Documentation

Mathlib.Algebra.Category.FGModuleCat.Basic

The category of finitely generated modules over a ring #

This introduces FGModuleCat R, the category of finitely generated modules over a ring R. It is implemented as a full subcategory on a subtype of ModuleCat R.

When K is a field, FGModuleCatCat K is the category of finite dimensional vector spaces over K.

We first create the instance as a preadditive category. When R is commutative we then give the structure as an R-linear monoidal category. When R is a field we give it the structure of a closed monoidal category and then as a right-rigid monoidal category.

Future work #

Show that FGModuleCat R is abelian when R is (left)-noetherian.

def ModuleCat.isFG (R : Type u) [Ring R] :

CategoryTheory.ObjectProperty (ModuleCat R)

Finitely generated modules, as a property of objects of ModuleCat R.

Equations

ModuleCat.isFG R V = Module.Finite R ↑V

theorem ModuleCat.isFG_iff {R : Type u} [Ring R] (V : ModuleCat R) :

isFG R V ↔ Module.Finite R ↑V

@[reducible, inline]

abbrev FGModuleCat (R : Type u) [Ring R] :

Type (u + 1)

The category of finitely generated modules.

Equations

FGModuleCat R = (ModuleCat.isFG R).FullSubcategory

def FGModuleCat.carrier {R : Type u} [Ring R] (M : FGModuleCat R) :

A synonym for M.obj.carrier, which we can mark with @[coe].

Equations

↑M = ↑M.obj

instance instCoeSortFGModuleCatType {R : Type u} [Ring R] :

CoeSort (FGModuleCat R) (Type u)

Equations

instCoeSortFGModuleCatType = { coe := FGModuleCat.carrier }

@[simp]

theorem FGModuleCat.obj_carrier {R : Type u} [Ring R] (M : FGModuleCat R) :

↑M.obj = ↑M

instance instAddCommGroupCarrier {R : Type u} [Ring R] (M : FGModuleCat R) :

AddCommGroup ↑M

Equations

instAddCommGroupCarrier M = id inferInstance

instance instModuleCarrier {R : Type u} [Ring R] (M : FGModuleCat R) :

Module R ↑M

Equations

instModuleCarrier M = id inferInstance

@[simp]

theorem FGModuleCat.hom_comp (R : Type u) [Ring R] (A B C : FGModuleCat R) (f : A ⟶ B) (g : B ⟶ C) :

ModuleCat.Hom.hom (CategoryTheory.CategoryStruct.comp f g) = ModuleCat.Hom.hom g ∘ₗ ModuleCat.Hom.hom f

@[simp]

theorem FGModuleCat.hom_id (R : Type u) [Ring R] (A : FGModuleCat R) :

ModuleCat.Hom.hom (CategoryTheory.CategoryStruct.id A) = LinearMap.id

instance FGModuleCat.finite (R : Type u) [Ring R] (V : FGModuleCat R) :

Module.Finite R ↑V

instance FGModuleCat.instInhabited (R : Type u) [Ring R] :

Inhabited (FGModuleCat R)

Equations

FGModuleCat.instInhabited R = { default := { obj := ModuleCat.of R R, property := ⋯ } }

@[reducible, inline]

abbrev FGModuleCat.of (R : Type u) [Ring R] (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] :

Lift an unbundled finitely generated module to FGModuleCat R.

Equations

FGModuleCat.of R V = { obj := ModuleCat.of R V, property := ⋯ }

@[simp]

theorem FGModuleCat.of_carrier (R : Type u) [Ring R] (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] :

↑(of R V) = V

@[reducible, inline]

abbrev FGModuleCat.ofHom {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (f : V →ₗ[R] W) :

of R V ⟶ of R W

Lift a linear map between finitely generated modules to FGModuleCat R.

