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    Title
    Algebraic groups : the theory of group schemes of finite type over a field
    Author
    Milne, J. S., 1942- author.

    Bibliographic Information

    MARC Record

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    Personal Author:
    Milne, J. S., 1942- author.
    Title:
    Algebraic groups : the theory of group schemes of finite type over a field / J.S. Milne, University of Michigan, Ann Arbor.
    Publication info:
    Cambridge, United Kingdom ; New York, New York : Cambridge University Press, [2017]
    Copyright info:
    ©2017
    Physical description:
    xvi, 644 pages : illustrations ; 24 cm.
    Content type:
    text
    Medium:
    unmediated
    Format:
    volume
    Series:
    Cambridge studies in advanced mathematics ; 170.
    Series:
    Cambridge studies in advanced mathematics ; 170
    General Note:
    Includes references (pages 627-636) 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 -- 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.
    Summary:
    "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
    Subject term:
    Geometry, Algebraic.
    Subject term:
    Group theory.
    ISBN:
    9781107167483
    ISBN:
    1107167485

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      QA 564 .M5x 2017 1 Book Harold B. Lee Library Bookshelves
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