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Algebraic Groups
The Theory of Group Schemes of Finite Type over a Field

$55.00 (P)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: June 2022
  • availability: Available
  • format: Paperback
  • isbn: 9781009018586

$ 55.00 (P)

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About the Authors
  • 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.

    • The first comprehensive introduction to the theory of algebraic group schemes over fields
    • This book is accessible to non-specialists, with few prerequisites
    • The book is written in the language of modern algebraic geometry
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    Reviews & endorsements

    'All together, this excellent text fills a long-standing gap in the textbook literature on algebraic groups. It presents the modern theory of group schemes in a very comprehensive, systematic, detailed and lucid manner, with numerous illustrating examples and exercises. It is fair to say that this reader-friendly textbook on algebraic groups is the long-desired modern successor to the old, venerable standard primers …' Werner Kleinert, zbMath

    'The author invests quite a lot to make difficult things understandable, and as a result, it is a real pleasure to read the book. All in all, with no doubt, Milne's new book will remain for decades an indispensable source for everybody interested in algebraic groups.' Boris È. Kunyavskiĭ, MathSciNet

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    Product details

    • Date Published: June 2022
    • format: Paperback
    • isbn: 9781009018586
    • length: 664 pages
    • dimensions: 229 x 152 x 37 mm
    • weight: 0.964kg
    • contains: 5 b/w illus. 95 exercises
    • availability: Available
  • Table of Contents

    1. Definitions and basic properties
    2. Examples and basic constructions
    3. Affine algebraic groups and Hopf algebras
    4. Linear representations of algebraic groups
    5. Group theory: the isomorphism theorems
    6. Subnormal series: solvable and nilpotent algebraic groups
    7. Algebraic groups acting on schemes
    8. The structure of general algebraic groups
    9. Tannaka duality: Jordan decompositions
    10. The Lie algebra of an algebraic group
    11. Finite group schemes
    12. Groups of multiplicative type: linearly reductive groups
    13. Tori acting on schemes
    14. Unipotent algebraic groups
    15. Cohomology and extensions
    16. The structure of solvable algebraic groups
    17. Borel subgroups and applications
    18. The geometry of algebraic groups
    19. Semisimple and reductive groups
    20. Algebraic groups of semisimple rank one
    21. Split reductive groups
    22. Representations of reductive groups
    23. The isogeny and existence theorems
    24. Construction of the semisimple groups
    25. Additional topics
    Appendix A. Review of algebraic geometry
    Appendix B. Existence of quotients of algebraic groups
    Appendix C. Root data

  • Author

    J. S. Milne, University of Michigan, Ann Arbor
    J. S. Milne is Professor Emeritus at the University of Michigan, Ann Arbor. His previous books include Etale Cohomology (1980) and Arithmetic Duality Theorems (2006).

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