ABSTRACT
In this paper, the concept of the projective cover of a module is generalized to the case of complexes of finitely generated modules over a Noetherian local ring. The projective cover of a complex possesses many but not all of the key properties of the projective cover in the module case. In this chapter, the author shall consider some of the important properties of the projective cover of a module and compare and contrast these with the properties of the projective cover of a complex. They see that the projective cover has an additional property which is not inherited from the module case: The projective cover of a complex retains the homology of the original complex. Furthermore, Roberts has proven that every free resolution is the direct sum of the minimal free resolution and an exact complex.
