Singularities of locally acyclic cluster algebras

Benito, Angélica; Muller, Greg; Rajchgot, Jenna; Smith, Karen E.

doi:10.2140/ant.2015.9.913

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Singularities of locally acyclic cluster algebras

Angélica Benito, Greg Muller, Jenna Rajchgot and Karen E. Smith

Vol. 9 (2015), No. 4, 913–936

DOI: 10.2140/ant.2015.9.913

Abstract

We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly $F$ -regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are $F$ -regular if the local acyclicity is dropped.

Keywords

cluster algebras, locally acyclic cluster algebras, singularities, $F$-regularity, Frobenius splitting

Mathematical Subject Classification 2010

Primary: 13F60

Secondary: 13A35, 14B05

Milestones

Received: 4 September 2014

Revised: 12 January 2015

Accepted: 18 March 2015

Published: 30 May 2015

Authors

Angélica Benito
Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States

Greg Muller
Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States

Jenna Rajchgot
Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States

Karen E. Smith
Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States

Vol. 9, No. 4, 2015