Derived categories and projective classes

Dan  Christensen

1991 Mathematics Subject Classification:  Primary 18E30;  Secondary
18G35, 55U35, 18G25.

Keywords:  Derived category, chain complex, relative homological algebra,
projective class, pure homological algebra.

Abstract:
An important example of a model category is the category of unbounded
chain complexes of R-modules, which has as its homotopy category the
derived category of the ring R.  This example shows that traditional
homological algebra is encompassed by Quillen's homotopical algebra.
The goal of this paper is to show how more general forms of
homological algebra also fit into Quillen's framework.  Specifically,
any set of objects in a complete and cocomplete abelian category A
generates a projective class on A, which is exactly the information
needed to do homological algebra in A.  The main result is that if the
generating objects are "small" in an appropriate sense, then the
category of chain complexes of objects of A has a model category
structure which reflects the homological algebra of the projective
class.  The motivation for the work is the construction of the "pure
derived category" of a ring R.  Pure homological algebra has
applications to phantom maps in the stable homotopy category and the
(usual) derived category of a ring, and these connections will be
described.  Finally, we explain how the category of simplicial objects
in a possibly non-abelian category can be equipped with a model
category structure reflecting a given projective class.