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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 | import OAI.Algebra.Finitistic.Rectification.ExternalTensor
namespace OAI
open scoped TensorProduct
noncomputable section
namespace ExternalTensor
attribute [local instance] bimodule rightModule
variable (k B : Type*) [Field k] [Ring B] [Algebra k B]
/-- The actual principal left ideal B e, the image of right multiplication by e. -/
def leftIdeal (e : B) : Submodule B B := (LinearMap.mulRight B e).range
@[simp] theorem mem_leftIdeal (e x : B) : x ∈ leftIdeal B e ↔ ∃ b : B, b * e = x :=
Iff.rfl
theorem leftIdeal_finiteDimensional [FiniteDimensional k B] (e : B) :
FiniteDimensional k (leftIdeal B e) :=
FiniteDimensional.of_injective ((leftIdeal B e).subtype.restrictScalars k)
Subtype.val_injective
attribute [local instance] leftIdeal_finiteDimensional
variable (N : Type*) [AddCommGroup N] [Module k N]
[Module Bᵐᵒᵖ N] [IsScalarTower k Bᵐᵒᵖ N]
/-- The induced column in Section 4, with its external actions. -/
abbrev Column (e : B) := (leftIdeal B e) ⊗[k] N
theorem column_projective (e : B) [Module.Projective Bᵐᵒᵖ N] :
Module.Projective Bᵐᵒᵖ (Column k B N e) :=
projective_right k B (leftIdeal B e) N
theorem column_finite_right [FiniteDimensional k B] (e : B)
[Module.Finite Bᵐᵒᵖ N] : Module.Finite Bᵐᵒᵖ (Column k B N e) :=
finite_right k B (leftIdeal B e) N
theorem column_finiteDimensional [FiniteDimensional k B] (e : B)
[Module.Finite Bᵐᵒᵖ N] : FiniteDimensional k (Column k B N e) := by
let : Module.Finite k N := Module.Finite.trans Bᵐᵒᵖ N
exact finiteDimensional k (leftIdeal B e) N
/-- The full induced-column assertion required for each summand of L'_0,L'_1,L'_2. -/
theorem induced_column [FiniteDimensional k B] (e : B)
[Module.Finite Bᵐᵒᵖ N] [Module.Projective Bᵐᵒᵖ N] :
FiniteDimensional k (Column k B N e) ∧
Module.Finite Bᵐᵒᵖ (Column k B N e) ∧
Module.Projective Bᵐᵒᵖ (Column k B N e) :=
⟨column_finiteDimensional k B N e, column_finite_right k B N e,
column_projective k B N e⟩
/-- The distinguished generator e of the principal left ideal. -/
def generator (e : B) : leftIdeal B e := ⟨e, ⟨1, one_mul e⟩⟩
/-- The right-linear insertion used in the vertex maps iota_i. -/
def columnInclusion (e : B) : N →ₗ[Bᵐᵒᵖ] Column k B N e where
toFun n := generator B e ⊗ₜ[k] n
map_add' first second := TensorProduct.tmul_add _ first second
map_smul' c n := (right_smul_tmul k B (leftIdeal B e) N c (generator B e) n).symm
@[simp] theorem columnInclusion_apply (e : B) (n : N) :
columnInclusion k B N e n = generator B e ⊗ₜ[k] n := rfl
variable {B}
/-- Right multiplication by a path a: i→j maps B e_j to B e_i.
Only the relation a e_i = a is needed for membership in the target ideal. -/
def pathMap (ei ej a : B) (ha : a * ei = a) :
leftIdeal B ej →ₗ[B] leftIdeal B ei where
toFun c := ⟨c.val * a, ⟨c.val * a, by simp [LinearMap.mulRight_apply, mul_assoc, ha]⟩⟩
map_add' c d := Subtype.ext (add_mul c.val d.val a)
map_smul' b c := Subtype.ext (mul_assoc b c.val a)
@[simp] theorem pathMap_apply (ei ej a : B) (ha : a * ei = a) (c : leftIdeal B ej) :
(pathMap ei ej a ha c : B) = (c : B) * a := rfl
variable {N}
/-- The path-multiplication component c tensor n ↦ ca tensor n of the rectification maps. -/
def columnPath (ei ej a : B) (ha : a * ei = a) :
Column k B N ej →ₗ[B ⊗[k] Bᵐᵒᵖ] Column k B N ei :=
map (k := k) (pathMap ei ej a ha) LinearMap.id
@[simp] theorem columnPath_tmul (ei ej a : B) (ha : a * ei = a)
(c : leftIdeal B ej) (n : N) :
columnPath k ei ej a ha (c ⊗ₜ[k] n) = (pathMap ei ej a ha c) ⊗ₜ[k] n := rfl
variable {N' : Type*} [AddCommGroup N'] [Module k N']
[Module Bᵐᵒᵖ N'] [IsScalarTower k Bᵐᵒᵖ N']
/-- The differential or relation-homotopy component c tensor n ↦ c tensor f(n). -/
def columnMap (e : B) (f : N →ₗ[Bᵐᵒᵖ] N') :
Column k B N e →ₗ[B ⊗[k] Bᵐᵒᵖ] Column k B N' e :=
map (k := k) LinearMap.id f
@[simp] theorem columnMap_tmul (e : B) (f : N →ₗ[Bᵐᵒᵖ] N')
(c : leftIdeal B e) (n : N) :
columnMap k e f (c ⊗ₜ[k] n) = c ⊗ₜ[k] f n := rfl
end ExternalTensor
end
end OAI
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