Algebraic Groups: the theory of group schemes of finite type over a field --- J.S. Milne
This is a comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, but with minimal prerequisites. The final manuscript was sent to CUP on February 28, 2017 and the final copy-edited manuscript on June 24, 2017. Expected publication date is September 22, 2017 (644 pages). A rough preliminary draft of the book is available iAG200.

butterfly
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.)

Blurb

Algebraic groups play much the same role for algebraists that Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields, including the structure theory of semisimple algebraic groups, written in the language of modern algebraic geometry.* When Borel, Chevalley, and others introduced algebraic geometry into the theory of algebraic groups, they used the algebraic geometry of the day, whose terminology conflicts with that of modern (post 1960) algebraic geometry. As Tits wrote in 1960, "the scheme viewpoint ... is not only more general but also, in many respects, more satisfactory." Indeed, allowing nilpotents gives a much richer and more natural theory.

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.

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*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.

Table of Contents (hover to see the sections)

  1. Definitions and basic properties.
  2. Examples and some basic constructions.
  3. Affine algebraic groups and Hopf algebras.
  4. Linear representations of algebraic groups.
  5. Group theory; the isomorphism theorems.
  6. Subnormal series; solvable and nilpotent algebraic groups.
  7. Algebraic groups acting on schemes.
  8. The structure of general algebraic groups.
  9. Tannaka duality; Jordan decomposition.
  10. The Lie algebra of an algebraic group.
  11. Finite group schemes.
  12. Groups of multiplicative type; linearly reductive groups..
  13. Tori acting on schemes.
  14. Unipotent algebraic groups.
  15. Cohomology and extensions.
  16. The structure of solvable algebraic groups.
  17. Borel subgroups; Cartan subgroups.
  18. The geometry of algebraic groups.
  19. Semisimple and reductive groups.
  20. Algebraic groups of semisimple rank one.
  21. Split reductive groups.
  22. Representations of reductive groups.
  23. The isogeny and existence theorems.
  24. Construction of the semisimple groups.
  25. Additional topics.
  1. Review of algebraic geometry
  2. Existence of quotients of algebraic groups.
  3. Root data.