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.

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Author: Victor P. Snaith

Publisher: Birkhäuser

ISBN: 9783034882071

Category: Mathematics

Page: 309

View: 198

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 theory And Its Applications Proceedings Of The School

QUATERNIONIC EXERCISES IN K-THEORY GALOIS MODULE STRUCTURE II T. CHINBURG Department of Mathematics, University of Pennsylvania, Philadelphia, Pa 19104, USA E-mail: tedd math.upenn.edu M. KOLSTER, Department of Mathematics, ...

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Author: Bass Hyman

Publisher: World Scientific

ISBN: 9789814544795

Category:

Page: 620

View: 753

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 ...

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Author: Victor Percy Snaith

Publisher: American Mathematical Soc.

ISBN: 0821871781

Category: Mathematics

Page: 207

View: 885

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.

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 ...

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Author: A. Fröhlich

Publisher: Springer Science & Business Media

ISBN: 9783642688164

Category: Mathematics

Page: 266

View: 271

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.

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.

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Author: John Greenlees

Publisher: Springer Science & Business Media

ISBN: 9789400709485

Category: Mathematics

Page: 392

View: 586

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.

D Modules Perverse Sheaves and Representation Theory

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

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Author: Ryoshi Hotta

Publisher: Springer Science & Business Media

ISBN: 9780817643638

Category: Mathematics

Page: 412

View: 971

D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.

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.

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Author: Martin R. Bridson

Publisher: Cambridge University Press

ISBN: 9780521757249

Category: Mathematics

Page: 320

View: 626

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

Algebraic K Theory and Algebraic Topology

... r € Z. Let K be a complete, discrete valuation field and suppose that K* is an algebraic closure of K. Let Z/n(1) denote the QK-module given by the n-th roots of unity in K” , where Qk = Gal(K*/K) is the absolute Galois group of K.

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Author: P.G. Goerss

Publisher: Springer Science & Business Media

ISBN: 9789401706957

Category: Mathematics

Page: 328

View: 612

A NATO Advanced Study Institute entitled "Algebraic K-theory and Algebraic Topology" was held at Chateau Lake Louise, Lake Louise, Alberta, Canada from December 12 to December 16 of 1991. This book is the volume of proceedings for this meeting. The papers that appear here are representative of most of the lectures that were given at the conference, and therefore present a "snapshot" of the state ofthe K-theoretic art at the end of 1991. The underlying objective of the meeting was to discuss recent work related to the Lichtenbaum-Quillen complex of conjectures, fro~ both the algebraic and topological points of view. The papers in this volume deal with a range of topics, including motivic cohomology theories, cyclic homology, intersection homology, higher class field theory, and the former telescope conjecture. This meeting was jointly funded by grants from NATO and the National Science Foun dation in the United States. I would like to take this opportunity to thank these agencies for their support. I would also like to thank the other members of the organizing com mittee, namely Paul Goerss, Bruno Kahn and Chuck Weibel, for their help in making the conference successful. This was the second NATO Advanced Study Institute to be held in this venue; the first was in 1987. The success of both conferences owes much to the professionalism and helpfulness of the administration and staff of Chateau Lake Louise.

Galois Representations in Arithmetic Algebraic Geometry

Snaith, V.P.: Galois Module Structure, Fields Institute Monographs 2, Am. Math. Soc., Providence, 1994. Snaith, V.P.: Local fundamental classes derived from higher dimensional K-groups, to appear in Proc. London Math. Soc.

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Author: A. J. Scholl

Publisher: Cambridge University Press

ISBN: 0521644194

Category: Mathematics

Page: 493

View: 395

Conference proceedings based on the 1996 LMS Durham Symposium 'Galois representations in arithmetic algebraic geometry'.

Algebraic K theory

Conference on Algebraic K-theory : September 4-8, 1995, the Adam Mickiewicz University, Poznán, Poland Grzegorz Banaszak Wojciech Gajda, ... This Grothendieck group is a fundamental object in the Galois module theory a la Fröhlich.

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Author: Grzegorz Banaszak

Publisher: American Mathematical Soc.

ISBN: 9780821805114

Category: Mathematics

Page: 210

View: 471

This book contains proceedings of the research conference on algebraic $K$-theory that took place in Poznan, Poland, in September 1995. The conference concluded the activity of the algebraic $K$-theory seminar held at the Adam Mickiewicz University in the academic year 1994-1995. Talks at the conference covered a wide range of current research activities in algebraic $K$-theory. In particular, the following topics were covered: $K$-theory of fields and rings of integers; $K$-theory of elliptic and modular curves; theory of motives, motivic cohomology, Beilinson conjectures; and, algebraic $K$-theory of topological spaces, topological Hochschild homology and cyclic homology. With contributions by some leading experts in the field, this book provides a look at the state of current research in algebraic $K$-theory.

On the Topology of Isolated Singularities in Analytic Spaces

... V.P. Algebraic K - Groups as Galois Modules ( 2002 ) ISBN 3-7643-6717-2 PM 191 : Kashiwara , M. / Miwa , T. ( Eds . ) Physical Combinatorics ( 2000 ) ISBN 0-8176-4175-0 PM 177 : Birkenhake , C. / Lange , H. Complex Tori ( 1999 ) ...

