标题： 局部循环簇代数的奇点 显示英文标题

标题： Singularities of locally acyclic cluster algebras

摘要： 我们证明了局部无环的簇代数具有（最坏情况下）规范奇点。 事实上，我们证明了正特征的局部无环簇代数是强F正的。 此外，我们证明了上簇代数总是通过一个规范定义的分裂而被Frobenius分裂，并且它们有一个秩为一的自由规范模。 我们还给出了例子来说明如果去掉局部无环性，不是所有的上簇代数都是F正的。 显示英文摘要

摘要： 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.