A short computation of the Rouquier dimension for a cycle of projective lines
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- by Andrew Hanlon and Jeff Hicks;
- Proc. Amer. Math. Soc. Ser. B 11 (2024), 653-663
- DOI: https://doi.org/10.1090/bproc/252
- Published electronically: December 11, 2024
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Abstract:
Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.References
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Bibliographic Information
- Andrew Hanlon
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
- MR Author ID: 1079907
- Email: andrew.hanlon@dartmouth.edu
- Jeff Hicks
- Affiliation: School of Mathematics, The University of Edinburgh, Old College, South Bridge, Edinburgh EH8 9YL, United Kingdom
- MR Author ID: 1393409
- Email: jeff.hicks@ed.ac.uk
- Received by editor(s): December 7, 2023
- Received by editor(s) in revised form: November 8, 2024
- Published electronically: December 11, 2024
- Additional Notes: The second author was supported by EPSRC Grant EP/V049097/1 (Lagrangians from Algebra and Combinatorics).
- Communicated by: Jerzy Weyman
- © Copyright 2024 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 11 (2024), 653-663
- MSC (2020): Primary 14F08, 18G80
- DOI: https://doi.org/10.1090/bproc/252