| import OAI.Algebra.Finitistic.Dimension.PDSquareZeroResidue |
| import OAI.Algebra.Finitistic.Dimension.PDResidueRing |
| import OAI.Algebra.Finitistic.Dimension.PDTor |
| import OAI.Algebra.Finitistic.Tensor.BalancedTensorMaps |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.PDResidueProjection |
| universe u |
| variable {k A D : Type u} [Field k] [Ring A] [Algebra k A] [Ring D] [Algebra k D] |
| variable (f : A →ₐ[k] D) (hf : Function.Surjective f) |
| variable (hsq : ∀ a ∈ RingHom.ker f.toRingHom, ∀ b ∈ RingHom.ker f.toRingHom, a*b=0) |
|
|
| def InflatedResidue (_f : A →ₐ[k] D) := PDTensor.Residue (A := D) |
| instance : AddCommGroup (InflatedResidue f) := inferInstanceAs (AddCommGroup (PDTensor.Residue (A := D))) |
| instance : Module k (InflatedResidue f) := inferInstanceAs (Module k (PDTensor.Residue (A := D))) |
| instance : Module Aᵐᵒᵖ (InflatedResidue f) := |
| Module.compHom (PDTensor.Residue (A := D)) f.toRingHom.op |
| instance (priority := 50) : Module Dᵐᵒᵖ (InflatedResidue f) := |
| inferInstanceAs (Module Dᵐᵒᵖ (PDTensor.Residue (A := D))) |
| instance (priority := 50) : IsScalarTower k Dᵐᵒᵖ (InflatedResidue f) := |
| inferInstanceAs (IsScalarTower k Dᵐᵒᵖ (PDTensor.Residue (A := D))) |
| instance : IsScalarTower k Aᵐᵒᵖ (InflatedResidue f) where |
| smul_assoc c a s := by |
| change MulOpposite.op (f (c • a.unop)) • (s : PDTensor.Residue (A := D)) = |
| c • (MulOpposite.op (f a.unop) • (s : PDTensor.Residue (A := D))) |
| rw [map_smul, MulOpposite.op_smul, smul_assoc] |
|
|
| def map : PDTensor.Residue (A := A) →ₗ[k] InflatedResidue f := |
| ((PDTensor.rightRadical (A := A)).restrictScalars k).liftQ |
| (((PDTensor.rightRadical (A := D)).restrictScalars k).mkQ.comp f.toLinearMap) (by |
| intro a ha |
| apply (Submodule.Quotient.mk_eq_zero _).mpr |
| have he := PDSquareZero.comap_jacobson f.toRingHom hf hsq |
| change a ∈ Ring.jacobson A at ha |
| rw [← he] at ha |
| exact ha) |
|
|
| @[simp] theorem map_mk (a : A) : map f hf hsq (Submodule.Quotient.mk a) = |
| (Submodule.Quotient.mk (f a) : PDTensor.Residue (A := D)) := rfl |
|
|
| / |
| def equiv : PDTensor.Residue (A := A) ≃ₗ[k] InflatedResidue f := |
| LinearEquiv.ofBijective (map f hf hsq) (by |
| have he : (fun s => map f hf hsq s) = |
| (fun s => PDSquareZero.residueQuotientEquiv f.toRingHom hf hsq s) := by |
| funext s |
| induction s using Submodule.Quotient.induction_on with |
| | _ a => exact (PDSquareZero.residueQuotientEquiv_mk f.toRingHom hf hsq a).symm |
| change Function.Bijective (fun s => map f hf hsq s) |
| rw [he] |
| exact (PDSquareZero.residueQuotientEquiv f.toRingHom hf hsq).bijective) |
|
|
| / |
| def rightEquiv : PDTensor.Residue (A := A) ≃ₗ[Aᵐᵒᵖ] InflatedResidue f where |
| __ := (equiv f hf hsq).toAddEquiv |
| map_smul' a s := by |
| induction s using Submodule.Quotient.induction_on with |
| | _ b => |
| change (Submodule.Quotient.mk (f (b*a.unop)) : PDTensor.Residue (A := D)) = |
| Submodule.Quotient.mk (f b * f a.unop) |
| rw [map_mul] |
|
|
| variable (N : Type u) [AddCommGroup N] [Module k N] [Module A N] |
|
|
| def tensorEquiv : BalancedTensor.Tensor k A (PDTensor.Residue (A := A)) N ≃ₗ[k] |
| BalancedTensor.Tensor k A (InflatedResidue f) N := |
| LinearEquiv.ofLinearMap |
| (BalancedTensor.mapLeft (k := k) N (rightEquiv f hf hsq).toLinearMap) |
| (BalancedTensor.mapLeft (k := k) N (rightEquiv f hf hsq).symm.toLinearMap) |
| (by |
| apply BalancedTensor.ext |
| intro s n |
| simp only [LinearMap.comp_apply, BalancedTensor.mapLeft_mk, LinearEquiv.coe_coe, |
| LinearEquiv.apply_symm_apply, LinearMap.id_apply]) |
| (by |
| apply BalancedTensor.ext |
| intro s n |
| simp only [LinearMap.comp_apply, BalancedTensor.mapLeft_mk, LinearEquiv.coe_coe, |
| LinearEquiv.symm_apply_apply, LinearMap.id_apply]) |
|
|
| / |
| def tensorNatIso : PDTensor.tensorFunctor k A (PDTensor.Residue (A := A)) ≅ |
| PDTensor.tensorFunctor k A (InflatedResidue f) := |
| NatIso.ofComponents (fun M => (tensorEquiv f hf hsq (PDTensor.ScalarModule k A M)).toModuleIso) |
| (by |
| intro M L h |
| apply ModuleCat.hom_ext |
| apply BalancedTensor.ext |
| intro s n |
| rfl) |
|
|
| variable (M : ModuleCat.{u} A) (P : ProjectiveResolution M) |
| def resolutionIso : |
| ((PDTensor.tensorFunctor k A (PDTensor.Residue (A := A))).mapHomologicalComplex _).obj P.complex ≅ |
| ((PDTensor.tensorFunctor k A (InflatedResidue f)).mapHomologicalComplex _).obj P.complex := |
| HomologicalComplex.Hom.isoOfComponents (fun n => (tensorNatIso f hf hsq).app (P.complex.X n)) |
| (fun i j _ => ((tensorNatIso f hf hsq).hom.naturality (P.complex.d i j)).symm) |
|
|
| / |
| def torIso (n : ℕ) : PDTensor.Tor (k := k) (PDTensor.Residue (A := A)) M n ≅ |
| PDTensor.Tor (k := k) (InflatedResidue f) M n := |
| P.isoLeftDerivedObj (PDTensor.tensorFunctor k A (PDTensor.Residue (A := A))) n ≪≫ |
| (HomologicalComplex.homologyFunctor _ _ n).mapIso (resolutionIso f hf hsq M P) ≪≫ |
| (P.isoLeftDerivedObj (PDTensor.tensorFunctor k A (InflatedResidue f)) n).symm |
| end LittleFinitistic.PDResidueProjection |
|
|
| end |
|
|
| end OAI |
|
|
