DAN CHRISTENSEN - Derived categories, projective classes and phantom maps

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.

This talk will 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 generates a projective class on , which is exactly the information needed to do homological algebra in . Our result is that if the generating objects are ``small'', then the category of chain complexes of objects of 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.