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CHAPTER I
Basic Theory of Algebraic Groups
The emphasis in this chapter is on affine algebraic groups over a base field, but, when it requires noextra effort, we often study more general objects: affine groups (not of finite type); base rings rather
than fields; affine algebraic monoids rather than groups; affine algebraic supergroups (very briefly);
quantum groups (even more briefly). The base field (or ring) is alway denoted k, and R is always a
commutative k-algebra.
I Basic Theory of Algebraic Groups 1
1 Introductory overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1a The building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1b Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1c Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81d Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Definitions; examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2b Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2c Affine monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2d Affine supergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2e Relaxing the hypothesis on k . . . . . . . . . . . . . . . . . . . . . . . . . 20
2f Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Some basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3a Products of affine groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3b Fibred products of affine groups . . . . . . . . . . . . . . . . . . . . . . . 23
3c Extension of the base ring (extension of scalars) . . . . . . . . . . . . . . . 24
3d Restriction of the base ring (restriction of scalars) . . . . . . . . . . . . . . 24
3e The Greenberg functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Affine groups and Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4a Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4b Co-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4c The duality of algebras and co-algebras . . . . . . . . . . . . . . . . . . . 29
4d Bi-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
This is Chapter 1 of Algebraic Groups, Lie Groups, and their Arithmetic Subgroups, available at www.jmilne.
org/math/. Version 2.21, April 27, 2010. Copyright c 2005, 2006, 2009, 2010 J.S. Milne.
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4e Affine monoids and bi-algebras . . . . . . . . . . . . . . . . . . . . . . . 31
4f Affine groups and Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . 32
4g Abstract restatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4h Explicit description of, , and S . . . . . . . . . . . . . . . . . . . . . . 344i Commutative affine groups . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4j Finite flat algebraic groups; Cartier duality . . . . . . . . . . . . . . . . . 36
4k Galois descent of affine groups . . . . . . . . . . . . . . . . . . . . . . . . 36
4l Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4m Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4n Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Algebraic groups and affine algebraic schemes . . . . . . . . . . . . . . . . . . . . 40
5a Affine k-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5b The max spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . . . . 40
5c Affine algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5d Properties of affine algebraic schemes . . . . . . . . . . . . . . . . . . . . 435e Algebraic groups as groups in the category of affine algebraic schemes . . 43
5f Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5g Reduced algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5h Smooth algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5i Smooth algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5j Algebraic groups in characteristic zero are smooth (Cartiers theorem) . . . 48
5k Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5l Relaxing the conditions on k and G . . . . . . . . . . . . . . . . . . . . . 52
5m Appendix: The faithful flatness of Hopf algebras . . . . . . . . . . . . . . 52
5n Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Group theory: subgroups and quotient groups. . . . . . . . . . . . . . . . . . . . . 566a A criterion to be an isomorphism . . . . . . . . . . . . . . . . . . . . . . . 56
6b Subgroups; injective homomorphisms . . . . . . . . . . . . . . . . . . . . 56
6c Kernels of homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 58
6d Dense subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6e Normalizers; centralizers; centres . . . . . . . . . . . . . . . . . . . . . . 62
6f Quotient groups; surjective homomorphisms . . . . . . . . . . . . . . . . 65
6g Existence of quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6h Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6i Algebraic groups as sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 69
6j Limits of affine groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6k Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Representations of affine groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7a Definition of a representation . . . . . . . . . . . . . . . . . . . . . . . . . 74
7b Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7c Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7d The category of comodules over C . . . . . . . . . . . . . . . . . . . . . . 79
7e Representations and comodules . . . . . . . . . . . . . . . . . . . . . . . 79
7f The category of representations ofG . . . . . . . . . . . . . . . . . . . . 82
7g Affine groups are inverse limits of algebraic groups . . . . . . . . . . . . . 83
7h Algebraic groups admit finite-dimensional faithful representations . . . . . 84
7i The regular representation contains all . . . . . . . . . . . . . . . . . . . . 85
7j Every faithful representation contains all . . . . . . . . . . . . . . . . . . 87
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