On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebras

A cluster is a finite set of generators of a cluster algebra. The Laurent Phenomenon of Fomin and Zelevinsky says that any element of a cluster algebra can be written as a Laurent polynomial in terms of any 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. In 2014 it was conjectured that the equality of the cluster algebra and upper cluster algebra is equivalent to a combinatorial property regarding the existence of a maximal green sequence. In this work we prove a stronger result for cluster algebras from mutation-finite quivers, and provide a counterexample to show that the conjecture does not hold in general. Finally, we propose a new conjecture about the upper cluster algebra on the relationship which replaces maximal green sequences with more general green-to-red sequences and incorporates Mueller's local-acyclicity.