An Introduction to Algebraic Geometry and Algebraic Groups

320 Pages·2003·1.81 MB·English

by  Meinolf Geck

OXFORD GRADUATE TEXTS IN MATHEMATICS
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OXFORD GRADUATE TEXTS IN MATHEMATICS
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An Introduction to Algebraic
Geometry and Algebraic
Groups
Meinolf Geck
Department of Mathematics, University of Stuttgart.
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Preface
Algebraic geometry, in its classical form, is the study of algebraic
sets in affine or projective space. By definition, an algebraic set in
kn (where k is a field) is the set of all common zeros of a collection
of polynomials in n variables. Algebraic groups are both groups and
algebraic sets, where the group operations are given by polynomial
functions. For example, the special linear group SL (k) consisting of
n
all n×n matrices with determinant 1 is an algebraic group.
Historically, these groups were first studied in an analytic con-
text, wherethegroundfieldkisRorC. Thisistheclassicaltheoryof
‘Liegroups’(seeChevalley(1946)orRossmann(2002), forexample),
which plays an important role in various branches of mathematics.
Through the fundamental work of Borel and Chevalley in the 1950s,
it is known that this theory also makes sense over an arbitrary alge-
braically closed field. This book contains an introduction to the
theoryof‘groupsofLietype’oversuchageneralgroundfieldk; con-
sequently, the main flavour of the exposition is purely algebraic. In
the last chapter of this book, we will even exclusively study the case
wherek isanalgebraicclosureofafinitefieldofcharacteristicp > 0.
Then the corresponding algebraic groups give rise to various fami-
lies of finite groups. By the classification of the finite simple groups,
every non-abelian finite simple group arises from an algebraic group
over a field of characteristic p > 0, except for the alternating groups
and 26 sporadic simple groups; see Gorenstein et al. (1994). This is
one reason why algebraic groups over fields of positive characteristic
also play an important role.
Although large parts of this book are developed in a general set-
ting, thechoiceofthematerialandtheorientationistowardsgroups
over fields of positive characteristic and the corresponding ‘finite
groups of Lie type’. Many research articles in this area begin with a
statement like
‘Let G be a connected reductive algebraic group over F ;
q
we assume that G is defined over F , with corresponding
q
Frobenius map F’
vi Preface
or variants thereof. Later on, it is very likely that the author will
fix an F-stable Borel subgroup, an F-stable maximal torus therein,
and so on. One of the aims of this book is to explain these notions
carefully, to prove some fundamental results about them, and to
illustrate the general theory with many examples which are worked
outingreatdetail. Infact, thisbookarosefromthedesiretoprovide
a general introduction into the background material from algebraic
geometry which is needed to enter this web of ideas. This introduc-
tion should be at the same time as elementary as possible and yet
lead to some substantial results.
One of the most striking advances in this area is the Deligne–
Lusztig theory of representations of finite groups of Lie type. In the
last three sections of this book, we will arrive at a point where we
can give a first introduction to these ideas and study one family of
finite groups of Lie type in detail. For those who wish to read more
on this subject, there should then be no difficulty in continuing with
Lusztig (1977), Carter (1985), or Digne and Michel (1991).
Each chapter of this book has an introduction which explains
in more detail the content and the main results; at the end, the
reader will find bibliographic remarks and exercises. Usually, there
are rather explicit hints to solutions, at least for those exercises
which are used in the text. As for the prerequisites, I assume a
good knowledge of the material in a standard algebra course (the
basics about groups, commutative rings, polynomials, and fields). In
Chapter 1, we will make use of elementary properties of tensor prod-
uctsofvectorspacesandseparablefieldextensions. InChapter3, we
shall need at one point the Schur–Zassenhaus theorem (on the exis-
tence and conjugacy of complements of abelian normal subgroups).
In Chapter 4, we assume that the reader is willing to accept a very
deep result (Grothendieck’s trace formula) without proof. In the last
parts of Chapter 4, we assume some familiarity with the charac-
ter theory of finite groups. Otherwise, the exposition is completely
self-contained and proceeds in a fairly elementary and concrete way,
giving ‘full proofs for everything’.
Among the potential readers I imagine a student who wishes
to get a starting point for reading more advanced textbooks or
more specialized research articles on groups of Lie type and their
representations. I could also imagine a lecturer or researcher with a
specific interest in finite simple groups, say, who wishes to have an
easily accessible reference for basic geometric facts about groups of
Preface vii
Lie type. And last but not least, I imagine someone who just wishes
