File size: 2,784 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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap
namespace OAI
namespace ArExplicit.BalancedTensor
variable {k R S F M N L : Type*} [CommRing k] [Ring R] [Ring S] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F]
variable [Module S F] [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
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 mapRight (b : M →ₗ[R] N) : BalancedTensor k R F M →ₗ[S] BalancedTensor k R F N :=
liftLeft
{ toFun := fun f => (mk f).comp (b.restrictScalars k)
map_add' := by intros; ext m; simp
map_smul' := by intros; ext m; simp }
(by
intro f r m
change tmul (MulOpposite.op r • f) (b m) = tmul f (b (r • m))
rw [b.map_smul, balance])
(by intro s f m; exact (smul_tmul s f (b m)).symm)
@[simp] theorem mapRight_tmul (b : M →ₗ[R] N) (f : F) (m : M) :
mapRight (k := k) (S := S) b (tmul f m) = tmul f (b m) := rfl
omit [Algebra k R] [IsScalarTower k R M] in
@[ext] theorem leftHom_ext {Q : Type*} [AddCommGroup Q] [Module S Q]
(f g : BalancedTensor k R F M →ₗ[S] Q)
(h : ∀ x y, f (tmul x y) = g (tmul x y)) : f = g := by
apply LinearMap.ext
intro z
induction z using induction_on with
| zero => simp
| tmul x y => exact h x y
| add x y hx hy => simp only [map_add, hx, hy]
@[simp] theorem mapRight_id : mapRight (k := k) (S := S) (LinearMap.id : M →ₗ[R] M) =
(LinearMap.id : BalancedTensor k R F M →ₗ[S] BalancedTensor k R F M) := by
apply leftHom_ext
intro f m
rfl
@[simp] theorem mapRight_comp (b : M →ₗ[R] N) (c : N →ₗ[R] L) :
mapRight (k := k) (F := F) (S := S) (c.comp b) = (mapRight c).comp (mapRight b) := by
apply leftHom_ext
intro f m
rfl
@[simp] theorem mapRight_zero : mapRight (k := k) (S := S) (0 : M →ₗ[R] N) =
(0 : BalancedTensor k R F M →ₗ[S] BalancedTensor k R F N) := by
apply leftHom_ext
intro f m
exact tmul_zero f
theorem mapRight_add (b c : M →ₗ[R] N) :
mapRight (k := k) (F := F) (S := S) (b + c) = mapRight b + mapRight c := by
apply leftHom_ext
intro f m
exact tmul_add f (b m) (c m)
theorem mapRight_surjective (b : M →ₗ[R] N) (hb : Function.Surjective b) :
Function.Surjective (mapRight (k := k) (F := F) (S := S) b) := by
intro z
induction z using induction_on with
| zero => exact ⟨0, map_zero _⟩
| tmul f n =>
obtain ⟨m, rfl⟩ := hb n
exact ⟨tmul f m, rfl⟩
| add x y hx hy =>
obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ x y⟩
end ArExplicit.BalancedTensor
end OAI
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