Algebraic groups. Volume I: Algebraic geometry. Generalities. Commutative groups. With an appendix ‘Local class fields’ by Michiel Hazewinkel.
(Groupes algébriques. Tome I: Géométrie algébrique. Généralités. Groupes commutatifs. Avec un appendice ‘Corps de classes local’ par Michiel Hazewinkel.)

*(English)*Zbl 0203.23401
Paris: Masson et Cie, Éditeur; Amsterdam: North-Holland Publishing Company. xxvi, 700 p. (1970).

Summary: Up to recent time the only sources for study of algebraic groups were the roto-printed edition of [Sémin. Claude Chevalley 1956–1958. Classification des groupes de Lie algébriques. (Paris: École Normale Supérieure) (1958; Zbl 0092.26301)] and J.-P. Serre’s book on commutative groups [Groupes algébriques et corps de classes. Paris: Hermann (1959; Zbl 0097.35604)]. The lacking of a book, presenting the general theory of algebraic groups (and more generally – group schemes) was the essential gap in the algebraic geometry literature. Almost at the same time appeared the books of M. Demazure and A. Grothendieck [Séminaire de Géométrie Algébrique du Bois Marie 1962/64. Schémas en Groupes I–III (= SGA 3), Lecture Notes in Mathematics. Vols. 151–153. Berlin etc.: Springer Verlag (1970; Zbl 0207.51401, Zbl 0209.24201, Zbl 0212.52810); see also the first edition: Bures-sur-Yvette (1963; Zbl 0212.52801 ff.)] and the book under review fills up this gap.

The main aim of the authors is to develop the classical theory of semisimple algebraic groups over an algebraically closed field in frames of the theory of schemes. However, this first volume contains only “preliminaries” and is devoted to the general facts concerning group schemes. It also contains a very detailed theory of commutative algebraic groups over a perfect field.In certain places the exposition follows the above cited cited seminar of Demazure-Grothendieck (SGAD) and treats algebraic groups from the general point of view of view of group schemes over any affine basic scheme. The essential distinction is that with the help of the algebro-geometric introduction the authors succeeded to avoid the references to A. Grothendieck’s “Éléments de géométrie algébrique”.

Chapter 1 “Introduction into algebraic geometry” presents the necessary facts and definitions from algebraic geometry and can be viewed as a very good introduction into the theory of schemes. It is done in a very original way. According to the general presentation of group schemes over a basic scheme \(S\) as a group functor on the category of \(S\)-schemes, the authors define a scheme as a certain “functor over a ring” and prove that the category of schemes is equivalent to the category of certain locally ringed spaces. the basic scheme is assumed always to be affine corresponding to rings from the fixed universum (models). The reader will find in this chapter the important properties of some classes of morphisms of schemes (affine, faithfully flat, finitely presented, smooth, étale, proper).

Chapter 2 “Algebraic groups” contains definitions and general facts about group schemes over a fixed model \(k\) and follows essentially the first two exposés of SGAD (cohomology of Hochschild, the differential calculus on groups and so on). It also contains the important criteria of smoothness of a group, scheme and some special properties of group schemes over a field of a characteristic (Lie \(p\)-algebras, morphism of Frobenius, etc.), which can be found also in exposés 6 and 7 of SGAD.

Chapter 3 “Application of the sheaf theory to algebraic groups” corresponds more or less to exposés 3 and 6 of SGAD. The main tool of this chapter is the theory of sheaves with respect to fpqf topology of a ring. The first application of this theory is the construction of factorsheaves. The representability theorems of this chapter are due to Grothendieck and adapt the “old” Chevalley construction of homogeneous spaces to the case of affine group schemes. Another application is the calculation of some groups of torseurs and groups of extensions of certain special group schemes.

