Ergebnisse der Mathematik und ihrer Grenzgebiete 34
A Series of Modern Surveys in Mathematics
Editorial Board: P R Halmos_ P J Hilton (Chairman)
R Remmert B Sz6kefalvi-Nagy
Advisors: L V Ahlfors F.L Bauer A Dold
Trang 3Professor David Mumford Department of Mathematics Harvard University 1 Oxford Street Cambridge, MA 02138 USA
Professor John Fogarty
Department of Mathematics and Statistics University of Massachusetts
Amherst, MA 01003 USA
AMS Subject Classification (1980): 14D xx
ISBN 3-540-11290-1 Springer-Verlag Berlin Heidelberg New York N ISBN 0-387-11290-1 Springer-Verlag New York Heidelberg Berlin
ISBN 3-540-03284-3 1 Aufl Springer-Verlag Berlin Heidelberg New York ISBN 0-387-03284-3 Ist ed Springer-Verlag New York Heideiberg Beriin Library of Congress Cataloging in Publication Data Mumford, David Geometric invariant theory (Ergebnisse der Mathematik und ihrer Grenzgebiete; 34) Bibliography: p Includes index 1 Geometry, Algebraic 2 Invariants 3 Moduli theory L Fogarty, John, 1934- Il Title HII Series QAS64.MB85 1982 516.3°5 $1-21370 ISBN 0-387-11290-1 (U.S.) AACR2
This work is subject to copyright All rights are reserved whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to “Verwertungsgeselischaft Wort”, Munich © Springer-Verlag Berlin Heidelberg 1965, 1982 Printed in Germany Offset printing: Beltz Offsetdruck, Hemsbach/Bergstr Bookbinding: J Schaffer oHG, Grũnstadt 2141/3140-543210 i
Preface to second edition
In the 16 years since this book was published, there has been an ex-
plosion of activity in algebraic geometry In particular, there has been great progress in invariant theory and the theory of moduli Reprinting this monograph gave us the opportunity of making various revisions The first edition was written primilarly as a research monograph on a geometric way to formulate invariant theory and its applications to the theory of moduli In this edition we have left the original text essentially intact, but have added appendices, which sketch the progress on the topics treated in the text, mostly without proofs, but with discussions and references to all the original papers
In the preface to the 15 edition, it was explained that most of the invariant theoretic results were proven only in char 0 and that there- fore the applications to the construction of moduli spaces were either valid only in char 0, or else depended on the particular invariant theo- retic results in Ch 4 which could be established by elementary methods in all characteristics However, a conjecture was made which would
extend all the invariant theory to char p Fortunately, W HaBousH [128] has proved this conjecture, hence the book is now more straight-
forward and instead of giving one construction of the moduli space
, of curves of genus g over @ by means of invariant theory, and one
construction of , and o/,,, (the moduli space of principally polarized abelian varieties) over Z by means of the covariant of points of finite order and Torelli’s theorem, we actually give or sketch the following constructions of these spaces:
1) , and of, are constructed by proving the stability of the Chow forms of both curves and abelian varieties (Ch 4, § 6; Appendix 3B)
2) M, and , are independently constructed by covariants of finite sets of points (Appendix 7C; Ch 7, § 3)
3) «7, is constructed by an explicit embedding by theta constants (Appendix 7B)
All of this is valid over Z except that the covariant approach to 4,
Trang 4VỊ Preface to second edition
This preface gives us the opportunity to draw attention to some basic open questions in invariant theory and moduli theory 3 questions raised in the 1* edition have been answered:
a) the geometric reductivity of reductive groups has been proven
by Hazousu, op cit.,
b) the existence of canonical destabilizing flags for unstable points has been proven by Kempr [171] and Rousseau [285],
c) the stability of Chow forms and Hilbert points of pluricanonically embedded surfaces of general type has been proven by GrEsEKER [116] Pursuing the ideas in c), leads one to ask:
d) which polarized elliptic and K3-surfaces have stable Chow forms? and what is more difficult probably: -
©) can one compactify the moduli spaces of smooth surfaces by allow-
ing suitable singular polarized surfaces which are still ,,asymptotically stable", i.e., have stable Chow forms when embedded by any complete linear system which is a sufficiently large multiple of the polarization? In a more classical direction, now that the reasons for the existence of A, and sf, are so well understood, the time seems ripe to try to under- stand their geometry more deeply, e.g
f) Can one calculate, or bound, some birational invariants* of , or ,? Investigate the cohomology ring and chow ring of M, or #4 and
g) Find explicit Siegel modular forms vanishing on the Jacobian locus or cutting out this locus, and relate the various known special Br perties of Jacobians
For those who want to learn something of the subject of moduli,
we want to say what they will mot find here and where more background on these topics may be found The subject of moduli divides at present into 3 broad areas: deformation theory, geometric invariant theory, and the theory of period maps Deformation theory deals with local questions: infinitesimal deformations of a variety, or analytic germs of deformations Period maps deal with the construction of moduli of Hodge structures and the construction of moduli spaces of vareties by attaching a family of Hodge structures to a family of vareties Both of these subjects are discussed only briefly in this monograph (see Appendix 5B) Unfortuna- tely, deformation theory has not received a systematic treatment by anyone:a general introduction to the theory of moduliasa whole includ- ing deformation theory is given in SesHaprt [804], deformations of singularities are treated in ARTIN [45] The origins of the algebraic
* For certain g, WM, and o#, are not unirational: Freitac [108], [109],
HARRIS—MUMEORD [337]
Preface to second edition VII
treatment of the subject are in GROTHENDIECK [13], exp 195 and SCHLESSINGER, LICHTENBAUM [289], [185] The theory of period maps
is largely the creation of GRIFFITHS [121], and a survey of the theory can
be found in GrIFFITHS-Scumip [124], Expository or part expository/part research articles on geometric invariant theory proper and related que- stions of moduli have been written by DIEUDONNE-CaRREI [85], GIESE-
KER [119], NewstEap [247], and Mumrorp [213], [218], [220] We
hope this monograph will help to make the subject accessible
Writing these appendices has brought home to us very clearly how many people have been involved in invariant theory and moduli pro- blems It has been exciting to try to express coherently all their results and their interconnections We are sure, however, that some people have been overlooked For this we can only offer the hackneyed excuse that even together we had only four hands and two heads
D MUMFORD
J Focarty Cambridge, Mass
Trang 5Pitt
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Preface to first edition
The purpose of this book is to study two related problems: when does an orbit space of an algebraic scheme acted on by an algebraic
group exist? And to construct moduli schemes for various types of
algebraic objects The second problem appears to be, in essence, a special and highly non-trivial case of the first problem From an Italian point of view, the crux of both problems is in passing from a birational to a biregular point of view To construct both orbit spaces and moduli “generically” are simple exercises The problem is whether, within the set of all models of the resulting birational class, there is one model whose geometric points classify the set of orbits in some action, or the set of algebraic objects in some moduli problem In both cases, it is quite possible that some orbits, or some objects are so exceptional, or, as we shall say, are not stable, so that they must be left out of the model The difficulty is to pin down the meaning of stability in a given case One of the most intriguing unsolved problems, in this regard, is that of the moduli of non-singular polarized surfaces Which such surfaces
are not stable, in the sense that there is no moduli scheme for them and
their deformations? This property is very delicate
One of my principles has been not to worry too much about the
difference between characteristic O, and finite characteristics A large
part of this book is, therefore, devoted to a theory developed only in
characteristic O.* I am convinced, however, that it is almost entirely
valid in all characteristics What is necessary is to find some property of semi-simple algebraic groups in all characteristics which takes the place of the full reducibility of representations which is valid only in
characteristic O I conjecture, in fact, that if a semi-simple algebraic group G is represented in a vector space V, and ‘if V, is an invariant subspace of codimension 1, then for some » the invariant subspace of
codimension 1
Vạ- S“XV) C SZ()
* The hypothesis of characteristic O is disguised in the assumption that
the group which is acting is reductive (by reductive we always mean that all its representations are completely reducible) But, in characteristic 2, only relatively unimportant groups are reductive (cf [28]), so the theory is uninteresting
Preface to first edition IX
is complemented by an invariant 1-dimensional subspace.** Nonetheless, this is unknown, and one consequence is that this book is divided fairly sharply into two halves Although both parts are closely analogous, they are logically independent One half consists in Chap-
ters 1, 2, 4 and 5 which deal essentially only with characteristic zero,
and yield a construction of the moduli scheme, for curves over Q The,
other half consists in Chapters 3, 6, and 7, which deal with the ,,arith-
metic case’, i.e., over Spec (Z), and yield a construction of the moduli
schemes for curves over Z From a standpoint of content, however,
Chapters 1, 2, 3 and 4 deal with orbit space problems, while Chapters 5, 6 and 7 deal with moduli
This book is written entirely in the language of schemes Of course,
the results, for most purposes, could have been stated and proven in a classical language However, it seems to me that algebraic geometry
fulfills only in the language of schemes that essential requirement of all contemporary mathematics: to state its definitions and theorems in their natural abstract and formal setting in which they can be considered independently of geometric intuition Moreover, it seems to me incorrect to assume that any geometric intuition is lost thereby: for example,
the underlying variety in an algebraic scheme is rediscovered, and per-
haps better understood through the concept of geometric points As another example, the theory of schemes has made it possible, in a very intuitive way, to finally dispose of that famous embarrassment to the Italian school: the lack of an algebraic proof of the completeness of the characteristic linear system of suitable complete continuous systems on a surface in characteristic O (cf (40, 18, 33 and 36])
It is my pleasure to acknowledge at this point the great encourage- ment and stimulation which J have received from Oscar ZARISKI,
jJoun Tate, and ALEXANDER GROTHENDIECK In addition, J want to
give credit to the many mathematics from whom J have taken a great deal This book is primarily an original monograph, but secondarily an exposition of a whole topic, so J have taken the liberty of including
anybody else’s results when relevant J am particularly conscious of my indebtedness to GROTHENDIECK, HILBERT, and NaGaTa It is impossible to enumerate all the sources from which J have borrowed, but this is
a,partial list:
Ch 1 and 2 owe a great deal to D Hitperr [ii],
§ 1.2 was developed independently by C CHEvaLLey, N Iwanort,
and M Nacata,
§ 2.2 is largely a theory of J Tits,
Ch 3 was worked out by J Tare and myself,
** Here S4V stands for the &t8 symmetric power of V
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x Preface to first edition
§ 4.3 includes an example of M Nacata,
§ 4.5 is a theorem of B Kostant
In Ch 5 and 7, the whole approach to moduli via functors is due to
A, GROTHENDIECK,
§ 5.3 and 5.4 follow suggestions of A GROTHENDIECK, Ch 6 is almost entirely the work of A GROTHENDIECK
Finally a word about references: the tremendous contributions made by GROTHENDIECK to both the technique and the substance of algebraic geometry have not always been paralleled by their publication in permanently available form In particular, for many of his results, we have only the barest outlines of proofs, as presented in the Bourbaki Seminar (reprinted in [13]) Nonetheless, since all the results which { want to use have been presented in detail in seminars at Harvard and will be published before too long by GROTHENDIECK, there seems
no harm in making full use of them For the convenience of the reader,
the results which are only to’ be found in [13], and some others for which no good reference is available, are reproduced in Ch 0, § 5 We
have not reproduced, however, the results which we need from the
semi-published Seminar Notes “Séminaire géométrie algébrique, IHES, 1960-—61” since full proofs appear there The results in exposés 3 and 8 of these notes are among the most vital tools which we use, and a fami- liarity with them is essential in order to read Chapters 6 and 7
Harvard University, March, 1965 Davip Mumrorp Contents ‘ Chapter O Preliminaries 2 6 6 we ee ee ee coe ee 1 1: Definitions 2 2 2 ee 2 2 First properties 6 6 6 6 kw ee ee 4 3 Good and bad actions rn 9 4, Further properties © 2 2 0 ee ee ee ee ee 13
5 Resumé of some results of GROTHENDIECK 19
Chapter 1 Fundamental theorems for the actions of reductive groups 24 1 Definitions 2 ee ki 24 2 The affine case 2 6 6 6 ee ee 27 3 Linearization of an invertible sheaf 2 2.1 30 4 The generalcase 2 2 2 ee 36 5 Functional properties 2 1 ee ee ee eee 44 Chapter 2 Analysis of stability © 6 1 ee ee ee ee 48 1 A numerical criterion co ch 48 2 The flag complex 2 6 1 ee ee ee ee ee es Bố 3 Applications Qua 63 Chapter 3 An elementary example 6 6 1 6 oe ee 67 1 Pre-stabilitv 2 0 ee 67 2 Stability 2 ee ee 72 Chapter 4 Further examples © 6 0 6 0 ee 74 1 Binary quantics © 2 2 2 ee ee ee ee ee 76 2 Hypersurfaces 2 2 ee ee eee ee 79 3 Counter-examples 2 2 6 6 hở 83
Trang 71 2Znhen#E Erênerkbesxaekkule MT mugt xe coưentreceotecdesveog ALL Contents 4 4 AAAg 115 ¬ 115 2 Polarizations 2 Q Q 120 3 Deformations 2 2 124 Chapter 7 The method of covariants —2%¢ construction 127 1 The technique 2.2 0 UL, 127 2 Moduli as an orbit space 2 0 129 3 The covariant 2 138 4 Application tocurves 2 2 142 Appendix to Chapter 1 2000022020 02, 145 Appendix to Chapter2 0 TQ QC TQ gu uc 156 Appendix to Chapter 3 .2002020202020 , 159 Appendix to Chapter4 0 020000,., 160 Appendix to Chapter 5 171 Appendix to Chapter? 2 188 References > ee 2 206 Index of definitions © II aA 221 Chapter 0 Preliminaries
We list first some notations and conventions which we will follow: (1) A “‘pre-scheme X/S” means a morphism from the pre-scheme X to the pre-scheme S If S == Spec (R), we shall abbreviate this to “a pre-scheme X/R”
(2) An S-valued point of a pre-scheme X means a morphism from S to X If S = Spec (R), we shall abbreviate this to “‘an R-valued point” of X If, moreover, R is an algebraically closed field, such a point will be referred to as a “geometric point” of X
(3) Given pre-schemes X/k, Y/k, where k is a field fixed in some
discussion, then all morphisms /: X—> Y will be understood to be &-
morphisms; and XX will be be abbreviated to Xx Y Moreover,
k will stand for an algebraic closure of k, and we shall abbreviate
X x Spec (&) to X In this case, an algebraic pre-scheme* X/k will be called a pre-variety if X is irreducible and reduced Finally, given a R-rational point x¢X, the image point, as reduced subscheme of X ; will be denoted {x}
(4) If Z is a closed subscheme of X, or a cycle on X, then supp (Z) will denote the closed subset of X which is the support of Z
(5) The symbols A", and P,, will denote affine n-space and projective #-space over Z, (i.e Proj Z [X, , X,]) unless, in a particular chapter, all considerations are over a ground field 2, in which case they will denote affine m-space and projective n-space over k PGL (n) will be the projective group acting on P,,GL (n) the general linear group acting on A*, and G,, =GL(1) The same conventions on the base scheme hold for these group schemes ,
(6) 1, is the identity morphism from X to X ở¡ and , are the projections from X x Y to X and Y; dy, etc are the projections from Xxx YxZ to xX Y, etc If: X,; > X,andg: Y,; > Y, are S-morphisms, then /Xg: AUX Yy> XX Y, is the product If f:X-—>Y, and
* ie a scheme of finite type over Spec(f)
+
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g:X-> Y, are S-morphisms, then (/, g): X > Ÿx Y, is the induced S-morphism
(7) If X is a pre-scheme, and x ¢ X, then x(x) will denote the residue field of 9, x If X is a scheme over a field k, X, will denote the set of points xe X such that ky x(x)
§ 1 Definitions
Definition 0.1 A group pre-scheme G/S is a morphism 2:G—> S of pre-schemes, plus S-morphisms :Œ XG >G, B:G->G, e:S>G satisfying the usual identities: (a) Assoctativity: GxGxG-***“.6@xG ss S |zxe lạ + G x G——*~—+G commutes (b) Law of inverse: The compositions G—*+ 6xe 24, 6x6—4+ 6 § Bxig s
both equal eo (here 4 is the diagonal) (c) Law of identity: The compositions SxG o> Ne G Gxe——*¬€ Gx $ both equal 1z,
Definition 0.2 An algebraic group G over a field È, is a group pre- scheme G/k which is an algebraic scheme, smooth over k
Definition 0.3 A group pre-scheme G/S acts or operates on a pre- scheme X/S if an S-morphism a: cx4 -> Ä is given, such that: (a) GxGxX 722 6xX Ss S$ Ss | was |e ơ + GxxX x Ss commutes {where 4 is the group law for G) ị ị § 1 Definitions 3 (b) The composition: XœSxx-”“*.6@xx .S S x equals 1y (where e is the identity morphism for G)
