  Let $\Lambda$ be a derived-discrete algebra. We show that the Krull-Gabriel
dimension of the homotopy category of projective $\Lambda$-modules, and
therefore the Cantor-Bendixson rank of its Ziegler spectrum, is $2$, thus
extending a result of Bobi\'nski and Krause. We also describe all the
indecomposable pure-injective complexes and hence the Ziegler spectrum for
derived-discrete algebras, extending a result of Z. Han. Using this, we are
able to prove that all indecomposable complexes in the homotopy category of
projective $\Lambda$-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.
