Introduction to Affine Group Schemes

Abstract

It is said that ideas which seemed startling or obscure to oldergenerations become natural and simple to younger generations. Abook such as this will doubtless promote the naturalisation ofsome ideas and methods of commutative algebra, modern algebraicgeometry and the theory of group schemes. The author conducts thereader, like a skilled guide, through these fields of knowledgeand allows him to pick out of the solution of exercises, someinteresting samples of group schemes over fields of positivecharacteristic or over nonintegral ground rings. I dare say thateven an experienced explorer acquainted with the classicalarchitecture of matrix groups will find pleasure in newelucidations of their structure.\par All attention isconcentrated on affine groups; thus, for example, such results asCartier's theorem on the nonsingularity of group schemes overfields of characteristic zero or Lang's theorem on the absence ofnontrivial homogeneous spaces of algebraic group schemes over afinite field are proved only for affine groups. Functorial andHopf algebraic interpretations of affine group schemes appear inthe opening pages of the book and, together with matrixrepresentations, interact in proofs, constructions andmotivations for new notions.\par The author also considersCartier duality for finite abelian group schemes, discusses thedecomposition of abelian group schemes into separable andunipotent factors (first for matrix groups and then for groupschemes over a perfect field), and describes results on thestructure of tori, unipotent group schemes and solvable matrixgroups. There is an infinitesimal theory which deals with amodule of differentials Ω\sb A of a Hopf algebra A, thesmoothness criterion \dim G=\text{rank}\, Ω\sb {k[G]}, andproperties of the Lie algebra of an affine group.\par To readthis introductory book, it is necessary to have a preliminaryknowledge of Galois theory and linear algebras over rings,including tensor products. The further algebraic needs of thereader are supplied in the course of the treatment. Thus, forexample, the notion of separable algebras and theircorrespondence to finite sets with a continuous action of aGalois group arises through the investigation of etale groups andthe functor π\sb 0 describing the set of connectedcomponents. Faithful flatness appears first as an instrument inthe proof of the smoothness criterion and, subsequently,information on flatness alternates with applications of flatness.Faithful flatness of a Hopf algebra over a Hopf subalgebra isestablished. The faithful flatness is used in a description ofthe properties of the quotient and in construction of thequotient group with a given closed normal subgroup as kernel. Inconnection with the sheaf property of quotients, coverings andsheaves in the faithfully flat topology are introduced and thenotion of the etale topology is given. The theory of faithfulflatness, so decorated with applications, looks more attractivethan the usual statement of standard platitudes.\par The lastpart of the book is entitled ``Descent theory''. The idea of thedescent of an algebraic structure is exemplified firstly by thedescent of bilinear multiplication and by an example of descentdata such as patching conditions on a Zariski covering. Then,explanations of twisted forms, H\sp 1 interpretations, Galoiscohomology and principal homogeneous spaces follow. There areapplications of these objects to central simple algebras and tothe construction of the Arf invariant.\par The book featureseight sections called ``vistas'', pointing out further resultsand related areas. In order to outline the limits of thetextbook, let me list the titles of the vistas: ``Schemes'',``Borel subgroups'', ``Differential algebra'', ``The algebrogeometric meaning of smoothness'', ``Formal groups'', ``Reductiveand semisimple groups'', ``The etale topology'', ``Invarianttheory''