File size: 3,591 Bytes
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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap
namespace OAI
namespace ArExplicit.BalancedTensor
variable {k R : Type*} [CommRing k] [Ring R] [Algebra k R]
section Left
variable {M : Type*} [AddCommGroup M] [Module k M] [Module R M]
variable [IsScalarTower k R M]
def leftAction : R →ₗ[k] M →ₗ[k] M where
toFun r :=
{ toFun := fun m => r • m
map_add' := smul_add r
map_smul' := fun c m => smul_comm r c m }
map_add' r s := by ext m; exact add_smul r s m
map_smul' c r := by ext m; exact smul_assoc c r m
def lunitMap : BalancedTensor k R R M →ₗ[R] M :=
liftLeft leftAction (by intro x r m; exact mul_smul x r m)
(by intro s r m; exact mul_smul s r m)
@[simp] theorem lunitMap_tmul (r : R) (m : M) :
lunitMap (tmul r m : BalancedTensor k R R M) = r • m := rfl
def lunitInv : M →ₗ[R] BalancedTensor k R R M where
toFun m := tmul 1 m
map_add' := tmul_add 1
map_smul' r m := by
change tmul (k := k) (R := R) 1 (r • m) = tmul (r * 1) m
rw [mul_one]
simpa using (balance (k := k) (R := R) (1 : R) r m).symm
def lunit : BalancedTensor k R R M ≃ₗ[R] M where
__ := lunitMap (k := k) (R := R) (M := M)
invFun := lunitInv
right_inv m := by simp [lunitInv]
left_inv z := by
change lunitInv (lunitMap z) = z
induction z using induction_on with
| zero => simp
| tmul r m =>
change tmul (k := k) (R := R) 1 (r • m) = tmul r m
simpa using (balance (k := k) (R := R) (1 : R) r m).symm
| add x y hx hy => simp only [map_add, hx, hy]
@[simp] theorem lunit_tmul (r : R) (m : M) :
lunit (tmul r m : BalancedTensor k R R M) = r • m := rfl
end Left
section Right
variable {F : Type*} [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F]
variable [IsScalarTower k Rᵐᵒᵖ F]
def rightAction : F →ₗ[k] R →ₗ[k] F where
toFun f :=
{ toFun := fun r => MulOpposite.op r • f
map_add' r s := by simp [add_smul]
map_smul' c r := by simp [MulOpposite.op_smul, smul_assoc] }
map_add' f g := by ext r; exact smul_add _ _ _
map_smul' c f := by ext r; exact smul_comm _ _ _
def runitMap : BalancedTensor k R F R →ₗ[k] F :=
lift rightAction (by
intro f r s
change MulOpposite.op s • MulOpposite.op r • f = MulOpposite.op (r * s) • f
rw [MulOpposite.op_mul, mul_smul])
@[simp] theorem runitMap_tmul (f : F) (r : R) :
runitMap (tmul f r : BalancedTensor k R F R) = MulOpposite.op r • f := rfl
def runit : BalancedTensor k R F R ≃ₗ[k] F where
__ := runitMap (k := k) (R := R) (F := F)
invFun f := tmul f 1
right_inv f := by simp [runitMap_tmul]
left_inv z := by
change tmul (runitMap z) 1 = z
induction z using induction_on with
| zero => simp
| tmul f r => simpa using (balance (k := k) (R := R) f r (1 : R))
| add x y hx hy => simpa only [map_add, add_tmul] using congrArg₂ (· + ·) hx hy
@[simp] theorem runit_tmul (f : F) (r : R) :
runit (tmul f r : BalancedTensor k R F R) = MulOpposite.op r • f := rfl
variable {S : Type*} [Ring S] [Module S F]
variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
def runitLeft : BalancedTensor k R F R ≃ₗ[S] F where
__ := (runit (k := k) (R := R) (F := F)).toAddEquiv
map_smul' s z := by
change runit (s • z) = s • runit z
induction z using induction_on with
| zero => simp
| tmul f r => simpa only [smul_tmul, runit_tmul] using (smul_comm s (MulOpposite.op r) f).symm
| add x y hx hy => simp only [smul_add, map_add] at *; rw [hx, hy]
end Right
end ArExplicit.BalancedTensor
end OAI
|