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.. Modern Birkhauser Classics
Linear Algebraic Groups
Second Edition

Modern Birkhauser Classics
Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modem c lassics, entirely uncorrected, are being re-released in paperback (and as eBookS) to ensure that these treasures rem ain accessible to new generations of students, scholars. and researchers.

Linear Algebraic Groups
Second Edition T.A. Springer
Reprint of the 1998 Second Edition
Birkhauser Boston • Basel • Berlin

T.A. Springer Rijksunivcrsitcit Utrecht Marhcmatisch lnsti1uut Budapesllann 6 3584 CD Utrechl ll1e Nelherlands sp1inger@math.uu.nl
Originally published as Volume 9 in 1he series Progres.~in Matlrema1ics

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Library ofCongress Comrol Number: 2008938475 Malhcmatics Subjcc1 Classification (2000): I4XX. 20Gx.x

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T.A. Springer
Linear Algebraic Groups
Second Edition
Birkhauser
Boston • Basel • Berlin

T.A. Springer Mathematisch Instituut 3584 CD Utrecht The Netherlands

Library of Congress Cataloging-in-Publication Data

Springer, T.A. (Tonny Albert), 1926-

Linear algebraic groups/ T.A. Springer. -- 2nd ed.

p. cm. -- (Progress in mathematics ; v. 9)

Includes bibliographical references and index.

ISBN 0-8176-4021-5 (alk. paper). -- ISBN 3-7643-4021-5 (alk.

paper)

1. Linear algebraic groups. I. Title. II. Series: Progress in

mathematics (Boston, Mass.); vol. 9.

QA179.S67 1998

98-9333

512' .55--DC21

CIP

1991 AMS Subject Classification: 14XX, 20Gxx

Printed on acid-free paper © 1998 Birkhauser Boston, 2nd edition © 1981 Birkhauser Boston, 1st edition

a,)® Birkhauser ll{J?)

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ISBN 0-8176-4021-5 ISBN 3-7643-4021-5

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9 8 7 6 5 4 3 2 1

Contents
Preface to the Second Edition .............................................xiii 1. Some Algebraic Geometry ............................................... 1 1. 1. The Zariski topology .................................................... 1 1.2. Irreducibility of topological spaces ....................................... 2 1.3. Affine algebras ......................................................... 4 1.4. Regular functions, ringed spaces ......................................... 6 1.5. Products .............................................................. 10 1.6. Prevarieties and varieties ............................................... 11 1.7. Projective varieties ..................................................... 14 1.8. Dimension ............................................................ 16 1.9. Some results on morphisms ............................................. 17 Notes ..................................................................... 20
2. Linear Algebraic Groups, First Properties ............................... 21 2.1. Algebraic groups ...................................................... 21 2.2. Some basic results ..................................................... 25 2.3. G-spaces ............................................................. 28 2.4. Jordan decomposition .................................................. 31 2.5. Recovering a group from its representations .............................. 37 Notes ..................................................................... 41
3. Commutative Algebraic Groups ......................................... 42 3.1. Structure of commutative algebraic groups ............................... 42 3.2. Diagonalizable groups and tori .......................................... 43

Vlll

Contents

3.3. Additive functions ..................................................... 49 3.4. Elementary unipotent groups ............................................ 51 Notes ..................................................................... 56

4. Derivations, Differentials, Lie Algebras .................................. 57 4.1. Derivations and tangent spaces .......................................... 57 4.2. Differentials, separability ............................................... 60 4.3. Simple points ......................................................... 66 4.4. The Lie algebra of a linear algebraic group ............................... 69 Notes ..................................................................... 77

5. Topological Properties of Morphisms, Applications ....................... 78 5.1. Topological properties of morphisms .................................... 78 5.2. Finite morphisms, normality ............................................. 82 5.3. Homogeneous spaces .................................................. 86 5.4. Semi-simple automorphisms ............................................ 88 5.5. Quotients ............................................................. 91 Notes ..................................................................... 97

6. Parabolic Subgroups, Borel Subgroups, Solvable Groups ................. 98 6.1. Complete varieties ..................................................... 98 6.2. Parabolic subgroups and Borel subgroups ............................... 101 6.3. Connected solvable groups ............................................. 104 6.4. Maximal tori, further properties of Borel groups .......................... 108 Notes .................................................................... 113

Contents

IX

7. Weyl Group, Roots, Root Datum ....................................... 114 7.1. The Weyl group ...................................................... 114 7.2. Semi-simple groups of rank one ........................................ 117 7.3. Reductive groups of semi-simple rank one .............................. 120 7.4. Root data ............................................................ 124 7.5. 1\vo roots ............................................................ 128 7.6. The unipotent radical ................................................. 130 Notes .................................................................... 131

8. Reductive Groups ..................................................... . 132 8.1. Structural properties of a reductive group ............................... 132 8.2. Borel subgroups and systems of positive roots ........................... 137 8.3. The Bruhat decomposition ............................................. 142 8.4. Parabolic subgroups .................................................. 146 8.5. Geometric questions related to the Bruhat decomposition ................. 149 Notes .................................................................... 153

