$τ$-tilting theory via the morphism category of projective modules I: ICE-closed subcategories

Abstract

This paper endeavors to explore certain distinguished modules and subcategories within mod$Λ$. Let $\mathrm{proj}\mbox{-}Λ$ denote the category of all finitely generated projective $Λ$-modules and define $\mathcal{P}(Λ):=\mathrm{Mor}(\mathrm{proj}\mbox{-}Λ)$. Due to the favorable homological properties of $\mathcal{P}(Λ)$, we initially examine several noteworthy objects and subcategories of $\mathcal{P}(Λ)$, subsequently relating these findings to ${\rm mod}Λ$. We demonstrate the existence of a bijection between tilting objects of $\mathcal{P}(Λ)$ and support $τ$-tilting $Λ$-modules. This bijection further suggests a correspondence between tilting objects of $\mathcal{P}(Λ)$ that possess a specific direct summand and $τ$-tilting $Λ$-modules. We establish a bijection between two-term silting complexes within $\mathbb{K}^{b}({\rm proj}\mbox{-}Λ)$ and tilting objects of $\mathcal{P}(Λ)$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\mathcal{P}(Λ)$, we demonstrate a bijection between rigid objects in $\mathcal{P}(Λ)$ and ICE-closed subcategories of $\mathcal{P}(Λ)$ with enough Ext-projectives. Subsequently, we present bijections linking rigid objects in $\mathcal{P}(Λ)$ with a designated direct summand, $τ$-rigid pairs within ${\rm mod}Λ$, and ICE-closed subcategories of $\mathcal{P}(Λ)$ that contain a special object and also have enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\mathcal{P}(Λ)$ to ${\rm mod}Λ$, it is necessary to introduce the framework of ICE-closed subcategories of ${\rm mod}Λ$ relative to a projective module.

Theorems & Definitions (100)

• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Proposition 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.8
• Proposition 2.9
• Definition 2.10
• Definition 2.11