| import Mathlib |
| import OAI.RingTheory.Multiplicity.ScalarHomologyError |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| open scoped TensorProduct |
| namespace Lech.TensorConductor |
| universe u |
| variable {D R C : Type u} [CommRing D] [CommRing R] [CommRing C] |
| [Algebra R C] [Algebra D C] |
| (B : Subalgebra R C) [Algebra D B] [IsScalarTower D B C] |
|
|
| def stageComplex (F : CochainComplex (ModuleCat.{u} D) ℤ) : |
| CochainComplex (ModuleCat.{u} R) ℤ := |
| ((ModuleCat.restrictScalars (algebraMap R B)).mapHomologicalComplex _).obj |
| (((ModuleCat.extendScalars (algebraMap D B)).mapHomologicalComplex _).obj F) |
|
|
| def ambientComplex (F : CochainComplex (ModuleCat.{u} D) ℤ) : |
| CochainComplex (ModuleCat.{u} R) ℤ := |
| ((ModuleCat.restrictScalars (algebraMap R C)).mapHomologicalComplex _).obj |
| (((ModuleCat.extendScalars (algebraMap D C)).mapHomologicalComplex _).obj F) |
|
|
| def complexInclusion (F : CochainComplex (ModuleCat.{u} D) ℤ) : |
| stageComplex B F ⟶ ambientComplex (R := R) (C := C) F := by |
| have hc := IsScalarTower.algebraMap_eq D B C |
| let fB := algebraMap D B |
| let fC := algebraMap D C |
| letI : Algebra D B := fB.toAlgebra |
| letI : Algebra D C := fC.toAlgebra |
| letI : IsScalarTower D B C := IsScalarTower.of_algebraMap_eq' hc |
| refine { f := fun i => ?_, comm' := ?_ } |
| · let X := (stageComplex B F).X i |
| let Y := (ambientComplex (R := R) (C := C) F).X i |
| letI : Module R X := X.isModule |
| letI : Module R Y := Y.isModule |
| refine ModuleCat.ofHom (X := X) (Y := Y) (show X →ₗ[R] Y from |
| { __ := (inclusion (D := D) B (F.X i)).toAddHom, map_smul' := ?_ }) |
| intro r x |
| change B ⊗[D] (F.X i) at x |
| change inclusion (D := D) B (F.X i) ((algebraMap R B r) • (x : B ⊗[D] (F.X i))) = |
| (algebraMap R C r) • inclusion (D := D) B (F.X i) x |
| simpa only [IsScalarTower.algebraMap_smul] using |
| (inclusion (D := D) B (F.X i)).map_smul r x |
| · intro i j hij |
| apply ModuleCat.hom_ext |
| ext x |
| induction x using TensorProduct.inductionOn with |
| | tmul b x => rfl |
| | add x y hx hy => |
| exact (map_add _ x y).trans ((congrArg₂ (·+·) hx hy).trans (map_add _ x y).symm) |
|
|
| def complexRetreat (F : CochainComplex (ModuleCat.{u} D) ℤ) |
| (g : R) (hg : ∀ c : C,g • c∈B) : |
| ambientComplex (R := R) (C := C) F ⟶ stageComplex B F := by |
| have hc := IsScalarTower.algebraMap_eq D B C |
| let fB := algebraMap D B |
| let fC := algebraMap D C |
| letI : Algebra D B := fB.toAlgebra |
| letI : Algebra D C := fC.toAlgebra |
| letI : IsScalarTower D B C := IsScalarTower.of_algebraMap_eq' hc |
| refine { f := fun i => ?_, comm' := ?_ } |
| · let X := (ambientComplex (R := R) (C := C) F).X i |
| let Y := (stageComplex B F).X i |
| letI : Module R X := X.isModule |
| letI : Module R Y := Y.isModule |
| refine ModuleCat.ofHom (X := X) (Y := Y) (show X →ₗ[R] Y from |
| { __ := (conductor (D := D) B (F.X i) g hg).toAddHom, map_smul' := ?_ }) |
| intro r x |
| change C ⊗[D] (F.X i) at x |
| change conductor (D := D) B (F.X i) g hg ((algebraMap R C r) • (x : C ⊗[D] (F.X i))) = |
| (algebraMap R B r) • conductor (D := D) B (F.X i) g hg x |
| simpa only [IsScalarTower.algebraMap_smul] using |
| (conductor (D := D) B (F.X i) g hg).map_smul r x |
| · intro i j hij |
| apply ModuleCat.hom_ext |
| ext x |
| induction x using TensorProduct.inductionOn with |
| | tmul c x => rfl |
| | add x y hx hy => |
| exact (map_add _ x y).trans ((congrArg₂ (·+·) hx hy).trans (map_add _ x y).symm) |
|
|
| lemma complex_conductor_composition (F : CochainComplex (ModuleCat.{u} D) ℤ) |
| (g : R) (hg : ∀ c : C,g • c∈B) : |
| complexInclusion B F ≫ complexRetreat B F g hg = g • 𝟙 (stageComplex B F) := by |
| have hc := IsScalarTower.algebraMap_eq D B C |
| let fB := algebraMap D B |
| let fC := algebraMap D C |
| let : Algebra D B := fB.toAlgebra |
| let : Algebra D C := fC.toAlgebra |
| let : IsScalarTower D B C := IsScalarTower.of_algebraMap_eq' hc |
| ext i : 1 |
| apply ModuleCat.hom_ext |
| ext x |
| change B ⊗[D] (F.X i) at x |
| have he := LinearMap.congr_fun (conductor_composition (D := D) B (F.X i) g hg) x |
| change conductor (D := D) B (F.X i) g hg (inclusion (D := D) B (F.X i) x) = |
| (algebraMap R B g) • (x : B ⊗[D] (F.X i)) |
| simpa only [IsScalarTower.algebraMap_smul,LinearMap.comp_apply,LinearMap.smul_apply,LinearMap.id_apply] using he |
|
|
| lemma stage_homology_length_le (T : NormalizedLength.Tower R) |
| (F : CochainComplex (ModuleCat.{u} D) ℤ) |
| (g : R) (hg : ∀ c : C,g • c∈B) (i : ℤ) |
| (hinj : ∀ j,Function.Injective (fun x : (stageComplex B F).X j => g • x)) |
| (hfin : T.length ((stageComplex B F).homology i) ≠ ⊤) : |
| T.length ((stageComplex B F).homology i) ≤ |
| T.length ((ambientComplex (R := R) (C := C) F).homology i) + |
| T.length ((((Lech.complexQuotient (Ideal.span {g}) (.up ℤ)).obj |
| (stageComplex B F)).homology i)) := |
| normalized_homology_length_le_of_scalar_retract T |
| (complexInclusion B F) (complexRetreat B F g hg) g |
| (complex_conductor_composition B F g hg) i hinj hfin |
| end Lech.TensorConductor |
|
|
| end |
|
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| end OAI |
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