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Indices and c-vectors in extriangulated categories

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Abstract

Let \({\cal C}\) be an extriangulated category and τ be any n-cluster tilting subcategory of \({\cal C}\). We consider the index with respect to τ and introduce the index Grothendieck group of τ. Using the index, we prove that the index Grothendieck group of τ is isomorphic to the Grothendieck group of \({\cal C}\), which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic. In particular, we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic. Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories. We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 12271257). The authors thank the reviewers for the careful reading, helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000, China

    Li Wang

  2. Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Jiaqun Wei & Haicheng Zhang

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  1. Li Wang
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  2. Jiaqun Wei
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  3. Haicheng Zhang
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Correspondence to Haicheng Zhang.

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Wang, L., Wei, J. & Zhang, H. Indices and c-vectors in extriangulated categories. Sci. China Math. 66, 1949–1964 (2023). https://doi.org/10.1007/s11425-022-2054-y

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