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When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

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Abstract

Let R be an Artin algebra and e be an idempotent of R. Assume that Tor eRei (Re, G) = 0 for any G ∈ Gproj eRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor Se to induce a triangle-equivalence ⅅdef(R) ≃ ⅅdef(eRe). Combining this with a result of Psaroudakis et al. (2014), we provide necessary and sufficient conditions for the singular equivalence ⅅsg(R) ≃ ⅅsg(eRe) to restrict to a triangle-equivalence GprojR ≃ GprojeRe. Applying these to the triangular matrix algebra \(T = \left( {\matrix{A & M \cr 0 & B \cr } } \right)\), corresponding results between candidate categories of T and A (resp. B) are obtained. As a consequence, we infer Gorensteinness and CM (Cohen-Macaulay)-freeness of T from those of A (resp. B). Some concrete examples are given to indicate that one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.

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Acknowledgements

This research was supported by National Natural Science Foundation of China (Grant Nos. 11626179, 12101474, 12171206 and 11701455), Natural Science Foundation of Jiangsu Province (Grant No. BK20211358), Natural Science Basic Research Plan in Shaanxi Province of China (Grant Nos. 2017JQ1012 and 2020JM-178) and Fundamental Research Funds for the Central Universities (Grant Nos. JB160703 and 2452020182). The authors are grateful to the referees for reading this paper carefully and for many suggestions on mathematics and English expressions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Huanhuan Li

  2. School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, 213001, China

    Jiangsheng Hu

  3. College of Science, Northwest A&F University, Yangling, 712100, China

    Yuefei Zheng

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  1. Huanhuan Li
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  2. Jiangsheng Hu
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  3. Yuefei Zheng
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Correspondence to Yuefei Zheng.

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Li, H., Hu, J. & Zheng, Y. When the Schur functor induces a triangle-equivalence between Gorenstein defect categories. Sci. China Math. 65, 2019–2034 (2022). https://doi.org/10.1007/s11425-021-1899-3

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