Introduction to algebraic geometry and algebraic groups, Volume 39

366 Pages·1980·26.1 MB·English

by  M. Demazure

NORTH-HOLLAND
MATHEMATICS STUDIES 39
Introduction to Algebraic Geometry
and Algebraic Groups
Michel DEMAZURE
Ecole Polytechnique
France
and
Peter GABRIEL
University ofZurich
Switzerland
1980
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM NEW YORK OXFORD
0 North- Holland Publishing Company, I980
AN rights reserved. No part of thispublication may be reproduced,s tored in o retrieval system,
or transmitted, in any form or by any means. electronic. mechanical, photocopying, recording
or otherwi.re. without the prior permixyion of the copyright owwcr.
ISBN: 0 444 85443 6
Tratirlation of
GROUPES ALGEBRIQUES, Tome I (Chapters I & 11)
Masson & Cie, Pam I970
North-Holland Publishing Company, Amsterdam 1970
7 randoted by J. Bell
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM.NEWYORK*OXFORD
Sole distrihurors for the U.SA. . and ('atiada:
ELSEVIER NORTH-HOLLAND, INC.
52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging In Publication Datn
Demazure, Michel.
Introduction to algebraic geometry and algebraic
groups.
(North-Holland mathematics stuaies ; 39)
Translation of Groupes algebriques, vol. 1.
Bibliography: p.
Includes indexes.
1. Geometry, Algebraic, 2. Linear algebraic
groups. I. Gabriel, Peter, 1933- joint
author. 11. Title.
QA5 64. lh 5 13 516.3': 79-28481
ISBN 0-444-85443 -6
PRINTED IN THE NETHERLANDS
A. GRDTHENDIECK has introduced tm very useful tools in algebraic geanetsy:
the functorial calculus and varieties with nilpotent "functions". These
tools supply a better understanding of the phenanena related to inseparabili-
ty, they rehabilitate differential calculus in characteristic p f 0 , and
they simplify in a significant way the general theory of algebraic groups;
hence we first intended to develop within the frame of schms the classical
theory of semi-simple algebraic groups over an algebraically closed field
due to XXEG and CKEWZEY; our purpose simply was to present the 1956-58
seminar notes of CHEVALLFY in a new light. But then we realized the in-
existence of a convenient reference for the general theory of algebraic
groups, and the impossibility to refer a non-specialized reader to the
"Elhts de Gdtrie ALg6brique" (M;A) by QOTWNDIECK. This led us to a
considerable modification of the original project and to the publication of
this introductory treatise.
In a first chapter, we develop what we need fran algebraic gecanetry. In fact,
chapter I contains mre than what is strictly necessary; it supplies a gener-
al introduction to the theory of schmes frm a functorial pint of view and
presents the fundamental notions, with the exception of those related to
ample bundles and projective mrphisns. The matter of the first chapter is
taken almost canpletely frcm EGA; but the presentation has been modified in
a way we would like to justify now.
There are essentially tm pints of view in mdern algebraic geanetry. Let
us take a simple example: If P1,.../P are canplex polynanials in n in-
r
8
determinates, we may assign to them, on the one hand the subset X of
consisting of the points x such that P (XI=. .. =P (XI= 0 , which m y be
1 r
given some other structures: Zariski toplogy, sheaf of plynmial func-
...
-
tions this is the geanetric point of view. On the other hand, we may
watch the functor assigning to every unital, cmtative, associative algebra
.
A the set X(A) fo& by all xEAn such that P1 (XI =. .=Pr (x)= 0 - this
I
is the functorial pint of view. The first pint of view is generally adopted
V
vi INTRODUCTION
in proper algebraic geanetry; in the theory of linear algebraic groups hm-
ever, the second outlook is often mre beneficial, because it fits better
the constructions of group theory (it supplies an embedding of the category
of group-schemes into the category of group-functors, which is closed unda
m y co nstructions). Therefore, instead of defining scherres as geametric
spaces (endwed with sheaves of local rings) , as M3A does it, we define them
as functors over scnae category of rings. We then show that the category of
our functors is equivalent to scme category of geanetric spaces. In this way,
sane functors h a p t o schemes, instead of defininq a scheme up to iso-
morphisns; this is beneficial fran a technical point of view.
Chapter I1 then develops the general features of algebraic groups, avoiding
the mre delicate problem of residue class groups and the specialized theo-
ries (cmtative affine groups, abelian varieties, semi-simple groups),
parts of which were included in the second part of the first edition.
