Ladder Functors with an Application to Representation-finite Artinian Rings

Ladders were introduced by Igusa and Todorov for the investigation of representation-finite artinian algebras and algebras over an algebraically closed field [7]. They prove a radical layers theorem [7] which exhibits the graded structure of Auslander-Reiten sequences. In a second article [8] they obtain a characterization of the Auslander-Reiten quivers of representation-finite artinian algebras. Their construction of ladders starts with an irreducible morphism f0: A0 → B0 in a module category A. So f0 factors through a right almost split map u: θB0 → B0. Assume that f0 = ug with a split monomorphism g. Then g can be written as g = ( 1 0 ) with respect to a decomposition θB0 = A0 ⊕ B1. This gives a pullback