Definition 0.4 Let #: T+ X be a T-valued point of X Then
ao° (igxf) is a morphism from Gx T to X Define the morphism G :GxT T
Ws GxT Xx
as (øe (lạ<ƒ),#s) If no confusion arises, yf will be shortened to y,
If f= ly, y, will be denoted % ¥ is simply: (ơ, #a): XÃ ~> XXX
The image of y, will be denoted O(/) and called the orbit of 7 Now x x f, as a scheme over T, has a canonical section, namely (/,17) Via this, we set up a fibre product defining S(f): Sứ) +T | Ì tam ‡ + Gx7-*.XxT7T Ss Ss
Now Gx T is, of course, a group pre-scheme over 7, and it is not hard
to show that S(f) is a subgroup pre-scheme over T It is called the
stabilizer of f
In case T = Spec (4), and / is a closed immersion of a point x eX into X, we shall also write O(x) for O(f), and S(x) for S(f) In case T = Spec (Q) and @ is an algebraically closed field, then { is a geo- metric point of X, and it will usually be denoted by a letter x, y, etc Then note that 0(x) is a subset of X and S(x) is a subgroup pre-scheme of G, where X,G are the geometric fibres of X,G respectively over the geometric point of S which is under /
Definition 0.5 Given an action o of G/S on X/S, a pair (Y, ¢) consisting of a pre-schéme Y/S and an S-morphism ¢: X — Y will be
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4 0 Preliminaries
ii) given any pair (Z, y) consisting of a pre-scheme Z over S, and an S-morphism y: X->Z such that yoo = yds, Le (i) holds for Z and y, then there is a unique S-morphism y: Y -> Z such that =#só Definition 0.6 Given an action o of G/S on X/S, a pair (Y, ¢) consisting of a pre-scheme Y over S and an S-morphism $: X-> Ÿ will be called a geometric quotient (of X by G) if
i) $°o = gop, (as in definition 0.5),
1) $ is surjective, and the image of Y is X xx (cf definition 0.4); [equivalently, the geometric fibres of ¢ are precisely the orbits of the geometric points of X, for geometric points over an algebraically closed field of sufficiently high transcendence degree].*
iii) ¢ is submersive, i.e a subset U CY is open if and only if $1(U) is open in X
U’ CY’ is open if and only if ¢’-1(U’) is open in X',
iv) the fundamental sheaf oy is the subsheaf of ¢, (ox) consisting of
invariant functions, ie if feI(U, oy (ox)) =P(¢7(U), ox), then
fel(U, oy) if and only if:
Gx#"1(U) —— #1(U)
R + lz
#1(0)——”
(where F is the morphism defined by f)
Definition 0.7 Given an action ¢ of G/S on X/S, a pair (Y, ¢) as above will be called a universal categorical quotient (resp universal geo- metric quotient) if, for all morphisms Y’ > Y, we put X' = X x Y’ and
A} commutes,
let $': X’-> Y’ denote py, then (Y’, 9) is a categorical quotient (resp
geometric quotient) of X’ by G If this holds only for flat morphisms
Y’ + Y, then (Y, $) will be called a uniform categorical quotient (resp uniform geometric quotient)
§ 2 First Properties
The above definitions are the basic concepts for everything that follows Their first properties will be given in this section
Proposition 0.1 Let o be an action of G/S on X/S and suppose (Y, ¢) is a geometric quotient of X by G Then (Y, ở) is a categorical
* If G and X are of finite type over S and Y respectively, this is true
for any algebraically closed field
§ 2 First Properties 5
quotient of X by G, hence it is unique up to isomorphism Moreover if (Y, 4) is a universal geometric quotient, then it is also a universal categorical quotient
Proof Suppose p : X —> Z is any S-morphism such that yoo = po Pa
as morphisms from cxx to Z To construct a morphism y: Y - Z,
let {V,} be any affine open covering of Z Then for each 7, y4(V,) is
an invariant open subset of X, hence by condition (ii) of definition 0.6, yt (Vi) = ¢4(U,) for some subset U; of Y* But then, by condition (iti) of definition 0.6, U; is necessarily open
' Now since ¢ is surjective, {U;} is an open covering of Y, and any morphism y:Y->Z such that y= yo must satisfy y(U,) CV;
Therefore it must be defined by a set of homomorphisms 4; such that
the diagram:
Tự, 9z) h T(0,, 0y)
n
là (w^(, 0x) =T (#-1(0), 9x)
commutes Since $* is injective by condition (iv) of definition 0.6, Z, is also uniquely determined — if it exists — and hence at most one x exists But for any geI°(V;, 92), one checks that p*(g) is an invariant element of '(¢-1(U;), ox) in the sense of condition (iv): hence it is in the sub-ring ¢*[I°(U;, oy)] Therefore such an h; does exist
This 4; defines y,: U; > V; It remains only to check that y; = y; on UNA U; and this is immediate This constructs y QED
In the-rest of this section, we shall analyze informally the various concepts of quotients, in a series of remarks Therefore, we fix the nota-
tions S,G, X, Y,o, ¢ as in Definitions 0.5, 0.6, and 0.7
(1) S plays no essential role in any of these definitions That is to say, we can replace S by Y, and G by the Y-group pre-scheme Gx Y if we wish
(2) Suppose (Y, ¢) is a categorical quotient of X by G Then, by means of the universal mapping property, it is easy to check the follow- ing implications: + x reduced = Y reduced X connected => Y connected X irreducible => Y irreducible
& locally integral => Y locally integral X locally integral = Y locally integral
and normal and normal
* In fact, by condition (ii), one proves easily that, if z, y are two points of X, then $% = ¢y implies yx = py
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On the other hand, it does not seem likely that if X is noetherian, then Y is noetherian, or even locally noetherian in all cases (this is an analog of Hilbert’s 14th problem) I only know one fact in this direction:
if S = Spec (4), & a field, X is a normal algebraic scheme over #, and
(Y, 4) is a geometric quotient, then Y is an algebraic scheme over R Since we have no use for this, I omit the proof
(3) Suppose (Y, $) is a universal categorical quotient Applying this assumption to the base extensions given by the inclusion of open sub- sets U in Y, one deduces condition (iv) of definition 0.6 Moreover, applying it to the inclusion of a single point y in Y, and noting that the categorical quotient of an empty scheme by any group is an empty scheme, one deduces that ¢ is surjective Therefore, for (Y, ¢) to be a universal geometric quotient, it is necessary and sufficient that (1) ¢ is universally submersive, and (2) the image of V is X xx
(4) In this remark we shall assume that all schemes are noetherian, all morphisms of finite type, and that the base S$ is even normal A very useful hypothesis is that G is universally open over S By Che- valley's criterion (EGA Ch 4,14.4)* this is equivalent to assuming either that G is open over S, or that all the group schemes which occur as fibres of G over S have the same dimension This implies, for example, that if (Y, ¢) is a geometric quotient of X by G, ¢ is also universally open:
To see that ¢ is open, let U C X be any open set But fy: Gxx> x is open since G is universally open over S Now a can be factored:
Gxx S - “th 6xx =®— X
and (f;, ơ) is an isomorphism Therefore ơ is open In particular o(G x U) = U' is open But $(U) = ¢(U’), and since $(U’) is a subset of X invariant under G, U’ == ¢-1(¢(U’)) Since ¢ is submersive, this proves that ¢(U’), hence #(U) is open The same argument proves that ¢ is even universally open
Another consequence concerns dimensions Suppose ¢: X > Y is a
dominating S-morphism such that
i) sơ =ó- be
uu) for every algebraically closed field k, the geometric fibres of ở over & contain at most one orbit under G = G x Spec (3)
* The abreviation SGA will always refer to [12], and the number will always be the exposé referred to Similarly the abreviation EGA will always refer to (11], and the number will then be the chapter referred to
§ 2 First Properties , 7 For all x € X, let
o (x) = dimension of stabilizer of x,
+ (+) = dimension of fibre ¢-1(¢(x)) at x ,
By standard theorems on upper semi-continuity of dimensions (applied
to the stabilizer S(1,) over S, and to $) ø and r are upper semi-con-
tinuous (cf EGA 4, 13.1) On the other hand, if g is the dimension of all the group schemes occurring as fibres in G over S, then by (ii)
a(x) -+t(*)=¢g
for all x Therefore @ and + are both constant on all topological com- ponents of X
Using this remark, we obtain the important criterion:
Proposition 0.2 Suppose X and Y are irreducible, normal noetherian pre-schemes over S, and suppose ¢: X—> Y is a dominating S-morphism of finite type Suppose that the residue field of the generic point of Y has characteristic O Suppose a group pre-scheme G, of finite type and universally open over S, acts on X via o:GxXx —> X Then if
i) Goo = po py,
ii) for every algebraically closed field k, the geometric fibres of ở over & contain at most one orbit under G = Gx Spec (5),
it follows that ¢ is a universally open morphism, and (¢X, ¢) is a geo- metric quotient of X by G
Proof First of all, since G is universally open over S, all the group schemes G have the same dimension Therefore, by ii) all components
of all geometric fibres of ¢ have the same dimension; hence ¢ is uni-
versally open by Chevalley’s criterion To prove the second statement, we need only verify condition (iv) of Definition 0.6 Therefore let U C $(X) be an open set Since ¢ is dominating and Y.is reduced, it is clear that:
TU, sy) —~T'(#1(U), sx) Py!
Now let f be an invariant section of Ệ ÿ F
of gx over ¢-1(U): we must show t fee =
that f is a section of gy Let f gt? eye Axy—=> TM?
define the morphism: ø ø
#: ¿1(U) — AI 2
: U
Let U’ be the reduced and irre- Fig.1 ducible subscheme of Ax U whose
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constructible subset): for the geometric fibres of ¢ already contain only one orbit each, and an invariant function F cannot split up these fibres any further In particular, since the generic characteristic is 0, Ø is birational But secondly, suppose we form the fibre product ¢1(U) x U" (cf Fig 1) Since @ is a separated morphism, #, : ¢~4(U) x U —> ¢71(U) is separated The morphism ¢’ from ¢-!(U) to U’ induces a section o of #,: by EGA 1, 5.4.6, its image, of¢1(U)], is closed But pe: F1(U) x U'-» U’ is obtained from ¢ by base extension Therefore it is open; in particular
% {91(U) xỮ' — ø[#^1(0)1)
is open in U’ But the geometric points y’ of this set are those such that there is a geometric point x of ¢-1(U) and (a) ¢’ (x) y’, (b) $(x) = G(y’) By our first remark, all geometric points x of ¢-1(U) for which $(#) = G(y’) are mapped by ¢’ to the same geometric point ¢' (x) of U*: therefore such a y’ is not in the image of ¢’ Since this image is dense, there are no such y’s Therefore o '¢71(U)] = $7! (U) x U, and ¢’ is surjective Therefore @ is geometrically injective without restriction, and by Zariski’s Main Theorem, @ is an isomorphism QED
Remark (5) The following are equivalent:
(a) (Y, ở) is a universal categorical quotient of X by G,
{b) for all affine schemes Y’, and morphisms Y’ -—> Ÿ, if $': X” —> Y7 is the base extension, then (Y’, ¢’) is a categorical quotient of X’ by G,
(c) there is an affine open covering {U;} of ¥ such that if ¢;: 6-#(Uj)
—> U;, is the restriction of ¢, then (U;,¢,) is a universal categorical
quotient of ¢-1(U;) by G, for all z,
(6) In Chapter 1, we will need a criterion for (Y, ¢) to be a categorical quotient which does not imply that (Y, ¢) is a geometric quotient In fact, the following 3 conditions suffice:
i) doo= bo fy,
ii) gy is the subsheaf of invariants of $, (ox),
iii) if W is an invariant closed subset of X, then ¢(JV) is closed in
Y; if W;, te J, is a set of invariant closed subsets of X, then:
sịn W) =n +
Moreover, if these conditions hold, ¢ is submersive
Proof Note first of all that ¢ is dominating by (ii); hence by (iii) ý is actually surjective Now suppose : XZ is any S-morphism such that poo = po , We proceed exactly as in the proof of Pro- position 0.1: choose an affine open covering {V7} of Z Once we show that there is an open covering {U;} of Y such that y1(V;) >Â2(U), _
Đ 3 Good and bad actions 9
we can conclude the proof as in Proposition 0.1 To show this, let W,= X — w1(V;) These are closed invariant subsets of X Therefore, by (iii), U;== Y — $W, is open; and ¢2(U,) C y}(V;) for all i Finally, {y1(V;)} cover X, hence n W; =9, hence by (iii) A oW,;=, hence {U;} cover Y Finally, let ZC Y be any subset such that ¢-1(Z) is closed in X Then ¢-1(Z) is invariant by (i), hence by (iii) $(¢7(Z)) is closed in Y But ¢ is surjective, so Z = $(¢7(Z)) This proves that ở is submersive QED
(7) Suppose (Y, ¢) is a geometric quotient of X by G Notice that conditions (i) and (ii) of Definition 0.6 are preserved by any base ex-
tension Y’-> Y But the more delicate condition (iv) might (and
indeed sometimes does) break down However, if Y’—> Y is flat, then condition (iv) is preserved To see this, let p = ¢° o = ¢° f, Then (iv) asserts the exactness of: o*~ pF (® 0 —> dy —> $¿ (9x) Ya 9 exx) But taking direct image sheaves commutes with flat base extension, i.e $,, (0x) @ Oy & $„ (0x) 0x xì ® Oy 0„(9sxx' Ya exx) Y Pa xx")
where X’ == X x Y’, etcN, and the exactness of a sequence is preserved under flat base extension — hence the analogous sequence (*)’ on Y’ is exact
(8) On the other hand, consider the converse I claim that if Y’ +> Y is faithfully flat and quasi-compact, and if the extended morphism
$': X”-> Y’ makes (Y’, ¢’) a universal categorical quotient or geometric
quotient or universal geometric quotient of X’ by G, then (Y, ¢) is the same type of quotient of X by G These assertions are all simple corollaries of the general theory of descent of SGA 8
§ 3 Good and bad actions
We shall now look somewhat more closely at the structure of an action of a group As above, let G/S act via o on X/S
Definition 0.8 The action o is said to be
i) closed if for all geometric points x of X, the orbit 0(x) C X is
closed, (Le X = X XSpec 2 if x is an Q-valued point),
li) separated if the image of
Ww = (0, b2) | GXX—> XXX
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10 0 Preliminaries is closed,
ili) proper if \W is proper,
iv) free if W is a closed immersion
Moreover, let ý = S(1x) be the stabilizer of 1y: X-> X (cf Defini-
tion 0.4) It is a sub-group pre-scheme of the group pre-scheme Gx X/X Let œ:đ—> X be the projection and let ex: X->& be the identity section Let o(x) be the dimension at ex (x) of w-1(x), as a scheme over Spec (x), for all points + e X If X is noetherian and G is of finite type
over S, this is finite and upper semi-continuous (cf EGA 4, 13.1; com-
pare remark 4, § 2) moe this, we make:
Definition 0.9 S,(X) = {xe X |o(x) >1} By the above, this is closed Moreover, say x '§ regular for the action of G if o is constant in some neighborhood of x
The set of regular points of X for the action of G forms an open set X™é in X According to remark 4 of § 2, if G is of finite type and uni- versally open over S and if a geometric quotient (Y, ¢) of X by G exists such that Y is noetherian and ¢ is of finite type, then every point of X must be regular for the action: X™ = X
A useful observation is that if G is of finite type over S and X** = X, then the action of G is closed — even if X is not of finite type over S To prove this, it suffices to take S = Spec (k), & algebraically closed; we may assume that G is irreducible too Note that if xe X,, then 0(x) is irreducible, and if y is its generic point, then ~(y) is a subfield of &(G),
the function field of G Moreover, it is not hard to verify that:
tr.d.x(y)/k + o(y) = dimG
Now if the action were not closed, at least over some & there would be two points +, x,¢ X, such that 0(x;) C 0() — 0x) Let y; be the generic point of 0(%,) Since x, is a regular point of the action, one finds that o(x,) = o(x,), hence
tr.d (y,)/k = tr.d x(yq)/k <0 But y, is a specialization of y,, hence this is absurd
Concerning restrictive conditions on the action of G, a notion inter-
mediate between the closedness of one orbit and the properness of the whole action is sometimes convenient:
Lemma 0.3 Let S = Spec (A), & an algebraically closed field, and assume X and G are algebraic pre-schemes over & Let o be an action of G on X, and let « e X; Then y, is proper if and only if 0(x) is closed in X and S(x) is proper over &
Proof If y, is proper, then its image 0(x) must be closed, and its fibre over x — which is S(x) — must be proper over & On the other hand, assume that 0(x) is closed and S(x) is proper over & To prove
§ 3 Good and bad actions 11 that „ is proper, we may as well replace G ‘by G,.q; then let Z be the teduced closed subscheme of X with support 0(x) Then G and Z are
homogeneous reduced pre-schemes, hence they are non-singular If we
define Ve to be py as a morphism from G to Z, then it suffices to prove that y;, is proper But ; is flat — since G and Z are non-singular and all components of all fibres of p, have the same dimension, i.e dim S(x)/k (cf EGA 4, 15.4) Therefore, to prove y, proper we may make a base extension by y, itself; i.e it suffices to prove that
Ww, = pe! CXG—>G
is proper (cb SGA 8, Cor 4.7) But let yu’ be the restriction of the group law in G to a morphism from S (x) XG to G Then (w’, p2)z is a morphism from S(x)xG to Gx G One checks formally that it is an isomorphism Therefore it suffices to prove that:
ys ° (w’, bez: S(x)XE>G
is proper But wy o (u', p2)z is the second projection p,: S(x)xG—>G This is proper since S(x) is proper over k QED
From the proof of this lemma, it is clear that one can play various tricks with group actions to obtain implications that are not entirely obvious However, the (global) properness of an action, i.e the properness of WY, is subtler than the properness of y,, for one x To illustrate this, we give an example and a ie which suggest opposite conclusions (the example shattered over-optimistic conjectures that the author had entertained !)