Since the publication of the first edition, several books on algebraic geo-
metry and algebraic groups appeared. Sane of then are mentioned in the cm-
plementiq bibliography. They all adopt the gmtric pint of view. There-
fore we hope that a text-book presenting the fundamentals of the functorial
approach may still be useful.
This second edition reproduces with s~nemi nor changes in chapter I, 8 2 the
first two chapters of the first French edition. The translation is due to
J. BELL, the typing to Mrs. R. WF@WN. To both we express our best thanks.
The following itens are supposed to be known:
a) The " E l h t s de mthhatique" by N. BOURBAKI, especially the chapters I
to V of his Cmnutative Algebra. W e refer to it by giving first the name of
.. .
the h k , then the number of the chapter, of the paragraph (for instance
Alg. cam., 111, 5 2, no 4).
b) A primer of the theory of sheaves, including the paragraphs 1 a d 2 fran
.
chapter 11 of [3]
c) A good groding in categories and hamlogical algebra, which may be
.
found in C2],[3],C4] arid C51 References to these treatises make mention
of the authors, chapters and paragraphs (for instance WAN-EIWERG,
chap. XVII, 5 7).
In order to refer to number X.Y of paragraph 2 of chaptex T of the present
treatise, we simply write X.Y if the reference takes place in paragraph 2
of chapter T ; otherwise, we write 5 2, X.Y if the reference takes place
witkin chapter T , ard T, 5 2,X.Y if it takes place in another chapter (the
reference (2.3 and 1,s 2, 5.6 and 5.7) means for instance: see number 2.3
of the present chapter arad number 5.6 and 5.7 of paragraph 2 of chapter I).
We collect in a functorial dictionary, to which we refer by means of "dict.",
sane stanlard definitions and notations of category theory freely used
throughout the kwk.
Modulo these references, and with the exception of some very peculiar comple-
ments, for which a reference is given within the text, all definitions and
proofs are canplete.
[l] N.BOURBAKI, ElCinents de n?atht;matique, [email protected]
[2] H.CARTAN-S .EILENBERG, Hamlogical algebra, Princeton University Press,
1956
[3] R.GODm, Ti-Gorie des faisceau, Hermann, 1958
[4] S.MACLANE, H m l q y , Springer, Grundlehren, aand 114, 1963
[5] B.MITCHELL, Theory cf categories, Acadenic Press, 1965
X
BOREL, A.
Groups alg6briques linCSaires, Ann. of Math. 64, 1956.
EOREL, A.
Linear algebraic groups, raig6 par H. Bass, Benjamin, 1969.
CARTIER, P.
Groups algsbriques et groups formels, Conf. au coll. sur la thbrie
des groups alggbriques, Bruxelles, 1962.
,c- c.
Thhrie des groups de Lie, tame 11, Hermann, 1951
,-c c.
Classification des groups de Lie alggbriques, Sgninaire 1956-58,
mltigraphi6, Paris, Secr6tariat matMmatique.
DEMAZURE, M.
Schhas en groups reductifs, Bull. Soc. Math. France 93, 1965.
DEMAZURE, M., A. GROEENDIECK et al.
Sch6mas en groups, S6minaire de g-trie alggbrique 1963-64, IHES,
Bures-=-Yvette.
DIEUDONNE J. et A. G R O I E C K
El&nents de g6cdtrie algsbrique, -1. Math. IHES, nos. 4,8,11,17,20,
...
24,28,32,
GABRIEL, P.
Ues catggories sMliennes, Bull. SOC. Math. France 90, 1962.
GRDTHENDIECK, A.
Sur quelques pints d'algare hmlcgique, Tobku Math. J. 9, 1957.
GROIXENLIIEK, A.
Fondements de la g6dtrie algGbrique, Extraits du s@minaire -1,
multigraphi6, Paris, Secr6tariat math@katicpe, 1962.
G€OE+NDIKK, A. et al.
SWnaires de g6&trie alg6briqe du Mis-Marie, multigraphi6s, IHES,
Bures-sur-Yvette.
xi
xii CaMPLEMENTARY LITrnWRE!
HAFTSHORNE, R.
Algebraic Geanetry, Springer-Verlag, 1977.
MlMpHRFyS, J.E.
Linear Algebraic Groups, Springer-Verlag, 1975.
MUMFORD, D.