to study some basic aspects in a beautiful area of mathematics,
wherealgebraicgeometryandfinitegroups(whichare,apriori,quite
unrelated areas!) get mixed.
In the elaboration of this book, I have used a number of sources.
Asfarasalgebraicgeometryandcommutativealgebraareconcerned,
my favourite references are Atiyah and Macdonald (1969), Mumford
(1988), andShafarevich(1994). Thestandardtextbooksonalgebraic
groups are Borel (1991), Humphreys (1991), and Springer (1998),
which contain much more material than we can cover here. Our
treatment owes a lot to the lecture notes of Steinberg (1974) and
the article by Steinberg (1977), in which Steinberg gave new and, in
manycases, moreelementaryproofsforstandardresultsonalgebraic
groups. For the more specific theory of finite groups of Lie type and
their characters (following Deligne–Lusztig), the standard reference
is Carter (1985). Finally, the work of George Lusztig on characters
of finite groups of Lie type has had the most important influence on
the development of my own understanding of all these matters.
For the experts, let me try to mention some points which I
consider to be the distinctive features of this book.
(1) The running examples in this book are the ‘classical groups’,
that is, groups of matrices which leave a non-degenerate bilinear or
quadratic form invariant. We will not discuss the classification of
such forms, for which one can consult Dieudonn´e (1971) or Grove
(2002), for example. Here, we introduce three families of groups
(which are precisely the ‘groups of Lie type’ B ,C ,D ) corre-
m m m
sponding to three fixed choices of a symmetric or alternating form.
We will give a complete treatment of these groups in the frame-
work of so-called ‘groups with a BN-pair’, a most useful concept
introduced by Tits in the 1960s. In this approach, for example, the
Dicksonpseudodeterminantfororthogonalgroupsincharacteristic2
is not needed at all; see Section 1.7.
(2) In this book, we exclusively work in the framework of affine
varieties. There is no mention at all of general (quasi-projective)
varieties, sheaves of functions, or such things. We do introduce pro-
jectivespace,butthiswillonlyplayanauxiliaryrole,asatopological
space on which an algebraic group acts. All the properties of pro-
jective varieties, grassmannian varieties, and flag varieties that are
needed can be quite naturally obtained by working directly with
viii Preface
closed cones in the underlying affine spaces. This is the approach
taken by Steinberg (1977), and it will be used to establish the main
results on Borel subgroups in Section 3.4.
(3) The general structure theory of connected reductive groups
showsthatthesegroupsalwayshaveaBN-pairinwhichB isaBorel
subgroup. This important result is achieved as the culmination of a
long series of arguments (see §14.15 of Borel (1991), for example),
and we will not endeavour to prove it. In this book, we turn these
matters upside down: We shall consider algebraic groups G with a
BN-pair satisfying an additional condition which guarantees that G
is reductive and that B is a Borel subgroup. As mentioned in (1),
we establish the existence of such BN-pairs for all groups of clas-
sical type. In this way, we can avoid various classification results
which form a tricky part in the general theory (for example, the
classification of algebraic groups of dimension one).
(4) There is a general theory of fields of definition and rationality
properties for algebraic varieties and groups. In this book, we only
discuss this in the case of fields of positive characteristic where the
rational structure is determined by a Frobenius map. As a highlight,
we obtain the general order formula for finite groups of Lie type;
see Section 4.2. Finally, following Carter (1985), we present some
basic properties of the theory of (cid:1)-adic cohomology in the form of
axioms, on which a number of concrete applications and explicit
computations can be based. This will be illustrated in detail for the
example of the finite Suzuki groups.
Most of the material in the first two chapters has been used
as a basis for my undergraduate courses on elementary algebraic
geometry at the University of Lyon 1 in 2000/2001/2002. The first
five sections of Chapter 1 might actually be used as the core of a
short course in algebraic geometry. For example, one could say more
onalgorithmicaspectsusingtheexcellenttextofCoxetal.(1992);or
treat in more detail algebraic curves in two-dimensional space, using
Fischer (1994) or Reid (1988), for example; or one could extend
the discussion of tangent spaces and Lie algebras in Section 1.4,
by including more examples concerning classical groups, following
Goodman and Wallach (1998). I used parts of Chapters 3 and 4 for
a recent graduate course on algebraic groups over a field of charac-
teristic p > 0 in Lyon and, much earlier, in 1995, in a two-semester
course on algebraic geometry and algebraic groups at the RWTH
Aachen.
Preface ix
Finally, IwishtothankDavidH´ezard, GerhardHiss, andGunter
Malle, who carefully read various chapters and sent me detailed lists
ofcorrections.Forthetypesettingofthistext,the‘LATEXCompanion’
by M. Goossens, F. Mittelbach, and A. Samarin (Addison–Wesley
1994) was an indispensible tool.
Lyon, February 2003