Stable Categories and Structured Ring Spectra

Stable Categories and Structured Ring Spectra
Title Stable Categories and Structured Ring Spectra PDF eBook
Author Andrew J. Blumberg
Publisher Cambridge University Press
Pages 441
Release 2022-07-21
Genre Mathematics
ISBN 1009123297

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A graduate-level introduction to the homotopical technology in use at the forefront of modern algebraic topology.

Structured Ring Spectra

Structured Ring Spectra
Title Structured Ring Spectra PDF eBook
Author Andrew Baker
Publisher Cambridge University Press
Pages 246
Release 2004-11-18
Genre Mathematics
ISBN 9780521603058

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This book contains some important new contributions to the theory of structured ring spectra.

Rings, Modules, and Algebras in Stable Homotopy Theory

Rings, Modules, and Algebras in Stable Homotopy Theory
Title Rings, Modules, and Algebras in Stable Homotopy Theory PDF eBook
Author Anthony D. Elmendorf
Publisher American Mathematical Soc.
Pages 265
Release 1997
Genre Mathematics
ISBN 0821843036

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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

Foundations of Stable Homotopy Theory

Foundations of Stable Homotopy Theory
Title Foundations of Stable Homotopy Theory PDF eBook
Author David Barnes
Publisher Cambridge University Press
Pages 432
Release 2020-03-26
Genre Mathematics
ISBN 1108672671

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The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of the subject together in one place for the first time, from the motivating phenomena to the modern theory, at a level suitable for those with only a first course in algebraic topology. Starting from stable homotopy groups and (co)homology theories, the authors study the most important categories of spectra and the stable homotopy category, before moving on to computational aspects and more advanced topics such as monoidal structures, localisations and chromatic homotopy theory. The appendix containing essential facts on model categories, the numerous examples and the suggestions for further reading make this a friendly introduction to an often daunting subject.

Global Homotopy Theory

Global Homotopy Theory
Title Global Homotopy Theory PDF eBook
Author Stefan Schwede
Publisher Cambridge University Press
Pages 847
Release 2018-09-06
Genre Mathematics
ISBN 110842581X

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A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups
Title Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups PDF eBook
Author John Rognes
Publisher American Mathematical Soc.
Pages 154
Release 2008
Genre Mathematics
ISBN 0821840762

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The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

H Ring Spectra and Their Applications

H Ring Spectra and Their Applications
Title H Ring Spectra and Their Applications PDF eBook
Author Robert R. Bruner
Publisher Springer
Pages 396
Release 2006-11-14
Genre Mathematics
ISBN 3540397787

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