Example 0.4 Of a semi-simple group G over C, the complex numbers, acting on a non-singular quasi-affine scheme of finite type over C, such that:
i) all stabilizers are reduced to ¢ itself, i.e the action is set-theoreti-
cally free,
li) a geometric quotient exists, which is also a scheme of finite type over C; hence (cf § 4) the action is separated,
iii) the action is not proper, and, in particular, not -free (algebro- geometrically) t
Proof Take G = SL(2) Let V, stand for the (ø + 1)-dimensional affine space whose closed points are homogeneous forms in X and Y of degree » over C: then SZ (2) acts on V, by means of substitutions in X and Y Define a reduced subscheme X C V;xV, by means of
(Fy Fy eXee (a) i #0
(b) F, is the square of a homogeneous quadratic form of discriminant 1 One checks that X is non-singular, invariant under SZ (2), and that no
a“ +
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12 0 Preliminaries
non-trivial ô Â SL (2)Â leaves fixed any point of X Define ¢: X—> Al by b(aX + BY, Fy(X, Y)) = F,(—8, x)
Then it is easy to check that (At, ¢) is a geometric quotient of X by SZ (2) Finally, the morphism ¥ is not even closed Let Z be the closed subscheme of Gx X whose closed points are the pairs:
— 1—i
(; ; ) xax+y, xevy
for 4eC — (0) The image under ¥ of the above point is:
(—AX + Y, X2Y*x(ÂX + Y, X?Y?), In the closure of this set is the extra point:
(Y, X?¥2)x(¥, X2Y2),
Lemma 0.5 Let o be an action of G,, 0 an algebraic scheme X,,
all over a field k Assume that X admits an immersion in projective space P,, for which the action o extends to an action of G,, on P, Then:
@ is proper © (i) o is separated, (1) S;(X) =o
Proof The implication = is clear Conversely, assume a is separated and S;(X) = To prove that the morphism Y is proper, we use the valuative criterion (EGA 2, § 7) Let R be a valuation ring over k, and let K be its quotient field We may assume that R contains an algebra- ically closed overfield of k, Q, which is isomorphic to its residue field
(Remark 7.3.9, EGA 2) Now suppose $xé is a K-valued but not R
valued point of Gx X such that (¿x£) is an R-valued point of Xx X
But ¥(bxé) = 0(6, 8) xé, ;
hence a(¢, #) and & are R-valued, but ¢ is not Let a(¢, ‡) and £ be the
induced Q-valued points of X obtained via the inclusion: Spec (Q) —> Spec (R) Since a is separated, ơ(ở, £} <£ is in the image of ¥, hence there is an Q-valued point $ of G such that:
o($, 4) =o(hy, 8
Since R contains £2, dy can be lifted to an R-valued point $ of G; replac- ing ¢ by $5 1+, we may assume from the beginning that ơ(, £) and £
induce the same Q-valued point € of X (more classically: specialize to the same geometric point of X)
Now let I: X-> P, be a G,,-linear immersion for a suitable action
of G,, on P, We require the following:
(*) If G,, acts on P, over the field 2, then for every point x P,,
there is an invariant affine subspace U containing x (i.e of the form,
§ 4 Further properties 13
P,, minus a hyperplane), and there are coordinates #; ,3„ on U, such that the action ¢:G,xU-—>U is defined by
o* (x) = a Ấy l<i<n
for suitable integers 7,, (identifying G,, with Spec Q[x, «*)), (For proof
of (*), cf [8])
We apply (*) to obtain a neighborhood U of I() where the action
has this form Since J(€) is a point of U , So are I(¢) and I(o(¢, £)) Using
the assumption S,(X) = 9, it follows that 7(€) is not left fixed by G,;
hence for some 7, the 7th coordinate I(£); is not zero and 7, % 0 Now’
¢ is a morphism:
Spec K > G,, == Spec & [x, on],
Suppose the function « on G,, induces A « K Then the ith coordinates of I(é) and I (o(¢, §)) =ơ (6, (é)) are related by:
1Íø(9, 8); = A* - 1);
Here 7(); and 7(ơ(ó, &)); are elements of R whose reductions in the tesidue field Q of R are both I (@); Since 7; 5£ 0, it follows that A must be a unit in R But then ¢ factors through Spec R, i.e ¢ is actually an R-valued point of G,, QED
§ 4 Further properties
In this section we wish to relate Yhe Properties of an action o of G/S on X/S to the Properties of the geometric quotient (Ÿ, $), assuming that it exists We fix these notations for the whole of this section
Lemma 0.6 If a geometric quotient (Y, ¢) exists at all, the action o is closed In this case, Y is a scheme over S if and only if vis separated Proof Let x be a geometric point of X over an algebraically closed
field Q If X= X x Spec (Q), Y = ¥xSpec (Q) and Â:XƠ+Ơ is
induced by Â, then
| 06)=j-'2(6)) — +
Therefore 0(x) is closed, since # is continuous, and an Q-rational point
(such as $(x)) of any pre-scheme over Q is a closed point,
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where A is the diagonal morphism, and this diagram makes X xx the
fibre product of Y and X XX over YX Y Moreover, $x ¢ is submersive
since ¢ is universally submersive Therefore
|4 closed ín Yx Y]œ êm (4(Ý)) closed in X xx]
= [*x*x closed in XxX] Y Ss
This proves that Y is an S-scheme if and only if o is separated QED A more subtle point is:
Proposition 0.7 Let S = Spec (f), ka field, and assume that G, X, and Y are algebraic schemes over k Assume that G is affine, and that the action o is proper Then ¢ is affine
Proof To prove this, suppose first of all that #hasasections: Y > X Then consider the following diagram, where the top triangle is obtained from the bottom by the base extension s: Wo t@xs (tx,sé) $ Gx X— eX XX Ps Ps xX
Since the action o is proper, the morphism ¥ is proper Therefore t is proper But £, is an affine morphism since G is affine Therefore by Chevalley’s Theorem (EGA 2, Theorem 6.7.1), ¢ is affine
In general, we wish to reduce to the case where ¢ has a section by
considering a base extension 2: Y’-» Y Suppose that after such an extension, the new morphism ¢’: X x Y’ — Y’ is affine In some cases,
this allows us to conclude that ¢ was affine This is so if x is finite and
surjective, again by Chevalley’s Theorem Since being affine is a local property on Y, it is also true if {U;} is an open covering of Y, and z:uU;-> Y is given by the inclusion morphisms Therefore, by Remark 4, § 2, the Proposition will follow from:
Lemma Let ¢: X— Y be a universally open surjective morphism of finite type of algebraic schemes Then the extended morphism $:X x Y’-» ¥’ will have a section after a suitable base extension zœ: Y ~> Ý where x is a composition of
i) finite surjective morphisms,
ii) the union of inclusion morphisms v U;—> Y, where {U;} is an open covering of Y
§ 4 Further properties 15
Proof As a first base extension, let ,: :Y¥,> Y be the canonical morphism from the normalization of Yreqa to Ÿ Let X, = X x Y, and let $,: X,-> Y, be the extended morphism Since ¢ is open, the i image of every component of A, is a component of Y,, and the fibres of ¢, in this component all have the same dimension (cf EGA 4, 14.2) Now for every closed point y ¢ Y,, let x be a closed point of X, over y First
of all, there exists a reduced and irreducible closed subscheme H C Xy
such that
i) dim H = dim Yj,
ii) x is an isolated point of Hn $r(y)
Such a subscheme can be constructed because the fibres of ¢, have
the same dimension, e.g., as the set of zeroes of f,, , /,€ Qz,x,, Where n== dim, ¢74(y), and ƒ¿, restricted to 9, yy generate an ideal primary
to the maximal ideal Then the function field 2(H) is a finite algebraic extension of &(Y,) (or of &(Y,), where Y, is the component of Y, con- taining y) Let L > &(H) be a further finite algebraic extension which is purely inseparable over a Galois extension of &(Y,) Let Y’ be the normalization of Y, in the field Z (or of that component of Y, containing y) Then J claim that Xx x Y’ has a section over Y’ locally i in a neigh-
borhood of each y’ « Y’ which lies over ye Y But since all y’ over y
are conjugate, it suffices to prove this for one such y’ Now consider the rational map Y'— H induced by L A({H) I claim that it is a morphism in some neighborhood U’ of some over y When composed with the inclusion morphism of H in Xj, this will define a section of Ay x Y’ over U’ But let H’ be the normalization of H in L Then since
H > Y, isa morphism, the birational map of mm
the normalizations H’—Y’ is a morphism Let x’ be a point of H’ over x « H Then since x is isolated in its fibre for the morphism H—> Y, (assumption ii), it follows that x’ is isolated in
its-fibre for the morphism H’—> Y’ Therefore,
by Zariski’s Main Theorem, an open set con- y taining x’ is isomorphic to its image U’ C 1” The rational map from Y’ to H must be a morphism on this U' This proves that X, x 1
NGỘ
Ji |
7 +-— 41
has sections over Y’ in a neighborhood of cách Fig 2 y’e Y' over ye Y,, (compare Fig 2)
Therefore there is an open set U,C Y, containing y such that if
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16 0 Preliminaries
of V,’s over these U,’s: this is a Y3 finite over Y, Finally, we replace Y, by an open covering Y,, such that in each component of Y,, a global section of XV 4 over Y, is defined QED
This result’ can be generalized with some more technique to more general base schemes S We omit this More interesting for the sequel is that, using the methods of chapters 1 and 2, we can prove that the converse of Proposition 0.7 is true in one case:
Proposition 0.8 Let S = Spec (2), & a field, and assume that G, X,
and Y are algebraic schemes over & Assume that G is a reductive algebraic group (cf § 1.1) and that the stabilizers of geometric points
of X under the action of G are finite Then if ¢ is affine, the action o
is proper
Proof Since W: GŒxX-> XxX isa Y-morphism, it suffices to prove that it is proper over every set U; of an affine open covering {U,} of Y
Therefore assume that Y is affine Then X is affine, and by Definiti.n 1.7:
X = Xi) (0x)
Therefore G acts properly on X by Corollary 2.5 QED
Putting these two results together, it follows that if G is reductive
and the stabilizers are finite, @ being affine is, in essence, a weak topo-
logical restiction, i.e it is equivalent to the properness of o But if the stabilizers are positive dimensional, ¢ being affine is a much stronger
restriction At least in characteristic 0, it implies that these stabilizers
are themselves reductive groups (cf [37])
The final “preliminary”' result is aimed at further motivating the concept of a geometric quotient by showing that, for a free action,
it implies all that one might hope for
Definition 0.10 Let (Y, $) be a geometric quotient of X bv G (over S) Assume that G is flat and of finite type over S Then X is a principal
fibre bundle over Y, with group G, if
i) ¢ is a flat morphism of finite type, ii) YW is an isomorphism of GxXxX and X xÃ
Proposition 0.9 Let S == Spec (k), & a field, assume that G is an algebraic group, and that X and Y are algebraic schemes over & If the action g is free, and (Y, ¢) is a geometric quotient of X by G, then X is a principal fibre bundle over Y with group G.*
Proof First of all, we may assume that & is algebraically closed: for by such a base extension, i) o remains free, ii) (Y, ¢) remains a geo- metric quotient (Remark 7, § 2), iii) the property of being a principal
* The assumption that Y is algebraic is not neccssary If g:X > Yis any geometric quotient, then X algebraic > Y algebraic (see [102])
§ 4, Further properties 1? fibre bundle descends (SGA 8, 5.4) Secondly, we may replace Y by one of its local rings Spec (g,,y), for some closed point y« Y: for all conditions of Definition 0.10 are expressible locally And thirdly, we may replace Spec (g,,y) by Spec (0,,y) (its completion) For this is also a faithfully flat quasi-compact base extension and, as before, i) ¢ remains free, ii) (Y, ¢) remains a geometric quotient, and iii) the property of being a principal fibre bundle descends
Therefore we have Y = Spec (A), where A is a complete local ring with algebraically closed residue field Let y be the closed point of Y, and let x be a closed, hence f-rational point of X over y Let Ï; CC 0xx be the ideal defining the orbit 0(x) of x Since o is free, the morphism y, defines an isomorphism:
9z,xÍ1„ > 9œ
Let g= dimG, and let í, ,/¿e m„eø be a basis of this maximal
iđeal Let z;, ,„ernt,x be elements whose reductions mod J, are
foecoh, respectively Then the ideal (*,, ,*,) C a: defines a germ ofa subscheme of X, at the point x Since this germ intersects ¢7}(y) in the isolated point x, it follows from Proposition 5.5.1, EGA 3, that there is a unique connected closed subscheme H in X extending this
germ, and that H is finite over Y
Consider Y as an X-morphism from GxXX to X xx Then by the
base extension H > X, we define `X
\J⁄':GxH-xX xH :
Since o is free, ¥ is a closed immersion, hence ¥” is a closed immersion
Therefore the morphism YW” = ~,° Y’ from GXH to X is a finite morphism I claim that ¥’’ is an isomorphism Note first of all that this will prove the Proposition For, by the definition of ¥”, if we let G act on GXH by the product of left multiplication on G, and the identity on H, then Y”’ is a G-linear morphism But (H, ;) is a geometric quotient of GXH by G By the uniqueness of geometric quotients, there is a commutative diagram: ~ ` GxH~.,* | ® | $ + + H ~ Y
Since f, is flat, ¢ is flat; and since Gx (GX H) is isomorphic to (Gx H)
x(GX#), it follows that GxX is isomorphic to X xX
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The union of the supports of these two coherent sheaves is a closed
subset Z in X Since 7“ commutes with the action of G, it follows that
Z is invariant under G But Y is a topological quotient space of X under G, since ¢ is submersive Therefore Z == ¢~1(Z,) for some closed subset Z,
in Y Therefore, if Z 54 ®, it follows that yeZ,, hence x «Z
We now prove that ô Â Z Note first that (Y”’)-1(x) consists of the single point (e, x) eGXxH For if («, x’) is a geometric point in (W)-! (x) ~~ possibly over a larger algebraically closed field 2k — then G(x, %’) =x Therefore $(x’) == $(x) = y, and xzeÖHAd 1(y) Thịs
implies that x’ = x by definition of H; and since the action o is free,
this implies that « =e Therefore the stalk of YY (og,x) at x is the
local ring of o¢xy at (e,x) Now since this stalk is finite over 0, x,
to show that it is isomorphic to 9, y it suffices to show that the complete local rings:
9.,x and 2.68 Qn,H
are isomorphic We now rephrase this whole situation in a lemma on local rings as follows, where & is dual to the group action o, 9 = 0, x,
ACO - +> fel] = nq and the f; and x; are the same as above:
Iemma Let o be a complete local ring, mC 9 its maximal ideal, and kCo a subfield such that &=¥ o/m Let o@ be a local homomor- phism from o to the formal power series ring o[{[f,, ,f,]] such that: ~~" i) The composition p om Os wo en fg] > ott Le + so fel] is surjective, ii) the _— oS ols os fel] + Stel ow is the identity
Let x; be elements of 9 such that a = };, Then the composition yp”
s>ð0o[Ùh :!;l> Cys fel]
©
„1s an isomorphism
Proof It follows easily that p’’ is surjective since the local rings are complete To prove that p’”’ is injective, it suffices to prove by induction
that if p’ (x)= 0, then we (%, ,x,)"+' Suppose ÿ“(x) = 0 and awe (%, ,%,)" Write = v a=, c4xt in symbolic notation A = (q4, ,4,), |A| = ~ a, But by (ii) and (i): G(x) ex; +4,+ sim (fp /g] § 56 Resumé of some results of GROTHENDIECK 19 hence ; a § ?” (x,) = = ashe, det (4,3) él + m g ca : Therefore woe) LỮ ., ƒ;]] is isomorphic to 2 lu sân zy (Gin: Now since
Os yas ate y (ca) (9,
it follows that ” (cy) e (p(x), oF Ge ) Therefore by (ii), the
original coefficients 4 are in (%, ,%,) Therefore ô Â (x,, , %,)"*" QED
§ 5 Resumé of some results of Grothendieck
In this section, we want to list some definitions and theorems which
are not specifically related to group actions, but for which there is no satisfactory reference available
a) The first topic is a theorem in EGA 3, §7, which is extremely
useful, but which is unfortunately buried there in a mass of general- izations* Let
xy
be a proper morphism of noetherian schemes Let F be a\coherent sheaf on X, flat over Y If ye Y, let X, (resp #,) denote the fibre of ƒ over y (resp the sheaf induced by ¥ on the fibre) Assume that for all ye Y, HUX,, Fy) = (0)
Then /,(#) is a locally free sheaf on Y, and ‘‘the formation of /, com- mutes with base extension’, i.e in all fibre product situations: x'.x rl [s + + y’ > Y the natural homomorphism: £” 0#) —> Íy (s” (2) is an isomorphism In particular, if Y” = Spec ⁄(y), for a point ye Ÿ, then: he F @ u(y) H9(X„, 2,)
This result follows immediately from Theorem 7.7.5 and Proposition 2.7.10 applied when 2 = 1, and Proposition 7.8.4 applied when ý = 0 (all in EGA 3)
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20 : 0 Preliminaries
b) The second topic is the functorial interpretation of PGL(n) As ị
usual, PGL(n + 1) is the open subset of Proj Z[agq, ., @a_] where ị det (a;;) 40 It is a group scheme over Spec (Z) and it acts on P,
over Spec (Z) in the usual way Now let PGL(n + 1) denote the functor defined by PGL(n + 1) in the category of noetherian schemes, i.e
Zớ ( + 1) (S) = Hom (S, PGL(n + 1))
Let Aut (P,) denote the functor which assigns to every S the group of automorphisms « of P,xS over S (i.e such that 4,0 = #,) The action of PGL(n + 1) on P, defines a morphism of functors:
PY L(n + 1) — Aut (P,) ` |
(*) This ts’ an isomorphism
Proof Most of the proof can be found in EGA 2, § 4.2 The key point which is not proven there is that when S is connected, every invertible sheaf on P, x S is isomorphic to
OF (04) @ AF (L)
for some integer k, and some invertible sheaf Z on S A proof of this (based on (a) above) can be found in [40], Lecture 13 (with trivia modifications) It follows that if
P,xS—— P.xs
Nh
is any S-morphism, then:
o™ [pt (0(1))] & pF (0 (A) @ pe (L) If @ is an isomorphism, then Pe, [PT (9(1))] and 2z,x [#Ÿ (ø )) @ AF (Z)] must.be isomorphic sheaves on S But + Pox BT (2(E)) @ £2 (Z)] = (0), if #< 0 * IR L, ifk=0 IR Pyfbeetraeak @ (Ag Xp)-L, >0, Therefore, k = 1; and by EGA 2, § 4.2, the group of automorphisms « is isomorphic to the group of
a) invertible sheaves L on S, plus b) isomorphisms of * @ X;-0f and © X,-L oo =O | tm
§ 5 Resumé of some results of GRoTHENDIECK 21
But such an isomorphism is given by an (a+ 1)X(n + 1) matrix of sections a, of L (if X; is mapped to < X; «), provided that đet (2y), as a section of L"*1, is nowhere zero But a unique S-valued point
S 4+ PGL(n + 1)
of PGL(n + 1) is defined by the conditions:
f*(o(1)) =L
i (a;;) = Ose
and one checks that this and only this / induces the automorphism «, c) The third topic is the definition and existence theorems connected with the Hilbert scheme Various details can be found in [13], exposé 221,
in SERRE’s talk at the International Congress in Stockholm, and in my
Notes [40].* For all locally noetherian schemes S, let
Hilbp,(S) = {Set of closed subschemes ZC P,xS, flat over S}
intoitivety {Set of families of subschemes of P,,, parametrized by S} This is a contravariant functor from the category of such S to the category of sets The fundamental existence theorem states that it is tepresentable, i.e there is a locally noetherian scheme Hilbp, and iso- morphisms Wilby, (S) s+ Hom (S, H bp) N
one for each S, which are functorial in S
Equivalently, this means that there is a closed subscheme
W CP,XHilbp,
flat over Hilbp,, which is “universal”, ie given any ZC P,xS, flat over S, there is a unique morphism /:S— Hilbp, such that
Z = (1z,x/*(1)
Xow given Z C P„xS, flat over S, for all s e S, let 2:C€ P„xSpec x(s) be the induced subscheme over s Then each 4, has a Hilbert polynomial
Po be P,(n) = x (02, (2)
Then if S is connected, all the polynomials P, are equal Therefore the universal family breaks up via Hilbert polynomials: i.e
Hilbp = IT Hủb?
polynomials " Pin)
Trang 18-22 0 Preliminaries
The strong form of the fundamental existence theorem states that
Hilbp, is projective over Spec (Z)
As corollaries of this theorem, one checks that many other functors
are representable:
i) Hilbp, represents
S— {Set of closed subschemes ZC P,,xS, flat over S, such that every
Z, has Hilbert polynomial P}
ii) If WC P,,xHilbg, is the universal subscheme then kX —————— HP, see rŠ P W represents S—> {Set of closed subscheme Z C P,,XS, as in (i), plus k sections of Z over S}
iii) If X is a closed subscheme of P,xT, then a suitable closed subscheme of Hilbp, xT, called Hilby;7, represents the functor:
S— {Set of closed subschemes ZC X XS, flat over S}
(in the category of locally noetherian T-schemes)
iv) If X and Y are closed subschemes of P,XT, and X is flat over T,
then a suitable subscheme of Hilby, yir, called Hom7(X, Y), represents T
the functor:
S~» {Set of S-morphisms from X xs to YxS}
d) The fourth topic is the definition and existence theorems con- cerned with the Picard Scheme Various details can also be found in [13], exposés 232 and 236, SERRE's talk, and [40}* Start with a locally noetherian base scheme 7, and a flat, projective morphism
aiX—>T
whose geometric fibres are varieties For all locally noetherian T-
schemes /: S > T, let
if {group of invertible sheaves L on X x S} (Piexir (5) = {subgroup of sheaves of the form p¥(K), for Kon 5} If X/T has a section ¢: T—> X, then one checks immediately:
group of isomorphism classes of invertible
Diex)r(S) < sheaves L on X XS, plus isomorphisms
(9 ef, 15)*(Z) & 05
* Complete proofs, plus the construction of the bigger scheme Picy;p pro- jective locally over T containing Pic'y,r as an open set, have been given by
ALTMAN and KLEIMAN [42] coe aes 4 lÀn9/4/+90/013T888892//120/ 9440 30202190200088 i 5 Ễ Ệ : :
§ 5 Resumé of some results of GROTHENDIECK 23
This is what GROTHENDIECK calls “trivializing” or “normalizing” a
family of invertible sheaves along the section o
In any case, ‘Diex;r is a functor from the category of locally noe- therian T-schemes to the category of abelian groups The fundamental existence theorem states that under the hypotheses above {including the existence of o), there is a group scheme over T, called Picxyr, representing Diex)r In case X is still flat and projective over T with
varieties as geometric fibres, but no‘section o exists, the fundamental
theorem states only: there is a unique locally noetherian group scheme over T, called Picy,7, and a homomorphism ¢ of functors in S:
Piexp(S) + Home (S, Piexm)
such that
a) $(S) is injective for all S,
b) $(S) is surjective when X xS admits a section over S Just like Hilb, Pic can be broken up via Hilbert polynomials: fix an invertible sheaf 9(1) on X, relatively ample for x For all invertible
sheaves L on X XS, and all seS, let L, be the sheaf induced on
x x Spec x(s), and let
P,(n) = x(L,(n))
Then P, is constant, for s in each component of S, and Picy;z is the
disjoint union of open and closed subsets Pic); where each piece represents sheaves Z with a given Hilbert polynomial And the strong form of the fundamental existence theorem, that Pic%,; is a quasi- projective scheme over T, appears to be true
More important than Pick), is an open subgroup scheme PtXm More generally start with any group scheme G, locally of finite type over J For all geometric points:
Spec 2 +, I,
let G; be the induced group scheme over Q Then let G? be the component of G; containing the identity, and let G7 be the union of the components
of finite order, i.e consisting of points x some multiple of which is inG?
