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Let $A$ be a dimer algebra and $Z$ its center. It is well known that if $A$is cancellative, then $A$ and $Z$ are noetherian and $A$ is a finitelygenerated $Z$-module. Here we show the converse: if $A$ is non-cancellative (asalmost all dimer algebras are), then $A$ and $Z$ are nonnoetherian and $A$ isan infinitely generated $Z$-module. Although $Z$ is nonnoetherian, we show thatit nonetheless has Krull dimension 3 and is generically noetherian.Furthermore, we show that the reduced center is the coordinate ring for aGorenstein algebraic variety with the strange property that it containsprecisely one 'smeared-out' point of positive geometric dimension. In ourproofs we introduce formalized notions of Higgsing and the mesonic chiral ringfrom quiver gauge theory.

Nonlocality and the central geometry of dimer algebras