The Category of Morphisms Between Projective Modules#

Abstract

Let Λ be an artin algebra over a local commutative artinian ring k. We consider the category P(Λ) whose objects are morphims f : P → Q with P and Q projective left Λ-modules. In P(Λ), we introduce an exact structure in the sense of Gabriel and Roiter [Gabriel, P., Roiter, A. V. (1992). Representations of finite-dimensional algebras. In: Kostrikin, A. I., Shafarevich, I. V. eds. Encyclopaedia of the Mathematical Sciences. Vol. 73. Springer. Algebra VIII.] or equivalently Quillen [Quillen, D. (1973). Higher Algebraic K-Theory, 1. SLNM 341. Berlin: Springer, pp. 85–147], then we describe the corresponding projectives and injectives. For p(Λ), the full subcategory whose objects are morphisms f : P → Q with P and Q finitely generated, we prove a relation between the Hom-functor and the Ext-functor. From here, we can prove the existence of almost split sequences in p(Λ) and its relation with the almost split sequences for left Λ-modules.