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.

Disclosure

“te algebras. The second author subsequently proposed that the orbit lengths of adjacent vertices in the AR-quiver under category self-equivalences can be used, via large primes, to define an invariant of the components of the AR-quiver. ChatGPT assisted with the technical construction and verification of the semilinear twist argument in Lemma 4.2. ChatGPT was used in extending the result from al- gebraically closed fields to arbitrary perfect fields. In particular, it assisted wi”

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2026-08-23T19:08:55.936866Z

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