# Axiomatic stable homotopy theory

### About this Title

**Mark Hovey**, **John H. Palmieri** and **Neil P. Strickland**

Publication: Memoirs of the American Mathematical Society

Publication Year
1997: Volume 128, Number 610

ISBNs: 978-0-8218-0624-1 (print); 978-1-4704-0195-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0610

MathSciNet review: 1388895

MSC (1991): Primary 55U35; Secondary 18E30, 18G35, 55N20, 55P42, 55P60, 55U15

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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 constructions—cellular
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 Devinatz-Hopkins-Smith and
the thick subcategory theorem of Hopkins-Smith in stable homotopy
theory, and the thick subcategory theorem of Benson-Carlson-Rickard 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

**Chapters**

- 1. Introduction and definitions
- 2. Smallness, limits and constructibility
- 3. Bousfield localization
- 4. Brown representability
- 5. Nilpotence and thick subcategories
- 6. Noetherian stable homotopy categories
- 7. Connective stable homotopy theory
- 8. Semisimple stable homotopy theory
- 9. Examples of stable homotopy categories
- 10. Future directions