Equations

FGModuleCat.ofHom f = ModuleCat.ofHom f

theorem FGModuleCat.hom_ext {R : Type u} [Ring R] {V W : FGModuleCat R} {f g : V ⟶ W} (h : ModuleCat.Hom.hom f = ModuleCat.Hom.hom g) :

f = g

theorem FGModuleCat.hom_ext_iff {R : Type u} [Ring R] {V W : FGModuleCat R} {f g : V ⟶ W} :

f = g ↔ ModuleCat.Hom.hom f = ModuleCat.Hom.hom g

instance FGModuleCat.instFiniteCarrier (R : Type u) [Ring R] (V : FGModuleCat R) :

Module.Finite R ↑V

instance FGModuleCat.instFullModuleCatForget₂ (R : Type u) [Ring R] :

(CategoryTheory.forget₂ (FGModuleCat R) (ModuleCat R)).Full

def FGModuleCat.isoToLinearEquiv {R : Type u} [Ring R] {V W : FGModuleCat R} (i : V ≅ W) :

↑V ≃ₗ[R] ↑W

Converts and isomorphism in the category FGModuleCat R to a LinearEquiv between the underlying modules.

Equations

FGModuleCat.isoToLinearEquiv i = ((CategoryTheory.forget₂ (FGModuleCat R) (ModuleCat R)).mapIso i).toLinearEquiv

def LinearEquiv.toFGModuleCatIso {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :

FGModuleCat.of R V ≅ FGModuleCat.of R W

Converts a LinearEquiv to an isomorphism in the category FGModuleCat R.

Equations

e.toFGModuleCatIso = { hom := ModuleCat.ofHom ↑e, inv := ModuleCat.ofHom ↑e.symm, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem LinearEquiv.toFGModuleCatIso_inv {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :

e.toFGModuleCatIso.inv = ModuleCat.ofHom ↑e.symm

@[simp]

theorem LinearEquiv.toFGModuleCatIso_hom {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :

e.toFGModuleCatIso.hom = ModuleCat.ofHom ↑e

instance FGModuleCat.instIsMonoidalModuleCatIsFG (R : Type u) [CommRing R] :

(ModuleCat.isFG R).IsMonoidal

@[simp]

theorem FGModuleCat.tensorUnit_obj (R : Type u) [CommRing R] :

(CategoryTheory.MonoidalCategoryStruct.tensorUnit (FGModuleCat R)).obj = CategoryTheory.MonoidalCategoryStruct.tensorUnit (ModuleCat R)

@[simp]

theorem FGModuleCat.tensorObj_obj (R : Type u) [CommRing R] (M N : FGModuleCat R) :

(CategoryTheory.MonoidalCategoryStruct.tensorObj M N).obj = CategoryTheory.MonoidalCategoryStruct.tensorObj M.obj N.obj

instance FGModuleCat.instAdditiveModuleCatForget₂ (R : Type u) [CommRing R] :

(CategoryTheory.forget₂ (FGModuleCat R) (ModuleCat R)).Additive

instance FGModuleCat.instLinearModuleCatForget₂ (R : Type u) [CommRing R] :

CategoryTheory.Functor.Linear R (CategoryTheory.forget₂ (FGModuleCat R) (ModuleCat R))

theorem FGModuleCat.Iso.conj_eq_conj (R : Type u) [CommRing R] {V W : FGModuleCat R} (i : V ≅ W) (f : CategoryTheory.End V) :

i.conj f = ofHom ((isoToLinearEquiv i).conj (ModuleCat.Hom.hom f))

theorem FGModuleCat.Iso.conj_hom_eq_conj (R : Type u) [CommRing R] {V W : FGModuleCat R} (i : V ≅ W) (f : CategoryTheory.End V) :

ModuleCat.Hom.hom (i.conj f) = (isoToLinearEquiv i).conj (ModuleCat.Hom.hom f)

instance FGModuleCat.instFiniteHom (K : Type u) [Field K] (V W : FGModuleCat K) :

Module.Finite K (V ⟶ W)

instance FGModuleCat.instIsMonoidalClosedModuleCatIsFG (K : Type u) [Field K] :

(ModuleCat.isFG K).IsMonoidalClosed

@[simp]

theorem FGModuleCat.ihom_obj (K : Type u) [Field K] (V W : FGModuleCat K) :

(CategoryTheory.ihom V).obj W = of K (V ⟶ W)

def FGModuleCat.FGModuleCatDual (K : Type u) [Field K] (V : FGModuleCat K) :

The dual module is the dual in the rigid monoidal category FGModuleCat K.

