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Silting Reduction in Exact Categories

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Abstract

Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently, which are generalizations of those concepts in triangulated categories. Exact categories and triangulated categories are extriangulated categories. In this paper, we prove that the Gabriel-Zisman localization \(\mathcal {B}/(\textsf{thick}\hspace{.01in}\mathcal W)\) of an exact category \(\mathcal {B}\) with respect to a presilting subcategory \(\mathcal W\) satisfying certain condition can be realized as a subfactor category of \(\mathcal {B}\). Afterwards, we discuss the relation between silting subcategories and tilting subcategories in exact categories, which gives us a kind of important examples of our results. In particular, for a finite dimensional Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang.

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Acknowledgements

The authors would like to thank the referee for many helpful suggestions on mathematics and English expressions.

Funding

Yu Liu is supported by the National Natural Science Foundation of China (Grant No. 12171397). Panyue Zhou is supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 2023JJ30008) and by the National Natural Science Foundation of China (Grant No. 12371034). Yu Zhou is supported by National Natural Science Foundation of China (Grant Nos. 12271279 and 12031007). Bin Zhu is supported by National Natural Science Foundation of China (Grant Nos. 12371034 and 12031007).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, 710062, Xi’an, Shaanxi, People’s Republic of China

    Yu Liu

  2. School of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, Hunan, People’s Republic of China

    Panyue Zhou

  3. Yau Mathematical Science Center, Tsinghua University, 100084, Beijing, People’s Republic of China

    Yu Zhou

  4. Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People’s Republic of China

    Bin Zhu

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  1. Yu Liu
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The authors Y. Liu, P. Zhou, Y. Zhou and B. Zhu contributed equally. The authors read and approved the final manuscript.

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Correspondence to Panyue Zhou.

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Presented by: Karin Baur

Dedicated to the memory of professor Helmut Lenzing.

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Liu, Y., Zhou, P., Zhou, Y. et al. Silting Reduction in Exact Categories. Algebr Represent Theor 27, 847–876 (2024). https://doi.org/10.1007/s10468-023-10238-6

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