Generalized Etale Cohomology Theories

Generalized Etale Cohomology Theories

Author: John Jardine

Publisher: Springer Science & Business Media

Published: 2010-12-15

Total Pages: 323

ISBN-13: 3034800657

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A generalized etale cohomology theory is a theory which is represented by a presheaf of spectra on an etale site for an algebraic variety, in analogy with the way an ordinary spectrum represents a cohomology theory for spaces. Examples include etale cohomology and etale K-theory. This book gives new and complete proofs of both Thomason's descent theorem for Bott periodic K-theory and the Nisnevich descent theorem. In doing so, it exposes most of the major ideas of the homotopy theory of presheaves of spectra, and generalized etale homology theories in particular. The treatment includes, for the purpose of adequately dealing with cup product structures, a development of stable homotopy theory for n-fold spectra, which is then promoted to the level of presheaves of n-fold spectra. This book should be of interest to all researchers working in fields related to algebraic K-theory. The techniques presented here are essentially combinatorial, and hence algebraic. An extensive background in traditional stable homotopy theory is not assumed. ------ Reviews (...) in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. (...) This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in K-theory applications (...). As an application of the techniques the author gives proofs of the descent theorems of R. W. Thomason and Y. A. Nisnevich. (...) The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason’s theorem without Bott periodicity). The recent proof of this conjecture, by V. Voevodsky, (...) makes this volume compulsory reading for all who want to be au fait with current trends in algebraic K-theory! - Zentralblatt MATH The presentation of these topics is highly original. The book will be very useful for any researcher interested in subjects related to algebraic K-theory. - Matematica


Book Synopsis Generalized Etale Cohomology Theories by : John Jardine

Download or read book Generalized Etale Cohomology Theories written by John Jardine and published by Springer Science & Business Media. This book was released on 2010-12-15 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: A generalized etale cohomology theory is a theory which is represented by a presheaf of spectra on an etale site for an algebraic variety, in analogy with the way an ordinary spectrum represents a cohomology theory for spaces. Examples include etale cohomology and etale K-theory. This book gives new and complete proofs of both Thomason's descent theorem for Bott periodic K-theory and the Nisnevich descent theorem. In doing so, it exposes most of the major ideas of the homotopy theory of presheaves of spectra, and generalized etale homology theories in particular. The treatment includes, for the purpose of adequately dealing with cup product structures, a development of stable homotopy theory for n-fold spectra, which is then promoted to the level of presheaves of n-fold spectra. This book should be of interest to all researchers working in fields related to algebraic K-theory. The techniques presented here are essentially combinatorial, and hence algebraic. An extensive background in traditional stable homotopy theory is not assumed. ------ Reviews (...) in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. (...) This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in K-theory applications (...). As an application of the techniques the author gives proofs of the descent theorems of R. W. Thomason and Y. A. Nisnevich. (...) The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason’s theorem without Bott periodicity). The recent proof of this conjecture, by V. Voevodsky, (...) makes this volume compulsory reading for all who want to be au fait with current trends in algebraic K-theory! - Zentralblatt MATH The presentation of these topics is highly original. The book will be very useful for any researcher interested in subjects related to algebraic K-theory. - Matematica

Transfer in Generalized Cohomology Theories

Transfer in Generalized Cohomology Theories

Author: Fred William Roush

Publisher: Akademiai Kiads

Published: 1999

Total Pages: 168

ISBN-13:

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&Quot;The standard invariant, homology, of topological spaces was generalized in the 1950s and 1960s to similar invariants into abelian groups. K. Theory, cobordism, and stable homotopy, and such theories were automatized under the name generalized cohomology theories, as having properties like exact sequences, homotopy invariance, and excision. If there is a map f from X to Y of topological spaces, there is an induced map on homology, H (X) to H (Y) (or backwards in cohomology). Transfer is a mapping in the reverse direction which exists for covering maps (and some other maps), special kinds of locally one to one maps. It is important in studying coverings and actions of finite groups. In this book after the necessary background on generalized cohomology and related topics, it is proved that transfer exists and is unique in all generalized cohomology theories having the properties that one would expect."--BOOK JACKET.


