| import OAI.Algebra.Finitistic.Tensor.IdempotentTensor |
| import Mathlib.Algebra.Module.Projective |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.IdempotentTensor |
| universe u v w |
| variable {A : Type v} [Ring A] {e e' : A} |
|
|
| /-- An element of e'Ae acts by left multiplication on the right projectives. -/ |
| def leftMultiply (b : A) (hb : e'*b=b) : rightIdeal e →ₗ[Aᵐᵒᵖ] rightIdeal e' where |
| toFun m := ⟨b*m.val, by change e'*(b*m.val)=b*m.val; rw [← mul_assoc,hb]⟩ |
| map_add' m n := Subtype.ext (mul_add b m.val n.val) |
| map_smul' a m := by apply Subtype.ext; exact (mul_assoc b m.val a.unop).symm |
|
|
| @[simp] theorem leftMultiply_apply (b : A) (hb : e'*b=b) (m : rightIdeal e) : |
| (leftMultiply b hb m).val = b*m.val := rfl |
|
|
| / |
| def splitProjection (he : e*e=e) : Aᵐᵒᵖ →ₗ[Aᵐᵒᵖ] rightIdeal e where |
| toFun a := ⟨e*a.unop, by change e*(e*a.unop)=e*a.unop; rw [← mul_assoc,he]⟩ |
| map_add' a b := Subtype.ext (mul_add e a.unop b.unop) |
| map_smul' a b := by apply Subtype.ext; exact (mul_assoc e b.unop a.unop).symm |
|
|
| def splitInclusion : rightIdeal e →ₗ[Aᵐᵒᵖ] Aᵐᵒᵖ where |
| toFun a := MulOpposite.op a.val |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| theorem rightIdeal_projective (he : e*e=e) : Module.Projective Aᵐᵒᵖ (rightIdeal e) := by |
| apply Module.Projective.of_split splitInclusion (splitProjection he) |
| ext a |
| exact a.property |
| end LittleFinitistic.IdempotentTensor |
|
|
| end |
|
|
| end OAI |
|
|
