Memoirs of the American Mathematical Society 1997; 114 pp; softcover Volume: 128 ISBN10: 0821806246 ISBN13: 9780821806241 List Price: US$44 Individual Members: US$26.40 Institutional Members: US$35.20 Order Code: MEMO/128/610
 This book gives an axiomatic presentation of stable homotopy theory. It starts with axioms defining a "stable homotopy category"; using these axioms, one can make various constructionscellular towers, Bousfield localization, and Brown representability, to name a few. Much of the book is devoted to these constructions and to the study of the global structure of stable homotopy categories. Next, a number of examples of such categories are presented. Some of these arise in topology (the ordinary stable homotopy category of spectra, categories of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the representation theory of groups or of Lie algebras, as well as the derived category of a commutative ring). Hence one can apply many of the tools of stable homotopy theory to these algebraic situations. Features:  Provides a reference for standard results and constructions in stable homotopy theory.
 Discusses applications of those results to algebraic settings, such as group theory and commutative algebra.
 Provides a unified treatment of several different situations in stable homotopy, including equivariant stable homotopy and localizations of the stable homotopy category.
 Provides a context for nilpotence and thick subcategory theorems, such as the nilpotence theorem of DevinatzHopkinsSmith and the thick subcategory theorem of HopkinsSmith in stable homotopy theory, and the thick subcategory theorem of BensonCarlsonRickard in representation theory.
This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics. It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics. Readership Graduate students and research mathematicians interested in algebraic topology, representation theory, and algebraic geometry. Table of Contents  Introduction and definitions
 Smallness, limits and constructibility
 Bousfield localization
 Brown representability
 Nilpotence and thick subcategories
 Noetherian stable homotopy categories
 Connective stable homotopy theory
 Semisimple stable homotopy theory
 Examples of stable homotopy theory
 Future directions
