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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedFourExt
namespace OAI
noncomputable section
namespace ArExplicit.ExternalBalanced
open scoped TensorProduct
universe u
variable {k A B E V F G X Y Z : Type u} [Field k] [Ring A] [Ring B]
[Algebra k A] [Algebra k B]
[AddCommGroup E] [AddCommGroup V] [AddCommGroup F] [AddCommGroup G]
[AddCommGroup X] [AddCommGroup Y] [AddCommGroup Z]
[Module k E] [Module k V] [Module k F] [Module k G] [Module k X] [Module k Y]
[Module (A ⊗[k] B)ᵐᵒᵖ E] [Module (A ⊗[k] B) V]
variable (e : E ≃ₗ[k] F ⊗[k] G) (v : V ≃ₗ[k] X ⊗[k] Y)
theorem fourfold_add_hom_ext (f g : BalancedTensor k (A ⊗[k] B) E V →+ Z)
(h : ∀ a b x y, f (BalancedTensor.tmul (e.symm (a ⊗ₜ[k] b)) (v.symm (x ⊗ₜ[k] y))) =
g (BalancedTensor.tmul (e.symm (a ⊗ₜ[k] b)) (v.symm (x ⊗ₜ[k] y)))) : f=g := by
apply AddMonoidHom.ext
intro z
induction z using BalancedTensor.induction_on with
| zero => simp
| add z w hz hw => simp only [map_add, hz, hw]
| tmul z w =>
obtain ⟨z, rfl⟩ := e.symm.surjective z
obtain ⟨w, rfl⟩ := v.symm.surjective w
induction z using TensorProduct.inductionOn with
| add z z' hz hz' => simp only [map_add, BalancedTensor.add_tmul, hz, hz']
| tmul a b =>
induction w using TensorProduct.inductionOn with
| add w w' hw hw' => simp only [map_add, BalancedTensor.tmul_add, hw, hw']
| tmul x y => exact h a b x y
end ArExplicit.ExternalBalanced
end
end OAI
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