File size: 2,932 Bytes
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 | 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
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