ABSTRACT
The process of extending homological algebra from modules to complexes started with the last Chapter of Cartan and Eilenberg’s book, Homological Algebra. The projective, injective, and flat complexes can be defined in a similar manner as for modules. Their existence allows defining projective, injective, and flat resolutions for complexes. In turn, this allows defining the injective, projective, and flat dimensions for complexes. This chapter shows that that the class of Gorenstein flat complexes is covering in the category Ch(R) over any left GF-closed ring R. It also shows that the Gorenstein flat complexes form a preenveloping class over two-sided noetherian rings such that the character modules of Gorenstein injectives are Gorenstein flat. The chapter proves a result about the existence of dg-injective covers over a noetherian ring R : the class of dg-injective complexes is covering if and only if the ring is regular.