Then there is a (unique) open subgroup scheme G’ C G whose intersection
with each G; is Gj The key results are:
(A) Pick;r is quasi-projective over T
(B) If X is smooth over T, then Pic’y,p is projective over T Finally, when T = Spec (Q), what is the connection between Picy;o and the classical Picard variety of X?
(C) The Picard variety of X is the reduced part of the connected component of Pic%j9 containing the identity And P¿c%;ø is reduced if
“—
_and only if _ :
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24 J Fundamental theorems for the actions of reductive groups
e) The last topic is the concept of a relative Cartier divisor For details, cf EGA IV, 21.15 and [40], Lecture 10 Suppose {:X + Y is a flat morphism and D < X is an effective Cartier divisor Then the following are equivalent:
i) D is flat over Y,
1) for all ye Y, the induced subscheme D, C X, is a Cartier divisor Such a divisor is called a relative Cartier divisor, and can be thought of as a family of Cartier divisors in the fibres of f over Y Moreover, given any fibre product: x’ x r| + M Y—>Y then g~1(D) is a relative Cartier divisor on X’, over Y', Chapter 1
Fundamental theorems for the actions of reductive groups
To set the stage, we shall be concerned in this chapter exclusively with schemes X over a fixed (but not necessarily algebraically closed) ` mg field È We also fix an algebraically closed over-field Q > k,
d write X for X x2 whenever X is a scheme over & The basic set
up will be an action of a reductive algebraic group G on a scheme X Our purpose is to investigate for which open sets U C X, invariant under the action of G, does a geometric quotient U/G exist? To this end, we will introduce the basic concept of a stable point of X The rest of this book is devoted to exploiting this concept
` § 1 Definitions
Due to the fact that ‘we are working over a non-algebraically closed field, we require a preliminary analysis of the notions of an action of -an algebraic group on a ring, of invariants, etc We have grouped all these trivial facts together in this section, Definition 1.1 Let G be an algebraic group By a representation of G we mean a morphism @:G-—>GL(n) such that, if «:GxG—+G and B:GL(n) XGL (n) > GL (n) denote multiplication in G and GL (0) respec- tively, then #o (xạ) =go (as morphisms from G xG to GLi(n))
Such a representation induces three types of actions: (a) a morphism
o:GXA"-> A" which is an action on the scheme A* in the sense of _ + 5S SEP URE a Er § 1 Definitions 25 Chapter 0; (b) a map of sets from G,x V -> V, where V is an n-dimensio- nal vector space over k; (c) a dual action in the following sense, (if G is linear): : ;
Definition 1.2 Let G be a linear algebraic group, let S = L’'G, 9), leta:S>S @ S be the homomorphism defining multiplication, and let B:S —> k be the homomorphism defining the inclusion of the identity in G Let V be a vector space (resp R a ring) over & Then a dual action of G on V (resp R) is a homomorphism of vector spaces (resp of rings): 6:V>S@V (resp 6: R> SQR # 2z Xe» S&@S@W_ commutes kk F sev Qe &
(resp.: same for R)}
such that (i)
and -
(ii) y———se y22™ 7 is the identity
(resp.: same for R)
In the case of a finite-dimensional vector space V, let {eds <i<y be a basis of V, and let đ(ø;) = ~Š a„ @ ¢; Then the elements a; eI'(G, Og) define a morphism from G to A* and conditions (i) and (ii) state that the image is contained in the open subset GL (n) of A” and that the morphism is a homomorphism In other words,a dual action on a finite-dimensional vector space is simply a translation of the concept of a representation In case of a ring R, on the other hand, a dual representation is simply a translation of the concept of an action of G on the scheme Spec (R) We shall see that this dual point of view is quite convenient
Definition 1.3 Let ¢ be a dual action of G on V (resp on R) Then a vector space W C V (resp W C R) is invariant under the action of G if o(W) CS OW Moreover, x« V (resp x ¢ R) is invariant if G(x) = 1@ x
A very important, although elementary, observation is:
Lemma’ If ¢ is a dual action of G on an arbitrary vector space V, then V is the union of finite-dimensional invariant subspaces
Proof Let S=I'G, 9¢), let &: S-+S@S be the law of group
roultiplication, and let B: S->k correspond to the inclusion of the
identity in G Then, to prove the lemma, let V,C V be any finite-
dimensional vector space It suffices to construct a finite-dimensional invariant subspace V,C V such that V, C V3
Trang 2026 1 Fundamental theorems for the-actions of reductive groups
Let S* = Hom,(S, &), and let y be the composition:
1g* QE $
S*eV-”®”, se sạ y-Q81, „
where <> stands for “contraction” Let P; = y(S* @ P))
(a) Vp is finite-dimensional For, if {v;} is a basis of V,, and 6(»,)
=+ 4;@ vy, then V, is contained in the subspace spanned by the elements {v,}
(b) V,C V2, For B is an element of S*, and y satisfies the identity y(8@v) =v, where ve V :
(c) Z(;¿) C S® Ứ¿ To prove this, it suffices to prove 6 [y(u* @v)] é5@ V, for every u* e S*, ve Vì But this follows if, for every u’* ¢ S*,
we have :
<u'*, 6 [y(u* @ v)) € Vy
But since ¢ is a dual action, (&@ ly) 0G = (1s @ ở) s G Therefore it
follows that: ,
<u'*, Gly (u* @ 0)]> = y [&* (u'* @ u*) @ v]
where &* is dual to & The latter is obviously in V, QED
Definition 1.4 An algebraic group G is reductive if its radical is a torus and linearly reductive if every representation of G is completely re- ducible, ie if the action of G on A® leaves invariant some A® c A* (where A*’ is a linear subspace through the origin), then it leaves in-
‘\varaint a complementary A*-*’,
` TÍ G is linearly reductive, and o:Gx A" > A® is a representation of G, then clearly the affine space A"can be decomposed into a product
B.xB,x - xB„
of affine spaces such that G leaves each factor invariant and such that B; is irreducible under the action of G Suppose the factors are so num- bered that G acts trivially on B,, , B,and non-trivially on Brat, - +, Bry Then G acts trivially on
“BX "- <8,x(0)x -:‹- x(0)c A*
and this subspace contains every other subspace on which G acts trivially The projection of A" onto this subspace which annihilates B,.4X+++XB, is all-important in the sequel Actually, it is more
convenient on the dual level:
Definition 1.5 Let G be a linearly reductive algebraic group, and let S = IG, 9g) Let ở be a dual action of G in a vector space V Then a Reynolds operator is a homomorphism E:V -» V such that:
(i) EZ commutes with 6, ie Go E = (1s@E)oG
(ii) EH? = E
(iii) Ex = x if and only if ¢(2) =1@x eins
If V is finite-dimensional, it follows immediately from the definition
of “reductive” that E exists and is unique The same holds for a general V é Ệ Ễ i E
2# Here we give the proof only when char (8) = 0 Modifications to đea with char (k) = are discussed in Appendix 1B
§ 2 The affine case 27 by virtue of the above lemma, (compare Cartier [6], exposé 7)
Because of the canonical nature of E, whenever we have dual actions
of G on two vector spaces V, and V, and a linear map ¢: V,> V,
commuting with the dual actions, then ¢e E, = E,° ¢ where E, and
E, are the Reynolds operators on V, and V, respectively
Suppose V is actually a ring R, and o is an action on the ring Then the image of E is the subspace of invariant elements of R, which is a subring R, of R Although E is not a ring homomorphism, it satisfies the well-known Reynolds identity:
(*) TfxeR, ye R, then E(x-y) = (Ex)-y
(cf CARTIER [6], 7—08) To prove this, set R,= ker (E) Then Ro R,@ R,, and (*) is equivalent to the 2 statements R,-R,C Ry and R,- R, C R, To prove the latter, let ge Ry, and let VC R, bea (finite dimensional) irreducible invariant subspace Then either g- V = 0 or multiplication by g sets up an isomorphism of V with an irreducible invariant subspace g- V Since g is an invariant, the representations of G on V and g- V are isomorphic and nontrivial Therefore g-V C R, The linearly reductive algebraic groups have been classified by M Nacata [28] His result is this; a) in characteristic p (34 0), the only linearly reductive groups G are those whose connected component Gy is a torus (G,)’ and such that the order of G/G, is prime to 2 (b) in
characteristic 0, then G is linearly reductive if and only if it is reductive
In that case, it is well-known that Gạ is isogenous to a direct product of a torus T, and asemi-simple algebraic group Gj which may be taken as the commutator subgroup of G, (cf [28], lemma 10, or [8], exposé 1,6
lemma 2)
§ 2 The affine case
With these preliminaries disposed of, we can now take up the first case of the theory — the action of a reductive algebraic group on an affine scheme X To keep straight the techniques in the proof, we do not assume that X is an algebraic scheme over &
Theorem 1.1 Let X be an affine scheme over &, let G be a reductive
algebraic group, and let o:GxX— X be an action of G on X Then a uniform categorical quotient (Y, ¢) of X by G exists, ¢ is universally
submersive, and Y is an affine scheme Moreover, if X is algebraic, then
Y is algebraic over &
If char (2) = 0, (Y, $) is a universal categorical quotient Moreover & noetherian implies Y noetherian
Proof.* Let R = I(X,0z); then G acts dually on R Let R, c R be
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Ortaca
28 i Fundamental theorems for the actions of reductive groups
by the inclusion of R, in R Then the first part of the theorem is a consequence of the algebraic facts:
(1) Tí Sạ is an Rạ-algebra, then S, is the ring of invariants in R @ Sys
(2) If (Mer is a set of invariant ideals in R, then °
(2) A Ry = J (YU; R)
In fact, we can use Remark 6, p 8 to prove that (Y, ¢) is a categorical
quotient and ¢ is submersive For (1) implies that for all affine open subsets UC Y, I'(U, oy) is the ring of invariants in (VU, $„(0x)):
hence oy is the sub-sheaf of invariant sections of ¢, (ox) — condition
(ti) of Remark 6, §0,2 To put (2) in geometric form, let W; c X be the closed subset defined by Y; Then (2) asserts:
Closure {#(9 1.) = Closure (¢(W,))
But, applying this to the case W, = any closed invariant subset of X, W, = $+ (y), where y is any closed point of Y, we conclude that $(W,) is closed This being so, (2) implies also:
$ (9 #) = fn $(W;)
which is condition (iii) of Remark 6, p 8, Now to conclude that (Y, 4) is actually a universal categorical quotient, we must consider base ex-
tensions Y’-» Y By remark 8, § 0.2, we need only consider the case
where Ys affine, and show that Y’ is a categorical quotient of X x Y’ by G But by (1), ['(Y’, oy) is still the ring of invariants in T@& x Y’,
ox y’), and we are reduced to the case just considered
To prove the statement (1), let E: R-» R be the Reynolds operator, and let R, = ker (Z) Then, by the Reynolds identity, Rx [R, © Ry]
as an R,-module Therefore [2 ® So| x~5,e [Ri g So]; and in particular,
Sis a subring of R @ So Now suppose ' 2; @ 6; is an invariant element of R ẹ Sy, where a;¢ R;b; e Sy Letting E also denote the Reynolds operator for R 2 So, we deduce: = & = (4,8 5) = E LE (4; 1) - (1@ )} = YE (2; 1)-(1@&) (via Reynolds identity) = S £a,@ b,; = 1@ (XEa,- bà € So This proves (1)
§ 2, The affine case 29 To prove statement (2), let (%);.7 be any set of invariant ideals
in R Suppose fe (YW) Rạ Then f= 3 í;,, where ƒ;e9(,, and all
$ẽ
but a finite number of f; are 0 It ‘follows that:
f=Ef=[ZEAle FON Ky) tel tel
We now pass to the second half of the theorem Suppose X is noe- therian Note that, if & CR, is any ideal, statement (1) applied to So = R,/U implies that (+R) A Ry == A This implies that the partially ordered set of ideals in R, is a sub partially ordered set of the set of ideals in R Hence the a.c.c for the ideals in R implies that the a.c.c
for ideals in R,, and we conclude that Y is noetherian
Now suppose X is of finite type over &, ie R is an algebra of finite type over k Suppose first of all that R is a graded algebra over k, and that the action of G preserves the gradation Then R, is a sub-graded ring of R, and it is automatically finitely generated over & since we have already shown it to be noetherian
Secondly we can reduce the general case to the graded case: choose a finite dimensional invariant subspace V C R which contains a set of generators of R Let R’ be the symmetric algebra on the vector space V Then the action of G in V extends to a gradation preserving action
of G on R’; and R together with the action of G on R can be identified
with the quotient R’/% of R’ by an invariant ideal Mf Then if Ro is the ring of invariants in R’, Rj is finitely generated: therefore it suffices to prove that R, is the image of Rg in R This follows.from the ge- neral algebraic fact:
(3) Hí 9 is an invariant ideal in R, then R,/%U A R, is the ring of
invariants in R/W
Proof of (3): let E: R-> R be the Reynolds operator on R, and let E: RJW-—> RỊW be the Reynolds operator on R/W Then one has:
(Ef) mod & = E(f mod M)
for all fe R In particular, if fe R is such that (f mod %) is an invariant in R/U, then Ef represents the same element of R/M as ƒ In other words, every invariant of R/Q is in the image of R, QED
Corollary 1.2 (of proof): If W, and W, are two closed disjoint invariant subsets of X, then they are separated by an invariant feI(X, ox) which is 0 on W, and 1 on W,
Proof Let M; correspond to W;, Using statement (2), we have 1e + HU) AR, = UA Ry + U0 Ry Therefore 1 = f+ g where feM A Ry, ge, Ry, and f is the required invariant QED
This corollary is, for many purposes, the only really important geo- metric property implied by the reductivity of G
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30 1 Fundamental theorems for the actions of reductive groups
Amplification 1.3 In the situation of Theorem 1.1, (Y, ¢) is a geo- metric quotient of X by G if and only if the action of G on X is closed
Moreover, in char 0, (Y, ¢) is actually a universal geometric quotient
Proof If (Y, ¢) is a geometric quotient, then g is closed by lemma 0.6 Now assume o is closed Suppose Y(GxX) were a proper subset of X x4 Then in a suitable algebraically closed field Q, there is a pair of geometric points x,, %, of X such that ó(xy) = ở(z;), but 0(x,), 0(x) are disjoint subsets of X But then 0(x,), 0(x,) are, by assumption,
closed invariant disjoint subsets of X: therefore, there is an invariant
fel (&, oz) = IX, 0x) @@ which is 0 at x, and 1 at 25 Since the ring of invariants in re, 0x) is of the form 1ạ@8, where R, is the ring of invariants in J"(X, ox), there is an invariant fe I"(X, ay) such that /(*) +4 /(x,.) Therefore $(%) % $(%) which is a contradiction
Now the whole situation is preserved by affine base extension Y’-» Y, because of (1) in the proof of Theorem 1.1, According to Remark 3 of § 0.2, this proves that (Y,¢) is a universal geometric quotient QED
§ 3 Linearization of an invertible sheaf
Before passing to the analysis of the action of a reductive algebraic group G on an arbitrary algebraic pre-scheme X, we must relate the actions of G_on X\with invertible sheaves L on X
Definition 1.6 Let G, X, L, o be an algebraic group, an algebraic pre-scheme, an invertible sheaf on X, and an action of G on X respec- tively Then a G-linearization of L consists of an isomorphism:
$:o*Ly?ÿL
of sheaves on GXX, satisfying the co-cycle condition:
(*) let u:GXG->G be the group law Note that ,, uX1x, and Igxo allmapGxGxX toGxX The condition is the commutativity of:
lơ s (Iexø)]*L Sexe 1#; + (IoXø)†*E
⁄ Pu®d
lơ s ()]*L———— ([#; s a]*L [ỡ s (wx1„)]*L —" #— Ip, 0 (ux) IPL
To understand this concept, suppose the ground field is algebraically closed and let G, be the group of geometric points of G, in & Then for all weG,, we can restrict ¢ to {a}xX CGxX: then if T,:X+~X “a + TS re ftSugt
§ 3 Linearization of an invertible sheaf 31
is the automorphism of X given by x—> ala, x), $ restricts to an isomorphism ¢,: Tỷ +^T The co-cycle condition then implies: É TP, I———mL PC Tag me, /% + AEE
commutes, for «, BeG,, ie dag = dạ s TS be
Another way to consider the definition is by means of the line bundle L corresponding to L Let x: L — X be the projection Then iso- morphisms such as ¢ correspond canonically to bundle isomorphisms of the line bundles over GXX:
©
(xX) XL x (GxX) xi product via product via
ơ:GxX—> Xj \?,:GŒxX—> X] `
(Recall that the transition from sheaves to bundles is a contra-variant functor) These correspond canonically to morphisms X = $,° ® such that: GxL-+L igxx | | ” + + GxX—+X
commutes, and such that Y is a bundle isomorphism of the line bundles GXL over GXX and L over X Then the co-cycle condition translates readily into the commutativity of the following cube: GxGx LH“ WK Nee + GxGxX— mm tlạxơ ø GxX-
That is to say, the morphism Z corresponds to a ¢ satisfying (*) if and only if X' is an action of G on L Therefore, a G-linearization of LZ is a lifting of the action of G on X to a “bundle action” of G on L
Trang 2332 1, Fundamental theorems for the actions of reductive groups
For example, suppose X = Spec (2), L =oy =k, L =A} Then a G-linearization of L is nothing other than a character 7: G-> G,,, since G,, is the group of automorphisms of A! as a line bundle over Spec (A) To be explicit, y defines the action Y of G on A! such that
2 (oe, 2) = x(a) +2
for « e G,, ze (A) On the other hand, it defines $: 0g—> dg given by
bf) =x -F
where 7eI°(G, 0%) is the function corresponding to the morphism y (Ihe inverse results from the contra-variant relation between’ sheaves and bundles.) -
There is still another kind of action induced by a G-linearization
of L, and, for the sequel, this is the most important: a dual action
of G on H°(X,L) This action is given by the composition:
H(X, L) > H°(Gx X, o* L) 2+ H°(GXX, pL)
dl
H°(G, 06) @ H9%(X, 1)
(The last isomorphism follows from the Kunneth formula.) The condi- tions for a dual action result from the co-cycle condition on ¢ Therefore ¢ allows us to speak of invariant sections of L
Notice that the tensor product of 2 G-linearized invertible sheaves, and that the inverse of 1 G-linearized invertible sheaf both carry canon- ical G-linearizations Therefor, the set of G-linearized invertible sheaves modulo isomorphism, forms an abelian group We will denote this group
by: Pic&(X)
Moreover, if /: X + Y is a G-linear morphism of pre-schemes on which
G acts, there is an induced homomorphism:
f*: Pic (Y) > Pic®(X)
In particular, if G is acting trivially on Y, then each invertible sheaf
on Y possesses a trivial G-linearization Therefore, we obtain an induced
map from Pic (Y) to Pic® (X) In case (Y, f} is a geometric quotient of X by G, and the action of G is free, it follows from Proposition 0.9, and the theory of descent of SGA 8, that Pic&(X) is isomorphic to Pic(Y)
An interesting example of this concept is given by the action of PGL(n + 1) on projective space P, Recall that PGL(n + 1) is the open subset of