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Author: José Seade

Publisher: Springer Science & Business Media

ISBN: 3764373229

Category: Mathematics

Page: 238

View: 789

This book has been awarded the Ferran Sunyer i Balaguer 2005 prize. The aim of this book is to give an overview of selected topics on the topology of real and complex isolated singularities, with emphasis on its relations to other branches of geometry and topology. The first chapters are mostly devoted to complex singularities and a myriad of results spread in a vast literature, which are presented here in a unified way, accessible to non-specialists. Among the topics are the fibration theorems of Milnor; the relation with 3-dimensional Lie groups; exotic spheres; spin structures and 3-manifold invariants; the geometry of quadrics and Arnold's theorem which states that the complex projective plane modulo conjugation is the 4-sphere. The second part of the book studies pioneer work about real analytic singularities which arise from the topological and geometric study of holomorphic vector fields and foliations. In the low dimensional case these turn out to be related to fibred links in the 3-sphere defined by meromorphic functions. This provides new methods for constructing manifolds equipped with a rich geometry. The book is largely self-contained and serves a wide audience of graduate students, mathematicians and researchers interested in geometry and topology.

Singular Sets of Minimizers for the Mumford Shah Functional

... M. Hyperbolic Manifolds and Discrete Groups ( 2000 ) ISBN 0-8176-3904-7 PM 167 : Facchini , A. Module Theory ... V.P. Algebraic K - Groups as Galois Modules ( 2002 ) ISBN 3-7643-6717-2 PM 191 : Kashiwara , M. / Miwa , T. ( Eds ...

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Author: Guy David

Publisher: Springer Science & Business Media

ISBN: 376437182X

Category: Mathematics

Page: 581

View: 887

The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. It is largely self-contained, and should be accessible to graduate students in analysis. The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.

Lie 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. Index Theory for Symplectic Paths ...

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Author: Jean-Philippe Anker

Publisher: Springer Science & Business Media

ISBN: 9780817681920

Category: Mathematics

Page: 331

View: 580

* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.

Taming Wild Extensions Hopf Algebras and Local Galois Module Theory

It follows that Oc is a free H*-module, and hence that H" is the associated order of Oc in the dual of the K-Hopf algebra K..[a]/[T"](a). These examples are examples of normal field extensions where the Hopf Galois structure on the ...

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Author: Lindsay Childs

Publisher: American Mathematical Soc.

ISBN: 9780821821312

Category: Mathematics

Page: 215

View: 292

This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree $p$ and $p^2$; and Kummer theory of formal groups. Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.

Rigid Analytic Geometry and Its Applications

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. Index Theory for Symplectic Paths ...

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Author: Jean Fresnel

Publisher: Springer Science & Business Media

ISBN: 9781461200413

Category: Mathematics

Page: 299

View: 821

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

Polynomial Convexity

Kac–Moody Groups, their Flag Varieties, and Representation Theory BOUWKNEGT/W U. Geometric Analysis and Applications to Quantum Field Theory SNAITH. Algebraic K-groups as Galois Modules LONG. Index Theory for Symplectic Paths with ...

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Author: Edgar Lee Stout

Publisher: Springer Science & Business Media

ISBN: 9780817645380

Category: Mathematics

Page: 439

View: 514

This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

Introduction to Vertex Operator Algebras and Their Representations

Kac-Moody Groups, their Flag Varieties, and Representation Theory BOUWKNEGT/WU. Geometric Analysis and Applications to Quantum Field Theory SNAITH, Algebraic K-groups as Galois Modules LONG. Index Theory for Symplectic Paths with ...

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Author: James Lepowsky

Publisher: Springer Science & Business Media

ISBN: 9780817681869

Category: Mathematics

Page: 318

View: 185

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Coxeter Matroids

Differential Galois Theory and Non-Integrability of Hamiltonian Systems PATERNAIN. ... Hyperbolic Manifolds and Discrete Groups du SAUTOY/SEGAL/SHALEV (eds). New Horizons in pro-p Groups ... Algebraic K-groups as Galois Modules LONG.

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Author: Alexandre V. Borovik

Publisher: Springer Science & Business Media

ISBN: 9781461220664

Category: Mathematics

Page: 266

View: 977

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.

Momentum Maps and Hamiltonian Reduction

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

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Author: Juan-Pablo Ortega

Publisher: Springer Science & Business Media

ISBN: 9781475738117

Category: Mathematics

Page: 501

View: 464

* Winner of the Ferran Sunyer i Balaguer Prize in 2000. * Reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds. * Currently in classroom use in Europe. * Can serve as a resource for graduate courses and seminars in Hamiltonian mechanics and symmetry, symplectic and Poisson geometry, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers.

Algebraic Number Theory and Diophantine Analysis

Galois module structure and Kummer theory for Lubin-Tate formal groups Nigel P. Byott Abstract. ... order in a Hopf algebra A arising from the formal group construction, even though it is not free over its associated order in K [F].

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Author: F. Halter-Koch

Publisher: Walter de Gruyter

ISBN: 9783110801958

Category: Mathematics

Page: 571

View: 248

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.