Gorenstein derived functors for extriangulated categories

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper $\xi ‐\mathcal{G}\text{projective}$ resolution for an object in $\mathcal{C}$ and define the functors $\xi {\text{xt}}_{\mathcal{G}\mathcal{P}(\xi )}$ and $\xi {\text{xt}}_{\mathcal{G}\mathcal{I}(\xi )}$. Under some assumptions, we give some equivalent characterizations for $\xi ‐\mathcal{G}\text{projective}$ and $\xi ‐\mathcal{G}\text{injective}$ dimensions of objects in $\mathcal{C}$ by vanishing of such functors. As an application, our main results generalize Ren and Liu’s work. Moreover, our proof is not far from the usual module categories or triangulated categories.

KEYWORDS:

• Extriangulated categories
• Gorenstein derived functors
• Gorenstein objects

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 18G05
• 18G10
• 18G25

Acknowledgments

The author would like to thank the referee for important comments, suggestions and corrections on improving this paper.