Algebra Abstrata Ufrj Luciane

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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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7k Stabilizers of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7l Sub-coalgebras and subcategories . . . . . . . . . . . . . . . . . . . . . . 90

7m Quotient groups and subcategories . . . . . . . . . . . . . . . . . . . . . . 91

7n Normal subgroups and subcategories . . . . . . . . . . . . . . . . . . . . 917o Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8 Group theory: the isomorphism theorems . . .

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