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.
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.
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).
Inhaltsangabe
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.
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.
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