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

Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites.

J. S. Milne (Author)

9781009018586, Cambridge University Press

Paperback / softback, published 30 June 2022

664 pages, 5 b/w illus. 95 exercises
22.7 x 15.1 x 4.1 cm, 1.06 kg

'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. Kunyavski?, MathSciNet

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.

Introduction
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
Bibliography
Index.

Subject Areas: Algebraic geometry [PBMW], Geometry [PBM], Algebra [PBF]