The homotopy category of monomorphisms between projective modules

TL;DR

This paper explores the homotopy category of monomorphisms between projective modules over a noetherian local ring, embedding it into the singularity category and analyzing its Auslander-Reiten theory.

Contribution

It introduces a triangulated homotopy category for monomorphisms, embeds it into the singularity category, and characterizes its Auslander-Reiten translation behavior.

Findings

Homotopy category is triangulated

Embedding into the singularity category established

Almost split sequences are proven to exist

Abstract

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.