$r$ICE-closed subcategories induced by the morphism category of projective modules

Abstract

Let $\Lambda$ be an Artin $R$-algebra, and ${\rm proj}\mbox{-}\Lambda$ denotes the category of all finitely generated projective $\Lambda$-modules. Define $\CP(\Lambda) := {\rm Mor}({\rm proj}\mbox{-}\Lambda)$. Due to the favorable homological properties of $\CP(\Lambda)$, we initially examine several noteworthy objects and subcategories of $\CP(\Lambda)$, subsequently relating these findings to $\mmod \Lambda$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\CP(\Lambda)$, among other bijections, we demonstrate a bijection between rigid objects in $\CP(\Lambda)$ and ICE-closed subcategories of $\CP(\Lambda)$ with enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\CP(\Lambda)$ to $\mmod \Lambda$, it is necessary to introduce the framework of rICE-closed subcategories of $\mmod \Lambda$. We then establish a bijection between $\tau$-rigid modules in $\mmod \Lambda$ and rICE-closed subcategories of $\mmod \Lambda$ that possess an rExt-progenerator. This is a generalization of a bijection given by Enomoto for hereditary algebras. Our morphism approach improves a bijection given by Buan and Zhou by introducing r-cotorsion-torsion triples. We conclude our paper with further applications for $\tau$-tilting theory.