Filtered objects in extriangulated categories

Abstract

Let R be an Artin ring and $\Theta =\{\Theta (1),\Theta (2),\dots ,\Theta (n)\}$ be a family of objects in an Artin extriangulated R-category $(\mathcal{C},\mathbb{E},\mathfrak{s})$ such that $\mathbb{E}(\Theta (j),\Theta (i))=0$ for all $j\ge i.$ In this article, we show that the class $\mathcal{P}(\Theta )$ of the Θ-projective objects is a precovering class and the class $\mathcal{I}(\Theta )$ of the Θ-injective objects is a pre-enveloping one in $\mathcal{C}.$ Furthermore, if $\mathcal{C}$ has enough projectives and enough injectives, we show that the subcategory $\mathfrak{F}(\Theta )$ of Θ-filtered objects is functorially finite in $\mathcal{C}.$ As an application, this generalizes the works by Ringel in a module category case and Mendoza–Santiago in a triangulated category case.

Keywords:

• Extriangulated category
• filtered object
• functorially finite

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E30
• 18E10

Acknowledgments

This work was carried out when the author was a postdoctoral fellow at Université de Sherbrooke. The author thanks Professor Shiping Liu for his helpful discussions and warm hospitality. He also wants to thank his colleagues at Université de Sherbrooke for their help. The author would like to thank the referee for the very kind and helpful comments and advice in shaping the article into its present form.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11901190 and 11671221), the Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3205) and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19B239).