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Rouquier dimension is Krull dimension for normal toric varieties

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Abstract

We prove that for any normal toric variety, the Rouquier dimension of its bounded derived category of coherent sheaves is equal to its Krull dimension. Our proof uses the coherent-constructible correspondence to translate the problem into the study of Rouquier dimension for certain categories of constructible sheaves.

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Notes

  1. We are not sure if this list is exhaustive.

  2. For completeness, we give the reference to the stacky case. The original proof (for schemes) is due to Rouquier [29, Proposition 7.17].

References

  1. Bai, S., Côté, L.: On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov. Compositio Math. 159(3), 437–487 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ballard, M.R., Duncan, A., McFaddin, P.K.: The toric Frobenius morphism and a conjecture of Orlov. Eur. J. Math. 5(3), 640–645 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ballard, M., Duncan, A., McFaddin, P.: Generation and the toric Frobenius (in preparation)

  4. Ballard, M., Favero, D.: Hochschild dimensions of tilting objects. Int. Math. Res. Not. IMRN 2012(11), 2607–2645 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Ballard, M., Favero, D., Katzarkov, L.: Orlov spectra: bounds and gaps. Invent. Math. 189(2), 359–430 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballard, M., Favero, D., Katzarkov, L.: A category of kernels for equivariant factorizations, II: further implications. J. Math. Pures Appl. 102(4), 702–757 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. 746, 235–303 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bondal, A.: Derived categories of toric varieties. In: Convex and Algebraic Geometry. Oberwolfach Reports, vol. 3, pp. 284–286. Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach (2006)

  9. Brown, M.K., Sayrafi, M.: A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2 (2022). arXiv:2208.00562

  10. Cooke, G.E., Finney, R.L.: Homology of Cell Complexes. Princeton University Press, Princeton (1967)

    Book  MATH  Google Scholar 

  11. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

  12. Elagin, A., Xu, J., Zhang, S.: On cyclic strong exceptional collections of line bundles on surfaces. Eur. J. Math. 7(1), 69–115 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. Invent. Math. 186(1), 79–114 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: T-duality and homological mirror symmetry for toric varieties. Adv. Math. 229(3), 1875–1911 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: The coherent-constructible correspondence for toric Deligne–Mumford stacks. Int. Math. Res. Not. IMRN 2014(4), 914–954 (2014)

  16. Favero, D., Huang, J.: Homotopy path algebras (2022). arXiv:2205.03730

  17. Geraschenko, A., Satriano, M.: Toric stacks I: The theory of stacky fans. Trans. Amer. Math. Soc. 367(2), 1033–1071 (2015)

  18. Ganatra, S., Pardon, J., Shende, V.: Microlocal Morse theory of wrapped Fukaya categories (2018). arXiv:1809.08807

  19. Hanlon, A., Hicks, J., Lazarev, O.: Relating categorical dimensions in topology and symplectic geometry (2023). arXiv:2308.13677

  20. Hanlon, A., Hicks, J., Lazarev, O.: Resolutions of toric subvarieties by lines bundles and applications (2023). arXiv:2303.03763

  21. Huang, J., Zhou, P.: Variation of GIT and variation of Lagrangian skeletons II: Quasi-symmetric case. Adv. Math. 408(A), 108597 (2022)

  22. Kawamata, Y.: Semi-orthogonal decomposition of a derived category of a 3-fold with an ordinary double point. In: Abban, H., et al. (eds.) Recent Developments in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 478, pp. 183–215. Cambridge University Press, Cambridge (2022)

    Chapter  Google Scholar 

  23. Kuwagaki, T.: The nonequivariant coherent-constructible correspondence for toric stacks. Duke Math. J. 169(11), 2125–2197 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nadler, D.: Wrapped microlocal sheaves on pairs of pants (2016). arXiv:1604.00114

  25. Olander, N.: The Rouquier dimension of quasi-affine schemes (2021). arXiv:2108.12005

  26. Orlov, D.: Remarks on generators and dimensions of triangulated categories. Moscow Math. J. 9(1), 153–159 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Polo, P.: On Cohen–Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties. Bull. London Math. Soc. 27(5), 425–434 (1995)

  28. Pirozhkov, D.: Rouquier dimension of some blow-ups. Eur. J. Math. 9(2), Art. No. 45 (2023)

  29. Rouquier, R.: Dimensions of triangulated categories. J. K-theory 1(2), 193–256 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shende, V.: Toric mirror symmetry revisited. C. R. Math. Acad. Sci. Paris 360, 751–759 (2022)

    MathSciNet  MATH  Google Scholar 

  31. Yanagawa, K.: Derived category of squarefree modules and local cohomology with monomial ideal support. J. Math. Soc. Japan 56(1), 289–308 (2004)

  32. Yanagawa, K.: Dualizing complex of the incidence algebra of a finite regular cell complex. Illinois J. Math. 49(4), 1221–1243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, S.: A note on the Rouquier dimensions of product varieties. J. Algebra Appl. 15(4), Art. No. 1650065 (2016)

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Acknowledgements

We thank Andrew Hanlon for notifying us about [20] and providing us with an early draft with an independent proof of Theorem 1.2 found therein. We are also grateful to Evgeny Shinder and Martin Kalck for providing us with the reference [22]. We thank Matt Ballard and Alex Duncan for their excellent explanations of the results in [3] and additional discussions at the Banff International Research Station.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street, Minneapolis, MN, 55455, USA

    David Favero

  2. Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    David Favero

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Central Academic Building 632, Edmonton, AB, T6G 2C7, Canada

    Jesse Huang

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  1. David Favero
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Correspondence to Jesse Huang.

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J. Huang is supported by a Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship and by NSERC through the Discovery Grant program. D. Favero is supported by NSERC through the Discovery Grant and Canada Research Chair programs.

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Favero, D., Huang, J. Rouquier dimension is Krull dimension for normal toric varieties. European Journal of Mathematics 9, 91 (2023). https://doi.org/10.1007/s40879-023-00686-1

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  • DOI: https://doi.org/10.1007/s40879-023-00686-1

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