Upper cluster algebras and choice of ground ring

  • Eric Bucher
  • John Machacek
  • Michael ShapiroEmail author
Articles Special Topic on Cluster Algebras


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.


cluster algebras upper cluster algebras locally acyclic cluster algebras 




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


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsXavier UniversityCincinnatiUSA
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  3. 3.Department of MathematicsMichigan State UniversityEast LansingUSA

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