The Ziegler spectrum for derived-discrete algebras

Let A be a derived-discrete algebra. We show that the Krull-Gabriel dimension of the homotopy category of projective A-modules, and therefore the Cantor-Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobinski and Krause [8]. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [17]. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective A-modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple. (C) 2017 The Authors. Published by Elsevier Inc.