Pisson &% Proj È [đoo, ., đạy; Brgy ess Bigs oo sd Aggy ee os Aun] where det (a;) 40, and that P, = Proj &[Xp, ,X,] Then the a“ + § 3 Linearization of an invertible sheaf 33 morphism : oo o:PGL(n+1)xP,>P, - can be defined (cf EGA 2, § 4.2) by the condition:
o* (ø,(U) = Z7es„,„„()] @ 2Zfep„ (11
o*(X) = FAT (ay) @ 23K)
Unfortunately, op,,,,, (1) is not trivial on the open set PGL(n + 1): in fact, its order in Pic[PGL(n + 1)] is m +1 Therefore, op (1) admits
no PGL(n + 1)-linearization! `
To obtain PGE(n + 1)-linearizations, let L be the line bundle on P„ corresponding to gp,(i) Then it is well-known that the nth exterior power of the cotangent bundle to P, is isomorphic to the (# + 1)* tensor power of L Any action on P,, lifts to an action on the cotangent bundle, hence to L"*! Therefore, op (7 + 1) does admit a PGL(n + 1)- linearization
On the other hand, let @:SL(m + 1) > PGL(n + 1) be the canonical isogeny and consider the induced action t = oo (wX1p,) of SL(n + 1) on P, Then gp,(1) does admit a SL(n + 1)-linearization For it is well-known that SL(n + 1) acts on the affine cone A** over P,, so that the projection
wm: A*tl — (0) > P
is SL (n + 1)-linear But L is obtained from A**! by blowing up (0), and the projection from L to P, is obtained from z: therefore SL (n + 1) acts on L, compatibly with its projection to P,
We now consider the general theory of G-linearizations:
Proposition 1.4 Let a connected algebraic group G act on an algebraic pre-scheme X Assume that there is no homomorphism of G
onto Gi, and that X is geometrically reduced Then each invertible sheaf ZL on X has at most one G-linearization
Proof If some L had 2 G-linearizations, then the induced sheaf 7, on X¥ = X xK , for an algebraically closed over-field K > k, would still
have 2 G-linearizations Therefore we can assume without loss of gener- ality that & is algebraically closed Also, since G is connected, G-lineariza- tions of an Z on the different components of X are independent of each other; therefore, in order to prove the Proposition, we can assume that
X is connected „
The Proposition is equivalent to the statement that the canonical map: Pic®(X) > Pic(X) is injective Since this is a homomorphism it
suffices to ask whether oy admits a non-trivial G-linearization Let
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34 1 Fundamental theorems for the actions of reductive groups
: o* (0x) -> p3 (ox) be aG-linearization Then let ¢ take the unit section
of ox to fe H°(agx x) Since pis an isomorphism, f ¢ H°(o%, x) Moreover
the co-cycle condition on ¢ implies that ƒ= 1on the subscheme {2xx
o£ GŒxX Now by a result of RosENLIcmr (Theorem 2, p 986, [32]), we know that H°(0%,) is spanned by the images of H' °(0%) and H(o%).t Moreover, by Theorem 3 of the'same paper, we know that
H (0g) a k* Therefore
HO(08, x) SHC)
It follows that /, being equal to 1 on {e}xX, must be identically 1 Therefore, ¢ is the identity QED
Proposition 1.5 Let a connected linear algebraic group G act on an algebraic variety X, proper over k Let L be an invertible sheaf on X, and let 2 be the k-rational point of the Picard scheme Pic(X/k) defined by L Then some power L* of L is G-linearizable if and only if some multiple A of 4 is left fixed by G
Proof The only if is clear Conversely, suppose nA is left fixed byG Then J claim first that for some m, the two sheaves o* (1) and 22 (L””) onGXX are isomorphic To see this, consider the see-saw exact sequence:
0~> HP (gi) —> HẺ (dễ, x) —> H°(G, E12 (dễ „x)) -
Since H1(0@) is a finite group (Séminaire Chevalley, [9], 5—-31), it is enough to show that the image of o* (L") @ Ø2 (L")~Lin #9%(G, R')+(0¢ 4x)
is zero But, by the functorial definition of Pic (X/k), (cf Ch 0, § 5, (d))
HO(G, R°p,e(08,cx)) C Hom, (6, Pic (X/3)
But, as in the proof of proposition 1.4, H°(o%,, y) sz H°(o%), and the latter is just &* x M, where M is the set of homomorphismus 7:G > Gp Choose an isomorphism ¢:o*(L*™) => p3(L"™) which is the identity on {e} x X Then 236 o (1g Xo)* ó and (w x 1z)* ¢differ by a factor of the form 4/1(x¡) Đ3(x:), 4 € k*, x¡€ M Restricting to {e} x G x X, we see that a= 1, 7, = 1 and restricting to G x {e} x X shows that y,=21 QED
t Rosrwricut only stated his result in the case where X is a variety The generalization to an arbitrary reduced algebraic pre-scheme is imme- diate
§ 8 Linearization of an invertible sheaf 85
Corollary 1.6 Let G, X, and L be as above Then if X is a normal
variety, some power L” of L is always linearizable
Proof According to a result oí CHEVALLEY [10], in this case all components of Pic (X/k) are proper over k, hence all reduced components are abelian varieties Therefore the connected’ linear group G, being birational to projective space as a variety, must act trivially on Pic (X/k) QED
The following connection between G-linearizations and projective embeddings will be needed:
Proposition 1.7 Let X be an algebraic pre-scheme, let G be an algebraic group acting on X, and let L be a G-linearized invertible sheaf on X such that the sections of L have no common zeroes Then there
exists a morphism I: X -> P,, an action of G on P,,, and a G-linearization
of op(1) such that I is a G-linear and such that £ together with its G-linearization is induced via J from op(1) and its G-linearization Moreover, if X is proper over k, one may take I to be the morphism associated with the complete linear system H°(X, L); and if L is very ample, we may take J to be an immersion
Proof.* Let $:a*L ~ p3L be the given linearization Then ¢ defines a dual action of G on H°(X, L)
Let V, C H°(X, L) be any finite-dimensional subspace such that the
sections in V, have no common zeroes If L is very ample, take V, so
that the morphism from X to P(V,) is an immersion; if X is proper over k, take V, = H°(X,L) In any case, by the lemma of §1, there is a finite-dimensional vector space V such that V,C V C H°(X,L), and such that V is invariant under the action of G Then we take I to be the morphism from X to P(V) Then, by definition, H°(P(V), op(1)) = V, so we have also a dual action of G on H°(P(V), op({1)) But J claim:
(*) There is a natural equivalence between the set of all dualactions of G on H°(P(V), op(1)), and the set of all actions of G on P(V) plus G-linearizations of op(1) (Proof omitted.)
This gives us all the required actions Their compatibility follows in a straightforward way also QED
We will require in Chapter 2 a strengthening of this result Although this involves concepts which we only define in the next section, we include it here since its proof is related to the proof just given:
* See KAMBAYASHI [338], SUMIHIRO [319]
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36 1 Fundamental theorems for the actions of reductive groups Amplification 1.8 Assume L is ample Then, for large N, the G-
linear immersion I of Proposition 1.7 such that L¥ cz I* (op, (1)) can
be so chosen that: :
Xo (LE) = IAPS oy (Op, (1))}
Proof By definition, Xo (Z) is covered by a finite number of affine open sets X;,, where s; is an invariant section of L™, and where the action of G on X,, has only 0-dimensional stabilizers Let
P(X 0x) = AY £2),
Replacing each s; by a suitable power of itself, we can assume: i) each s; is a section of LY, some fixed N, -
1) for all 7 between 1 and m,, f s, is a section of LY
Then suppose, in the previous proof, we select the vector space V, so that s;e V, and / s;¢ V, for all i, j It follows that if I: X > P, is the resulting immersion, then s, = I* (X;) for a suitable invariant section X; of op, (1): hence X;, = I-"(P,,x,] Moreover, by (ii) we conclude that J maps X,, isomorphically onto a closed subscheme of (P,)x, Therefore, the orbit of every geometric point of X,, is closed in
(P2)x¿ By Amplification 1.11, this implies that every point of I(X,,)
is properly stable in P, Hence
Xo (LZ) = U Xs, C I {Pio (op, (1))}
The other inclusion “>” is proven in Proposition 1.18 QED
§ 4 The general case
We now proceed to analyze the general case: G is a reductive algebraic
group, acting via o, on an arbitrary algebraic pre-scheme X The key concepts will be these:
Definition 1.7, Let x be a geometric point of X Then:
(a) + is pre-stable (with respect to o) if there exists an invariant
affine open subset U €C & such that x is a point of U, and the.action
of G on U is closed `
Now suppose L is an invertible sheaf on X, and ¢ is a G-linearization
of L Then:
(b) x is semi-stable (with respect to o, L, ¢) if there exists a section se, +”) for some ø, such that s(x) £0, X, is affine, and ¢ is ee Le if $,:0*(L") > p¥(L") is induced by ¢, then ¢, (o*(s)) = £3 (s) (c) x is stable (with respect to o, L, $) if there exists a section s e H°(X, L") for some n, such that s(x) % 0, X, is affine, s is invariant,
and the action of G on X is closed
§ 4 The general case 37
Note that the set of geometric points with any of these 3 properties is the set of geometric points of an open subset of X The corresponding 3 open subsets will be written:
A‘ (Pre)
xs (LZ)
XS(L)*,
(1) We note immediately that, for each invariant open U C X such that U is connected and affine and the action of G on U is closed, the stabilizers of all the geometric points of U have the same dimension — cf Amplification 1.3 and Remark 5 of § 0.2 Therefore we can write
X* (Pre) = Xj (Pre) U -U Xp (Pre)
X*(L) = Xi (L) UU iy)
where the right hand side represents a disjoint union of open sets, and where the dimension of the stabilizer of every geometric point of X « (Pre)
and of Xiy(L) is 7
Definition 1.8 X{o(L) is the set of properly stable points (with
respect to a, L, ¢)
(2) Note that if X is proper over 2, and L is ample, then the condition
that X, be affine is redundant Moreover, in any case, L will be ample
when restricted to X*“(L) (cf condition (b), Theorem 4.5.2 of EGA, Chapter 2)
(3) Recall that the set of all sections se H®(X, LZ") such that X, is affine is a linear subspace of H°(X, L"): this follows, for example,
from condition (b) of Theorem 4.5.1, EGA 2 Therefore so is the set of
be
co
invariant sections such that X, is affine Call this V,, Moreover =, Vv, im
is a ring, since X= X,/\ X;, and this is affine if both X, and X;
are affine.t This ring will take the place of the affine ring of invariants in the general case
(4) The 4th concept “pre-semi-stable’’, is not so useful, but we note
SUMIHIRO’s result [319]: if a torus T acts on a normal variety X, then
every x € X has a T-invariant affine open neighborhood
Proposition 1.9 Let X be an algebraic pre-scheme over &, and let G be a reductive algebraic group acting on X Then a uniform geometric
quotient (Y, ¢) of X*(Pre) by G exists Moreover, ¢ is affine and Y is an algebraic pre-scheme Conversely, if U C X is any invariant open set
such that a geometric quotient (Z, p) of U by G exists, and such that y is affine, then U < X® (Pre) If char (k) = 0, then (Y, 4) is a universal geometric quotient :
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38 1 Fundamental theorems for the actions of reductive groups
The proof of this is omitted, as it is easy from what has already been proved and is never used in the sequel Incidentally, in view of Proposition 0.5, the condition that w is affine in the converse can be replaced by the stronger hypothesis: “G acts properly on U”, which may seem a more reasonable assumption since it is topological in nature The problem with this theorem is that even when X is a projective scheme, Y can be a very “‘pre’”’ pre-scheme For this reason, we will find much more useful:
Theorem 1.10 Let X be an algebraic pre-scheme over 2, and let G be a reductive algebraic group acting on X Suppose L is a G-linearized invertible sheaf on X Then a uniform categorical quotient (Y, $) of
X*(L) by G exists Moreover: Ỷ (i) ó is affne and universally submersive;
(ii) there is an ample invertible sheaf M on Y such that ¢*(M) ~ L* for some n; hence Y is a quasi-projective algebraic scheme;
(ili) there is an open subset Y C Y such that X*(L) = ¢71(Y) and such that (Y, ¢ | X*(L)) is a uniform geometric quotient of X*(L) by G
Proof Since X is noetherian, there exists an N, and a finite set
S,+++, Sy of invariant sections of L¥ such that U; = X,, is affine and n A™(L) =U Uy By Theorem 1.1, there exists a uniform categorical quotient ( Vụ, $;) of U; by G Our first step is to patch the affine schemes V; together into a pre-scheme But for every pair of integers (1, 7) (L<%,7< 1), consider the quotient s,/s; in the open set U; where
s; 0: it is an invariant element of the ring I'(U;, 9x) Now since V;
is a categorical quotient of U; by G, I"(V;, oy,) is exactly the ring of invariants in I"(U;, 9x) Therefore, s,/s; is induced by a function
ơy£ Ï'(U, ®yị) Let tụ = V; — fy | (3) = 0} Then: $; (Vis) = Ữ, U; = $*(Vạ)
Since V; (resp Vj) is a uniform categorical quotient of U; (resp Uj), it follows that both V;; and V;; are categorical quotients of U;/\ U im
therefore, there is a unique isémorphism yy: Vj cy Vj such that:
Ũ, ^ U;
s⁄ `
Vy a Vis
commutes From the uniqueness, it follows easily that this set of identi- fications patches {V;} into a pre-scheme Y containing each V; as an affine open subset Moreover, it is clear that the ¢; patch together into an affine and universally submersive morphism ¢ from X*(L) to Y, such that ¢1(V,) = U; :
§ 4 The general case 39
The second step is to notice that the collections of functions ơy | Vz
forms a Cech i1-cocycle for the covering {V;} of Y and in the sheaf of
Therefore these functions define an invertible sheaf M on Y Moreover,
it is clear that LY —¢$*(M) I claim that M is an ample sheaf on Y in the sense of Theorem 4.5.2 Chapter 2, EGA To prove this, we use condition a’) of that theorem, ie the collection of functions oy for
fixed 7, and variable 7, is a collection of functions, one on each V; such that on Vi; AV;, o
Tạo, j= 9; tuỷ Ø,
Therefore, it defines a section t; of M Moreover, it is clear that s; = Ở* ()
Therefore, for every point x ¢ X*(L),
8;(x) = 0 & t,($(x)) = 0
Therefore, Vj = Y,, = Ÿ — {y|ø(y) = 0} Since V; is affine, the con-
dition a’) referred to is verified By the remark following Theorem 4.5.2,
this implies that Y is in fact a scheme, and, of course, M is ample on Y
Finally, by enlarging the set of sections s; is necessary, we may
assume that the action o of G on U; is closed if ¢e I C {1, , n}, and
that X*(L) = UU,
Then X*(L) = ở 1(U V;), Then if y= U V;, the rest of the Theorem
to the effect that Y (resp ¥) is a uniform categorical quotient (resp uniform geometric quotient) of X* (L) (resp X*(LL)) by G, follows from remark 7, § 0.2 QED
Amplification 1.11 In the situation of the above theorem, let x be
a geometric point of X*(L) Then the following are equivalent: 1) x is a point of X‘(Z),
2) x is regular for the action of G, and its orbit 0(x) is closed in
X*(L) = X*(L) xQ,
3) x is regular for the action of G and there is an invariant section se H°(L") such that s(x) + 0, X, is affine, and 0(x) is closed in X,
Moreover, if x, , %, is a finite set of geometric points of X*(L), then there is an invariant se H°(X,L™) (for some m) such that X, is
affine and s(x;) + 0, all 7 If #,, ,%, are all stable, then we can even
assume that the action of G on X, is closed
Proof 1) => 2) First of all, since a geometric quotient of X*(L) by G exists, it follows that + is regular for the action of G (cf Remark 4, § 0.2) Secondly, in X**(Z), I claim 0 (x) = $1 ($(x)): forX*(L) = Â1(Ơ),
hence ¢1(¢(z)) C X*(L); and if y e X*(L), then $(x) = $(y) if and only
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40 i Fundamental theorems for the actions of reductive groups
2) => 3) obvious
3) => 1) Let s ¢ H°(L") be an invariant section as in (3) We shall construct an invariant function / ¢I"(X,, 0x) such that f(x) 34 0, and such that the action of G on (X,), is closed Then, for some , s” - ƒ will be an invariant section of L*” and (X,)r== X,m.z will be contained in X*(L), which proves (1) To construct /, introduce the closed subsets S, of X, consisting of points whose stabilizer has dimension at least +
Suppose x eS, —~ S,,4 Put
Z, = S,,,V {Closure of Sy — S,}
2; = 0(2)
Since x is regular, Z, and Z, are disjoint, closed, invariant subsets of X, Therefore there is an invariant f eI°(X,, 0x) which is 0 on Z, and 1 on Z,, Write 7 = Zh @a;, where o,, , ay are elements of Q, linearly independent over k, and where f,, ,/ are elements of T,, 0x) It follows that each /; is invariant Moreover, since Z, is the
extension to 2 of the closed subset Z, of X,, it follows that every /;
is 0 on Z, Finally, since #(x) = 1, it follows that for some 7, f,(x) 5£ 0 This /; can be taken as the sought-for # One need only notice that, since {= 0 on Z,, every orbit of G in (X,); has the same dimension, hence the action of G on (X,), is closed
To prove the last statement, note that by definition there is a set of invariant sections s; e H°(X, L™) such that X,, is affine, and s;(2\4 0 If the ground field & is infinite, then there are constants ô;Âk so that the section J «;s; is not zero at any of the points x;; if & is finite, there is still some homogeneous polynomial P(X,, ,X,) such that the section P(s,, ,s,) is not zero at any of the points x, In any case, Am 0T Ẩpq, , ) 18 still affine by Remark 3 at the beginning of the
section Finally, if all the x; are stable, then we can assume that the
action o on X,, is closed: then the dimension of the stabilizers of points of X,, is constant on the connected components of each X,, Therefore, this i is also true on v X;,, hence on Xyyj5, or X PQsu „sạ)- Therefore the
action of these sets is closed also QED
Another interesting point in connexion with Theorem 1.10 is that,
if X ts proper over k and if L is ample on X, then the categorical quotient
Q of X*(L) by G is actually projective.over & Therefore, this categorical
quotient is a compactification of the topologically more significant geo- metric quotient of X*(L) by G We omit the proof, except to say that it follows from interpreting Q as Proj (R,), where Ry is the subring of SX, L*) of invariant sections In fact (see [30ð]),X*(L)/G can be sễn, regarded as the quotient of X*(L ) by the equivalence relation : ELS OR) 100) zó ee enn
§ 4 The general case 41 Converse 1.12 Let G be a reductive algebraic group acting on an algebraic pre-scheme X Then if a categorical quotient (Y, ¢) of X byG exists, and if ¢ is affine and Y is quasi-projective, it follows that for some Le Pi”(X), X = X“(L) Moreover, if (Y, 4) is a geometric quotient of X by G, then X = X°(L) b
Proof Let M be an ample invertible sheaf on Y Then, as pointed out in § 3, L = ¢*(M) carries a canonical G-linearization with the pro- perty that all sections ¢*(s), s e H°(Y,M"), are invariant sections of L* Since Y is covered by affine open sets Y,, for suitable sections s of M*, X is covered by the affine open sets ¢+(Y,) = Xe) Therefore X = X*(L) If (Y, 4) isa geometric quotient, the action of G on X is closed by lemma 0.6; therefore X == X°(L) QED
