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The Norm Residue Theorem in Motivic Cohomology

2019-06-11 Mathematics
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Release : 2019-06-11
Genre : Mathematics
Kind : eBook
Book Rating : 042/5 ( reviews)

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Book Synopsis The Norm Residue Theorem in Motivic Cohomology by : Christian Haesemeyer

Download or read book The Norm Residue Theorem in Motivic Cohomology written by Christian Haesemeyer. This book was released on 2019-06-11. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky’s proof and introduce the key figures behind its development. They proceed to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations. Comprehensive and self-contained, The Norm Residue Theorem in Motivic Cohomology unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

Lecture Notes on Motivic Cohomology

2006 Mathematics
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Release : 2006
Genre : Mathematics
Kind : eBook
Book Rating : 471/5 ( reviews)

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Book Synopsis Lecture Notes on Motivic Cohomology by : Carlo Mazza

Download or read book Lecture Notes on Motivic Cohomology written by Carlo Mazza. This book was released on 2006. Available in PDF, EPUB and Kindle. Book excerpt: The notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, to develop its main properties, and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, etale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The first part presents the definition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (etale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part five. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000. In addition, many of the original proofs have been simplified and improved so that this book will also be a useful tool for research mathematicians. Information for our distributors: Titles in this series are copublished with the Clay Mathematics Institute (Cambridge, MA).

The Bloch–Kato Conjecture for the Riemann Zeta Function

2015-03-19 Mathematics
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Release : 2015-03-19
Genre : Mathematics
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Book Rating : 300/5 ( reviews)

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Book Synopsis The Bloch–Kato Conjecture for the Riemann Zeta Function by : John Coates

Download or read book The Bloch–Kato Conjecture for the Riemann Zeta Function written by John Coates. This book was released on 2015-03-19. Available in PDF, EPUB and Kindle. Book excerpt: There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

The Arithmetic and Geometry of Algebraic Cycles

2000-02-29 Mathematics
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Release : 2000-02-29
Genre : Mathematics
Kind : eBook
Book Rating : 947/5 ( reviews)

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Book Synopsis The Arithmetic and Geometry of Algebraic Cycles by : B. Brent Gordon

Download or read book The Arithmetic and Geometry of Algebraic Cycles written by B. Brent Gordon. This book was released on 2000-02-29. Available in PDF, EPUB and Kindle. Book excerpt: The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This book offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods. Topics include a discussion of the arithmetic Abel-Jacobi mapping, higher Abel-Jacobi regulator maps, polylogarithms and L-series, candidate Bloch-Beilinson filtrations, applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection, motivic cohomology, Chow groups of singular varieties, and recent progress on the Hodge and Tate conjectures for Abelian varieties.

The $K$-book

2013-06-13 Mathematics
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Release : 2013-06-13
Genre : Mathematics
Kind : eBook
Book Rating : 324/5 ( reviews)

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Book Synopsis The $K$-book by : Charles A. Weibel

Download or read book The $K$-book written by Charles A. Weibel. This book was released on 2013-06-13. Available in PDF, EPUB and Kindle. Book excerpt: 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

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