Infinitely Many Components in Auslander--Reiten Quivers of Representation-Infinite Algebras over Perfect Fields

Abstract

Let $k$ be a perfect field and let $A$ be a representation-infinite finite-dimensional $k$-algebra. We prove that the Auslander--Reiten quiver of $A$ has infinitely many connected components. This establishes, for finite-dimensional algebras over perfect fields, a conjecture of Auslander, Reiten, and Smalø concerning Artin algebras. Over an algebraically closed field, the proof combines a localized polynomial representation embedding with semilinear twists induced by field automorphisms. The passage from a perfect field to its algebraic closure is obtained by separable base change: we prove that if the Auslander--Reiten quiver of $A$ has only finitely many components, then the same holds for the scalar extension to the algebraic closure.

19 pages, comments welcome!