In applications, however, this converse is usually too weak Suppose one is given a pre-scheme X, on which G is acting The interesting question is to classify those invariant open subsets U in X such that a quotient U/G exists The above converse relates this to G-linearized " sheaves L on U, and the open sets U*(L) Much more interesting is to relate this with the open sets X‘(L), which may sometimes be computed without a prior analysis of all possible invariant open sets U In this
direction, the most useful result is:
Converse 1.13 Let X be a connected algebraic pre-scheme smooth over k, and let G be a connected reductive algebraic group acting on X If U is an invariant open subset of X, then the following are equivalent:
i) for some L e Pic®(X), U C X*(L),
ii) there is a geometric quotient (Y, ¢) of U by G, ¢ is affine and Y is quasi-projective
If the stabilizer of the generic point of X is 0-dimensional, then i) and ii) are also equivalent to:
iii) the action of G on U is proper, a geometric quotient (Y, ¢) of U by G exists and Y is quasi-projective
Proof iii) implies ii) by Proposition 0.7; ii) implies iii) by Converse 1.12 and Corollary 2.5 (still to be proven) i) implies ii) by Theorem 1.10 It remains to prove that ii) implies i) But first of all, by Converse 1.12, there is a G-linearized invertible sheaf L on U such that U = U%(L)
Then, in any case, since X is smooth, there is some invertible sheaf
on X extending L: choose one and write it L also Let D,, , D, be
the components of X — U of codimension 1 For a very large integer N, to be fixed later, let
L=L ÍM, =D)
This is also an invertible sheaf, since X is smooth Note that the G-
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42 1 Fundamental theorems for the actions of reductive groups
over GXU Since G is connected, the only irreducible divisors in GxXX —GXU are GXDy,, .,G@XD, Suppose the order of p at Gx D; is k; Then p extends to a non-zero section of pF L @ (ø*L} 1 [- 2 R(G 6xÐ)|| Therefore ở extends to an isomorphism of oF L and aL E = § A(GXD)| But, restricting to {e}xX, it follows that it ter]
È
extends to an isomorphism of L and 7 Ẹ 3 Di Since ¢ satisfies < the co-cycle condition, its restriction to {2}x Ứ must be the indentity homomorphism from L to L Therefore all the integers 2; equal 0 tt follows that the original ¢ extends to an isomorphism of o* L and PEL and, as ¢ satisfies the co-cycle condition generically, ¢ must satisfy it
over all of Gx X For similar reasons, ¢ extends to a G-linearization
of L’ also
We claim now that if N is sufficiently large, X*(L’} > U Let x be a geometric point of U Then there is an invariant section s « H°(U, L”),
for some n, such that s(x) + 0, U, is affine, and the action of G on U ,
is closed But if N is large enough, s extends to a section ¢ e 0(X, 7")
And, increasing N further, ¢ must be 0 on all the divisors D; Since s is invariant, it follows that ¢ is invariant Finally, J claim that X, = U,
Certainly, we have
k
U,CX,CX~ Ủ D,
But, in general, if V is an affine open sbuset of a scheme Z smooth/k then the components of Z — V all have codimension 1 Therefore, all the components of X,— U, are subsets of X — U of codimension 1 Since all the D; are outside X;, there are no such subsets, ie X; = U, Therefore, x is a stable point of X with respect to L’ QED
In the last three Propositions of § 4, we shall show that stability is independent of some alterations of X, G, L, and š
Proposition 1.14 Let X be an algebraic pre-scheme over &, let G be a reductive group acting on X, and let L be a G-linearized invertible
sheaf on X Let K > k be any over-field Then if X = X xK ,and if £
is the sheaf induced by L on X, we have:
X*(L) = X*(L)
X*(L) = X8(L)
Proof By amplification 1.11, X*(L), as a subset of X*{Z), is charac- terized by a geometric condition, i.e part (2) Therefore, if the result is proven for X*(L), it follows for X*(L)
To prove it for X**(L), it is clearly sufficient to treat the two cases? —
§ 4 The general case 43
(i) K algebraically closed, and K/k separable, (ii) K/k purely inseparable We treat first case (i) Let V,, C H°(X, E”) be the subspace of invariant sections s such that X, is affine; let 7, be the analogous subspace of H°(X, 1") Since K/k is flat, it follows that ,
HO(X, L") a H(X, L") @ K
Then the subspace V,,, has a least field of definition L, where K > L > &
But if r is any automorphism of K over k, then t commutes with the
action of G, and the G-linearization of L, hence certainly maps invariant
sections (XÝ, E7) into invariant.sections Also, if X, is affine, then
(X)¿—= * is affine Therefore, ? maps „ into itself According to Cư tị Chapter 1, this implies that t leaves L pointwize fixed By our assumptions on K/k, this implies that L = k Therefore, for some subspace VC H°(X, L”), V,= Vi, @K But if se H°(X, EL"), then s is invariant and X, is affine if and “only if s is invariant as a section of H9(X, Z*) and (X), = ( Xs) XK is affine, (cf SGA 8, Cor 5.6) Therefore Và = V,, Finally, X — X*(L) (resp X — X*(L)) is the set of common zeroes of all sections of all the spaces V, (resp V,,) Since
V,, generates V,, the result follows
Now suppose K/& is purely inseparable The result is then immediate because if s ¢ H°(X, Z*) then for some », s*” is a section of L*?” over X, and because:
(X) = Xz)xK QED
Proposition 1.15 Let the reductive algebraic group G act via o on an algebraic pre-scheme X, and let L be a G-linearized invertible sheaf on X Then, if G, is the connected component of eG, the open set of stable points (resp semi-stable points) is the same for the action of G
and of Gạ
Proof By Proposition 1.14, it suffices to look at the case where the
ground field & is algebraically closed Moreover, by amplification 1.11, it suffices to prove this for semi-stability Let U (resp Uy) be the open set of semi-stable points with respect to G (resp G)) Clearly U C U4 Secondly, note that U, is invariant under the action of G as well as of Go In fact, if x eG,, and if s e H°(X, L") is G,-invariant, we may define s* by means of the G-linearization of L, and s* will still be G,-invariant since Gy is a normal subgroup of G Then, since o(a, X,) = Xe, it follows that U, is invariant under «, hence under G
Now let x be any &-rational point of Uo Let a1, ., «y be represent- atives for the cosets of G/G) Applying the second part of amplification
1.11 to the finite set of points o(«,, x) ,we find that there is a Gy-invariant
section s of L* (for some ), which is not zero at any of the points o(«;, x),
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oe “char: ra "are egiven in Appendix 1Be-——
44 1, Fundamental theorems for the actions of reductive groups N and such that X, is affine Then put s’ = H $“, s” Ís a G-invariant = section of L™, s'(x) 54 0, and N Xy = a ơ(%, X;)
is affine Therefore xe U QED
Proposition 1.16 Let the reductive algebraic group G act via o on
an algebraic pre-scheme X, and let L be a G-linearized invertible sheaf on X, Let /: Xua-> X be the canonical immersion Then
A™(L) = Xtea(*L), AO(L) = Xpea(f*L)
Proof.* The ‘“¢”’ is obvious, To prove the inclusion “>”, start with an invariant section se H°(X,.4, /*L") such that [X,.a], is affine It
will suffice to prove that s* lifts to an invariant section #in H°(X, L*),
for some &, since it follows automatically that X, is affine But by lemma
4.5.13.1 of EGA 2, there is a positive integer & such that s* lifts to some section ¢ of L™ over X Now we are given a dual action of G on H®(X,L™): therefore there is a Reynolds operator E on this vector space, and Et is an invariant section that still restricts to s* on Ä „sa QED
Corollary 1.17 In the situation of the above Proposition, if there
are no homomorphisms of the connected component of G onto G,,, then Ä+#(L) and X°(L) are independent of the G-linearization of L
Proof By Propositions 1.4, 1.14, 1.15, and 1.16
§ 5 Functorial properties
In this section, we are concerned with the following situation:
{:X-— Y is a G-linear morphism between two algebraic pre-schemes on which G is acting Moreover, M ¢ Pic?(Y) Then what is the relation
between:
Xo FM) and £1Y0(M))
Many of our results are also valid for semi-stability, and stability but for the sake of simplicity we will only consider the open sets of properly stable points
Proposition 1.18 In the above situation, if / is quasi-affine, then
Xo*M) 5ƒ*{Y@(M)
Proof.* By Proposition 1.14, it suffices to prove this under the added assumption that the ground field & is algebraically closed Now let x
* Modifications to extend this proof, and the Proofs | of 1.18, 1.19, to
_
§ 56 Functorial properties 45
be a closed point of /-1{Y%o (M!)}: we must prove that 2 & Xp) *M) Since ƒ(+) is properly stable, there is an invariant secHon / e H°(Y, M”), for some ø, such that Y; is affine, ƒ() e Y¿, and all the stabilizers of the k-rational points of Y; are 0-dimensional Let s= /*t be the induced element of H°(X, (/*M)") s is still invariant, xe X,,and all stabilizers of k-rational points of X, are 0-dimensional — but X, is not necessarily affine In fact, X, == f-'(Y;) is merely quasi-affine in general But let: R=TI(X,, ox) X = Spec (R) Then we have the commutative diagram: xe X, ——># ƒ
where J is an open immersion (Prop 5.1.2, part (b) in EGA 2) Moreover,
since G acts on X,, there is a dual action of G on R, hence an action
of G on X It is clear that all the above morphisms are G-linear Notice first that as / is G-linear, it follows that all stabilizers of all geometric points of X are 0-dimensional: therefore the action of G on X is closed Put:
4, =Ä—1 (X,),
Z; = 00)
Then Z, and Z, are disjoint closed invariant subsets of Xx Therefore,
by Corollary 1.2, there is an invariant ƒ e R which is 0 on Z,, and 1 at x It follows that (X);C X;, and that (X)y is affine But by Theorem 1.3.1, EGA 1, there is an integer & such that the section s* - f of (/*4)"* over X, extends toa section s’ of ({*)"* over X Finally, let E be the Rey-
nolds operator on H°(X, ( W*M) )“) Then #s” is invariant, and Es’ still
equals the invariant section s*.# over the open set X, Then (s+ Es’) is 0 both (a) outside X,, and (b) at points of X, where / = 0 Therefore
Xiszs) is an affine neighborhood of x contained in X,: since s- Es’
is invariant and every stabilizer in X, is 0-dimensional, x is properly stable QED
The next question is: when does Xj(/*3f) actually equal PU Yio) (M)}? A reasonable suggestion would be that they are equal at least if f is finite Unfortunately, this is false One need only consider the following case: xe X, and f = yp,:G— X Then if 0(x) is closed, and S(x) is O-dimensional, it follows from lemma 0.3 that fis is finite
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“46 1 Fundamental theorems for the actions of reductive groups
But we will see from almost every example in Chapter 4 that there are
plenty of closed orbits, with trivial stabilizers, that are not stable The
following is true, however:
Theorem 1.19 In the above situatién, assume fis finite, X is proper over k, and M is ample on Y (but Y is not assumed proper over #!) en
TM) = Ƒ1{Y(M)}
Proof By Proposition 1.7, there is an embedding I: Y—> P„, an action of G on P, and a G-linearization of 9p(1) such that I is G-linear, and M’ = I* (op(1)) (as a G-linearized sheaf) Using Proposition 1.18, it is clear that if we prove the Theorem for Je /, then it follows for /
Now suppose Y= P,, M= Op(1) Let R be the homogeneous co- ordinate ring of P,, let S be the homogeneous coordinate ring of X,
and let {: X-> P,, be defined by a graded R-algebra structure on S
Since / is finite, S is a finite R-module, The actions of G on X and P,,
and the G-linearizations of /* (@(1)) and gp(1) define dual actions of G on
Rand S, compatible with the R-module structure on S Let E and F
be the Reynolds operatorson R and S$ respectively Now suppose x is a properly stable geometric point of X Then there is an invariant element
se HX, f*(op(n)) = S n
such that s(x) # 0, the orbit of x is closed in X,, and the stabilizer of % is 0-dimensional Since S is a finite R-module, there is an equation of integral dependence: S™ + a, -s™ ee 4G 0 where a; is a homogeneous element of R, and a; denotes its image in S Applying F, we obtain: Om F{s*+a,-s™-1 4 + đ„} =$” + Fát-s”TỦL + cFấy,
=s”+ Ea, n1 ese + Ea
Since s(x) + 0, it follows that for some t, Ea;(}(x)) 0 Therefore, f(z) € (Pp) za But this implies that
vE ao == fl [Pa)zal :
By amplification 1.11, the orbit of x in_X, '{*(Ea,) 18 Closed; since / is proper,
the orbit of /@) im (P,) za, is closed.Since fis actually finite, the dimension of the orbits of x and /(x) is the same; therefore the stabilizer of T(x)
is 0-dimensional By amplification 1.11, f(x) is properly stable This
proves Xo (f* op (1)) cfr P, N 9 1))} Usi Pr iti 1
result follows QED {Cal or(0)} Using Proposition 1.18, th Corollary 1.20 Let a reductive algebraic group G act on a scheme X, proper over & If there are no homomorphisms of the connected compo-
-”
§ 5 Functorial properties 47
nent of G onto G,,, and if Le Pic®(X) is ample,.then Xf)(L) depends
only on the polarization containing L ~ Proof This means two things: /
(i) X((Z4) = Xi (£9, for positive integers and g, and ample L,
in Pic? (X)
(ii) If L, and L, are ample sheaves in Pic* (X), which are algebraically
equivalent (in Pic(X)), then X()(L,) = Xo (Z,)
The first is obvious To prove the second, we can make preliminary reductions to the case: k algebraically closed, X reduced, and G connected
by Propositions 1.14, 1.15 and 1.16 In fact, we may assume that X
is normal: for let 7: X’ > X be the finite morphism from the normaliza- tion of X to X Then applying Theorem 1.19 to øz, and noting the alge- braically equivalent sheaves on X are algebraically equivalent on X’, we are reduced to proving the result for X”
Now let L; define the k-rational point 4;« Pic(X/k), and let P be the reduced scheme of the connected component of Pic(X/R) containing A, and 4,, Let £ be an invertible sheaf on XP which restricts on each closed fibre X x {p} to the sheaf L, corresponding to p Replacing £ by £ @ pf (M), for a sufficiently ample sheaf M on P, we may assume that £is ample on Xx P By a result of Chevalley [10], P is a principal homogeneous space over an abelian variety; therefore the action of the linear group G on P induced from its action on X is trivial Then G acts on XxX P by the product of its action on X and the trivial action on P By Corollary 1.6, some power £* of £ admits a G-linearization For every k-rational point ¢ P, we can apply Theorem 1.19 to the G-linear morphism
í,:XZXxÿ}CXxP We conclude that
[Xx?)e(4)1/^ [Xx}) = Xã (Ly)
Tn particular, for any point ze X;, it follows that the set of pe P, such that x is properly stable for E„ is the set of #-rational points of an open súbset ỨC ? Now suppose that for +, Ữ is non-empty Then I
claim that U= P: for,if p is a &-rational point of P — U, then there
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48 2 Analysis of stability
Since x X%(L,) for every geU,, it follows that xe Xf(L,), ie
pe U, Therefore U = P
This implies that X{o)(L,) is independent of #; in particular, Xo (Ly)
= Xo (Li) = Xo (La) = Xf) (Le) QED
For the sake of applications in the next chapter, we must make one mention of semi-stability We note that Proposition 1.18 is trivially
valid, if f is affine, for X* and Y*; and that Theorem 1.19 is likewise valid, if Y is proper over k too, for X* and Y*, The proof of the latter
can be read word for word from the above proof, omitting the final
steps Actually, Theorem 1.19 is valid in all cases for X* and Y*®, using
a small additional argument: but we will not use this fact
Chapter 2
Analysis of stability
It might seem that the various concepts of stability, introduced in the previous chapter, are merely accidental and unworkable notions We hope to show here that this is not so In the first place, there is a strong numerical criterion for stability Almost all of our later examples
will use this criterion And in the second place, this criterion leads to
a description of structures in the group which are associated naturally to points where stability breaks down: these structures are “convex sets in the flag complex”’, and this association is the explanation of a well-known intuition which runs as follows:
if PGL(n) is acting on a set of cycles or subschemes in P,,, and we attempt to form projective invariants for these objects, ive to construct the quotient of the set of objects by PGL (nm), then we cannot do this until we first discard those cycles or subschemes
which have a singularity at a jlag or order of contact with a flag which is too “bad” `
Examples of this will be seen in Chapters 3 and 4
Since we are dealing with the questions of stability for a fixed geo- metric point, there is no loss in assuming from the start that the ground field 2 is algebraically closed Since all pre-schemes X will be of finite type over 2, the set X, will be the set of closed points of X
§ 1 A numerical criterion
Definition 2.1 A 1-parameter subgroup of G is a homomorphism 4: G,-—>G We abbreviate this to: J is a 1-PS of G
Now suppose we are given an action o of an algebraic group G on
let Abe a LPS ofG co
~~an algebraic schemeé"X, proper Over R Let x be a closed point of X, and
§ 1 A numerical criterion 49 Consider the morphism y,° 4 from G,, to X Identifying G,, with Spec k[«, x1], we may embed G,, in the affine line A! = Spec 2 [a] Then y,°4 extends uniquely to a morphism /: A!— X This follows from the valuative criterion for the properness of X over k (cf EGA 2,
§ 7), since the local ring of A! at the origin (0) is a valuation ring The
closed point /(0) in X will be called the specialization of o(A (a), x) when œ —> 0 Clearly /(0) is fixed under the action of G,, on X induced by A Now if L ¢ Pic®(X), we can consider the induced G,,-linearization of L restricted to the fixed point /(0) As remarked in § 3 of Chapter 1, this ‘is given by a character of G,,: say y(«) = o7, for « e(G,,), With all
this preparation, we can make the key definition:
Definition 2.2 If G acts on the algebraic scheme X, proper over , if x is a closed point of X, A is a 1-PS of G, and L e Pic®(X), then
pe (2, a) = —?
The functorial properties of are:
(i) p* (oe, *), 4) =p" (x, 071 -A- a), if «eG,
ii) for fixed x and 4, u(x, A) defines a homomorphism from Pic*(X)
to Z as L varies
ili) If /: X-> Y is a G-linear morphism of schemes on which G acts,
Le PicS(Y) and xe X,, then
BI? (x, A) = pw" (fx, a)
iv) If o(A(a), x) >y as «> 0, then p(x, /) = p*(y, 4) (Proofs
' Immediate.)
