| import OAI.Algebra.AuslanderReiten.Modules.CoinducedEmbedding |
| import OAI.Algebra.AuslanderReiten.Modules.SplitModules |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit |
|
|
| @[ext] structure CoinducedModule (k R M : Type*) [Field k] [Ring R] [Algebra k R] |
| [AddCommGroup M] [Module k M] where |
| functional : R →ₗ[k] M |
|
|
| namespace CoinducedModule |
|
|
| variable {k R M : Type*} [Field k] [Ring R] [Algebra k R] |
| variable [AddCommGroup M] [Module k M] |
|
|
| def coordinates : CoinducedModule k R M ≃ (R →ₗ[k] M) where |
| toFun := functional |
| invFun := CoinducedModule.mk |
| left_inv _ := rfl |
| right_inv _ := rfl |
| |
| instance : AddCommGroup (CoinducedModule k R M) := coordinates.addCommGroup |
| instance : Module k (CoinducedModule k R M) := coordinates.addEquiv.module k |
| |
| @[simp] theorem functional_zero (r : R) : (0 : CoinducedModule k R M).functional r = 0 := rfl |
| @[simp] theorem functional_add (f g : CoinducedModule k R M) (r : R) : |
| (f + g).functional r = f.functional r + g.functional r := rfl |
| @[simp] theorem functional_smul (a : k) (f : CoinducedModule k R M) (r : R) : |
| (a • f).functional r = a • f.functional r := rfl |
| |
| instance : SMul R (CoinducedModule k R M) := |
| ⟨fun r f => ⟨{ toFun := fun t => f.functional (t * r) |
| map_add' := fun s t => by simp [add_mul] |
| map_smul' := fun a t => by simp }⟩⟩ |
| |
| @[simp] theorem functional_action (r t : R) (f : CoinducedModule k R M) : |
| (r • f).functional t = f.functional (t * r) := rfl |
| |
| instance : Module R (CoinducedModule k R M) where |
| one_smul f := by ext t; simp |
| mul_smul r s f := by ext t; simp [mul_assoc] |
| smul_zero r := by ext t; simp |
| smul_add r f g := by ext t; simp |
| zero_smul f := by ext t; simp |
| add_smul r s f := by ext t; simp [mul_add] |
| |
| instance : IsScalarTower k R (CoinducedModule k R M) where |
| smul_assoc a r f := by ext t; simp |
| |
| instance : SMulCommClass k R (CoinducedModule k R M) where |
| smul_comm a r f := by ext t; rfl |
|
|
| def scalarEquiv : CoinducedModule k R M ≃ₗ[k] (R →ₗ[k] M) := |
| coordinates.addEquiv.linearEquiv k |
| |
| instance [Module.Finite k R] [Module.Finite k M] : Module.Finite k (CoinducedModule k R M) := |
| Module.Finite.equiv scalarEquiv.symm |
| |
| section Embedding |
| |
| variable [Module R M] [IsScalarTower k R M] |
| |
| def embedding : M →ₗ[R] CoinducedModule k R M where |
| toFun m := ⟨{ toFun := fun t => t • m |
| map_add' := fun r s => add_smul r s m |
| map_smul' := fun a r => smul_assoc a r m }⟩ |
| map_add' m n := by ext t; exact smul_add t m n |
| map_smul' r m := by ext t; exact (mul_smul t r m).symm |
|
|
| @[simp] theorem embedding_apply (m : M) (t : R) : |
| (embedding (k := k) m).functional t = t • m := rfl |
|
|
| theorem embedding_injective : Function.Injective (embedding (k := k) (R := R) (M := M)) := by |
| intro m n h |
| have h' := congrArg (fun f : CoinducedModule k R M => f.functional 1) h |
| simpa using h' |
|
|
| end Embedding |
|
|
| variable [Module.Finite k M] |
|
|
| private def coefficients : M ≃ₗ[k] (Fin (Module.finrank k M) → k) := |
| (Module.finBasis k M).equivFun |
|
|
| def coordinateEquiv : CoinducedModule k R M ≃ₗ[R] |
| (Fin (Module.finrank k M) → Coinduced k R) where |
| toFun f i := ⟨(LinearMap.proj i).comp (coefficients.toLinearMap.comp f.functional)⟩ |
| invFun g := ⟨coefficients.symm.toLinearMap.comp |
| (LinearMap.pi fun i => (g i).functional)⟩ |
| left_inv f := by |
| ext t |
| exact coefficients.symm_apply_apply (f.functional t) |
| right_inv g := by |
| funext i |
| apply Coinduced.ext |
| apply LinearMap.ext |
| intro t |
| exact congrFun (coefficients.apply_symm_apply (fun j => (g j).functional t)) i |
| map_add' f g := by funext i; apply Coinduced.ext; ext t; simp |
| map_smul' r f := by funext i; apply Coinduced.ext; ext t; rfl |
|
|
| def freeEquiv [Module.Finite k R] (τ : FrobeniusTrace k R) : |
| CoinducedModule k R M ≃ₗ[R] (Fin (Module.finrank k M) → R) := |
| coordinateEquiv.trans (LinearEquiv.piCongrRight fun _ => τ.leftEquiv.symm) |
|
|
| theorem finite [Module.Finite k R] (τ : FrobeniusTrace k R) : |
| Module.Finite R (CoinducedModule k R M) := Module.Finite.equiv (freeEquiv τ).symm |
|
|
| theorem free [Module.Finite k R] (τ : FrobeniusTrace k R) : |
| Module.Free R (CoinducedModule k R M) := Module.Free.of_equiv (freeEquiv τ).symm |
|
|
| theorem projective [Module.Finite k R] (τ : FrobeniusTrace k R) : |
| Module.Projective R (CoinducedModule k R M) := by |
| let := free (M := M) τ |
| exact Module.Projective.of_free |
|
|
| end CoinducedModule |
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
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|
