The homotopy category of monomorphisms between projective modules

Abstract

Let $(S, \n)$ be a commutative noetherian local ring and $ω\in\n$ be non-zerodivisor. This paper deals with the behavior of the category $\mon(ω, \cp)$ consisting of all monomorphisms between finitely generated projective $S$-modules with cokernels annihilated by $\omega$. We introduce a homotopy category $\HT\mon(ω, \cp)$, 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} $\mon(ω, \cp)$ is proven, but also the Auslander-Reiten translation, $τ_{\mon}(-)$, is completely recognized. Particularly, it will be observed that any non-projective object of $\mon(ω, \cp)$ with local endomorphism ring is invariant under the square of the Auslander-Reiten translation.