$τ$-tilting theory via the morphism category of projective modules I: ICE-closed subcategories
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| Format: | Preprint |
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2024
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| _version_ | 1866916449603289088 |
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| author | Hafezi, Rasool Nasr-Isfahani, Alireza Wei, Jiaqun |
| author_facet | Hafezi, Rasool Nasr-Isfahani, Alireza Wei, Jiaqun |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $τ$-tilting theory via the morphism category of projective modules I: ICE-closed subcategories Hafezi, Rasool Nasr-Isfahani, Alireza Wei, Jiaqun Representation Theory Rings and Algebras 18A25, 18E40, 16G20, 16G99 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. |
| title | $τ$-tilting theory via the morphism category of projective modules I: ICE-closed subcategories |
| topic | Representation Theory Rings and Algebras 18A25, 18E40, 16G20, 16G99 |
| url | https://arxiv.org/abs/2410.17965 |