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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingBimodule
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct ModuleCat.Algebra
namespace ArExplicit.EnvelopingBimodule
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R]
variable {M N P : ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ)}
variable [Module k M] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) M]
variable [Module k N] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) N]
variable [Module k P] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) P]
def map (f : M ⟶ N) : EnvelopingBimodule k R M →ₗ[k] EnvelopingBimodule k R N where
toFun x := ⟨f.hom x.value⟩
map_add' x y := by
ext
exact f.hom.map_add x.value y.value
map_smul' c x := by
ext
exact (f.hom.restrictScalars k).map_smul c x.value
@[simp] theorem map_value (f : M ⟶ N) (x : EnvelopingBimodule k R M) :
(map f x).value = f.hom x.value := rfl
@[simp] theorem map_left_smul (f : M ⟶ N) (a : R)
(x : EnvelopingBimodule k R M) : map f (a • x) = a • map f x := by
ext
exact f.hom.map_smul (a ⊗ₜ[k] (1 : Rᵐᵒᵖ)) x.value
@[simp] theorem map_right_smul (f : M ⟶ N) (b : Rᵐᵒᵖ)
(x : EnvelopingBimodule k R M) : map f (b • x) = b • map f x := by
ext
exact f.hom.map_smul ((1 : R) ⊗ₜ[k] b) x.value
def mapLeft (f : M ⟶ N) : EnvelopingBimodule k R M →ₗ[R] EnvelopingBimodule k R N where
toFun := map f
map_add' := (map f).map_add
map_smul' := map_left_smul f
@[simp] theorem map_id : map (𝟙 M) = LinearMap.id := by
ext x
rfl
@[simp] theorem map_comp (f : M ⟶ N) (g : N ⟶ P) :
map (f ≫ g) = (map g).comp (map f) := by
ext x
rfl
@[simp] theorem map_zero : map (0 : M ⟶ N) = 0 := by
ext x
rfl
@[simp] theorem map_add (f g : M ⟶ N) : map (f + g) = map f + map g := by
ext x
rfl
variable (X : ModuleCat.{u} R) [Module k X] [IsScalarTower k R X]
def tensorMap (f : M ⟶ N) :
BalancedTensor k R (EnvelopingBimodule k R M) X →ₗ[R]
BalancedTensor k R (EnvelopingBimodule k R N) X where
__ := (BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X)
(map_right_smul f) (fun _ _ => rfl)).toAddHom
map_smul' a z := by
change BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X)
(map_right_smul f) (fun _ _ => rfl) (a • z) =
a • BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X)
(map_right_smul f) (fun _ _ => rfl) z
induction z using BalancedTensor.induction_on with
| zero => simp
| tmul m x => simp only [BalancedTensor.smul_tmul, BalancedTensor.map_tmul,
LinearMap.id_apply, map_left_smul]
| add z w hz hw => simp_all
omit [IsScalarTower k R X]
@[simp] theorem tensorMap_tmul (f : M ⟶ N)
(m : EnvelopingBimodule k R M) (x : X) :
tensorMap X f (BalancedTensor.tmul m x) = BalancedTensor.tmul (map f m) x := rfl
@[simp] theorem tensorMap_id : tensorMap X (𝟙 M) = LinearMap.id := by
apply LinearMap.ext
intro z
induction z using BalancedTensor.induction_on with
| zero => simp
| tmul m x => rfl
| add z w hz hw => simp_all
@[simp] theorem tensorMap_comp (f : M ⟶ N) (g : N ⟶ P) :
tensorMap X (f ≫ g) = (tensorMap X g).comp (tensorMap X f) := by
apply LinearMap.ext
intro z
induction z using BalancedTensor.induction_on with
| zero => simp
| tmul m x => rfl
| add z w hz hw => simp_all
@[simp] theorem tensorMap_zero : tensorMap X (0 : M ⟶ N) = 0 := by
apply LinearMap.ext
intro z
induction z using BalancedTensor.induction_on with
| zero => simp
| tmul m x => simp
| add z w hz hw => simp_all
@[simp] theorem tensorMap_add (f g : M ⟶ N) :
tensorMap X (f + g) = tensorMap X f + tensorMap X g := by
apply LinearMap.ext
intro z
induction z using BalancedTensor.induction_on with
| zero => simp
| tmul m x => simp [BalancedTensor.add_tmul]
| add z w hz hw =>
simp only [_root_.LinearMap.map_add, hz, hw]
end ArExplicit.EnvelopingBimodule
end
end OAI
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