Stable Categories And Structured Ring Spectra

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Stable Categories And Structured Ring Spectra

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Author : Andrew J. Blumberg
language : en
Publisher: Cambridge University Press
Release Date : 2022-07-21

Stable Categories And Structured Ring Spectra written by Andrew J. Blumberg and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-21 with Mathematics categories.

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

Rings Modules And Algebras In Stable Homotopy Theory

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Author : Anthony D. Elmendorf
language : en
Publisher: American Mathematical Soc.
Release Date : 1997

Rings Modules And Algebras In Stable Homotopy Theory written by Anthony D. Elmendorf and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.

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

Structured Ring Spectra

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Author : Andrew Baker
language : en
Publisher: Cambridge University Press
Release Date : 2004-11-18

Structured Ring Spectra written by Andrew Baker and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-11-18 with Mathematics categories.

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

Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups

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Author : John Rognes
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups written by John Rognes and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

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 $E$.

Global Homotopy Theory

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Author : Stefan Schwede
language : en
Publisher: Cambridge University Press
Release Date : 2018-09-06

Global Homotopy Theory written by Stefan Schwede and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-06 with Mathematics categories.

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

Triangulated Categories

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Author : Thorsten Holm
language : en
Publisher: Cambridge University Press
Release Date : 2010-06-24

Triangulated Categories written by Thorsten Holm and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-06-24 with Mathematics categories.

A 2010 collection of survey articles by leading experts covering fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. This is a valuable reference for experts and a useful introduction for graduate students entering the field.

Equivariant Stable Homotopy Theory

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Author : L. Gaunce Jr. Lewis
language : en
Publisher: Springer
Release Date : 2006-11-14

Equivariant Stable Homotopy Theory written by L. Gaunce Jr. Lewis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.

This book is a foundational piece of work in stable homotopy theory and in the theory of transformation groups. It may be roughly divided into two parts. The first part deals with foundations of (equivariant) stable homotopy theory. A workable category of CW-spectra is developed. The foundations are such that an action of a compact Lie group is considered throughout, and spectra allow desuspension by arbitrary representations. But even if the reader forgets about group actions, he will find many details of the theory worked out for the first time. More subtle constructions like smash products, function spectra, change of group isomorphisms, fixed point and orbit spectra are treated. While it is impossible to survey properly the material which is covered in the book, it does boast these general features: (i) a thorough and reliable presentation of the foundations of the theory; (ii) a large number of basic results, principal applications, and fundamental techniques presented for the first time in a coherent theory, unifying numerous treatments of special cases in the literature.

H Ring Spectra And Their Applications

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Author : Robert R. Bruner
language : en
Publisher: Springer
Release Date : 2006-11-14

H Ring Spectra And Their Applications written by Robert R. Bruner and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.

Formal Geometry And Bordism Operations

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Author : Eric Peterson
language : en
Publisher: Cambridge University Press
Release Date : 2019

Formal Geometry And Bordism Operations written by Eric Peterson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Mathematics categories.

Delivers a broad, conceptual introduction to chromatic homotopy theory, focusing on contact with arithmetic and algebraic geometry.

Homotopical Algebraic Geometry Ii Geometric Stacks And Applications

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Author : Bertrand Toën
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Homotopical Algebraic Geometry Ii Geometric Stacks And Applications written by Bertrand Toën and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

This is the second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category $C$, and prove that this notion satisfies the expected properties.