Chapter 4 “Nilpotent and solvable affine commutative groups” contains results which can be used in the classification of linear groups and considers the case when the base ring is a field. Here are defined and presented fundamental properties of nilpotent, solvable, unipotent and multiplicative algebraic groups. The theorem asserting that any affine commutative \(k\)-group is the extension (trivial in case \(k\) is perfect) of a unipotent group with the help of a multiplicative group allows to reduce the classification of affine commutative groups to one of unipotent groups.

This is done in the last chapter 5 “The structure of affine commutative groups” which is the most interesting in the book. The classification of affine commutative groups over any perfect ground field is presented here in full details. For the first time the theory earlier presented only in the original papers of Dieudonné, Cartier, and others is treated in detail in a monograph. The main tools of this theory are modules of Dieudonné, Witt group schemes, modules of Greenberg, etc. The results obtained here are closely related with the local class field theory, which is exposed in the Appendix due to M. Hazewinkel.

The main aim of the authors is to develop the classical theory of semisimple algebraic groups over an algebraically closed field in frames of the theory of schemes. However, this first volume contains only “preliminaries” and is devoted to the general facts concerning group schemes. It also contains a very detailed theory of commutative algebraic groups over a perfect field.In certain places the exposition follows the above cited cited seminar of Demazure-Grothendieck (SGAD) and treats algebraic groups from the general point of view of view of group schemes over any affine basic scheme. The essential distinction is that with the help of the algebro-geometric introduction the authors succeeded to avoid the references to A. Grothendieck’s “Éléments de géométrie algébrique”.

Chapter 1 “Introduction into algebraic geometry” presents the necessary facts and definitions from algebraic geometry and can be viewed as a very good introduction into the theory of schemes. It is done in a very original way. According to the general presentation of group schemes over a basic scheme \(S\) as a group functor on the category of \(S\)-schemes, the authors define a scheme as a certain “functor over a ring” and prove that the category of schemes is equivalent to the category of certain locally ringed spaces. the basic scheme is assumed always to be affine corresponding to rings from the fixed universum (models). The reader will find in this chapter the important properties of some classes of morphisms of schemes (affine, faithfully flat, finitely presented, smooth, étale, proper).

Chapter 2 “Algebraic groups” contains definitions and general facts about group schemes over a fixed model \(k\) and follows essentially the first two exposés of SGAD (cohomology of Hochschild, the differential calculus on groups and so on). It also contains the important criteria of smoothness of a group, scheme and some special properties of group schemes over a field of a characteristic (Lie \(p\)-algebras, morphism of Frobenius, etc.), which can be found also in exposés 6 and 7 of SGAD.

Chapter 3 “Application of the sheaf theory to algebraic groups” corresponds more or less to exposés 3 and 6 of SGAD. The main tool of this chapter is the theory of sheaves with respect to fpqf topology of a ring. The first application of this theory is the construction of factorsheaves. The representability theorems of this chapter are due to Grothendieck and adapt the “old” Chevalley construction of homogeneous spaces to the case of affine group schemes. Another application is the calculation of some groups of torseurs and groups of extensions of certain special group schemes.

Chapter 4 “Nilpotent and solvable affine commutative groups” contains results which can be used in the classification of linear groups and considers the case when the base ring is a field. Here are defined and presented fundamental properties of nilpotent, solvable, unipotent and multiplicative algebraic groups. The theorem asserting that any affine commutative \(k\)-group is the extension (trivial in case \(k\) is perfect) of a unipotent group with the help of a multiplicative group allows to reduce the classification of affine commutative groups to one of unipotent groups.

This is done in the last chapter 5 “The structure of affine commutative groups” which is the most interesting in the book. The classification of affine commutative groups over any perfect ground field is presented here in full details. For the first time the theory earlier presented only in the original papers of Dieudonné, Cartier, and others is treated in detail in a monograph. The main tools of this theory are modules of Dieudonné, Witt group schemes, modules of Greenberg, etc. The results obtained here are closely related with the local class field theory, which is exposed in the Appendix due to M. Hazewinkel.

Reviewer: Igor V. Dolgachev