Introduction to Affine Group Schemes

William C. Waterhouse (auth.)0387904212, 9780387904214

Ah Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre’ KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con­ struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.

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Table of contents :
Front Matter….Pages i-xi
Front Matter….Pages 1-1
Affine Group Schemes….Pages 3-12
Affine Group Schemes: Examples….Pages 13-20
Representations….Pages 21-27
Algebraic Matrix Groups….Pages 28-35
Front Matter….Pages 37-37
Irreducible and Connected Components….Pages 39-45
Connected Components and Separable Algebras….Pages 46-53
Groups of Multiplicative Type….Pages 54-61
Unipotent Groups….Pages 62-67
Jordan Decomposition….Pages 68-72
Nilpotent and Solvable Groups….Pages 73-79
Front Matter….Pages 81-81
Differentials….Pages 83-91
Lie Algebras….Pages 92-100
Front Matter….Pages 101-101
Faithful Flatness….Pages 103-108
Faithful Flatness of Hopf Algebras….Pages 109-113
Quotient Maps….Pages 114-120
Construction of Quotients….Pages 121-127
Front Matter….Pages 129-129
Descent Theory Formalism….Pages 131-139
Descent Theory Computations….Pages 140-150
Back Matter….Pages 151-164