Book Synopsis Transfer in Generalized Cohomology Theories by : Fred William Roush

Download or read book Transfer in Generalized Cohomology Theories written by Fred William Roush and published by Akademiai Kiads. This book was released on 1999 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: &Quot;The standard invariant, homology, of topological spaces was generalized in the 1950s and 1960s to similar invariants into abelian groups. K. Theory, cobordism, and stable homotopy, and such theories were automatized under the name generalized cohomology theories, as having properties like exact sequences, homotopy invariance, and excision. If there is a map f from X to Y of topological spaces, there is an induced map on homology, H (X) to H (Y) (or backwards in cohomology). Transfer is a mapping in the reverse direction which exists for covering maps (and some other maps), special kinds of locally one to one maps. It is important in studying coverings and actions of finite groups. In this book after the necessary background on generalized cohomology and related topics, it is proved that transfer exists and is unique in all generalized cohomology theories having the properties that one would expect."--BOOK JACKET.

Etale Cohomology Theory

Etale Cohomology Theory

Author: Lei Fu

Publisher: World Scientific

Published: 2011

Total Pages: 622

ISBN-13: 9814307726

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Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.


Book Synopsis Etale Cohomology Theory by : Lei Fu

Download or read book Etale Cohomology Theory written by Lei Fu and published by World Scientific. This book was released on 2011 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

Etale Cohomology Theory (Revised Edition)

Etale Cohomology Theory (Revised Edition)

Author: Lei Fu

Publisher: World Scientific

Published: 2015-02-27

Total Pages: 624

ISBN-13: 9814675105

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Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ℓ-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.


Book Synopsis Etale Cohomology Theory (Revised Edition) by : Lei Fu

Download or read book Etale Cohomology Theory (Revised Edition) written by Lei Fu and published by World Scientific. This book was released on 2015-02-27 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ℓ-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

Canadian Mathematical Bulletin

Canadian Mathematical Bulletin

Author:

Publisher:

Published: 1993-12

Total Pages: 132

ISBN-13:

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Book Synopsis Canadian Mathematical Bulletin by :

Download or read book Canadian Mathematical Bulletin written by and published by . This book was released on 1993-12 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Axiomatic, Enriched and Motivic Homotopy Theory

Axiomatic, Enriched and Motivic Homotopy Theory

Author: John Greenlees

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 396

ISBN-13: 940070948X

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The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.


Book Synopsis Axiomatic, Enriched and Motivic Homotopy Theory by : John Greenlees

Download or read book Axiomatic, Enriched and Motivic Homotopy Theory written by John Greenlees and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.

Local Homotopy Theory

Local Homotopy Theory

Author: John F. Jardine

Publisher: Springer

Published: 2015-05-27

Total Pages: 508

ISBN-13: 1493923005

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This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory. Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory. A detailed treatment of the formalism of the subject is interwoven with explanations of the motivation, development, and nuances of ideas and results. The coherence of the abstract theory is elucidated through the use of widely applicable tools, such as Barr's theorem on Boolean localization, model structures on the category of simplicial presheaves on a site, and cocycle categories. A wealth of concrete examples convey the vitality and importance of the subject in topology, number theory, algebraic geometry, and algebraic K-theory. Assuming basic knowledge of algebraic geometry and homotopy theory, Local Homotopy Theory will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences.


Book Synopsis Local Homotopy Theory by : John F. Jardine

Download or read book Local Homotopy Theory written by John F. Jardine and published by Springer. This book was released on 2015-05-27 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory. Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory. A detailed treatment of the formalism of the subject is interwoven with explanations of the motivation, development, and nuances of ideas and results. The coherence of the abstract theory is elucidated through the use of widely applicable tools, such as Barr's theorem on Boolean localization, model structures on the category of simplicial presheaves on a site, and cocycle categories. A wealth of concrete examples convey the vitality and importance of the subject in topology, number theory, algebraic geometry, and algebraic K-theory. Assuming basic knowledge of algebraic geometry and homotopy theory, Local Homotopy Theory will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences.

