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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].

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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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