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Abstract
Cluster algebras are commutative rings inside a rational function field with a distinguished set of generators, that are obtained from an iterative combinatorial procedure, and grouped into overlapping finite subsets of a fixed cardinality. These subsets are called clusters and due to the Laurent Phenomenon of Fomin and Zelevinsky any generator of a cluster algebra can be written as Laurent polynomial in terms of a given cluster. The upper cluster algebra of a cluster algebra is the ring of rational functions that can be written as a Laurent polynomial in every cluster of the cluster algebra. By the Laurent phenomenon a cluster algebra is always contained in its upper cluster algebra, but they are not always equal. We conjecture that the equality of the cluster algebra and upper cluster algebra is equivalent to the algebraic property that the cluster algebra is locally-acyclic and to a combinatorial property regarding the existence of a maximal green sequence. In this work we prove this conjecture for cluster algebras from mutation-finite quivers and minimal-mutation infinite quivers.
