Directing projective modules

Abstract

Let A be an Artin algebra. The A-modules which we consider are always left modules of finite length. If X, Y, Z are A-modules, the composition of maps f : X ~ Y and g : Y ~ Z is denoted by f 9 : X ~ Z. The category of (finite length) A-modules is denoted by A-mod. If X, Y are indecomposable A-modules, we denote by rad (X, Y) the set of non-invertible maps from X to Y. A path in A-mod is a sequence (X o . . . . . Xs) of (isomorphism classes of) indecomposable A-modules X i, 0 1, and X o = Xs, then the path (Xo, . . . , X~) is called a cycle. A indecomposable A-module is called directing if X does not occur in a cycle. Our first aim will be to extend the definition of a directing module to decomposable modules. We show that an indecomposable projective A-module P is directing if and only if the radical of P is directing. In case the top of P is injective it follows that P is directing if and only if the radical of P is directing as a module over the factor algebra of A by the trace ideal of P.