Quillen model structures for relative homological algebra.
by
J. Daniel Christensen and Mark Hovey
Univ. of Western Ontario Wesleyan University
London, ON Middletown, CT
jdc@julian.uwo.ca hovey@member.ams.org
AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15
Submitted. 28 pages.
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 that more general forms of homological
algebra also fit into Quillen's framework. Specifically, a projective
class on a complete and cocomplete abelian category A is exactly the
information needed to do homological algebra in A. The main result is
that, under weak hypotheses, the category of chain complexes of objects
of A has a model category structure that reflects the homological
algebra of the projective class in the sense that it encodes the Ext
groups and more general derived functors. Examples include the "pure
derived category" of a ring R, and derived categories capturing relative
situations, including the projective class for Hochschild homology and
cohomology. We characterize the model structures that are cofibrantly
generated, and show that this fails for many interesting
examples. 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, and give examples that
include equivariant homotopy theory and bounded below derived
categories.