Overview
- Provides a complete theory of triangulated rational mixed motives satisfying Grothendieck’s six operations, including the state of the art for integral coefficients
- Gives a systematic, self-contained, account of Grothendieck’s six functor formalism
- Includes a modern presentation of descent theory with applications to K-theory
- Explains the close relationship between mixed Weil cohomologies and coefficient systems
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (4 chapters)
Keywords
About this book
Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson’s program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky’s entire work and Grothendieck’s SGA4, our main sources are Gabber’s work on étale cohomology and Ayoub’s solution to Voevodsky’s cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky.
Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck’ six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory.
This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given.
Reviews
“It is obviously aimed at readers who are intimately familiar with the motivic literature. … this work is likely to become a valuable reference for researchers in the area.” (Igor A. Rapinchuk, Mathematical Reviews, September, 2020)
Authors and Affiliations
About the authors
Denis-Charles Cisinski is Professor of Mathematics at the Universität Regensburg, Germany. He previously held positions at the University of Toulouse and at the University Paris XIII. His research focuses on homotopical algebra, category theory, K-theory and the cohomology of schemes. He is also the author of two monographs entitled Les préfaisceaux comme modèles des types d'homotopie (2006) and Higher Categories and Homotopical Algebra (2019).
Frédéric Déglise has been Chargé de recherche in the CNRS, at the University Paris XIII from 2003 to 2011 and at the ENS Lyon from 2011 to 2016. He is since then Directeur de recherche in the CNRS, at the University of Burgundy. He has been Marie Curie Fellow at the FRIAS institute in 2017. He is a specialist of homotopy theory, algebraic geometry, motivic homotopy theory and motives.
Bibliographic Information
Book Title: Triangulated Categories of Mixed Motives
Authors: Denis-Charles Cisinski, Frédéric Déglise
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-33242-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-33241-9Published: 20 November 2019
Softcover ISBN: 978-3-030-33244-0Published: 20 November 2020
eBook ISBN: 978-3-030-33242-6Published: 09 November 2019
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XLII, 406
Number of Illustrations: 1 b/w illustrations
Topics: Algebraic Geometry, Category Theory, Homological Algebra, K-Theory