Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PRERESOLVING SUBCATEGORIES IN EXTRIANGULATED CATEGORIES

  • Songsong Liu (School of Mathematical Sciences Nanjing Normal University) ;
  • Jiaqun Wei (School of Mathematical Sciences Nanjing Normal University)
  • Received : 2022.09.24
  • Accepted : 2022.12.30
  • Published : 2023.07.01
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Abstract

In this paper, we introduce and study preresolving subcategories in an extriangulated category ${\mathfrak{C}}$. Let ${\mathcal{Y}}$ be a ${\mathcal{Z}}$-preresolving subcategory of ${\mathfrak{C}}$ admitting a ${\mathcal{Z}}$-proper ξ-generator ${\mathcal{X}}$. We give the characterization of ${\mathcal{Z}}$-proper ${\mathcal{Y}}$-resolution dimension of an object in ${\mathfrak{C}}$. Next, for an object A in ${\mathfrak{C}}$, if the ${\mathcal{Z}}$-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 ${\mathcal{G}{\mathcal{X}}_{\mathcal{Z}}({\xi})$ of ${\mathfrak{C}}$ and prove that ${\mathcal{G}{\mathcal{X}}_{\mathcal{Z}}({\xi})$ is both ${\mathcal{Z}}$-resolving subcategory and ${\mathcal{Z}}$-coresolving subcategory of ${\mathfrak{C}}$.

Keywords

Acknowledgement

This work was supported by the National Science Foundation of China (Grant no. 12271249) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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