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Algebraic groups : the theory of group schemes of finite type over a field / J.S. Milne, University of Michigan, Ann Arbor.
Author
Milne, J. S., 1942-
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Format
Book
Language
English
Published/Created
Cambridge, United Kingdom ; New York, New York : Cambridge University Press, [2017]
©2017
Description
xvi, 644 pages : illustrations ; 24cm.
Availability
Copies in the Library
Location
Call Number
Status
Location Service
Notes
Lewis Library - Stacks
QA564 .M525 2017
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Details
Subject(s)
Geometry, Algebraic
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Group theory
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Series
Cambridge studies in advanced mathematics ; 170.
[More in this series]
Summary note
"Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti-Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel-Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry." -- publisher
Notes
Includes references (pages 627-636) and index.
Bibliographic references
Includes bibliographical references and index.
Contents
Definitions and basic properties
Examples and basic constructions
Affine algebraic groups and Hopf algebras
Linear representations of algebraic groups
Group theory: the isomorphism theorems
Subnormal series; solvable and nilpotent algebraic groups
Algebraic groups acting on schemes
The structure of general algebraic groups
Tannaka duality; Jordan decompositions
The Lie algebra of an algebraic group
Finite group schemes
Groups of multiplicative type; linearly reductive groups
Tori acting on schemes
Unipotent algebraic groups
Cohomology and extensions
The structure of solvable algebraic groups
Borel subgroups and applications
The geometry of algebraic groups
Semisimple and reductive groups
Algebraic groups of semisimple rank one
Split reductive groups
Representations of reductive groups
The isogeny and existence theorems
Construction of the semisimple groups
Additional topics
Appendix A: Review of algebraic geometry
Appendix B: Existence of quotients of algebraic groups
Appendix C: Root data.
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ISBN
9781107167483 ((hardback))
1107167485 ((hardback))
OCLC
992433996
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