The homotopy categories of injective modules of derived discrete algebras

We study the homotopy category K(InjA) of all injective A-modules InjA and derived category D(ModA) of the category ModA of all A-modules, where A is finite dimensional algebra over an algebraically closed field. We are interested in the algebra with discrete derived category (derived discrete algebra. For a derived discrete algebra A, we get more concrete properties of K(InjA) and D(ModA). The main results we obtain are as following. Firstly, we consider the generic objects in compactly generated triangulated categories, specially in D(ModA). We construct some generic objects in D(ModA) for A derived discrete and not derived hereditary. Consequently, we give a characterization of algebras with generically trivial derived categories. Moreover, we establish some relations between the locally finite triangulated category of compact objects of D(ModA), which is equivalent to the category K(projA) of perfect complexes and the generically trivial derived category D(ModA). Generic objects in K(InjA) were also considered. Secondly, we study K(InjA) for some derived discrete algebra A and give a classification of indecomposable objects in K(InjA) for A radical square zero selfinjective algebra. The classification is based on the fully faithful triangle functor from K(InjA) to the stable module category Mod  of repetitive algebra  of A. In general, there is no explicit description of this functor. However, we use the covering technique to describe the image of indecomposable objects in K(InjA). This leads to a full classification of indecomposable objects in K(InjA). Moreover, these indecomposable objects are endofinite. Thus we give a description of the Ziegler spectrum of K(InjA) according to the classification.