Algebraic K Theory The Homotopy Approach Of Quillen And An Approach From Commutative Algebra

Author : Satya Mandal
Publisher : World Scientific
Page : 680 pages
File Size : 53,9 Mb
Release : 2023-06-22
Category : Mathematics
ISBN : 9789811269400

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Algebraic K-theory: The Homotopy Approach Of Quillen And An Approach From Commutative Algebra by Satya Mandal Pdf

In this book the author takes a pedagogic approach to Algebraic K-theory. He tried to find the shortest route possible, with complete details, to arrive at the homotopy approach of Quillen [Q] to Algebraic K-theory, with a simple goal to produce a self-contained and comprehensive pedagogic document in Algebraic K-theory, that is accessible to upper level graduate students. That is precisely what this book faithfully executes and achieves.The contents of this book can be divided into three parts — (1) The main body (Chapters 2-8), (2) Epilogue Chapters (Chapters 9, 10, 11) and (3) the Background and preliminaries (Chapters A, B, C, 1). The main body deals with Quillen's definition of K-theory and the K-theory of schemes. Chapters 2, 3, 5, 6, and 7 provide expositions of the paper of Quillen [Q], and chapter 4 is on agreement of Classical K-theory and Quillen K-theory. Chapter 8 is an exposition of the work of Swan [Sw1] on K-theory of quadrics.The Epilogue chapters can be viewed as a natural progression of Quillen's work and methods. These represent significant benchmarks and include Waldhausen K-theory, Negative K-theory, Hermitian K-theory, 𝕂-theory spectra, Grothendieck-Witt theory spectra, Triangulated categories, Nori-Homotopy and its relationships with Chow-Witt obstructions for projective modules. In most cases, the proofs are improvisation of methods of Quillen [Q].The background, preliminaries and tools needed in chapters 2-11, are developed in chapters A on Category Theory and Exact Categories, B on Homotopy, C on CW Complexes, and 1 on Simplicial Sets.

Author : Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 53,9 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9781447143932

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The Local Structure of Algebraic K-Theory by Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy Pdf

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.

Author : Hvedri Inassaridze
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 47,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401585699

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Algebraic K-Theory by Hvedri Inassaridze Pdf

Algebraic K-theory is a modern branch of algebra which has many important applications in fundamental areas of mathematics connected with algebra, topology, algebraic geometry, functional analysis and algebraic number theory. Methods of algebraic K-theory are actively used in algebra and related fields, achieving interesting results. This book presents the elements of algebraic K-theory, based essentially on the fundamental works of Milnor, Swan, Bass, Quillen, Karoubi, Gersten, Loday and Waldhausen. It includes all principal algebraic K-theories, connections with topological K-theory and cyclic homology, applications to the theory of monoid and polynomial algebras and in the theory of normed algebras. This volume will be of interest to graduate students and research mathematicians who want to learn more about K-theory.

Author : Vasudevan Srinivas
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 44,6 Mb
Release : 2013-11-21
Category : Science
ISBN : 9781489967350

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Algebraic K-Theory by Vasudevan Srinivas Pdf

Author : Emilio Lluis-Puebla,Jean-Louis Loday,Henri Gillet,Christophe Soule,Victor Snaith
Publisher : Springer
Page : 172 pages
File Size : 53,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540466390

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Higher Algebraic K-Theory: An Overview by Emilio Lluis-Puebla,Jean-Louis Loday,Henri Gillet,Christophe Soule,Victor Snaith Pdf

This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.

Author : Charles A. Weibel
Publisher : American Mathematical Soc.
Page : 634 pages
File Size : 52,6 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9780821891322

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The $K$-book by Charles A. Weibel Pdf

Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr

Author : Hyman Bass,Aderemi Oluyomi Kuku,C Pedrini
Publisher : World Scientific
Page : 622 pages
File Size : 49,6 Mb
Release : 1999-03-12
Category : Electronic
ISBN : 9789814544795

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Algebraic K-theory And Its Applications - Proceedings Of The School by Hyman Bass,Aderemi Oluyomi Kuku,C Pedrini Pdf

The Proceedings volume is divided into two parts. The first part consists of lectures given during the first two weeks devoted to a workshop featuring state-of-the-art expositions on 'Overview of Algebraic K-theory' including various constructions, examples, and illustrations from algebra, number theory, algebraic topology, and algebraic/differential geometry; as well as on more concentrated topics involving connections of K-theory with Galois, etale, cyclic, and motivic (co)homologies; values of zeta functions, and Arithmetics of Chow groups and zero cycles. The second part consists of research papers arising from the symposium lectures in the third week.

Author : William Browder
Publisher : Princeton University Press
Page : 576 pages
File Size : 51,5 Mb
Release : 1987-11-21
Category : Mathematics
ISBN : 9780691084268

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Algebraic Topology and Algebraic K-theory by William Browder Pdf

This book contains accounts of talks held at a symposium in honor of John C. Moore in October 1983 at Princeton University, The work includes papers in classical homotopy theory, homological algebra, rational homotopy theory, algebraic K-theory of spaces, and other subjects.

Author : Jonathan Rosenberg
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 41,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461243144

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Algebraic K-Theory and Its Applications by Jonathan Rosenberg Pdf

Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

Author : Daniel G. Quillen
Publisher : Springer
Page : 165 pages
File Size : 53,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540355236

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Homotopical Algebra by Daniel G. Quillen Pdf

Author : Paul Gregory Goerss,Stewart Priddy
Publisher : American Mathematical Soc.
Page : 520 pages
File Size : 42,7 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821832851

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Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory by Paul Gregory Goerss,Stewart Priddy Pdf

As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections between homotopy theory and other areas of mathematics. This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic $K$-theory, and $\mathbb{A 1$ homotopy theory. Among the contributors to the volume were Alejandro Adem,Ralph L. Cohen, Jean-Louis Loday, and many others. The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.

Author : Paul Frank Baum,Guillermo Cortiñas,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën
Publisher : Springer
Page : 308 pages
File Size : 54,5 Mb
Release : 2010-10-28
Category : Mathematics
ISBN : 9783642157080

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Topics in Algebraic and Topological K-Theory by Paul Frank Baum,Guillermo Cortiñas,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën Pdf

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Author : Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 40,5 Mb
Release : 2007-07-11
Category : Mathematics
ISBN : 9783540458975

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Motivic Homotopy Theory by Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Author : Eric Friedlander,Daniel R. Grayson
Publisher : Springer Science & Business Media
Page : 1148 pages
File Size : 44,5 Mb
Release : 2005-07-18
Category : Mathematics
ISBN : 9783540230199

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Handbook of K-Theory by Eric Friedlander,Daniel R. Grayson Pdf

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.

Author : Fabien Morel
Publisher : American Mathematical Soc.
Page : 116 pages
File Size : 41,8 Mb
Release : 2006
Category : Mathematics
ISBN : 082183164X

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Homotopy Theory of Schemes by Fabien Morel Pdf

In this text, the author presents a general framework for applying the standard methods from homotopy theory to the category of smooth schemes over a reasonable base scheme $k$. He defines the homotopy category $h(\mathcal{E} k)$ of smooth $k$-schemes and shows that it plays the same role for smooth $k$-schemes as the classical homotopy category plays for differentiable varieties. It is shown that certain expected properties are satisfied, for example, concerning the algebraic$K$-theory of those schemes. In this way, advanced methods of algebraic topology become available in modern algebraic geometry.