Erratum Please send corrections and comments to me at jmilne at umich.edu.
A rough preliminary draft of the book is available iAG200 Caution: the final version is substantially rewritten, and the numbering has changed.
The picture illustrates Grothendieck's vision of a pinned reductive group: the body is a maximal torus T, the wings are the opposite Borel subgroups B, and the pins rigidify the situation.
"Demazure nous indique que, derrière cette terminologie [épinglage], il y a l'image du papillon (que lui a fournie Grothendieck): le corps est un tore maximal T, les ailes sont deux sous-groupes de Borel opposées par rapport à T, on déploie le papillon en étalant les ailes, puis on fixe des éléments dans les groupes additifs (des épingles) pour rigidifier la situation (c.-à-d., pour éliminer les automorphismes)." (SGA 3, XXIII, p.177, new edition.)
The first eight chapters of the book study general algebraic group schemes over a field. They culminate in a proof of the Barsotti-Chevalley theorem stating that every algebraic group is an extension of an abelian variety by an affine algebraic group. The remaining chapters treat only affine algebraic groups. After a review of the Tannakian philosophy, there are short accounts of Lie algebras and finite group schemes. Solvable algebraic groups are studied in detail in Chapters 12-16. The final eight chapters treat the Borel-Chevalley structure theory of reductive algebraic groups over arbitrary fields. Three appendices review the algebraic geometry needed, the construction of very general quotients of algebraic groups, and the theory of root data.
The exposition incorporates simplifications to the theory by Springer, Steinberg, and others. Although the theory of algebraic groups can be considered a branch of algebraic geometry, most of those using it are not algebraic geometers. In the present work, prequisites have been kept to a minimum. The only requirement is a first course in algebraic geometry including basic commutative algebra.
*The only previous attempt to write such a book that I know of was that of Demazure and Gabriel. As they wrote in the foreword of "Groupes Algébriques (1971)" (my translation):
A. Grothendieck introduced two particularly useful tools into algebraic geometry: the functorial calculus and varieties with nilpotent elements. These tools allow us in particular to better understand phenomena arising from inseparability, by restoring differential calculus in nonzero characteristic, and they considerably simplify the general theory of algebraic groups. Hence we originally intended to develop the now classical theory of semisimple algebraic groups over algebraically closed fields (Borel-Chevalley) within the framework of schemes; it was simply a question of updating the 1956-58 seminar of Chevalley. But we soon realized that there did not exist a suitable book to serve as a reference for the general theory of algebraic groups, and moreover that it was impossible to refer a nonspecialist reader to the Grothendieck's Eléments de Géométrie Algébrique (EGA). Thus we were led to considerably modify and develop our original project, and we now present to the "mathematical community" a first volume devoted to the general theory of algebraic groups, where the case of semisimple groups is not entered into.Alas, no second volume was published.