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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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