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Relative homological dimensions in recollements of triangulated categories

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Abstract

Let E be a proper class of triangles in a triangulated category C; and let (A, B, C) be a recollement of triangulated categories. Based on Beligiannis's work, we prove that A and C have enough E-projective objects whenever B does. Moreover, in this paper, we give the bounds for the E-global dimension of B in a recollement (A, B, C) by controlling the behavior of the E-global dimensions of the triangulated categories A and C: In particular, we show that the finiteness of the E-global dimensions of triangulated categories is invariant with respect to the recollements of triangulated categories.

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Acknowledgements

The authors would like to thank the referees for reading carefully the manuscript and valuable suggestions which improve the exposition a lot. In particular, they are grateful to one of the referees for showing them a simpler proof of Corollary 3. This work was supported by the National Natural Science Foundation of China (Grant No. 11671126).

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  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Yonggang Hu & Hailou Yao

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  1. Yonggang Hu
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  2. Hailou Yao
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Correspondence to Hailou Yao.

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Hu, Y., Yao, H. Relative homological dimensions in recollements of triangulated categories. Front. Math. China 14, 25–43 (2019). https://doi.org/10.1007/s11464-019-0751-2

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