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Upper cluster algebras and choice of ground ring

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Abstract

We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller’s theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.

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Acknowledgements

This work was supported by National Science Foundation of USA (Grant No. DMS-1702115).

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Authors and Affiliations

  1. Department of Mathematics, Xavier University, Cincinnati, OH, 45207, USA

    Eric Bucher

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

    John Machacek

  3. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Michael Shapiro

Authors
  1. Eric Bucher
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  2. John Machacek
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  3. Michael Shapiro
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Correspondence to Michael Shapiro.

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Bucher, E., Machacek, J. & Shapiro, M. Upper cluster algebras and choice of ground ring. Sci. China Math. 62, 1257–1266 (2019). https://doi.org/10.1007/s11425-018-9486-6

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  • DOI: https://doi.org/10.1007/s11425-018-9486-6

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