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Classification of pseudo-reductive groups

Author: Brian Conrad; Gopal Prasad
Publisher: Princeton : Princeton University Press, 2016.
Series: Annals of mathematics studies, no. 191.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
In the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
(DLC) 2015023803
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Brian Conrad; Gopal Prasad
ISBN: 1400874025 9781400874026
OCLC Number: 930040951
Language Note: In English.
Description: 1 online resource (pages cm.)
Contents: Cover; Title; Copyright; Contents; 1 Introduction; 1.1 Motivation; 1.2 Root systems and new results; 1.3 Exotic groups and degenerate quadratic forms; 1.4 Tame central extensions; 1.5 Generalized standard groups; 1.6 Minimal type and general structure theorem; 1.7 Galois-twisted forms and Tits classification; 1.8 Background, notation, and acknowledgments; 2 Preliminary notions; 2.1 Standard groups, Mevi subgroups, and root systems; 2.2 The basic exotic construction; 2.3 Minimal type; 3 Field-theoretic and linear-algebraic invariants; 3.1 A non-standard rank-1 construction 3.2 Minimal field of definition for Ru(Gk̅)3.3 Root field and applications; 3.4 Application to classification results; 4 Central extensions and groups locally of minimal type; 4.1 Central quotients; 4.2 Beyond the quadratic case; 4.3 Groups locally of minimal type; 5 Universal smooth k-tame central extension; 5.1 Construction of central extensions; 5.2 A universal construction; 5.3 Properties and applications of G̃; 6 Automorphisms, isomorphisms, and Tits classification; 6.1 Isomorphism Theorem; 6.2 Automorphism schemes; 6.3 Tits-style classification B Clifford constructionsB.1 Type B; B.2 Type C; B.3 Cases with [k : k^2] ≤ 8; B.4 Type BC; C Pseudo-split and quasi-split forms; C.1 General characteristic; C.2 Quasi-split forms; C.3 Rank-1 cases; C.4 Higher-rank and non-reduced cases; D Basic exotic groups of type F4 of relative rank 2; D.1 General preparations; D.2 Forms of k-rank 2; Bibliography; Index
Series Title: Annals of mathematics studies, no. 191.
Responsibility: Brian Conrad, Gopal Prasad.

Abstract:

In the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudo-reductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

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"This book is beautiful and will be at the origin of many advances in the general theory of arbitrary algebraic groups."--Bertrand Remy, MathSciNet

 
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