Commutative group schemes
We restrict ourselves to two aspects of the field of group schemes, in which the results are fairly complete: commutative algebraic group schemes over an algebraically closed field (of characteristic different from zero), and a duality theory concern ing
- PDF / 7,433,874 Bytes
- 140 Pages / 595.276 x 680.315 pts Page_size
- 31 Downloads / 271 Views
15
F. Dort Universiteit van Amsterdam
Commutative group schemes 1966
Springer-Verlag· Berlin· Heidelberg· New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photo mechanical means (photostat, microfilm and/or micro card) or by other procedure withoilt written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1966. Title No. 7335. Library of Congress Catalog Card Number 66- 20635.
Introduction We restrict ourselves to two aspects of the field of group schemes, in which the results are fairly complete: commutative algebraic group schemes over an algebraically closed field (of characteristic different from zero), and a duality theory concerning abelian schemes over a locally noetherian prescheme. The preliminaries for these considerations are brought together in chapter I. SERRE described properties of the category of commutative quasi-algebraic groups by introducing pro-algebraic groups. In char8teristic zero the situation is clear. In characteristic different from zero information on finite group schemee is needed in order to handle group schemes; this information can be found in work of GABRIEL. In the second chapter these ideas of SERRE and GABRIEL are put together. Also extension groups of elementary group schemes are determined. A suggestion in a paper by MANIN gave crystallization to a fee11ng of symmetry concerning subgroups of abelian varieties. In the third chapter we prove that the dual of an abelian scheme and the linear dual of a finite subgroup scheme are related in a very natural way. Afterwards we became aware that a special case of this theorem was already known by CARTIER and BARSOTTI. Applications of this duality theorem are: the classical duality theorem ("duality hypothesis", proved by CARTIER and by NISHI); calculation of Ext(~a,A), where A is an abelian variety (result conjectured by SERRE); a proof of the symmetry condition (due to MANIN) concerning the isogeny type of a formal group attached to an abelian variety. As we said before, our results originate from work of SERRE and GABRIEL. Besides that of course the ideas and results of GROTHENDIECK were indispensable. 1 am greatly indebted to
J.-~.Serre,
from whom I received valuable suggestions and helpful correspondence, to P.Gabriel, who, having read the manuscript, proposed many
improvements and who gave precious information, and to J.P.Murre for his continuous interest in my work.
Terminology and notations.
We say that a diagram is exact, if all
columns and all rows are exact sequences. The terms injective, surjective and bijective are to be taken in the set-theoretical sense, while monomorphic, epimorphic and isomorphic are to be taken in the categorical sense (an epimorphic ring-homomorphism needs not to be surjective, a surjective morphism of schemes needs not to be an epimorphism). Working with preschemes, we use Mor( - , -) in order to indicate a set of morphisms (this notation differs from the
Data Loading...
