Abstract for Axiomatic Stable Homotopy Theory, by Hovey, Palmieri,
Strickland (this is a preprint; the paper has been accepted for
publication in the Memoirs of the AMS):

 We define and investigate a class of categories with formal
 properties similar to those of the homotopy category of spectra.
 This class includes suitable versions of the derived category of
 modules over a commutative ring, or of comodules over a commutative
 Hopf algebra, and is closed under Bousfield localization.  We study
 various notions of smallness, questions about representability of
 (co)homology functors, and various kinds of localization.  We prove
 theorems analogous to those of Hopkins and Smith about detection
 of nilpotence and classification of thick subcategories.  We define
 the class of Noetherian stable homotopy categories, and investigate
 their special properties.  Finally, we prove that a number of
 categories occurring in nature (including those mentioned above)
 satisfy our axioms.