Abstract
This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A ‘locally acyclic cluster algebra’ is a cluster algebra which admits a finite cover (in a geometric sense) by acyclic cluster algebras. Many important results about acyclic cluster algebras extend to locally acyclic cluster algebras (such as being finitely generated, integrally closed, and equaling their upper cluster algebra), as well as a result which is new even for acyclic cluster algebras (regularity over Q when the exchange matrix has full rank).Several techniques are developed for determining whether a cluster algebra is locally acyclic. Cluster algebras of marked surfaces with at least two boundary marked points are shown to be locally acyclic, providing a large class of examples of cluster algebras which are locally acyclic but not acyclic. Some specific examples are worked out in detail.
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