| import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingBalancedMap |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorFirst |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct ModuleCat.Algebra |
|
|
| namespace ArExplicit.EnvelopingBimodule |
|
|
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
| variable {M N Q : ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ)} |
| variable [Module k M] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) M] |
| variable [Module k N] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) N] |
| variable [Module k Q] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) Q] |
|
|
| def mapRight (f : M ⟶ N) : |
| EnvelopingBimodule k R M →ₗ[Rᵐᵒᵖ] EnvelopingBimodule k R N where |
| toFun := map f |
| map_add' := (map f).map_add |
| map_smul' := map_right_smul f |
|
|
| theorem mapRight_injective (f : M ⟶ N) (hf : Function.Injective f.hom) : |
| Function.Injective (mapRight f) := by |
| intro x y h |
| apply EnvelopingBimodule.ext |
| apply hf |
| exact congrArg EnvelopingBimodule.value h |
|
|
| theorem mapRight_surjective (f : M ⟶ N) (hf : Function.Surjective f.hom) : |
| Function.Surjective (mapRight f) := by |
| rintro ⟨y⟩ |
| obtain ⟨x, hx⟩ := hf y |
| refine ⟨⟨x⟩, ?_⟩ |
| apply EnvelopingBimodule.ext |
| exact hx |
|
|
| theorem mapRight_exact (f : M ⟶ N) (g : N ⟶ Q) |
| (he : LinearMap.range f.hom = LinearMap.ker g.hom) : |
| LinearMap.range (mapRight f) = LinearMap.ker (mapRight g) := by |
| ext y |
| constructor |
| · rintro ⟨x, rfl⟩ |
| change mapRight g (mapRight f x) = 0 |
| apply EnvelopingBimodule.ext |
| have h : f.hom x.value ∈ LinearMap.range f.hom := ⟨x.value, rfl⟩ |
| rw [he] at h |
| exact h |
| · intro hy |
| have hy' : y.value ∈ LinearMap.ker g.hom := congrArg EnvelopingBimodule.value hy |
| rw [← he] at hy' |
| obtain ⟨x, hx⟩ := hy' |
| refine ⟨⟨x⟩, ?_⟩ |
| apply EnvelopingBimodule.ext |
| exact hx |
|
|
| variable (X : ModuleCat.{u} R) [Module k X] |
|
|
| @[simp] theorem mapFirst_mapRight_apply (f : M ⟶ N) |
| (z : BalancedTensor k R (EnvelopingBimodule k R M) X) : |
| BalancedTensor.mapFirst X (mapRight f) z = tensorMap X f z := rfl |
|
|
| theorem tensorMap_shortExact (f : M ⟶ N) (g : N ⟶ Q) |
| (hf : Function.Injective f.hom) (hg : Function.Surjective g.hom) |
| (he : LinearMap.range f.hom = LinearMap.ker g.hom) |
| [Module.Projective Rᵐᵒᵖ (EnvelopingBimodule k R Q)] : |
| Function.Injective (tensorMap X f) ∧ Function.Surjective (tensorMap X g) ∧ |
| LinearMap.range (tensorMap X f) = LinearMap.ker (tensorMap X g) := by |
| obtain ⟨hi, hs, hx⟩ := BalancedTensor.mapFirst_shortExact (k := k) (X := X) |
| (mapRight f) (mapRight g) (mapRight_injective f hf) |
| (mapRight_surjective g hg) (mapRight_exact f g he) |
| refine ⟨hi, hs, ?_⟩ |
| ext z |
| exact Iff.of_eq (congrArg (fun S : Submodule k (BalancedTensor k R (EnvelopingBimodule k R N) X) => |
| z ∈ S) hx) |
|
|
| end ArExplicit.EnvelopingBimodule |
|
|
| end |
|
|
| end OAI |
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|
