File size: 3,196 Bytes
ae00908 | 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 | import OAI.Algebra.Finitistic.Tensor.BalancedTensorMaps
namespace OAI
noncomputable section
namespace LittleFinitistic.IdempotentTensor
universe u v w
variable {k : Type u} [Field k] {A : Type v} [Ring A] [Algebra k A]
variable (e : A)
/-- The actual projective right ideal eA, presented as the fixed points of left multiplication. -/
def rightIdeal : Submodule Aᵐᵒᵖ A where
carrier := {a | e*a=a}
zero_mem' := mul_zero e
add_mem' ha hb := by change e*(_+_)=_+_; rw [mul_add,ha,hb]
smul_mem' b a ha := by
change e*(a*b.unop)=a*b.unop
rw [← mul_assoc,ha]
variable (N : Type w) [AddCommGroup N] [Module k N] [Module A N]
[IsScalarTower k A N]
/-- The vertex part eN, without assuming e is central. -/
def leftSpace : Submodule k N where
carrier := {n | e • n = n}
zero_mem' := smul_zero e
add_mem' ha hb := by change e • (_+_) = _+_; rw [smul_add,ha,hb]
smul_mem' c n hn := by change e • (c • n) = c • n; rw [smul_comm e c,hn]
def productEval : rightIdeal e →ₗ[k] N →ₗ[k] leftSpace (k := k) e N where
toFun a :=
{ toFun := fun n => ⟨a.val • n, by change e • (a.val • n) = a.val • n; rw [← mul_smul,a.property]⟩
map_add' n m := Subtype.ext (smul_add a.val n m)
map_smul' c n := Subtype.ext (smul_comm a.val c n) }
map_add' a b := by ext n; exact add_smul a.val b.val n
map_smul' c a := by ext n; exact smul_assoc c a.val n
/-- The natural map eA tensor_A N → eN. -/
def tensorEval : BalancedTensor.Tensor k A (rightIdeal e) N →ₗ[k] leftSpace (k := k) e N :=
BalancedTensor.lift (productEval (k := k) e N) (by
intro a m n
apply Subtype.ext
exact mul_smul m.val a n)
@[simp] theorem tensorEval_mk (m : rightIdeal e) (n : N) :
(tensorEval (k := k) e N (BalancedTensor.mk k A (rightIdeal e) N m n)).val = m.val • n := rfl
variable (he : e*e=e)
def tensorInv : leftSpace (k := k) e N →ₗ[k] BalancedTensor.Tensor k A (rightIdeal e) N :=
(BalancedTensor.mk k A (rightIdeal e) N ⟨e,he⟩).comp (leftSpace (k := k) e N).subtype
@[simp] theorem tensorEval_tensorInv (n : leftSpace (k := k) e N) :
tensorEval (k := k) e N (tensorInv (k := k) e N he n) = n := by
apply Subtype.ext
exact n.property
@[simp] theorem tensorInv_tensorEval (z : BalancedTensor.Tensor k A (rightIdeal e) N) :
tensorInv (k := k) e N he (tensorEval (k := k) e N z) = z := by
induction z using BalancedTensor.induction_on with
| hz => simp
| ht m n =>
change BalancedTensor.mk k A (rightIdeal e) N ⟨e,he⟩ (m.val • n) =
BalancedTensor.mk k A (rightIdeal e) N m n
rw [← BalancedTensor.balance]
have hm : MulOpposite.op m.val • (⟨e,he⟩ : rightIdeal e) = m := by
apply Subtype.ext
exact m.property
rw [hm]
| ha a b h₁ h₂ => simp [h₁,h₂]
/-- Evaluation on a vertex projective is its actual vertex space. -/
def tensorEquiv : BalancedTensor.Tensor k A (rightIdeal e) N ≃ₗ[k] leftSpace (k := k) e N where
__ := tensorEval (k := k) e N
invFun := tensorInv (k := k) e N he
left_inv := tensorInv_tensorEval (k := k) e N he
right_inv := tensorEval_tensorInv (k := k) e N he
end LittleFinitistic.IdempotentTensor
end
end OAI
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