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chore(RepresentationTheory): generalize most of FDRep to commutative rings #22235

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chore(RepresentationTheory): generalize most of FDRep to commutative rings #22235

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73 changes: 38 additions & 35 deletions Mathlib/RepresentationTheory/FDRep.lean
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Expand Up @@ -45,85 +45,88 @@ open CategoryTheory.Limits


/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
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Suggested change
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
/-- The category of finite dimensional `R`-linear representations of a monoid `G`. -/

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I pointed out that line but there's similarly some ks that should be Rs on lines 83 and 144. I'm not sure about the doc string at the beginning of the file, since the case where R is a field is the most important case.

abbrev FDRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
abbrev FDRep (R G : Type u) [Ring R] [Monoid G] :=
Action (FGModuleCat.{u} R) (MonCat.of G)

namespace FDRep

variable {k G : Type u} [Field k] [Monoid G]
variable {R k G : Type u} [CommRing R] [Field k] [Monoid G]

-- Porting note: `@[derive]` didn't work for `FDRep`. Add the 4 instances here.
instance : LargeCategory (FDRep k G) := inferInstance
instance : ConcreteCategory (FDRep k G) (Action.HomSubtype _ _) := inferInstance
instance : Preadditive (FDRep k G) := inferInstance
instance : LargeCategory (FDRep R G) := inferInstance
instance : ConcreteCategory (FDRep R G) (Action.HomSubtype _ _) := inferInstance
instance : Preadditive (FDRep R G) := inferInstance
instance : HasFiniteLimits (FDRep k G) := inferInstance

instance : Linear k (FDRep k G) := by infer_instance
instance : Linear R (FDRep R G) := by infer_instance

instance : CoeSort (FDRep k G) (Type u) :=
instance : CoeSort (FDRep R G) (Type u) :=
⟨fun V => V.V⟩

instance (V : FDRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FDRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FDRep R G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FDRep R G) (FGModuleCat R)).obj V).obj; infer_instance

instance (V : FDRep k G) : Module k V := by
change Module k ((forget₂ (FDRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FDRep R G) : Module R V := by
change Module R ((forget₂ (FDRep R G) (FGModuleCat R)).obj V).obj; infer_instance

instance (V : FDRep R G) : Module.Finite R V := by
change Module.Finite R ((forget₂ (FDRep R G) (FGModuleCat R)).obj V); infer_instance

instance (V : FDRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FDRep k G) (FGModuleCat k)).obj V); infer_instance
infer_instance

/-- All hom spaces are finite dimensional. -/
instance (V W : FDRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FDRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FDRep k G) (FGModuleCat k)))

/-- The monoid homomorphism corresponding to the action of `G` onto `V : FDRep k G`. -/
def ρ (V : FDRep k G) : G →* V →ₗ[k] V :=
def ρ (V : FDRep R G) : G →* V →ₗ[R] V :=
(ModuleCat.endRingEquiv _).toMonoidHom.comp (Action.ρ V).hom

@[simp]
lemma endRingEquiv_symm_comp_ρ (V : FDRep k G) :
lemma endRingEquiv_symm_comp_ρ (V : FDRep R G) :
(MonoidHomClass.toMonoidHom (ModuleCat.endRingEquiv V.V.obj).symm).comp (ρ V) =
(Action.ρ V).hom :=
rfl

@[simp]
lemma endRingEquiv_comp_ρ (V : FDRep k G) :
lemma endRingEquiv_comp_ρ (V : FDRep R G) :
(MonoidHomClass.toMonoidHom (ModuleCat.endRingEquiv V.V.obj)).comp (Action.ρ V).hom = ρ V := rfl

@[simp]
lemma hom_action_ρ (V : FDRep k G) (g : G) : (Action.ρ V g).hom = ρ V g := rfl
lemma hom_action_ρ (V : FDRep R G) (g : G) : (Action.ρ V g).hom = ρ V g := rfl

/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FDRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
def isoToLinearEquiv {V W : FDRep R G} (i : V ≅ W) : V ≃ₗ[R] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat R) (MonCat.of G)).mapIso i)

theorem Iso.conj_ρ {V W : FDRep k G} (i : V ≅ W) (g : G) :
theorem Iso.conj_ρ {V W : FDRep R G} (i : V ≅ W) (g : G) :
W.ρ g = (FDRep.isoToLinearEquiv i).conj (V.ρ g) := by
rw [FDRep.isoToLinearEquiv, ← hom_action_ρ V, ← FGModuleCat.Iso.conj_hom_eq_conj, Iso.conj_apply,
← ModuleCat.hom_ofHom (W.ρ g), ← ModuleCat.hom_ext_iff,
Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
Iso.eq_inv_comp ((Action.forget (FGModuleCat R) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm

/-- Lift an unbundled representation to `FDRep`. -/
@[simps ρ]
abbrev of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FDRep k G :=
⟨FGModuleCat.of k V, MonCat.ofHom ρ ≫ MonCat.ofHom
(ModuleCat.endRingEquiv (ModuleCat.of k V)).symm.toMonoidHom⟩
abbrev of {V : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]
(ρ : Representation R G V) : FDRep R G :=
⟨FGModuleCat.of R V, MonCat.ofHom ρ ≫ MonCat.ofHom
(ModuleCat.endRingEquiv (ModuleCat.of R V)).symm.toMonoidHom⟩

instance : HasForget₂ (FDRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
instance : HasForget₂ (FDRep R G) (Rep R G) where
forget₂ := (forget₂ (FGModuleCat R) (ModuleCat R)).mapAction (MonCat.of G)

theorem forget₂_ρ (V : FDRep k G) : ((forget₂ (FDRep k G) (Rep k G)).obj V).ρ = V.ρ := by
theorem forget₂_ρ (V : FDRep R G) : ((forget₂ (FDRep R G) (Rep R G)).obj V).ρ = V.ρ := by
ext g v; rfl

-- Verify that the monoidal structure is available.
example : MonoidalCategory (FDRep k G) := by infer_instance
example : MonoidalCategory (FDRep R G) := by infer_instance

example : MonoidalPreadditive (FDRep k G) := by infer_instance
example : MonoidalPreadditive (FDRep R G) := by infer_instance

example : MonoidalLinear k (FDRep k G) := by infer_instance
example : MonoidalLinear R (FDRep R G) := by infer_instance

open Module

Expand All @@ -139,13 +142,13 @@ theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FDRep k G) [Simple V] [
CategoryTheory.finrank_hom_simple_simple k V W

/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FDRep k G) :
((forget₂ (FDRep k G) (Rep k G)).obj X ⟶
(forget₂ (FDRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y where
def forget₂HomLinearEquiv (X Y : FDRep R G) :
((forget₂ (FDRep R G) (Rep R G)).obj X ⟶
(forget₂ (FDRep R G) (Rep R G)).obj Y) ≃ₗ[R] X ⟶ Y where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
invFun f := ⟨(forget₂ (FGModuleCat R) (ModuleCat R)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; rfl

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