| import OAI.Algebra.AuslanderReiten.Tensor.ExternalBalancedEquiv |
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| namespace OAI |
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| 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 k Z] |
| [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) |
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| theorem fourfold_hom_ext (f g : BalancedTensor k (A ⊗[k] B) E V →ₗ[k] 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 LinearMap.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 |
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| end ArExplicit.ExternalBalanced |
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| end |
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| end OAI |
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