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

Abstract

Let $Λ$ be an Artin $R$-algebra, and ${\rm proj}\mbox{-}Λ$ denotes the category of all finitely generated projective $Λ$-modules. Define $\CP(Λ) := {\rm Mor}({\rm proj}\mbox{-}Λ)$. Due to the favorable homological properties of $\CP(Λ)$, we initially examine several noteworthy objects and subcategories of $\CP(Λ)$, subsequently relating these findings to $\mmod Λ$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\CP(Λ)$, among other bijections, we demonstrate a bijection between rigid objects in $\CP(Λ)$ and ICE-closed subcategories of $\CP(Λ)$ with enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\CP(Λ)$ to $\mmod Λ$, it is necessary to introduce the framework of rICE-closed subcategories of $\mmod Λ$. We then establish a bijection between $τ$-rigid modules in $\mmod Λ$ and rICE-closed subcategories of $\mmod Λ$ 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 $τ$-tilting theory.