The Gabriel-Roiter measure for radical-square zero algebras

Abstract

Let Λ be a radical-square zero algebra over an algebraically closed field k with radical 𝔯, and let Γ = ( Λ / τ 0 τ Λ / τ ) be the associated hereditary algebra. There is an explicit functor F: mod Λ → mod Γ, which induces a stable equivalence. In this paper, it will be proved that the functor F preserves the Gabriel–Roiter (GR) measures and the GR factors. Thus the GR measure for Λ can be studied by the use of F and known facts for hereditary algebras. In particular, the middle terms of the Auslander–Reiten sequences ending at the GR factors and the relationship between the preprojective partition for Λ and the take-off Λ-modules will be investigated.