Algebraic groups: the theory of group schemes of finite type over a field/ J S Milne
By: Milne, J S [author].
Series: Cambridge Studies in Advanced Mathematics. Cambridge Studies in Advanced Mathematics 170.Publisher: UK: CUP, 2017Description: xvi, 644 pages 23cm.ISBN: 9781107167483.Subject(s): Mathematics  Algebraic Groups  Lie Group  Finite Group  Algebraic GeometryDDC classification: 516.35 Online resources: Click here to access onlineItem type  Current location  Call number  Status  Date due  Barcode  Item holds  

Books 
ISI Library, Kolkata

516.35 M659 (Browse shelf)  Available  138447 
Includes bibliographical reference and index
Preface,
Introduction,
Conventions and notation,
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 theorem,
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, Finite group schemes,
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 Roof data,
References,
Index
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 nonspecialists in algebraic geometry.
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