  Given a locally coherent Grothendieck category G, we prove that the homotopy
category of complexes of injective objects (also known as the coderived
category of G) is compactly generated triangulated. Moreover, the full
subcategory of compact objects is none other than D^b(fp G). If G admits a
generating set of finitely presentable objects of finite projective dimension,
then also the derived category of G is compactly generated and Krause's
recollement exists. Our main tools are (a) model theoretic techniques and (b) a
systematic study of the pure derived category of an additive finitely
accessible category.
