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