| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorProjective |
| import Mathlib.LinearAlgebra.TensorProduct.Opposite |
| import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct |
|
|
| namespace ArExplicit.ExternalTensor |
| |
| attribute [local instance] RingHomInvPair.of_ringEquiv RingHomInvPair.of_ringEquiv_symm |
|
|
| universe u |
| variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B] |
| |
| abbrev ProductEnveloping := (A ⊗[k] Aᵐᵒᵖ) ⊗[k] (B ⊗[k] Bᵐᵒᵖ) |
| abbrev EnvelopingProduct := (A ⊗[k] B) ⊗[k] (A ⊗[k] B)ᵐᵒᵖ |
| |
| def envelopingEquiv : ProductEnveloping k A B ≃ₐ[k] EnvelopingProduct k A B := |
| (Algebra.TensorProduct.tensorTensorTensorComm k k k k A Aᵐᵒᵖ B Bᵐᵒᵖ).trans |
| (Algebra.TensorProduct.congr (AlgEquiv.refl : (A ⊗[k] B) ≃ₐ[k] (A ⊗[k] B)) |
| (Algebra.TensorProduct.opAlgEquiv k k A B)) |
| |
| @[simp] theorem envelopingEquiv_tmul (a c : A) (b d : B) : |
| envelopingEquiv k A B ((a ⊗ₜ[k] MulOpposite.op c) ⊗ₜ[k] (b ⊗ₜ[k] MulOpposite.op d)) = |
| (a ⊗ₜ[k] b) ⊗ₜ[k] MulOpposite.op (c ⊗ₜ[k] d) := by |
| simp [envelopingEquiv] |
|
|
| @[simp] theorem envelopingEquiv_symm_tmul (a c : A) (b d : B) : |
| (envelopingEquiv k A B).symm ((a ⊗ₜ[k] b) ⊗ₜ[k] MulOpposite.op (c ⊗ₜ[k] d)) = |
| (a ⊗ₜ[k] MulOpposite.op c) ⊗ₜ[k] (b ⊗ₜ[k] MulOpposite.op d) := by |
| apply (envelopingEquiv k A B).injective |
| simp |
|
|
| abbrev envelopingFunctor : ModuleCat.{u} (ProductEnveloping k A B) ⥤ |
| ModuleCat.{u} (EnvelopingProduct k A B) := |
| ModuleCat.restrictScalars (envelopingEquiv k A B).symm.toRingHom |
| |
| abbrev envelopingObject (M : ModuleCat.{u} (ProductEnveloping k A B)) : |
| ModuleCat.{u} (EnvelopingProduct k A B) := (envelopingFunctor k A B).obj M |
|
|
| private instance forwardInverse : RingHomInvPair (envelopingEquiv k A B).toRingHom |
| (envelopingEquiv k A B).symm.toRingHom where |
| comp_eq := by apply RingHom.ext; intro x |
| comp_eq₂ := by apply RingHom.ext; intro x |
|
|
| private instance backwardInverse : RingHomInvPair (envelopingEquiv k A B).symm.toRingHom |
| (envelopingEquiv k A B).toRingHom where |
| comp_eq := by apply RingHom.ext; intro x |
| comp_eq₂ := by apply RingHom.ext; intro x |
|
|
| def envelopingSemilinear (M : ModuleCat.{u} (ProductEnveloping k A B)) : |
| LinearEquiv (envelopingEquiv k A B).toRingHom |
| (σ' := (envelopingEquiv k A B).symm.toRingHom) M (envelopingObject k A B M) where |
| toFun x := x |
| invFun x := x |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' a x := by |
| change a • x = ((envelopingEquiv k A B).symm ((envelopingEquiv k A B) a)) • x |
| rw [AlgEquiv.symm_apply_apply] |
| |
| theorem enveloping_projective (M : ModuleCat.{u} (ProductEnveloping k A B)) |
| [Module.Projective (ProductEnveloping k A B) M] : |
| Module.Projective (EnvelopingProduct k A B) (envelopingObject k A B M) := |
| Module.Projective.of_equiv (envelopingSemilinear k A B M) |
| |
| instance envelopingBaseModule (M : ModuleCat.{u} (ProductEnveloping k A B)) [Module k M] : |
| Module k (envelopingObject k A B M) := inferInstanceAs (Module k M) |
| |
| instance envelopingTower (M : ModuleCat.{u} (ProductEnveloping k A B)) [Module k M] |
| [IsScalarTower k (ProductEnveloping k A B) M] : |
| IsScalarTower k (EnvelopingProduct k A B) (envelopingObject k A B M) := |
| IsScalarTower.of_algebraMap_smul (A := EnvelopingProduct k A B) fun a x => by |
| change (envelopingEquiv k A B).symm (algebraMap k (EnvelopingProduct k A B) a) • |
| (show M from x) = a • (show M from x) |
| rw [AlgEquiv.commutes, algebraMap_smul] |
| |
| instance envelopingFinite (M : ModuleCat.{u} (ProductEnveloping k A B)) [Module k M] |
| [Module.Finite k M] : Module.Finite k (envelopingObject k A B M) := |
| inferInstanceAs (Module.Finite k M) |
| |
| end ArExplicit.ExternalTensor |
| |
| end |
| |
| end OAI |
| |
