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ae00908 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import OAI.Algebra.Finitistic.Rectification.TensorProjection
import OAI.Algebra.Finitistic.Encoding.RelationSequence
namespace OAI
noncomputable section
open scoped TensorProduct
open CategoryTheory
namespace LittleFinitistic.TensorComplex
attribute [local instance] ExternalTensor.rightModule
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
abbrev ScalarModule (N : ModuleCat.{u} Bᵐᵒᵖ) :=
(ModuleCat.restrictScalars (algebraMap k Bᵐᵒᵖ)).obj N
instance scalarTower (N : ModuleCat.{u} Bᵐᵒᵖ) :
IsScalarTower k Bᵐᵒᵖ (ScalarModule k B N) where
smul_assoc c b n := by
change (c • b) • (n : N) = algebraMap k Bᵐᵒᵖ c • (b • (n : N))
rw [Algebra.smul_def,mul_smul]
variable {k B}
/-- The actual tensor differential, with the original right B-action. -/
def mapRight (M : ModuleCat.{u} k) {N N' : ModuleCat.{u} Bᵐᵒᵖ} (g : N ⟶ N') :
(M ⊗[k] ScalarModule k B N) →ₗ[Bᵐᵒᵖ] (M ⊗[k] ScalarModule k B N') where
__ := (TensorProduct.map LinearMap.id
((ModuleCat.restrictScalars (algebraMap k Bᵐᵒᵖ)).map g).hom).toAddMonoidHom
map_smul' c x := by
induction x using TensorProduct.inductionOn with
| tmul m n =>
change TensorProduct.mk k M (ScalarModule k B N') m (g.hom (c • (n : N))) = TensorProduct.mk k M (ScalarModule k B N') m (c • g.hom n)
exact congrArg (TensorProduct.mk k M (ScalarModule k B N') m) (g.hom.map_smul c (n : N))
| add x y hx hy => simp_all [smul_add]
@[simp] theorem mapRight_tmul (M : ModuleCat.{u} k) {N N' : ModuleCat.{u} Bᵐᵒᵖ}
(g : N ⟶ N') (m : M) (n : ScalarModule k B N) :
mapRight M g (m ⊗ₜ[k] n) = TensorProduct.mk k M (ScalarModule k B N') m (g.hom n) := rfl
variable (D : CochainComplex (ModuleCat.{u} Bᵐᵒᵖ) ℤ)
/-- Tensor a vector space with every term and differential of the given right-module complex. -/
def complex (M : ModuleCat.{u} k) : CochainComplex (ModuleCat.{u} Bᵐᵒᵖ) ℤ where
X i := ModuleCat.of Bᵐᵒᵖ (M ⊗[k] ScalarModule k B (D.X i))
d i j := ModuleCat.ofHom (mapRight M (D.d i j))
shape i j hij := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n =>
have h : (D.d i j).hom (n : D.X i) = 0 := by rw [D.shape i j hij]; rfl
exact (congrArg (TensorProduct.mk k M (ScalarModule k B (D.X j)) m) h).trans
(map_zero (TensorProduct.mk k M (ScalarModule k B (D.X j)) m))
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
d_comp_d' i j l hij hjl := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n =>
change TensorProduct.mk k M (ScalarModule k B (D.X l)) m ((D.d j l).hom ((D.d i j).hom n)) = 0
have h := congrArg (fun f : D.X i ⟶ D.X l => f.hom n) (D.d_comp_d i j l)
change (D.d j l).hom ((D.d i j).hom n) = 0 at h
exact (congrArg (TensorProduct.mk k M (ScalarModule k B (D.X l)) m) h).trans
(map_zero (TensorProduct.mk k M (ScalarModule k B (D.X l)) m))
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
/-- The first-factor tensor map is a chain map. -/
def complexMap {M M' : ModuleCat.{u} k} (f : M ⟶ M') : complex D M ⟶ complex D M' where
f i := ModuleCat.ofHom (TensorProjection.rightMap (B := B)
(N := ScalarModule k B (D.X i)) f.hom)
comm' i j hij := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n => rfl
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
/-- The genuine additive tensor-with-D functor required by the path exact sequence. -/
def tensorFunctor : ModuleCat.{u} k ⥤ CochainComplex (ModuleCat.{u} Bᵐᵒᵖ) ℤ where
obj := complex D
map := complexMap D
map_id M := by
apply HomologicalComplex.Hom.ext
funext i
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n => rfl
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
map_comp f g := by
apply HomologicalComplex.Hom.ext
funext i
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n => rfl
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
instance tensorFunctor_additive : (tensorFunctor (k := k) D).Additive where
map_add {M M'} {f g} := by
apply HomologicalComplex.Hom.ext
funext i
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| tmul m n => exact TensorProduct.add_tmul _ _ _
| add x y hx hy =>
change TensorProjection.rightMap (B := B) (f.hom + g.hom) (x+y) =
TensorProjection.rightMap (B := B) f.hom (x+y) + TensorProjection.rightMap (B := B) g.hom (x+y)
simp only [map_add]
change TensorProjection.rightMap (B := B) (f.hom + g.hom) x =
TensorProjection.rightMap (B := B) f.hom x + TensorProjection.rightMap (B := B) g.hom x at hx
change TensorProjection.rightMap (B := B) (f.hom + g.hom) y =
TensorProjection.rightMap (B := B) f.hom y + TensorProjection.rightMap (B := B) g.hom y at hy
rw [hx,hy]
abel
/-- The adjacent-vertex graded piece is the cone of the path-space identity. -/
theorem adjacent_total_acyclic (M : ModuleCat.{u} k) :
(CochainComplex.mappingCone (𝟙 (complex D M))).Acyclic :=
RectificationAcyclic.cone_acyclic _
/-- The source-0 path-relation graded piece is acyclic for the actual tensor functor. -/
theorem relation_total_acyclic {d t : ℕ} (P : QuadraticEncoding.Relations k d t) :
(RectificationTotal.total ((RelationSequence.sequence P).map (tensorFunctor (k := k) D)).g
((RelationSequence.sequence P).map (tensorFunctor (k := k) D)).f
(Homotopy.ofEq ((RelationSequence.sequence P).map (tensorFunctor (k := k) D)).zero)).Acyclic :=
RelationSequence.realized_sequence_total_acyclic P (tensorFunctor (k := k) D)
end LittleFinitistic.TensorComplex
end
end OAI
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