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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | 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
end OAI
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