Title: Classification of stable model categories
Authors:
Stefan Schwede
Fakultat fur Mathematik
Universitat Bielefeld
33615 Bielefeld, Germany
schwede@mathematik.uni-bielefeld.de
and
Brooke Shipley
Department of Mathematics
Purdue University
W. Lafayette, IN, USA 47907
bshipley@math.purdue.edu
AMS Classification numbers: 55U35, 55P42
Abstract:
A stable model category is a setting for homotopy theory where the suspension
functor is invertible. The prototypical examples are the category of spectra
in the sense of stable homotopy theory and the category of unbounded chain
complexes of modules over a ring. In this paper we develop methods for
deciding when two stable model categories represent `the same homotopy theory'.
We show that stable model categories with a single compact generator are
equivalent to modules over a ring spectrum. More generally stable model
categories with a set of generators are characterized as modules over a
`ring spectrum with several objects', i.e., as spectrum valued diagram
categories. We also prove a Morita theorem which shows how equivalences
between module categories over ring spectra can be realized by smashing with
a pair of bimodules. Finally, we characterize stable model categories which
represent the derived category of a ring. This is a slight generalization of
Rickard's work on derived equivalent rings. We also include a proof of the
model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR
and (unbounded) chain complexes of R-modules for a ring R.
Remark: Our use of lamsarrows may make the .dvi file less portable than
the .ps or .pdf files.