| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleTensorTotal |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
|
|
| 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 (Z : CochainComplex (ModuleCat.{u} A) ℤ) |
| variable {l : ℕ} (p : Fin l → ℤ) (hp : StrictMono p) |
|
|
| abbrev forgetTotal : CochainComplex (ModuleCat.{u} k) ℤ := |
| ((ModuleCat.restrictScalars (algebraMap k S)).mapHomologicalComplex (.up ℤ)).obj (total k S A Q Z p hp) |
|
|
| def forgetTermIso (n : ℤ) : (forgetTotal k S A Q Z p hp).X n ≅ |
| (ComplexTensor.finiteTotal k A (rightComplex k S A Q) Z p hp).X n := by |
| let e : (forgetTotal k S A Q Z p hp).X n ≃ₗ[k] totalTerm k S A Q Z p n := |
| { 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 |
| have h (y : totalTerm k S A Q Z p n) : (algebraMap k S r) • y = r • y := |
| algebraMap_smul S r y |
| exact h x } |
| exact (e.trans (forgetTotalEquiv k S A Q Z p n)).toModuleIso |
|
|
| /-- The genuine outside-action total forgets to the actual previously verified k-total. -/ |
| def forgetTotalIso : forgetTotal k S A Q Z p hp ≅ |
| ComplexTensor.finiteTotal k A (rightComplex k S A Q) Z p hp := |
| HomologicalComplex.Hom.isoOfComponents (forgetTermIso k S A Q Z p hp) (by |
| intro n m h |
| change n+1=m at h |
| subst m |
| have h₁ : (total k S A Q Z p hp).d n (n+1) = ModuleCat.ofHom (differential k S A Q Z p n) := |
| CochainComplex.of_d (fun n => ModuleCat.of S (totalTerm k S A Q Z p n)) |
| (fun n => ModuleCat.ofHom (differential k S A Q Z p n)) n |
| have h₂ : (ComplexTensor.finiteTotal k A (rightComplex k S A Q) Z p hp).d n (n+1) = |
| ModuleCat.ofHom ((FiniteColumnComplex.totalDifferential |
| (ComplexTensor.columns k A (rightComplex k S A Q) Z p) |
| (ComplexTensor.tensorHorizontal k A (rightComplex k S A Q) Z p hp)).d n) := |
| CochainComplex.of_d (fun n => ModuleCat.of k |
| (FiniteColumnComplex.term (ComplexTensor.columns k A (rightComplex k S A Q) Z p) n)) |
| (fun n => ModuleCat.ofHom ((FiniteColumnComplex.totalDifferential |
| (ComplexTensor.columns k A (rightComplex k S A Q) Z p) |
| (ComplexTensor.tensorHorizontal k A (rightComplex k S A Q) Z p hp)).d n)) n |
| change (forgetTermIso k S A Q Z p hp n).hom ≫ _ = |
| (ModuleCat.restrictScalars (algebraMap k S)).map ((total k S A Q Z p hp).d n (n+1)) ≫ |
| (forgetTermIso k S A Q Z p hp (n+1)).hom |
| rw [h₁,h₂] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro z |
| exact (forget_differential k S A Q Z p hp n z).symm) |
| end LittleFinitistic.BimoduleComplexTensor |
|
|
| end |
|
|
| end OAI |
|
|