Equations

FGModuleCat.FGModuleCatDual K V = { obj := ModuleCat.of K (Module.Dual K ↑V), property := ⋯ }

@[simp]

theorem FGModuleCat.FGModuleCatDual_obj (K : Type u) [Field K] (V : FGModuleCat K) :

(FGModuleCatDual K V).obj = ModuleCat.of K (Module.Dual K ↑V)

@[simp]

theorem FGModuleCat.FGModuleCatDual_coe (K : Type u) [Field K] (V : FGModuleCat K) :

↑(FGModuleCatDual K V) = Module.Dual K ↑V

def FGModuleCat.FGModuleCatCoevaluation (K : Type u) [Field K] (V : FGModuleCat K) :

CategoryTheory.MonoidalCategoryStruct.tensorUnit (FGModuleCat K) ⟶ CategoryTheory.MonoidalCategoryStruct.tensorObj V (FGModuleCatDual K V)

The coevaluation map is defined in LinearAlgebra.coevaluation.

Equations

FGModuleCat.FGModuleCatCoevaluation K V = ModuleCat.ofHom (coevaluation K ↑V)

theorem FGModuleCat.FGModuleCatCoevaluation_apply_one (K : Type u) [Field K] (V : FGModuleCat K) :

(ModuleCat.Hom.hom (FGModuleCatCoevaluation K V)) 1 = ∑ i : ↑(Basis.ofVectorSpaceIndex K ↑V), (Basis.ofVectorSpace K ↑V) i ⊗ₜ[K] (Basis.ofVectorSpace K ↑V).coord i

def FGModuleCat.FGModuleCatEvaluation (K : Type u) [Field K] (V : FGModuleCat K) :

CategoryTheory.MonoidalCategoryStruct.tensorObj (FGModuleCatDual K V) V ⟶ CategoryTheory.MonoidalCategoryStruct.tensorUnit (FGModuleCat K)

The evaluation morphism is given by the contraction map.

Equations

FGModuleCat.FGModuleCatEvaluation K V = ModuleCat.ofHom (contractLeft K ↑V)

theorem FGModuleCat.FGModuleCatEvaluation_apply (K : Type u) [Field K] (V : FGModuleCat K) (f : ↑(FGModuleCatDual K V)) (x : ↑V) :

(ModuleCat.Hom.hom (FGModuleCatEvaluation K V)) (f ⊗ₜ[K] x) = f.toFun x

@[simp]

theorem FGModuleCat.FGModuleCatEvaluation_apply' (K : Type u) [Field K] (V : FGModuleCat K) (f : ↑(FGModuleCatDual K V)) (x : ↑V) :

(ModuleCat.Hom.hom (FGModuleCatEvaluation K V)) (f ⊗ₜ[K] x) = f.toFun x

@[simp]-normal form of FGModuleCatEvaluation_apply, where the carriers have been unfolded.

instance FGModuleCat.exactPairing (K : Type u) [Field K] (V : FGModuleCat K) :

CategoryTheory.ExactPairing V (FGModuleCatDual K V)

Equations

One or more equations did not get rendered due to their size.

instance FGModuleCat.rightDual (K : Type u) [Field K] (V : FGModuleCat K) :

CategoryTheory.HasRightDual V

Equations

FGModuleCat.rightDual K V = { rightDual := FGModuleCat.FGModuleCatDual K V, exact := FGModuleCat.exactPairing K V }

instance FGModuleCat.rightRigidCategory (K : Type u) [Field K] :

CategoryTheory.RightRigidCategory (FGModuleCat K)

Equations

FGModuleCat.rightRigidCategory K = { rightDual := FGModuleCat.rightDual K }

@[simp] lemmas for LinearMap.comp and categorical identities.

@[simp]

theorem LinearMap.comp_id_fgModuleCat {R : Type u} [Ring R] {G : FGModuleCat R} {H : Type u} [AddCommGroup H] [Module R H] (f : ↑G →ₗ[R] H) :

f ∘ₗ ModuleCat.Hom.hom (CategoryTheory.CategoryStruct.id G) = f

@[simp]

theorem LinearMap.id_fgModuleCat_comp {R : Type u} [Ring R] {G : Type u} [AddCommGroup G] [Module R G] {H : FGModuleCat R} (f : G →ₗ[R] ↑H) :

ModuleCat.Hom.hom (CategoryTheory.CategoryStruct.id H) ∘ₗ f = f