Algebraic K Groups as Galois Modules

Great Lakes K-theory Conf., Fields Institute Communications Series #16 (A. M. Soc. Publications) 1–29 (1997). [38] T. Chinburg, M. Kolster, G. Pappas and V. P. Snaith: Galois module structure of K-groups of rings of integers; K-Theory ...

Author: Victor P. Snaith

Publisher: Birkhäuser

ISBN: 9783034882071

Category: Mathematics

Page: 309

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This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.

Algebraic K Groups as Galois Modules

This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993.

Author: Victor P. Snaith

Publisher: Springer Science & Business Media

ISBN: 3764367172

Category: Mathematics

Page: 332

View: 824

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This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.

Galois Module Structure

CHAPTER 7 Higher Algebraic K - theory 7.1 A new Chinburg invariant 7.1.1 Ever since the appearance of Tate [ 1976 ] it has been clear that the higher algebraic K - groups of local and global fields and their rings of integers are ...

Author: Victor Percy Snaith

Publisher: American Mathematical Soc.

ISBN: 0821871781

Category: Mathematics

Page: 220

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This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems.

Algebraic K theory

Volume 16 , 1997 Quaternionic Exercises in K - Theory Galois Module Structure Ted Chinburg Department of ... In [ 5 ] invariants of the Galois module structure of algebraic K- groups of rings of algebraic integers were introduced .

Author: Victor Percy Snaith

Publisher: American Mathematical Soc.

ISBN: 0821871234

Category: Mathematics

Page: 380

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The conference proceedings volume is produced in connection with the second Great Lakes K-theory Conference that was held at The Fields Institute for Research in Mathematical Sciences in March 1996. The volume is dedicated to the late Bob Thomason, one of the leading research mathematicians specializing in algebraic K-theory. In addition to research papers treated directly in the lectures at the conference, this volume contains the following: i) several timely articles inspired by those lectures (particularly by that of V. Voevodsky), ii) an extensive exposition by Steve Mitchell of Thomason's famous result concerning the relationship between algebraic K-theory and etale cohomology, iii) a definitive exposition by J-L. Colliot-Thelene, R. Hoobler, and B. Kahn (explaining and elaborating upon unpublished work of O. Gabber) of Bloch-Ogus-Gersten type resolutions in K-theory and algebraic geometry. This volume will be important both for researchers who want access to details of recent development in K-theory and also to graduate students and researchers seeking good advanced exposition.

Galois Module Structure of Algebraic Integers

J. of Algebra 50 (1978), 463-487 Taylor, M. J.: Adams operations, local root numbers and Galois module structure of rings of ... S. V.: Character action on the class group of Fröhlich, preprint 1977, to appear in “Algebraic K-theory” in ...

Author: A. Fröhlich

Publisher: Springer Science & Business Media

ISBN: 9783642688164

Category: Mathematics

Page: 266

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In this volume we present a survey of the theory of Galois module structure for rings of algebraic integers. This theory has experienced a rapid growth in the last ten to twelve years, acquiring mathematical depth and significance and leading to new insights also in other branches of algebraic number theory. The decisive take-off point was the discovery of its connection with Artin L-functions. We shall concentrate on the topic which has been at the centre of this development, namely the global module structure for tame Galois extensions of numberfields -in other words of extensions with trivial local module structure. The basic problem can be stated in down to earth terms: the nature of the obstruction to the existence of a free basis over the integral group ring ("normal integral basis"). Here a definitive pattern of a theory has emerged, central problems have been solved, and a stage has clearly been reached when a systematic account has become both possible and desirable. Of course, the solution of one set of problems has led to new questions and it will be our aim also to discuss some of these. We hope to help the reader early on to an understanding of the basic structure of our theory and of its central theme, and to motivate at each successive stage the introduction of new concepts and new tools.

Hopf Algebras and Galois Module Theory

Background: Hopf-Galois theory and Galois module theory Hopf-Galois theory was first described by Chase and ... The idea is that if L/K is a Galois extension with Galois group G, then L becomes a module over the group algebra K[G] where ...

Author: Lindsay N. Childs

Publisher: American Mathematical Soc.

ISBN: 9781470465162

Category: Education

Page: 311

View: 881

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Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.

Axiomatic Enriched and Motivic Homotopy Theory

Chinburg T., Pappas G., Kolster M. and Snaith V.P. (1997) Quaternionic exercises in K-theory Galois module structure; Proc. Great Lakes Ktheory Conf., Fields Institute Communications Series #16 (A.M.Soc. Publications) 1-29.

Author: John Greenlees

Publisher: Springer Science & Business Media

ISBN: 9789400709485

Category: Mathematics

Page: 392

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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.

Geometric Methods in Algebra and Number Theory

Kac - Moody Groups , their Flag Varieties , and Representation Theory 205 BOUWKNEGT / WU . Geometric Analysis and Applications to Quantum Field Theory 206 SNAITH . Algebraic K - groups as Galois Modules 207 LONG .

Author: Fedor Bogomolov

Publisher: Springer Science & Business Media

ISBN: 0817643494

Category: Mathematics

Page: 384

View: 284

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* Contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory * The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry * Text can serve as an intense introduction for graduate students and those wishing to pursue research in algebraic and arithmetic geometry

Geometric and Cohomological Methods in Group Theory

D.G. Quillen: On the cohomology and K-theory of the general linear groups over a finite field; Annals of Math. ... V.P.Snaith: Algebraic K-groups as Galois Modules; Birkhauser Progress in Mathematics series #206 (2002). AA.

Author: Martin R. Bridson

Publisher: Cambridge University Press

ISBN: 9780521757249

Category: Mathematics

Page: 331

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An extended tour through a selection of the most important trends in modern geometric group theory.