Character Map In Non-abelian Cohomology, The: Twisted, Differential, And Generalized

Character Map In Non-abelian Cohomology, The: Twisted, Differential, And Generalized

Author: Domenico Fiorenza

Publisher: World Scientific

Published: 2023-08-11

Total Pages: 248

ISBN-13: 9811276714

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This book presents a novel development of fundamental and fascinating aspects of algebraic topology and mathematical physics: 'extra-ordinary' and further generalized cohomology theories enhanced to 'twisted' and differential-geometric form, with focus on, firstly, their rational approximation by generalized Chern character maps, and then, the resulting charge quantization laws in higher n-form gauge field theories appearing in string theory and the classification of topological quantum materials.Although crucial for understanding famously elusive effects in strongly interacting physics, the relevant higher non-abelian cohomology theory ('higher gerbes') has had an esoteric reputation and remains underdeveloped.Devoted to this end, this book's theme is that various generalized cohomology theories are best viewed through their classifying spaces (or moduli stacks) — not necessarily infinite-loop spaces — from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby not only uniformly subsuming the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applicable in the hitherto elusive generality of (twisted, differential, and) non-abelian cohomology.In laying out this result with plenty of examples, this book provides a modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories; 2. generalized cohomology in its homotopical incarnation; 3. rational homotopy theory seen via homotopy Lie theory, whose fundamental theorem we recast as a (twisted) non-abelian de Rham theorem, which naturally induces the (twisted) non-abelian character map.


Book Synopsis Character Map In Non-abelian Cohomology, The: Twisted, Differential, And Generalized by : Domenico Fiorenza

Download or read book Character Map In Non-abelian Cohomology, The: Twisted, Differential, And Generalized written by Domenico Fiorenza and published by World Scientific. This book was released on 2023-08-11 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a novel development of fundamental and fascinating aspects of algebraic topology and mathematical physics: 'extra-ordinary' and further generalized cohomology theories enhanced to 'twisted' and differential-geometric form, with focus on, firstly, their rational approximation by generalized Chern character maps, and then, the resulting charge quantization laws in higher n-form gauge field theories appearing in string theory and the classification of topological quantum materials.Although crucial for understanding famously elusive effects in strongly interacting physics, the relevant higher non-abelian cohomology theory ('higher gerbes') has had an esoteric reputation and remains underdeveloped.Devoted to this end, this book's theme is that various generalized cohomology theories are best viewed through their classifying spaces (or moduli stacks) — not necessarily infinite-loop spaces — from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby not only uniformly subsuming the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applicable in the hitherto elusive generality of (twisted, differential, and) non-abelian cohomology.In laying out this result with plenty of examples, this book provides a modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories; 2. generalized cohomology in its homotopical incarnation; 3. rational homotopy theory seen via homotopy Lie theory, whose fundamental theorem we recast as a (twisted) non-abelian de Rham theorem, which naturally induces the (twisted) non-abelian character map.

Handbook of K-Theory

Handbook of K-Theory

Author: Eric Friedlander

Publisher: Springer Science & Business Media

Published: 2005-07-18

Total Pages: 1148

ISBN-13: 354023019X

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This handbook offers a compilation of techniques and results in K-theory. Each chapter is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. It offers an exposition of our current state of knowledge as well as an implicit blueprint for future research.


Book Synopsis Handbook of K-Theory by : Eric Friedlander

Download or read book Handbook of K-Theory written by Eric Friedlander and published by Springer Science & Business Media. This book was released on 2005-07-18 with total page 1148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This handbook offers a compilation of techniques and results in K-theory. Each chapter is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. It offers an exposition of our current state of knowledge as well as an implicit blueprint for future research.

Homotopy Theory: Tools and Applications

Homotopy Theory: Tools and Applications

Author: Daniel G. Davis

Publisher: American Mathematical Soc.

Published: 2019-05-30

Total Pages: 268

ISBN-13: 1470442442

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This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and K-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.


Book Synopsis Homotopy Theory: Tools and Applications by : Daniel G. Davis

Download or read book Homotopy Theory: Tools and Applications written by Daniel G. Davis and published by American Mathematical Soc.. This book was released on 2019-05-30 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and K-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.