalization signs. |
| The total augmentation quasi-isomorphism must still be proved. -/ |
| def barTotal (Z : ChainComplex (ModuleCat.{u} A) ℕ) : |
| ChainComplex (ModuleCat.{u} A) ℕ := |
| (barBicomplex k A Z).total (ComplexShape.down ℕ) |
|
|
| end DerivedBar |
|
|
| end |
|
|
| end OAI |
|
|
