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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularInduction
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorPi
namespace OAI
noncomputable section
namespace ArExplicit.TriangularInduction
variable {k R F : Type*} [CommRing k] [Ring R] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F]
variable {M N L : Type*}
variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M]
variable [AddCommGroup N] [Module k N] [Module R N] [IsScalarTower k R N]
variable [AddCommGroup L] [Module k L] [Module R L] [IsScalarTower k R L]
def eMap (f : M →ₗ[R] N) : TriangularE k R F M →ₗ[Triangular R F] TriangularE k R F N := eHom f
def jMap (f : M →ₗ[R] N) : TriangularJ k R F M →ₗ[Triangular R F] TriangularJ k R F N := jHom f
omit [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] in
@[simp] theorem eMap_top (f : M →ₗ[R] N) (x : TriangularE k R F M) :
(eMap f x).top = f x.top := rfl
omit [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] in
@[simp] theorem eMap_bottom (f : M →ₗ[R] N) (x : TriangularE k R F M) :
(eMap f x).bottom = BalancedTensor.mapRight (S := R) f x.bottom := rfl
omit [Algebra k R] [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F]
[SMulCommClass k Rᵐᵒᵖ F] [Module k M] [IsScalarTower k R M]
[Module k N] [IsScalarTower k R N] in
@[simp] theorem jMap_bottom (f : M →ₗ[R] N) (x : TriangularJ k R F M) :
(jMap f x).bottom = f x.bottom := rfl
@[simp] theorem eMap_id : eMap (k := k) (F := F) (LinearMap.id : M →ₗ[R] M) = LinearMap.id := by
apply eHomEquiv.injective
ext m
rfl
@[simp] theorem eMap_comp (f : M →ₗ[R] N) (g : N →ₗ[R] L) :
eMap (k := k) (F := F) (g.comp f) = (eMap g).comp (eMap f) := by
apply eHomEquiv.injective
ext m
rfl
@[simp] theorem jMap_id : jMap (k := k) (F := F) (LinearMap.id : M →ₗ[R] M) = LinearMap.id := by
apply jHomEquiv.injective
ext m
rfl
omit [Module k M] [IsScalarTower k R M] [Module k N] [IsScalarTower k R N] in
@[simp] theorem jMap_comp (f : M →ₗ[R] N) (g : N →ₗ[R] L) :
jMap (k := k) (F := F) (g.comp f) = (jMap g).comp (jMap f) := by
apply jHomEquiv.injective
ext m
rfl
@[simp] theorem eMap_zero : eMap (k := k) (F := F) (0 : M →ₗ[R] N) = 0 := by
apply eHomEquiv.injective
ext m
rfl
omit [Module k M] [IsScalarTower k R M] in
@[simp] theorem jMap_zero : jMap (k := k) (F := F) (0 : M →ₗ[R] N) = 0 := by
apply jHomEquiv.injective
ext m
rfl
variable (ι : Type*) [Fintype ι] [DecidableEq ι]
def ePi : TriangularE k R F (ι → M) ≃ₗ[Triangular R F] (ι → TriangularE k R F M) where
toFun x i := ⟨x.top i, BalancedTensor.piRight (S := R) x.bottom i⟩
invFun y := ⟨fun i => (y i).top, (BalancedTensor.piRight (S := R)).symm (fun i => (y i).bottom)⟩
left_inv x := by ext <;> simp
right_inv y := by ext <;> simp
map_add' x y := by ext <;> simp
map_smul' a x := by
ext i
· rfl
· change BalancedTensor.piRight (S := R)
(BalancedTensor.tmul a.off x.top + a.bottom • x.bottom) i =
BalancedTensor.tmul a.off (x.top i) + a.bottom • BalancedTensor.piRight (S := R) x.bottom i
simp only [map_add, map_smul, Pi.add_apply, Pi.smul_apply]
rfl
def jPi : TriangularJ k R F (ι → M) ≃ₗ[Triangular R F] (ι → TriangularJ k R F M) where
toFun x i := ⟨0, x.bottom i⟩
invFun y := ⟨0, fun i => (y i).bottom⟩
left_inv x := by ext; rfl
right_inv y := by ext; rfl
map_add' x y := by ext; rfl
map_smul' a x := by ext; simp
end ArExplicit.TriangularInduction
end
end OAI
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