The Homotopy Category of Monomorphisms Between Projective Modules.

Let (S , n) be a commutative noetherian local ring and ω ∈ n be non-zerodivisor. This paper deals with the behavior of the category Mon (ω , P) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by ω . We introduce a homotopy category H Mon (ω , 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 / (ω) . As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of Mon (ω , P) is proved, but also the Auslander–Reiten translation, τ Mon (-) , is completely recognized. Particularly, it will be observed that any non-projective object of Mon (ω , P) with local endomorphism ring is invariant under the square of the Auslander–Reiten translation. [ABSTRACT FROM AUTHOR]