| import OAI.Algebra.Finitistic.Rectification.ExternalTensor |
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| namespace OAI |
| open scoped TensorProduct |
| noncomputable section |
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| namespace ExternalTensor |
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| attribute [local instance] bimodule rightModule |
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| variable (k B : Type*) [Field k] [Ring B] [Algebra k B] |
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| /-- 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 |
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| @[simp] theorem mem_leftIdeal (e x : B) : x ∈ leftIdeal B e ↔ ∃ b : B, b * e = x := |
| Iff.rfl |
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| 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 |
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| attribute [local instance] leftIdeal_finiteDimensional |
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| variable (N : Type*) [AddCommGroup N] [Module k N] |
| [Module Bᵐᵒᵖ N] [IsScalarTower k Bᵐᵒᵖ N] |
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|
| /-- The induced column in Section 4, with its external actions. -/ |
| abbrev Column (e : B) := (leftIdeal B e) ⊗[k] N |
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| 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 |
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| 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 |
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| 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 |
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|
| /-- 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⟩ |
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| /-- The distinguished generator e of the principal left ideal. -/ |
| def generator (e : B) : leftIdeal B e := ⟨e, ⟨1, one_mul e⟩⟩ |
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| /-- 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 |
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| @[simp] theorem columnInclusion_apply (e : B) (n : N) : |
| columnInclusion k B N e n = generator B e ⊗ₜ[k] n := rfl |
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| 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) |
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| @[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 |
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| 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 |
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| @[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 |
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| variable {N' : Type*} [AddCommGroup N'] [Module k N'] |
| [Module Bᵐᵒᵖ N'] [IsScalarTower k Bᵐᵒᵖ N'] |
|
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| /-- 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 |
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| @[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 |
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| end |
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| end OAI |
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