Preresolving subcategories in extriangulated categories

다국어 초록 (Multilingual Abstract)

In this paper, we introduce and study preresolving subcategories in an extriangulated category~$\mathscr{C}$. Let $\mathcal{Y}$ be a $\mathcal{Z}$-preresolving subcategory of $\mathscr{C}$ admitting a $\mathcal{Z}$-proper $\xi$-generator $\mathcal{X}$. We give the characterization of $\mathcal{Z}\text{-}{\rm proper}~\mathcal{Y}$-resolution dimension of an object in $\mathscr{C}$. Next, for an object $A$ in $\mathscr{C}$, if the $\mathcal{Z}\text{-}{\rm proper}~\mathcal{Y}$-resolution~dimension of $A$ is at most $n$, then all ``$n$-$\mathcal{X}$-syzygies" of $A$ are objects in $\mathcal{Y}$. Finally, we prove that $A$ has a $\mathcal{Z}$-proper $\mathcal{X}$-resolution if and only if $A$ has a $\mathcal{Z}$-proper $\mathcal{Y}$-resolution. As an application, we introduce $(\mathcal{X},\mathcal{Z})$-Gorenstein~subcategory $\mathcal{GX}_{\mathcal{Z}}(\xi)$ of $\mathscr{C}$ and prove that $\mathcal{GX}_{\mathcal{Z}}(\xi)$ is both $\mathcal{Z}$-resolving subcategory and $\mathcal{Z}$-coresolving subcategory of $\mathscr{C}$.

참고문헌 (Reference)

1 C. Klappoth, "n-extension closed subcategories of n-exangulated categories"

2 P. Zhou, "Triangulated quotient categories revisited" 502 : 196-232, 2018

3 M. Auslander, "Stable module theory, Memoirs of the American Mathematical Society" American Mathematical Society 1969

4 X. Ma, "Resolving subcategories of triangulated categories and relative homological dimension" 33 (33): 1513-1535, 2017

5 X. Zhu, "Resolving resolution dimensions" 16 (16): 1165-1191, 2013

6 J. Hu, "Proper resolutions and Gorensteinness in extriangulated categories" 16 (16): 95-117, 2021

7 J. Hu, "Proper classes and Gorensteinness in extriangulated categories" 551 : 23-60, 2020

8 L. Tan, "One-sided Frobenius pairs in extriangulated categories" 50 (50): 5345-5370, 2022

9 Z. Huang, "Homological dimensions relative to preresolving subcategories II" 34 (34): 507-530, 2022

10 Z. Huang, "Homological dimensions relative to preresolving subcategories" 54 (54): 727-757, 2014

11 Y. Liu, "Hearts of twin cotorsion pairs on extriangulated categories" 528 : 96-149, 2019

12 J. Asadollahi, "Gorenstein objects in triangulated categories" 281 (281): 264-286, 2004

13 W. Ren, "Gorenstein homological dimensions for triangulated categories" 410 : 258-276, 2014

14 J. Hu, "Gorenstein homological dimensions for extriangulated categories" 44 (44): 2235-2252, 2021

15 H. Nakaoka, "Extriangulated categories, Hovey twin cotorsion pairs and model structures" 60 (60): 117-193, 2019