Algebraic Groups

The Theory of Group Schemes of Finite Type over a Field

J. S. Milne (University of Michigan, Ann Arbor)

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English

Cambridge University Press
21 September 2017

Groups & group theory; Algebraic geometry

Series: Cambridge Studies in Advanced Mathematics

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.

By: J. S. Milne (University of Michigan Ann Arbor)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 170
Dimensions: Height: 233mm, Width: 160mm, Spine: 41mm
Weight: 1.050kg
ISBN: 9781107167483
ISBN 10: 1107167485
Series: Cambridge Studies in Advanced Mathematics
Pages: 660
Publication Date: 21 September 2017
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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).

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

'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 mentioned above.' Werner Kleinert, zbMath '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 mentioned above.' Werner Kleinert, zbMath