The complementary projection for a string-arrow cokernel radical

noncomputable def MagnitudeConjecture.BoundQuiver.StringPresentation.arrowRadicalComplementProjection {k A Q : Type u} [Field k] [Ring A] [Algebra k A] [Fintype Q] [Quiver Q] [(x y : Q) → Fintype (x ⟶ y)] (P : StringPresentation k A Q) {x y : Q} (a : x ⟶ y) : ↥(Module.jacobson P.quotientCategoryAlgebraᵐᵒᵖ ↑(P.representedVertexModule y)) →ₗ[P.quotientCategoryAlgebraᵐᵒᵖ] ↥(Module.jacobson P.quotientCategoryAlgebraᵐᵒᵖ ↑(P.representedVertexModule y))

Projection of the projective radical away from the selected arrow branch.

theorem MagnitudeConjecture.BoundQuiver.StringPresentation.arrowRadicalComplementProjection_quotient {k A Q : Type u} [Field k] [Ring A] [Algebra k A] [Fintype Q] [Quiver Q] [(x y : Q) → Fintype (x ⟶ y)] (P : StringPresentation k A Q) {x y : Q} (a : x ⟶ y) (z : ↥(Module.jacobson P.quotientCategoryAlgebraᵐᵒᵖ ↑(P.representedVertexModule y))) : (P.arrowCokernelRadicalLinearMap a) ((P.arrowRadicalComplementProjection a) z) = (P.arrowCokernelRadicalLinearMap a) z