Tilting objects in triangulated categories

Abstract

Based on Beligiannis’s theory in [Beligiannis, A. (2000). Relative homological algebra and purity in triangulated categories. J. Algebra 227(1):268–361], we introduce and study $\mathcal{E}$-tilting objects in a triangulated category, where $\mathcal{E}$ is a proper class of triangles. We show that each $\mathcal{E}$-tilting object cogenerates an $\mathcal{E}$-cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the $\mathcal{E}$-tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be $\mathcal{E}$-1-Gorenstein. Finally, we give a one to one correspondence between the class of $\mathcal{E}$-tilting objects and the class of tilting subcategories in a suitable functor category.

Keywords:

• $\mathcal{E}$-tilting object
• functor category
• proper class of triangles
• tilting subcategory
• triangulated category

MSC:

• 18E05
• 18E10
• 18E30

Acknowledgments

The authors would like to thank the referee for very useful advices and suggestions.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (Grant No. 11671126).