| import Mathlib.LinearAlgebra.TensorProduct.Basic |
| import Mathlib.LinearAlgebra.Quotient.Basic |
| import Mathlib.Algebra.Algebra.Tower |
|
|
| namespace OAI |
| noncomputable section |
| open scoped TensorProduct |
| namespace LittleFinitistic.BalancedTensor |
| universe u v w w' z |
| variable (k : Type u) [Field k] (A : Type v) [Ring A] |
| variable (M : Type w) [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] |
| variable (N : Type w') [AddCommGroup N] [Module k N] [Module A N] |
|
|
| /-- Balancing relations for a right A-module and a left A-module, over k. -/ |
| def relations : Submodule k (M ⊗[k] N) := Submodule.span k |
| (Set.range fun q : A × M × N => |
| (MulOpposite.op q.1 • q.2.1) ⊗ₜ[k] q.2.2 - q.2.1 ⊗ₜ[k] (q.1 • q.2.2)) |
|
|
| /-- Tensor over the possibly noncommutative algebra A. -/ |
| abbrev Tensor := (M ⊗[k] N) ⧸ relations k A M N |
|
|
| def mk : M →ₗ[k] N →ₗ[k] Tensor k A M N := |
| (TensorProduct.mk k M N).compr₂ (relations k A M N).mkQ |
|
|
| variable {k A M N} |
|
|
| theorem balance (a : A) (m : M) (n : N) : |
| mk k A M N (MulOpposite.op a • m) n = mk k A M N m (a • n) := by |
| apply sub_eq_zero.mp |
| change (relations k A M N).mkQ ((MulOpposite.op a • m) ⊗ₜ[k] n) - |
| (relations k A M N).mkQ (m ⊗ₜ[k] (a • n)) = 0 |
| rw [← map_sub] |
| exact (Submodule.Quotient.mk_eq_zero _).mpr |
| (Submodule.subset_span ⟨(a,m,n),rfl⟩) |
|
|
| variable {V : Type z} [AddCommGroup V] [Module k V] |
|
|
| /-- Universal property for k-bilinear maps satisfying the actual A-balancing equation. -/ |
| def lift (f : M →ₗ[k] N →ₗ[k] V) |
| (hf : ∀ (a : A) m n, f (MulOpposite.op a • m) n = f m (a • n)) : Tensor k A M N →ₗ[k] V := |
| (relations k A M N).liftQ (TensorProduct.lift f) (by |
| rw [relations, Submodule.span_le] |
| rintro _ ⟨⟨a,m,n⟩,rfl⟩ |
| change TensorProduct.lift f ((MulOpposite.op a • m) ⊗ₜ[k] n - m ⊗ₜ[k] (a • n)) = 0 |
| simp only [map_sub, TensorProduct.lift.tmul, hf, sub_self]) |
|
|
| @[simp] theorem lift_mk (f : M →ₗ[k] N →ₗ[k] V) |
| (hf : ∀ (a : A) m n, f (MulOpposite.op a • m) n = f m (a • n)) (m : M) (n : N) : |
| lift f hf (mk k A M N m n) = f m n := rfl |
|
|
| @[elab_as_elim] theorem induction_on {C : Tensor k A M N → Prop} (x : Tensor k A M N) |
| (hz : C 0) (ht : ∀ m n, C (mk k A M N m n)) |
| (ha : ∀ a b, C a → C b → C (a+b)) : C x := by |
| classical |
| by_cases zero_tensor : x = 0 |
| · simpa [zero_tensor] using hz |
| clear zero_tensor |
| induction x using Submodule.Quotient.induction_on with |
| | _ x => |
| induction x using TensorProduct.inductionOn with |
| | tmul m n => exact ht m n |
| | add a b h₁ h₂ => exact ha _ _ h₁ h₂ |
|
|
| @[ext] theorem ext {f g : Tensor k A M N →ₗ[k] V} |
| (h : ∀ m n, f (mk k A M N m n) = g (mk k A M N m n)) : f = g := by |
| apply LinearMap.ext |
| intro x |
| induction x using induction_on with |
| | hz => simp |
| | ht m n => exact h m n |
| | ha a b h₁ h₂ => simp [h₁,h₂] |
| end LittleFinitistic.BalancedTensor |
|
|
| end |
|
|
| end OAI |
|
|
