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Rings, Modules, and Algebras in Stable Homotopy Theory
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Mathematical Surveys and Monographs
1997; 249 pp; softcover
Volume: 47
ISBN-10: 0-8218-4303-6
ISBN-13: 978-0-8218-4303-1
List Price: US$71 Member Price: US$56.80
Order Code: SURV/47.S

This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $$S$$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of "$$S$$-modules" whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of "$$S$$-algebras" and "commutative $$S$$-algebras" in terms of associative, or associative and commutative, products $$R\wedge _SR \longrightarrow R$$. These notions are essentially equivalent to the earlier notions of $$A_{\infty }$$ and $$E_{\infty }$$ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $$R$$-modules in terms of maps $$R\wedge _SM\longrightarrow M$$. When $$R$$ is commutative, the category of $$R$$-modules also has an associative, commutative, and unital smash product, and its derived category has properties just like the stable homotopy category. These constructions allow the importation into stable homotopy theory of a great deal of point-set level algebra.

Graduate students and research mathematicians interested in algebraic topology.

Reviews

"Very well organized ... The exposition is quite clear, with just the right amount of motivational comments. All algebraic topologists should obtain some familiarity with the contents of this book."

-- Mathematical Reviews

• Introduction
• Prologue: the category of $${\mathbb L}$$-spectra
• Structured ring and module spectra
• The homotopy theory of $$R$$-modules
• The algebraic theory of $$R$$-modules
• $$R$$-ring spectra and the specialization to $$MU$$
• Algebraic $$K$$-theory of $$S$$-algebras
• $$R$$-algebras and topological model categories
• Bousfield localizations of $$R$$-modules and algebras
• Topological Hochschild homology and cohomology
• Some basic constructions on spectra
• Spaces of linear isometries and technical theorems
• Epilogue: The category of $${\mathbb L}$$-spectra under $$S$$