Algebra & Number Theory

Singularities of locally acyclic cluster algebras

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

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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.

Article information

Algebra Number Theory, Volume 9, Number 4 (2015), 913-936.

Received: 4 September 2014
Revised: 12 January 2015
Accepted: 18 March 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13F60: Cluster algebras
Secondary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

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


Benito, Angélica; Muller, Greg; Rajchgot, Jenna; Smith, Karen E. Singularities of locally acyclic cluster algebras. Algebra Number Theory 9 (2015), no. 4, 913--936. doi:10.2140/ant.2015.9.913.

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