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Rings, modules, and algebras in stable homotopy theory / A.D. Elmendorf [and three others] ; with an appendix by M. Cole.
- Format:
- Book
- Series:
- Mathematical surveys and monographs ; no. 47.
- Mathematical surveys and monographs, 0076-5376 ; volume 47
- Language:
- English
- Subjects (All):
- Homotopy theory.
- Point set theory.
- Physical Description:
- 1 online resource (265 p.)
- Place of Publication:
- Providence, Rhode Island : American Mathematical Society, [1997]
- Summary:
- 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 a
- Contents:
- ""Contents""; ""Introduction""; ""Chapter I. Prologue: the category of L-spectra""; ""Chapter II. Structured ring and module spectra""; ""Chapter III. The homotopy theory of R-modules""; ""Chapter IV. The algebraic theory of R-modules""; ""Chapter V. R-ring spectra and the specialization to MU""; ""Chapter VI. Algebraic K-theory of S-algebras""; ""Chapter VII. R-algebras and topological model categories""; ""Chapter VIII. Bousfield localizations of R-modules and algebras""; ""Chapter IX. Topological Hochschild homology and cohomology""; ""Chapter X. Some basic constructions on spectra""
- ""Chapter XI. Spaces of linear isometries and technical theorems""""Chapter XII. The monadic bar construction""; ""Chapter XIII. Epilogue: The category of L-spectra under S""; ""Appendix A. Twisted half-smash products and function spectra""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""J""; ""K""; ""L""; ""M""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""W""
- Notes:
- Description based upon print version of record.
- Includes bibliographical references (pages 243-245) and index.
- Description based on print version record.
- ISBN:
- 1-4704-1278-0
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