Constructing minimal P<∞-approximations over left serial algebras

Let Λ be a finite dimensional left serial algebra over an algebraically closed field K. In this case, Burgess and Zimmermann Huisgen have shown that P<∞, the full subcategory of Λ-mod consisting of the finitely generated Λ-modules of finite projective dimension, is contravariantly finite in Λ-mod. Moreover, they show that the minimal right P<∞-approximations of the simple Λ-modules can be obtained by glueing together uniserials to form modules known as saguaros, and they state without proof an algorithm for constructing these approximations. We will review this algorithm and then demonstrate how a new notion of graphical morphisms between saguaros can be used to prove it.