| 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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|
