Vol. 9, No. 4, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Singularities of locally acyclic cluster algebras

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

Vol. 9 (2015), No. 4, 913–936
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