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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