Algebraic Geanetry I, Canplex projective varieties, Springer-Verlagf1976.
msCHLER, R. et P. GABRIEL
Sur la dimension des annemx et ensembles ordods, C.R. Acad. Sc.
Paris 265, 1967.
SERRE, J.P.
Groups algariques et corps de classes, Hermann, 1959.
SHAFARENITCH, I.R.
Basic Algebraic Geometry, Springer-Verlag, 1974
GENERAL CONVENTIONS
In the present treatise two fixed universes 9 and Such that INEJE and
.,,UEX are supposed to be given. we replace the tenn "set" by the tenn "class",
reserving the name "set" to the elenents of the universe : for instance,
V is a class, not a set, whereas IN and ,V are bth classes and sets. A
*y
set will be called mall if it has the saw cardinality as sane element of
wU : for instance, is a ma11 set, whereas 2 is not.
If C is a category, ObC and F1C represent respectively the class of
.
objects and the class of mrphisns of C We simply write cEC instead of
cfObC ; if a,MC , we denote by C(a,b) the class of arrws or mrphisns
.
fran a to b Similarly, when C is an abelian category, ?(arb) is the
.
group of Yon&-extensions of b by a of order n We denote by
,E,~n,~r,Ab,~,To'o.t.h.e categories of sets, momids, groups, cmtative
... E .
groups, unital cmtative rings, topological spaces belonging to
...
Unless otherwise stated, we reserve the appellation mnoid, group to the
...
objects of gn,$r In particular, unless we expressly state the contrary,
we suppse all the considered rings to be ccnmrutative and unital. If A% ,
Ivbd represents the category of A-modules belonging to E ; if M W A , we
-A .
51 gA(~,~)
set =
..
A mnoid, group, ring, module . is called s ~ lilf the underlying set is
so. We give a special name to the (unital, cmnutative) rings belonging to
;, calling then dels. Consequently, a model is a mall ring, and every
mll ring is imrphic to m e model, without necessarily being a d e l it-
s .
self. The full subcategory of % formed by the mdels is denoted by
If k 9 , we write + for the category of associative, camnutative, uni-1
k-algebras; similarly, if @k J , &I represents the full subcategory of %
formed by the k-algebras having a model as underlying ring.
NOW let us reassure the readers frightened by universes: the part played by
V is canpletely secondary, and we could easily do witbut by using the
**
axicmatic of Bernays-G?del. The part played by 2 is samewhat more subtle:
xiii
XiV GENERAL CONVEDTIONS
on the one hand, we intend to study the category g of functors frm I$ to
...
E ard the morphisns between two such functors should form a set; for that
reason, 5 should not be to large. But on the other hand, we w l d like to
apply to rnodels the usual constructions of catmutative algebra: residue class
.
rings, ririgs of fractions, ccarcpletions.. For this purpose it wrxlld be enough
to assume that for any model R , every ring with cardinality sMller or
equal to (Card R) IN is imrphic to sane model. We could have ensured this
condition by fixing an infinite set E and calling model any rhg supported
.
by a subset of E~ we have not chosen this way, mause many mathmaticians
are accustomed to universes by now, and also because we wuld like to use
freely direct limits in the category of dels.
Section 1 GeQnetric spaces
0
1.1 Definition: -A geametric space E = (x, consists of a
X
topological space X together with a sheaf of rings QX such that, for
4
-each XEX , the stalk flx,x (or simply Qx) ~f at x is a local
ring.
.
By abuse of notation, we shall o€ten write X instead of E The unique
ox
oJmx
maximal ideal of will be denoted by mx and the residue field
ox
.
by K(X) If s is a sectoixon of over a neighbourhooa of x , the
canonical image of s in will be denoted by sx and called the germ
of s at x ; mreover, the canonical hage of s in K(x) will be called
.
the value s(x) of s at x This value is thus zero iff the germ of s
.
at x lies in mx
flX
1.2 E l e : Let X be a topological space, and let be the
.
sheaf of genns of continuous cmplex valued functions on X For each x€X ,
ux
the stalk is local and its maximal ideal is the set of g e m of func-
.
tions which vanish at x
4)
1.3 -1e: Let (x, be a gecmetric space, and let P be a
subset of X , endowed with the iladuced toplogy. Let i : P -+ X be the
inclusion mp; then the restrictioonx of) to P , written Qx/P , is by
.
definition the inverse image i' ( (dict. )
1