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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 | 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
/-- eA is a retract of the rank-one free right A-module. -/
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
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