Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 4 (2015), 913-936.
Singularities of locally acyclic cluster algebras
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 -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 -regular if the local acyclicity is dropped.
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
Permanent link to this document
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]
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. https://projecteuclid.org/euclid.ant/1510842340