Tilting objects in triangulated categories

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 -tilting objects in a triangulated category, where is a proper class of triangles. We show that each -tilting object c...

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Bibliographic Details
Published in:Communications in algebra 2020-01, Vol.48 (1), p.410-429
Main Authors: Hu, Yonggang, Yao, Hailou, Fu, Xuerong
Format: Article
Language:English
Subjects:
functor category
Homology
proper class of triangles
tilting object
tilting subcategory
triangulated category
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Summary: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 -tilting objects in a triangulated category, where is a proper class of triangles. We show that each -tilting object cogenerates an -cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the -tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be -1-Gorenstein. Finally, we give a one to one correspondence between the class of -tilting objects and the class of tilting subcategories in a suitable functor category.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2019.1648646