The Homotopy Category of Monomorphisms Between Projective Modules

Abstract

Let (S, n) be a commutative noetherian local ring and omega is an element of n be non-zerodivisor. This paper deals with the behavior of the category Mon(omega, P) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by omega. We introduce a homotopy category HMon(omega, P), which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring R = S/(omega). As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of Mon(omega, P) is proved, but also the Auslander-Reiten translation, tau Mon(-), is completely recognized. Particularly, it will be observed that any non-projective object of Mon(omega, P) with local endomorphism ring is invariant under the square of the Auslander-Reiten translation.