Introduction to Affine Group Schemes [Repost]

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 construct 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. I have tried hard to keep the book introductory. There is no prerequisite beyond a training in algebra including tensor products and Galois theory. Some scattered additional results (which most readers may know) are included in an appendix. The theory over base rings is treated only when it is no harder than over fields. Background material is generally kept in the background: affine group schemes appear on the first page and are never far from the center of attention. Topics from algebra or geometry are explained as needed, but no attempt is made to treat them fully. Much supplementary information is relegated to the exercises placed after each chapter, some of which have substantial hints and can be viewed as an extension of the text. There are also several sections labelled" Vista," each pointing out a large area on which the text there borders. Though non-affine objects are excluded from the text, for example, there is a heuristic discussion of schemes after the introduction of Spec A with its topology. There was obviously not enough room for a full classification of semisimple groups, but the results are sketched at one point where the question naturally arises, and at the end of the book is a list of works for further reading. Topics like formal groups and invariant theory, which need (and have) books of their own, are discussed just enough to indicate some connection between them and what the reader will have seen here. It remains only for me to acknowledge some of my many debts in this area, beginning literally with thanks to the National Science Foundation for support during some of my work. There is of course no claim that the book contains anything substantially new, and most of the material can be found in the work by Demazure and Gabriel. My presentation has also been influenced by other books and articles, and (in Chapter 17) by mimeographed notes of M. Artin. But I personally learned much of this subject from lectures by P. Russell, M. Sweedler, and J. Tate; I have consciously adopted some of their ideas, and doubtless have reproduced many others.