Upper cluster algebras and choice of ground ring

TL;DR

This paper explores how the choice of ground ring affects whether a cluster algebra equals its upper cluster algebra, providing conditions, examples, and a maximal green sequence.

Contribution

It introduces a new perspective on the ground ring dependence in cluster algebras and applies Muller’s localization theory to establish criteria for equality.

Findings

Identifies conditions for cluster algebra and upper cluster algebra equality

Provides an explicit example showing ground ring dependence

Constructs a maximal green sequence for the example

Abstract

We initiate a study of the dependence on the choice of ground ring on the question of 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.