The homotopy category of monomorphisms between projective modules

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