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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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorAction
namespace OAI
namespace ArExplicit.BalancedTensor
open TensorProduct
variable {k R F M N : Type*} [CommRing k] [Ring R]
variable [AddCommGroup F] [AddCommGroup M] [Module k F] [Module k M]
variable [Module Rᵐᵒᵖ F] [Module R M]
@[elab_as_elim] theorem induction_on {motive : BalancedTensor k R F M → Prop}
(z : BalancedTensor k R F M) (zero : motive 0)
(tmul : ∀ f m, motive (tmul f m))
(add : ∀ x y, motive x → motive y → motive (x + y)) : motive z := by
classical
by_cases hz : z = 0
· simpa only [hz] using zero
clear hz
induction z using Submodule.Quotient.induction_on with
| H z =>
induction z using TensorProduct.inductionOn with
| tmul f m => exact tmul f m
| add x y hx hy => exact add _ _ hx hy
variable {F' M' : Type*} [AddCommGroup F'] [AddCommGroup M']
variable [Module k F'] [Module k M'] [Module Rᵐᵒᵖ F'] [Module R M']
def map (a : F →ₗ[k] F') (b : M →ₗ[k] M')
(ha : ∀ (r : Rᵐᵒᵖ) f, a (r • f) = r • a f)
(hb : ∀ (r : R) m, b (r • m) = r • b m) :
BalancedTensor k R F M →ₗ[k] BalancedTensor k R F' M' :=
lift
{ toFun := fun f => (mk (a f)).comp b
map_add' := by intros; ext m; simp
map_smul' := by intros; ext m; simp }
(by
intro f r m
change tmul (a (MulOpposite.op r • f)) (b m) = tmul (a f) (b (r • m))
rw [ha, hb, balance])
@[simp] theorem map_tmul (a : F →ₗ[k] F') (b : M →ₗ[k] M')
(ha : ∀ (r : Rᵐᵒᵖ) f, a (r • f) = r • a f)
(hb : ∀ (r : R) m, b (r • m) = r • b m) (f : F) (m : M) :
map a b ha hb (tmul f m) = tmul (a f) (b m) := rfl
theorem map_surjective (a : F →ₗ[k] F') (b : M →ₗ[k] M')
(ha : ∀ (r : Rᵐᵒᵖ) f, a (r • f) = r • a f)
(hb : ∀ (r : R) m, b (r • m) = r • b m)
(asurj : Function.Surjective a) (bsurj : Function.Surjective b) :
Function.Surjective (map a b ha hb) := by
intro z
induction z using induction_on with
| zero => exact ⟨0, map_zero _⟩
| tmul f m =>
obtain ⟨f, rfl⟩ := asurj f
obtain ⟨m, rfl⟩ := bsurj m
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⟩
section LeftLinear
variable {S : Type*} [Ring S] [AddCommGroup N] [Module k N]
variable [Module S F] [Module S N]
variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
def liftLeft (b : F →ₗ[k] M →ₗ[k] N)
(hb : ∀ (f : F) (r : R) (m : M), b (MulOpposite.op r • f) m = b f (r • m))
(hs : ∀ (s : S) f m, b (s • f) m = s • b f m) :
BalancedTensor k R F M →ₗ[S] N where
__ := (lift b hb).toAddHom
map_smul' s z := by
change lift b hb (s • z) = s • lift b hb z
induction z using induction_on with
| zero => simp
| tmul f m => simpa only [smul_tmul, lift_tmul, RingHom.id_apply] using hs s f m
| add x y hx hy => simp only [smul_add, map_add] at *; rw [hx, hy]
@[simp] theorem liftLeft_tmul (b : F →ₗ[k] M →ₗ[k] N)
(hb : ∀ (f : F) (r : R) (m : M), b (MulOpposite.op r • f) m = b f (r • m))
(hs : ∀ (s : S) f m, b (s • f) m = s • b f m) (f : F) (m : M) :
liftLeft b hb hs (tmul f m) = b f m := rfl
end LeftLinear
end ArExplicit.BalancedTensor
end OAI
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