The theorem which we shall ultimately prove is:
Theorem 2.1 Let a reductive group G act on a scheme X, proper over & Let L ¢ Pic®(X), and assume L is ample Then if xe X,:
xe X*(L) <> p*(x, 4) > 0 for all 1-PS’sA,
xe Xf(L) © m”(x, 4) > 0 for all 1-PS’s A
The essential idea behind this proof stems from Hilbert [14], where the case G = SL(n),.X = P* is analyzed
There are two approaches to the proof of this theorem Either we
choose a G-linear immersion XCPy 1, and reduce to the case X == P,_4
by Theorem 1.19; or we can put Y equal to the normalization of the
closure of 0(x), and via the canonical 7: Y-> X, reduce to the case
where 0(x) is dense in X by Theorem 1.14 We shall follow the former which is more down to earth Therefore we may assume X =P,_1 Now let
V = E*P,_„„ ø(1)
A”= P(V): the affine cone over P„_¡
There is a natural projection from 4” — (0) to P„_¡ We shall say that 7¬
Trang 3250 2 Analysis of stability
P,,y As remarked in the proof of Proposition 1.7, the action of G on P,_; and the G-linearization of op, ,(1) together define also:
i) a dual action of G on V,
ii) a linear action o* of G on A*, compatible with o [with respect to the projection A* — (0) > P,_4]
The first step is:
Proposition 2.2 x is semi-stable if and only if (0) is not in the closure of the orbit 0(x*) for one (and hence all) closed points x* lying over x; x is properly stable if and only if yp,» is proper for one (and hence all) closed points x* lying over x
Proof By definition, x is semi-stable if and only if there is an in- variant s¢H°(P,_1, 0p(k)) such that s(x) 54 0 This is the same as asking whether there is an invariant homogeneous polynomial function F on A® such that F(x*) + 0, But if such an F exists, then F equals a non-zero constant on 0(x*), hence (0) cannot be in the closure of 0(«*) Conversely, suppose (0) is not in the closure of 0(x*) Then if:
Z, = closure of 0(x*), Z, = {(0)},
2, and Z, are disjoint invariant closed subsets of A* By corollary 1.2,
as G is reductive, there must be some invariant function F’ on A* such
that F’ = 0 on Z,, and F’ =: 1 on Z,, Suppose we write:
Fak, +F, +++ Fy
where each F,, is homogeneous of degree h; Then each &; is positive, ie each F,, is 0 on Z, Moreover some F,, is not zero at x* Let F be
that F, kẹt
By definition, x is properly stable if and only if its stabilizer is finite, and for one (and hence all) s¢ H°(P,_1, op(k)) as above, 0(x) is closed in [P,_,], (cf amplification 1.11 and lemma 0.3) That is to say, y y,:G— [P,_4],
is proper But let s correspond, as above, to the invariant homogeneous
function F on A* Then, for some non-zero « &., 0(x*) is contained in
the closed subscheme Z, of A” defined by F = « Therefore Pei G—> A”
is proper if and only if:
YieiG—>Z,
is proper But let the projection from A* — (0) to P,_, define the
morphism z: Z„-> [P„.;] Then, in fact: z(Z„) C [P„-;], and if x’
denotes œ with image taken as [P,.,],, then øœ is proper [ï.e let
§ 1 A numerical criterion B1
R = I'(A", 04), and let & be the degree of F Then Z, = Spec (R/(F — «))
and [P„_;]; = Spec (#(#)/(F — «)) But R/(F — «) is a finite module
over R(k)/(F — «).] Finally, py, = 2’ © yx, hence y, is proper if and only if py» is proper QED
The next step is to interpret the function u in the case where the ambient space X is P,_, We recall that a linear action of a torus on affine space can be diagonalized (cf Seminaire Chevalley, [8], exposé 4): i.e for a suitable coordinate system in A”, the closed points («) e (G,,)% act via diagonal matrices [y;(«) - 6], for characters y,, , 7, of (G,,)’ Moreover, recall that if (x) = (a,, ,«,), then every character + of (G,,)’ is of the form
x(a) = Tư? ¿=1
for suitable integers m; In particular, suppose we are given any action of G,, on P,_4, plus a G,,-linearization of (1) Then this gives a linear
action of G,, on the cone A", and, for suitable coordinates, the action
of ô Â (G,,), is given by the matrix («” - 6,), for fixed integers 7, ., 74 Proposition 2.3 Let x be a closed point of P,W1, let 4 be a 1-PS of G, and let o (A(x), x) specialize to y when a-> 0 Let x* be a closed homogeneous point over x Fix coordinates in A” so that the action of G,, induced via 4 is diagonalized as above Then
pO (x, A) == max {—7,| 7 such that xf + 0}
where x* = (z, , Z2} Moreover, [ø* (Â(œ), x*) has no specialization,
resp some specialization, resp specialization (0), in A*, when a —> 0]
<> [u(x; A) > 0, resp w(x, A) = 0, resp u(x, A) < 0]
Proof Let p, = (x;A) and ,ạ = max {—7;| xf 0} Note that
o* (A(x), x*) has coordinates
(0% + xf, 0 xX),
Therefore, x -x* has a finite non-zero specialization y* in A” when «> 0 Therefore, the last assertion is obvious, once we have proven
that „„ = uạ On the other hand, since a” z* has a non-zero speciali- zation in A”, this specialization lies over some point of P,_,, and this must be the specialization of o (A(«), x) Therefore y* lies over y Now
if y*=(y¥, , yZ), one sees that yf =0 if either x = 0 or 7;> —yp
Therefore,
ơ*(Â(x), y*) =x~ „*,
In other words, the trivial action of G,, on y, plus the G„-hnearization
of o(1) restricted to y, correspond to the linear representation on the cone {8+ y* | Bek} over y given by
ay > ey,
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52 2 Analysis of stability
- Here the affine line through y* is canonically the line bundle over y corresponding to the invertible sheaf 9(1) @x(y) Therefore, as we saw in §3 of Chapter 1, the character of G,, corresponding to the G-lineari- zation of o{1) over the fixed point y is:
(oe) = a
Therefore, 4 == j4, according to Definition 2.2 QED We fix the following notation:
R= *#{ữ]| (== formal power series ring in /)
K=k(()) (= quotient field of R)
We shall be interested in the groups of R and K-valued points of the reductive group G Let:
G(R) = Hom, (Spec R, G) = group of R-valued points, G(K) = Hom, (Spec K, G) = group of K-valued points Note that G(R) is a subgroup of G(K), i.e via the natural morphism Spec K — Spec R; moreover, there is a natural map @ from G(R) to G, induced by the morphism Spec k-» Spec R In classical language, G(R) is the subgroup of points ¢eG(K) which have a specialization
in G, when te K specializes to 0, and @(¢) is this specialization
If Ais a 1-PS of G, then a canonical K-valued point of G is defined by 4; it can be defined as the composition: Spec K _— G,, —+.¢ al Spec k [x, x3], where A, in turn, is defined by the &-homomorphism Ã: b[x, a1] => K A(x) =#
Note that, when ¢ specializes to 0, « specializes to 0 This point of G(K) will be denoted <A> We are now in a position to state:
Theorem (Iwaxort) Let G be a semi-simple algebraic group over R of adjoint type Every double coset of G(K) with respect to the sub- group G(R) is represented by a point of the type (4); for some 1-PS 2
In fact, Iwauori has proved this [16] for more general rings R, for e “Tohoku” or adjoint groups of CHEVALLEY On the other hand, suppose G is any reductive group and the characteristic is 0, and 2:G > G’ is the homomorphism of G to its adjoint group Since the characteristic
is 0, x is smooth; therefore the induced map G(R) — G'(R) is surjective,
since R isa hensel local ring Now starting with a K-valued point of G,
§ 1, A numerical criterion 53 we see that multiplying on the left and right by R-valued points, we can assume that its image in G’ becomes a K-valued point of a subgroup A(G„) C Œ” Then the point in G becomes a K-valued point of the sub- group x-1(A(Gm)): this is an extension of torus by a finite group By the
result for such a subgroup, the theorem is proven’ for G.f This result will be used in the last step of the proof:
(i) py is not proper if and only if for some non-trivial 1-PS A of G, a*(A(«), x*) has a specialization when « — 0
(ii) (0) is in the closure of 0(x*) if and only if for some 1-PS A of G, o* (A(x), x*) specializes to (0) when «> 0
Proof The two “‘if” statements are obvious Next consider the “only if” in (i): assume y,s is not proper Then by the valuative criterion for properness (EGA 2, § 7), there is a K-valued point ¢: Spec (K) >G such that ¢ is not induced by an R-valued point of G, but such that y.+°¢ is an R-valued point of A* By Iwahori’s theorem, then, we
know that ¢ has the form:
$= 41° A +H
for some 1-PS A of G, and for »,, y, ¢ G(R) Moreover, A is non-trivial
since ¢ is not itself an R-valued point of G Let b; eG, be the &-valued point
Ø(;) obtained from y; when ¢ specializes to 0 By choosing suitable coordinates in A*, we may assume that the action of by 1-2-8, on A” is diagonalized Suppose 071+ A(x) +b, acts via the matrix («*- 6,),
for «e (Gm); Now
Pye? $= o*( ,##)
= ø*[(0 - bạ) - (b1 ‹ <Â) - bạ) © (02 ° - ;), z*)
(Here k-valued points of G and of 4” are iđentified with the K-valued points obtained by the base extension K > &)
Therefore, since o* is a group action:
(*) a (4-225 yee © 6) = 0 [(by* + AD + bg); oF (BE cụy; x9)]1
But (p,-5,)-! is an R-valued point of G, and Pat ° ¢@ is an R-valued
point of A"; therefore the term on the left is an -R-valued point of A*
Now, if is a Spec (A)-valued point of A”, for ‘some ring A, let us denote by X;(/) the element of A which is its i coordinate Then,
recalling the definition of <A>, and that dy1-A- 6, is diagonalized,
} For the case of char (#) = p, see Appendix 2A Note that when G = GL(n), the result simply says that every nxn matrix A over A((é)) can be transformed to B = (6;¢%) by elementary row and column operations
Trang 3454 2 Analysis of stability equation (*) reads, in coordinates:
ŒỲ Xi*(pbj 2y ~s9)=/°Xi*0‹yz x9)
and, in particular:
(A) Xo" Og? py he OR,
But 61, is an R-valued point of G, whose specialization, when {->0,
is the identity e eG; therefore o* (by !y,, x*) is an R-valued point of A*
whose specialization, when t-> 0, is x*; therefore
(B) X,{o* (b5* yo, x*)} = X,(x*) + £(Z,)
for some Z;e R
Combining (A) and (B), it follows that 7; > 0, whenever X; (x*) 3 0 Therefore, u(x, b+ - by) < 0 and (i) is proven, by Proposition 2.3
The “only if” in (ii) is proven in the same way: assume (0) ¢ closure 0(x*) As a first step, we obtain a K-valued point ¢$ of G such that (0) is the specialization of y,ô Â when {-> 0 We decompose ¢ exactly as before, and obtain (*)’ But now the term on the left hand side of (*)’
is actually in ¢- R since o*((y,0,)-1, yy» ° ¢) actually specializes to (0)
when ¢-> 0 Therefore, instead of (A), we obtain:
(A) X,{o* (0g typ, e)} & HOF R,
(B) is the same as before, hence combining (A) and (B) it follows that r;> 0 whenever X;(x*) 54 0 Therefore, p(x, by} + A+ d,) <0 and (ii) is proven QED
A second application of Iwahori’s theorem is:
Proposition 2.4 Let a reductive algebraic group G over & act via ¢ on an algebraic scheme X over & Then the action of G is proper if and
only if for every non-trivial 1-PS1:G,,->G, the induced action of
G,, on X is proper
Proof The only if is clear To prove the if, suppose the action of G is not proper, ie ¥:GxX—+XxX is not proper Then, by the valuative criterion for properness, there are K-valued points ¢ and é of G and X respectively such that Ơ(Âxộ) is an R-valued point of
XXX, but $xé is not an R-valued point of GxX In other words,
š and ơ(ó, ‡) are R-valued points of X, but ¢ is not an R-valued point
of G By Iwahori’s theorem, we can write $ =ụ ' A> Ye
for R-valued points ựị and ; of Œ, and some 1-PS Moreover 2 is
non-trivial since ¢ is not an R-valued point of G, But then
&' = (y», Ê) a (<A>, £') = ơ(wr, ơ (ở, È))
and
§ 2 The flag complex 55
are both R-valued points of X In other words, the K-valued, but not
R-valued point {A> xé’ of GX X is mapped by Y to an R-valued point
of XXX Since by definition, <A> is induced from a K-valued point
of G, via A4:G_—>G, this means that the composite morphism Wo (AXI1x): GyxX—> XXX is not proper, ie: the induced action of G,, is not proper QED
Corollary 2.5.* Let a reductive algebraic group G act on an algebraic pre-scheme X Let Le Pic*(X) Then G acts properly on X{,)(L)
Proof Without loss of generality, we may replace X by X{)(L): then L is ample on X, and X is a scheme By the above proposition,
it suffices to prove that for every non-trivial 1-PS J of G, the induced
action of G,, is proper Fix some 4 Let U be the open set in X of points properly stable for the induced action of G,,: then J claim U = X For if x is a geometric point of X, then x has an affine neighborhood X,,
where s is an invariant section of L¥, and such that all stabilizers of
points of X, are 0-dimensional But then s is invariant under G,,, and
the action of G,, on X, is still closed Therefore, by definition, x is a
point of U
Therefore, a quasi-projective geometric quotient X/G,, exists But this implies that the action of G,, on X is separated Therefore, by lemma 0.5, the action of G,, on X is proper QED
2 The flag complex
In the last section, we ‘have related the concept of stability with respect to actions of G to the function w*(x; 4) involving a 1-parameter subgroup A of G In this section, we shall consider the dependence of ø on A The analysis leads naturally to an ungainly but remarkable metric space investigated extensively by J Tirs, which we will call the flag complex of G The first step is:
Definition 2.3/Proposition 2.6 Let G be a reductive algebraic group and let A be a 1-PS of G Then there is a unique algebraic subgroup
P(A) CG such that:
ye P(A), > A) -y + Ala) has a specialization in G when « € (G,,), specializes to 0 Moreover, P(A) is a parabolic subgroup of G, 4 is a 1-PS of the radical of P(a), and the specialization y’ of A(x) « y -4(«7), as a —> 0, centralizes A, for all ye P(A),
Proof Most of these facts can be seen most easily by means of a faithful representation ¢:G-—>GL(n) Recall that the image $(G) is
closed in GL(n) Therefore, P(A) = ở 1(P(d + À)), at least as a set of
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56 2 Analysis of stability
of GL(n), we may assume that $04 is diagonalized, i.e
Po Alor) = {a - 3,3
for « ¢ (G,,), and for suitable integers 7; We may even assume that 1 2%, 2 +++ > ry But now, if y = {a} eGL (n),,
[po Ae)]-y- [oA] = as a3
This has a finite specialization as «-> 0 if and only if a;; == 0 when
?¿<<?¡, ie y is of the form:
This is well-known to be a parabolic subgroup P C GL {n) Therefore, at least P(A) exists Secondly, the radical of P is precisely the set of
{ay} of the above form for which the diagonal blocks are multiples of
the identity, ie ay = a, - dj = ay + Oy; 1Í r; = rị Then Ais a 1-PS of this subgroup, hence it is also ‘a 1-PS of the radical of P(A) Thirdly, the specialization of [@s Â(œ)] -y - [@s Â(x-1)] as «> 0 is precisely
?ˆ ={ay} where
a, = đụ, if r= ? = 0, if 1? z ?‹
This certainly centralizes Ã
It remains to verify that P(A) is parabolic in G Let T be a maximal torus of G containing A(G,,) We use the notation of CHEVALLEY [8], exposé 9, § 5:
Definition 2.4 If T is a torus, then:
, I? (T) = Hom (G„, 7) @ Q
I*(T) = Hom (G,,,T) @ R
Now A defines a point A ¢ ['®(T ) Suppose — is in the closure of the Weyl chamber W, CI? (T) There is a canonical correspondence between the Weyl chambers WC I°®(T) and the Borel subgroups B of G con- taining T: namely, each W corresponds to the B spanned by T and those subgroups G, C G defining roots of T which are positive on W — cf Séminaire Chevalley [8], exposé 11, Theorem 1 Let the above W,
§ 2 The flag complex 57 correspond to B, Then it is easy to check that B,C P(A), which proves that P(A) is parabolic: namely, T itself is certainly in P(A) since it centralizes A Suppose G,C B, corresponds to the root 7: T-> G,,, ie
(a yo) =H (a) -y
for «se Ty, ye (G,), Let y define the linear functional 7: *(T)—> R Then ¥(w) > 0 if w ¢ W,, by definition of the correspondence between W, and B, Therefore —¥(A) => 0 But
Ala) +y Alo) = x (Ao) -y = gH y
for y e(G,), Therefore A(x) - y - A(x) has a finite specialization when œ->0, le G,C P(Â) Therefore B,C P(A) QED
Proposition 2.7 Let a reductive group G act via o on a scheme X,
proper over & Then for all xe X,, Le Pic? (X), and 1-PS’sA of G,
BH (%, a) = pi (x,y? Ay)
if ye P(A),
Proof Let y be the specialization of o (A(x), x) as x«~> 0 Suppose
we write y = im a (A(x), x) In this notation, we calculate:
Him o(y* Ala) «75 #) = lim @((7*- Ale) + Ao™); o Aa), 21) = o(y* -lim [A(@) «7 - Ao}; 9)
Let lim A(a)-y- Alo) =’ According to Proposition 2.6, y’ central-
izes Now (zs, yt -2-y) is calculated by looking at lim a(y - A()
xy; x) Therefore:
“@,yt+Ä‹y) =0(6iy 1y ;v),y *‹Â‹?)-
By property (i) of w (§1 above), we have:
BOY? 7,977 a+r) = w(ø0 9) 3) tứ
=0,y!:Ä-?) = p(y, A)
But p(y, A) =u(*,A) by property {iv) of uw Putting these together,
B(x, yt -A-y) = p(x, a) QED
In view of the last Proposition, we have reason to define the complex referred to above as follows:
Definition 2.5 Let G be a reductive algebraic group The rational
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58 2 Analysis of stability
flag complex A(G) is the set of non-trivial 1-PS’s 1 of G modulo the equivalence relation:
(*) 4, ~ 4, if there are positive integers », and n, and a point ye P(A), such that
| Aga) = yt «Ay (a!) -y
for all xe (G„j);
The point of 4(G) defined by  wHl be denoted 4(Â)
Note first of al that if ¡ — â;, then PA) = P(2¿) Therefore we
can talk of P(é) for a point 6 ¢A(G) Now, by means of a simple nor- malization of u, we can obtain from y a function on A(G) In fact, note
that if: A, («”) = Ag (a)
for all « ¢ (G,,),, and for positive integers n, and #, then the specializa- tions of o (A, (a), x) and of ø (2;(x), +) as œ —> 0 are the same Therefore we can readily show:
HE (a, Ay) 32 U(x, A)
Now suppose T is a maximal torus in G, and let N(T) be the normalizer of T Then N(T)/T is a finite group — the Weyl group — and it acts on J"*(T) via inner automorphisms There certainly exists at least one positive definite symmetric bilinear form <x, y> on I"®(T) which is invariant under this group and is rational on the rational subspace P9(T) Fix such a form We can then define a norm for a 1-PS 4 of G: for each A, there is a yeG, such that y-A-y! isa 1-PS of T
Let y-A-yt define y-A-yteD*(T) Then put
= Vy- arya)
We must check that this is independent of the choice of y By the invariance of <x, y> under the Weyl group, this amounts to:
Lemma 2.8 Suppose y ¢G,, and suppose 4 and y-A-y-1 are both 1-PS’s of T Then there is ay’ e N(T)}, such thaty -A-pob = y’-A-y' Proof* Let Z be the connected component of the centralizer of yA-y Since Ais a 1-PS of T, y-A-ytisa 1PS of y- Tey Therefore both T and y - T - y-! are maximal tori in Z By the conjugacy
