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Softcover ISBN:  9780821843031 
Product Code:  SURV/47.S 
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Book DetailsMathematical Surveys and MonographsVolume: 47; 1997; 249 ppMSC: Primary 19; 55
This book introduces a new pointset 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 pointset level algebra.
ReadershipGraduate students and research mathematicians interested in algebraic topology.

Table of Contents

Chapters

I. Prologue: the category of $\mathbb {L}$spectra

II. Structured ring and module spectra

III. The homotopy theory of $R$modules

IV. The algebraic theory of $R$modules

V. $R$ring spectra and the specialization to $MU$

VI. Algebraic $K$theory of $S$algebras

VII. $R$algebras and topological model categories

VIII. Bousfield localizations of $R$modules and algebras

IX. Topological Hochschild homology and cohomology

X. Some basic constructions on spectra

XI. Spaces of linear isometries and technical theorems

XII. The monadic bar construction

XIII. Epilogue: The category of $\mathbb {L}$spectra under $S$


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


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This book introduces a new pointset 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 pointset level algebra.
Graduate students and research mathematicians interested in algebraic topology.

Chapters

I. Prologue: the category of $\mathbb {L}$spectra

II. Structured ring and module spectra

III. The homotopy theory of $R$modules

IV. The algebraic theory of $R$modules

V. $R$ring spectra and the specialization to $MU$

VI. Algebraic $K$theory of $S$algebras

VII. $R$algebras and topological model categories

VIII. Bousfield localizations of $R$modules and algebras

IX. Topological Hochschild homology and cohomology

X. Some basic constructions on spectra

XI. Spaces of linear isometries and technical theorems

XII. The monadic bar construction

XIII. Epilogue: The category of $\mathbb {L}$spectra under $S$

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