theorem for maximal tori, there is a Be Z, such that
8-[y-T-y11-8 ?= [TI
Put y' =(8-y) QED
Definition 2.6 Let G be a reductive algebraic group Fix a norm
||4j| on the 1-PS’s of G as above Then ifG acts on the scheme X, proper
* I have taken this proof from a lecture of A Bork
§ 2 The flag complex 59
over k, if Le PicS(X), if xe X,, and if 6 = A(A), put: 7 (x; 6) = p(x, a)/||Al]
It follows immediately from the above that the left hand side is a func- tion only of 4(2), so that this definition is meaningful
The rest of this section will be devoted to examining the structure of A(G) Intuitively, 4(G) may be considered as the set of rational points at co on G It depends only on the Dynkin diagram of G, and can be constructed purely formally A(G) is extremely rich in structure For example, although we will not use this, via the norms |jA]| on the set of 1-PS’s of G, we can define the canonical metrics on A(G) In fact, we shall see below that if 6 and ¢ are any 2 points of A(G), then there are two 1-PS’s 4 and y of G such that 6 = 4(Â), e= Á(w) and
A and # commute with each other Then A- yu is also a 1-PS of G, and
we can define:
= +Í HA mHP _ H4H _ llzii
68,9) = =eses [Ta — Tin ~ II"
To bring out the structure of A(G), we require:
Lemma 2.9 If 6¢4(G), and T CG is a maximal torus, then
TC P(d) 4 Fa 1-PS A of T such that ơ =4Í|Â)
Therefore, if 3,, ô; are 3 points of 4(G), there is sorne maximal torus 7Ì such that both 6, and ở; are represented HA Snh of 7
Proof If A is a 1-PS of T, then T C P(A) by the definition of Đội) This establishes the implication «@ Conversely, if 7C P(ð) and 7 is a maximal torus, then any 1-PS of P(é) is conjugate by some o « P(d), to a 1-PS of T Since any A representing 6 is a 1-PS of P(6), this estab- lishes the implication = The last claim now follows from the fact that the intersection of any 2 parabolic subgroups contains some maximal torus (cf [34]) QED
Now suppose G and H are 2 reductive groups, and suppose that
$: H-+>G is a homomorphism with finite kernel Composition with ¢
maps non-trivial 1-PS’s of H to non-trivial 1-PS’s of G It is easy to check that this induces a map ¢,:4(H) > A(G)
Proposition 2.10 $4 is injective
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60 2 Analysis of stability
bs & = b4& Then P(p 0 Ay) = P($4e)) = P (bye) = P($4,) Call this group P, and let T’’ be the intersection of ¢(T) with the radical of P
On the one hand, notice that go 4, and $o 4, are 1-PS’s in $(T) and in the radical of P, hence in 7’ On the other hand, the set of all
characters of P defines homomorphism
y: P-> +
(T“” a torus) which is an isogeny when restricted to any maximal torus in the radical of P (compare the remarks in Chapter 1, § 1) In parti-
cular T"” has a character y’”’ such that y'"o yon T” is a power (y')#⁄ of zy’ Now by assumption,
A($o4,) =A(poA,), hence there is a y€P, and positive integers ?, f2 such that
Bo Ay (0) = y-Go dy (x) - yt
all « € (G,,), Therefore
pogpod, (a™) = po Po dy (a)
K opohod, = (y)Mo god, = 7% 04, = 0
1 poh ede = () các Ây = (0) s 2y sẽ 0
This is a contradiction, so therefore Pee, F $4lq QED
Corollary 2.11 If ¢ is an isogeny, „ is an isomorphism
Proof It remains to show that ¢, is surjective But if 2 is any 1- parameter subgroup of G, then for some positive integer n, and for some l-parameter subgroup # of H, $0 u(x) = A(x"), «eG, QED,
Suppose we apply Proposition 2.10 to the inclusion of a maximal torus fF in G This induces an inclusion of A (Z) in A(G) But A(T) is very simple — it is essentially the set of rational points on a sphere In fact, it follows immediately from the definition that A (17) is the set of rays in ['*(T) which contain a point of 9 (T) Therefore, in view of lemma 2.9, 4 (G) can be viewed as the result of pasting together spheres, one for each TCG, with sufficient identifications so that any two points are both on at least one sphere The subsets A (TZ) will be called skeletons of A(G) We can push this structure further in 2 ways: first the spheres A(T) can be broken up by considering the function P():
Definition 2.7 For all parabolic subgroups P, let Ap(G) be the set of d¢A(G) such that P(d) > P
To describe 4p(G) explicitly, let T be a maximal torus of P Then Ap(G) C A(T) by lemma 2.9 Suppose P is spanned by T and by additive subgroups G, C P such that
But while
ary a= Xa) -y
§ 2 The flag complex 61
for ae T,, y € (G,), and a suitable character y of T Then one checks
that if de A(T) is induced by Ae I*(T), and if x induces the linear functional ý on I’? (7), then
P()5P œ pysuch x,%(A) <0
Therefore, Áp(Œ) is simply the set of rational points on an intersection of hemi-spheres, (each determined by a rational linear functional, in fact) If G is actually semi-simple, it can be shown that Ap(G) is even the set of rational points on a spherical simplex, and hence the collection of Ap(G) constitutes a “triangulation”, and A(G) is the complex formed from the simplices Ap(G) This is the motivation of the terminology “flag complex’’.*
A second direction in which we can make explicit this structure on
A(G) is to show that the skeletons are pasted together in a sufficiently nice way so that various constructions which can be carried out in each skeleton can also be carried out in A(G) The principal one we have in mind is drawing a great circle through two points One approach to this is to complete the complex A(G) with respect to the metric 9 men- tioned above: then Tits has shown that the geodesic segments on this completion are precisely the segments of great circles of length <x, on the true spheres obtained by completing the skeletons A(T) Since
we are avoiding this metric, we define these lines, and other auxiliary
concepts, directly:
Definition 2.8 A pair of points 6,, 6,¢ A(T) is antipodal if there is a 1-PSA of G such that 6, = A(d) and 6, = A(A)
If G is a torus, then every point of A(G) has a uniqte antipodal point, i.e, its antipodal on the sphere is the usual sense More generally, if 6, is in the skeleton A(T) of A(G), for some maximal torus T, then the antipodal 6, of 6, as a point of A(T), is one of its antipodal points _in AG) In fact, this is the only point antipodal to 6, in A(T):
Lemma 2.12 If 6,, 6, are two points in the skeleton A(T) of A(G), for some maximal torus T, then 6, and 6, are antipodal as points of A(G) if and only if they are antipodal as points of the sphere 401 Proof Suppose 6, == A(A) and 6, == A(A~1) for some 1-PSA of G - Then Aisa 1-PS of P(A) N P(A) = P(6,) A P(6,); and T is a maximal torus of P(d,) \ P(d,) by lemma 2.9 Therefore there is some y ¢ [P(6,) ‘\ P(6,)], such that y-!-4-y is a 1-PS of T Then 6, = 40-+-Â-y) and ở; = A(y-}- 4-1 - y), hence 6, and 6, are antipodal in the group T QED
* In fact, if G is semi-simple, Tits defines his complex as follows: start with a single point 4, for each maximal parabolic subgroup P< G Then, if P,, , P, are maximal parabolic subgroups such that P,/\. NP, =P is parabolic, join the points 4p, , 42, with an ( — 1)-dimensional sim- plex 4,
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62 2 Analysis of stability
Now suppose 6, and 6, are not antipodal Suppose 6, and 6, are both in A(T) Then we make:
Definition 2.9/Proposition 2.13 Suppose 6; = A(d,) for a 1-PS A; of T Then the set of points A(ap - Af), where 7, , are non-negative integers, is called the line joining 6, and 6, It is independent of the choice of T
Before proving the last statement, we make precise the notation
Ap + Ag:
Ap? + Ag# (ox) = Ay (ce) + Ay (ox)
for all ô Â (G,,), Note that, since 6, and 6, are not antipodal, none of the 1-PS’s J} - A} is trivial Therefore A(A% - 43) is meaningful More- over, this “‘line”’ is also the set of rational points on the geodesic segments between 6, and 6, as points on the sphere obtained by completing A(T)
(in a standard metric)
Proof of Prop 2.13 First note the obvious remark: -
(*) If d,, 5, are in A(T) \ A(T,), then there is ay ¢ [P(6,) \ P(d,)}, such that T, = y?- T,-y
Proof T, and T, are both maximal tori in P(6,) \ P(6,) by lemma 2.9 Therefore they are conjugate in P(6,)\ P(é,) QED
Now suppose 6,, 6, are, in fact, in 4(7) 4(T;): we must show that the lines joining them in A(7,) and in A(T,) are the same Let T,=y1-T,-y where y € (P(6,) 1 P(6,)], and suppose 4,4, are 1-PS’s of T, representing 6,, 6,, respectively Then y~?-4,-y, y4-A, + are 1-PS’s of T, representing 6,, 6, respectively Therefore, we must show that the sets of points
(A (aps Ag)} and (4(07°:4-y)*°‹0@-°:&-y)®)
are the same But this is clear provided that A(Ap - A}*) = A(y“Ap-AR y); hence it follows if P(A.) \ P(A.) C P(Ap - Ap) — so that ye P(Ap - Ap) Therefore, we are reduced to proving that if 4,, A, are 1-PS’s of G
which centralize each other, then
(**) P(A, + 4g) 5 Pu) ^ P)
To prove this, it is convenient to represent G in GL{n) by some closed immersion $:G— GL(n) Then, exactly as in the proof of Pro- position 2.6, we have reduced the proof of (**) to the case G = GL (n) In this case, we can simultaneously diagonalize two commuting 1-PS’s:
say A,{«) is the matrix {a%-6,}, and A,(«) is the matrix {«%- 6,},
for ô Â (G,,),- Then, as in Proposition 2.6,
P(A) = {(a) [ag = 0 if 4,< 4} PL) = {(@,) |ag=0 if s;< s} P(A, + 4g) = {(4;) [ag = 90 if 1+ 5;< 7; + 5} § 3 Applications 63
Since 7; + s;< 7; + s; implies that 7; < 7; ors; < s;, it is immediate that P(A,-4,) > P(A)APA,) QED
Definition 2.10 A subset C C A(G) is semi-convex if it contains the line joining any pair of points 6,, 6, ¢ C provided 6, and 6, are not anti-
podal; C is convex if, in addition, it contains no pair of antipodal points
According to (**) above, it follows for example that the subsets A(T) and A p(G) are all semi-convex, for maximal tori T, and parabolic P Actually, if G is semi-simple, it can be shown that A p(G) is always convex
§ 3 Applications
In this section we return to the functions ø#(z, Ã} of 1-PS's  and y’ (x, 6) of be A(G) in order to apply the general theory to group actions
First of all, consider w§(x, 4) for a fixed action of G on some proper X, a fixed ample L ¢ Pic(C), a fixed xe X,, but where A varies among the 1-PS’s of a maximal torus T C G We use Proposition 2.3 to describe fe: namely, there exists a morphism
$:Z—>P„_ạ
for some ø%, pÏlus an action of G on P„_¡ and a G-linearization of op(1)
such that ¢ is G-linear and, for some N,
LẺ = g* (e()) NN
Therefore: p*(z, 4) = # pe) (px, a)
As in § 1, the action of G on P,_1 is induced by a linear representa- tion of G in the affine cone A" over P,_1 Suppose we choose coordinates in A* so that the representation of T is diagonalized: ie a ¢ T, acts via the matrix {y;(«) - 6,} for suitable characters y,, , 7%, of T Let zi induce the linear functional %; on ['*(T), and as usual let a 1-PSA
of T induce the point A eI? (T) Then if (px pees ox*) are coordinates of a homogeneous point lying over $x, Proposition 2.3 asserts:
pe) (px, A) == max {—7,(4) | ‡ such that af z 0}
We conclude:
Proposition 2.14 Let a reductive G act on X, proper over & Let
T CG be a maximal torus, and let L ¢ Pic®(X) be ample Then there
is a finite set of linear functionals 4, , J,onJ"*(T) which are rational on I'9(T) with the following property:
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64 : 2 Analysis of stability
Corollary 2.15 If 4,,4, are 1-PS’s of T and xe X,, then
B(x, Ay» Ag) <p" (x, dy) + (He, ds) Proof Immediate, since 4,- A, = 4, + A,
Corollary 2.16 Let a reductive G act on S, proper over & Let
L e Pic*(X) be ample, and let x X; Then {6 | »*(x, 6) < 0} is semi-
convex, and {6 | »*(x, 6) < 0} is convex Proof Immediate
It follows from this Corollary and from Theorem 2.1, that if «eX; is not semt-siable, then we can associate to x the convex set C of 6 ¢ A(G) for which *(x, 6) < 0 Roughly speaking, we can say that a parabolic subgroup P is “responsible” for this breakdown of semi-stability if C 1\ Ap(G) # ¢ Since C is convex, not too many parabolic subgroups P are responsible; according to a conjecture of Tits, there is even a natural way to find one P which is most responsible, i.e P(d) for 6 = the “center” of C
In fact, Tits conjectures that any convex subset C of a flag complex has a natural center 6 For the convex set C = {8|r2(x, ô) < 0}, KEMPE and Rousseau discovered quite simply that r(x, 6) is strictly convex on the line joining any 2 points 6,, 6, € C, and takes on a unique mini- mum at a single point dy € € (see Appendix 2B) This P(5,) may be considered the “worst” parabolic subgroup As a Corollary, they de- duce:
if x is rational over k, and not semi-stable for the action of G, then
there is a 1-PS A, rational over 2, such that u(x, 4) < 0
(This is a natural generalization of Godement’s conjecture on com-
pact fundamental sets.)
Another consequence of Proposition 2.14 is:
Proposition 2.17, Let a reductive G act via ¢ on X, proper over & Let L ¢ PicS(x) be ample Then there is a constant K such that
, | (x,ð)|<K
for ali xe X, and ôeA (G) Moreover, for every subset SC X}, there is a de A(G) and x,ÂS such that
Ơ (9 by) < ¥* (x, 8)
for every other 6 ¢ A(G)
Proof Let T C G be a maximal torus Let <x, y> be the form on
P*(T) which defines the norm ||A|| Let 7ạ, , J, be the linear func-
tionals on '*(T) given by Proposition 2.14 For any xe X;, let I (x) be the subset {1, , m} given by Proposition 2.14 To estimate v” (x, 6), § 3 Applications 65 1-PS of T Then: - wh(z,Ã) (x, 8) = Tai we (oly, 2), 7:4: 77) i ee a | L(y: A-y7}) VE a Ppa PD |
But, for each i, |2;(z)|//<z, z> is bounded when z varies over all non-
zero points of I**(T) Therefore »*(x, 6) is bounded when x and 6 vary arbitrarily Now suppose that for xeS C X;, we seek to minimize
v(x, 6) According to the above,
: : : 1, (z)
Bhool pt it [mores | eS zaS
But the first “inf'” ¡s essentially a minimum over the finite set of subsets
Ic {1, , 2} which occur as subsets I (o(y,x)) Therefore, this “inf”
will always be attained for some x and y As for the inner “inf’’, if we replace the variable by zeI"(T), then the “‘inf’’ will certainly be attained for some z — for the expression to be minimized is invariant under the transformation z—> «z, « > 0, and the set of rays in J"(7) is compact We must check that this “inf” is actually attained by a rational point z But each linear functional /; is rational, and the inner product <x, y> on [°*(T) is rational: then it is an exercise in clementasy calculus to check that this is so QED
An application of this is:
Proposition 2.18 Let G be a reductive group acting on algebraic
schemes X and Y Let {: X Y beaG-linear morphism, let Z ¢ Pic°({Y) and M « Pic*(X) Then if M is relatively ample for f and L is ample on Y, there is an m, such that:
n> Ny => Xo) (M@#£1 51 {Yin (L)}-
Proof We shall first reduce this Proposition to the case where X and Y are proper over k First of all, there is an open immersion Ty: Y C Y,, where Y, is proper over &, and there is an action of G on Y, extending the given one of G on Y, and there is an L, e Pic*(Y,), ample on Y, which extends L™ for some N, such that:
Ym (ZL) = Te {¥ 5 (L1)}
(cf Proposition 1.8) Secondly, for some Ny, M @ /*L™ is ample on X
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extends M™ @ /*L%" for some N (This N may be taken as the same N as above.) Then we have the diagram: Tx, T pe, xls ref) X,x¥, "mm Y CW Yy tr
Now if the result is proven when X and Y are proper, we can apply it to p,: X;x Y,-> Yj, and the sheaves M, @ oy, on X,xY, and L, on Y, Then f* oO} =r? UF (Vig (L)]} = (Ix, Ty ° * (77 [Yi (Li) } Cx Tye AT (GX Yo Wh @L)}, if n> np C Xm [(Lx, Ly ° f)* (My @ LY)] by Prop 1.18, = Xo) [M* @ /ƑRLNNH+Rm = X0 [M @ ƒfL +],
Now, if X and Y are proper over k, we note again that Ä @ /*L_" is ample on X for n > m Therefore, we can apply Theorem 2.1 to the computation of X()(M @ /*L") But if xe X,, de AG), then:
pM BIL" (xe, §) = yMBML™ (x, 8) + fh (x, ð)
= MOSEL" (x, 6) + (n — m)v* (F(x), 6)
First of all, apply the first part of Proposition 2.17 to »#@/*L", We conclude that, for some K:
z⁄@/*L”(x, ð) > —K
secondly, apply the second part of Proposition 2.17 to v*(y, 6), for „eS= Yậy (L); By Theorem 2.1, all these numbers are strictly posi-
tive Therefore, by the Proposition: vy, 6) >e>0 for all ye Yio)(Z),, and 6 ¢ A(G) Therefore, if n > [ m +4 + 1], we conclude yM@Me” (x, 6) > 0 whenever x & f~1{Y((L)}, Therefore, by Theorem 2.1, all such x are in Xiy(M @ f*L") QED 8 An elementary example 67 Chapter 3 An elementary example
The purpose of this chapter is to give, independently of the forgoing theory, an exhaustive analysis of a single special class of actions This has two objectives: first of all, the concepts of pre-stable and stable are worked out in a representative non-trivial case Secondly, by attack- ing this case directly, we can circumvent the difficulties involved in extending the previous work to semi-simple groups in characteristic p (cf Preface) And, in fact, we obtain the result over the ring of integers, hence, a fortiori, over any field This will enable us later to construct the so-called arithmetic schemes of moduli of abelian varieties
We have no use for a ground field in this chapter: all the schemes that we will consider will be of finite type over Spec (Z) A*, P,, and PGL(n) stand for the usual schemes over Z The action we shall study is the canonical action:
ơm„»:PGL(w + 1) x (P„)"?1 = (P„)n"t
ie of projective transformations on sequences of points in projective space
§ 1 Pre-stability
The first step is to obtain a large number of invariant affine open subsets in (P,,)"*+ To this end, let #9, ,m denote the projections of (P,)"*+' onto its m+ 1 factors Let L; =p} (op(1)) Now unfortunately, op(1) admits no PGL(n)-linearization with respect to the action of PGL(n + 1) on P, However ,if
w:SL(n + 1) > PGL(n + 1)
is the canonical isogeny, then with respect to the induced action of SL(n +1), op(1) admits an SZ(n + 1)-linearization This follows because SL(n + 1) acts on the affine cone over P,, compatibly with the given action of PGL (n) on P,, This linearization induces an SL (n+ 1)- linearization of L; for each 7 (for the induced action of SE + 1)), and therefore such a linearization for each product
@ Li
i=0
On the other hand, the {# + 1)“ power of the SZ(m + 1)-linearization
of op (1), which is a linearization of op(n + 1), is induced, via @, froma