9. The Isomorphism Theorem ............................................ . 154 9.1. 1\vo dimensional root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.2. The structure constants ................................................ 156 9.3. The elements na ...................................................... 162 9.4. A presentation of G ................................................... 164 9.5. Uniqueness of structure constants ....................................... 168 9.6. The isomorphism theorem ............................................. 170 Notes .................................................................... 174

X

Contents

10. The Existence Theorem ............................................... 175 10.1. Statement of the theorem, reduction .................................... 175 10.2. Simply laced root systems ............................................ 177 10.3. Automorphisms, end of the proof of 10.1.1 ............................. 181 Notes .................................................................... 184

11. More Algebraic Geometry ............................................ 185 11.1. F-structures on vector spaces ......................................... 185 11.2. F-varieties: density, criteria for ground fields .......................... 191 11.3. Forms .............................................................. 196 11.4. Restriction of the ground field ........................................ 198 Notes .................................................................... 207 12. F-groups: General Results ............................................ 208 12.1. Field of definition of subgroups ....................................... 208 12.2. Complements on quotients ........................................... 212 12.3. Galois cohomology .................................................. 216 12.4. Restriction of the ground field ......................................... 220 Notes .................................................................... 222

13. F -tori ................................................................ 223 13.1. Diagonalizable groups over F ......................................... 223
13.2. F-tori .............................................................. 225
13.3. Tori in F-groups .................................................... 227 13.4. The groups P(>..) .................................................... 233 Notes .................................................................... 236

Contents

xi

14. Solvable F-groups .................................................... 237 14.1. Generalities ......................................................... 237 14.2. Action of Ga on an affine variety, applications .......................... 239 14.3. F-split solvable groups .............................................. 243 14.4. Structural properties of solvable groups ................................ 248 Notes .................................................................... 251

15. F -reductive Groups . .................................................. 252 15.1. Pseudo-parabolic F-subgroups ........................................ 252 15.2. A fixed point theorem ................................................ 254 15.3. The root datum of an F-reductive group ................................ 256 15.4. The groups Uca) . 262 ..................................................... 15.5. The index ........................................................... 265 Notes .................................................................... 268

16. Reductive F-groups .................................................. 269 16.1. Parabolic subgroups ................................................. 269 16.2. Indexed root data .................................................... 271 16.3. F-split groups ...................................................... 274 16.4. The isomorphism theorem ............................................ 278 16.5. Existence ........................................................... 281 Notes .................................................................... 284

17. Classification ......................................................... 285 17.1. 'fype An-1 · .......................................................... 285

xii

Contents

17.2. Types Bn and Cn ..................................................... 289 17.3. Type Dn ............................................................ 293 17.4. Exceptional groups, type G2 .......................................... 300 17.5. Indices for types F4 and E8 ........................................... 302 17.6. Descriptions for type F4 .............................................. 305 17.7. 'Iype E6 ............................................................. 310 17.8. Type E1 ............................................................ 312 17.9. Trialitarian type D4 .................................................. 315 17.10. Special fields ....................................................... 317 Notes .................................................................... 319

Table of Indices .......................................................... 320 Bibliography ............................................................. 323 Index .................................................................... 331

Preface to the Second Edition
This volume is a completely new version of the book under the same title, which appeared in 1981 as Volume 9 in the series "Progress in Mathematics," and which has been out of print for some time. That book had its origin in notes (taken by Hassan Azad) from a course on the theory of linear algebraic groups, given at the University of Notre Dame in the fall of 1978. The aim of the book was to present the theory of linear algebraic groups over an algebraically closed field, including the basic results on reductive groups. A distinguishing feature was a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra.
The present book has a wider scope. Its aim is to treat the theory oflinear algebraic groups over arbitrary fields, which are not necessarily algebraically closed. Again, I have tried to keep the treatment of prerequisites self-contained.
While the material of the first ten chapters covers the contents of the old book, the arrangement is somewhat different and there are additions, such as the basic facts about algebraic varieties and algebraic groups over a ground field, as well as an elementary treatment of Tannaka's theorem in Chapter 2. Errors - mathematical and typographical - have been corrected, without (hopefully) the introduction of new errors. These chapters can serve as a text for an introductory course on linear algebraic groups.
The last seven chapters are new. They deal with algebraic groups over arbitrary fields. Some of the material has not been dealt with before in other texts, such as Rosenlicht's results about solvable groups in Chapter 14, the theorem of Borel of Tits on the conjugacy over the ground field of maximal split torus in an arbitrary linear algebraic group in Chapter 15 and the Tits classification of simple groups over a ground field in Chapter 17. The prerequisites from algebraic geometry are dealt with in Chapter 11.
I am grateful to many people for comments and assistance: P. Hewitt and Zhe-Xian Wang sent me several years ago lists of corrections of the second printing of the old book, which were useful in preparing the new version. A. Broer, Konstanze Rietsch and W. Soergel communicated lists of comments on the first part of the present book
and K. Bongartz, J. C. Jantzen. F. Knop and W. van der Kallen commented on points of
detail. The latter also provided me with pictures, and W. Casselman provided Dynkin and Tits diagrams. A de Meijer gave frequent help in coping with the mysteries of computers.
Lastly. I thank Birkhauser - personified by Ann Kostant- for the help (and patience) with the preparation of this second edition.
T. A. Springer

Chapter 1
Some Algebraic Geometry
This preparatory chapter discusses basic results from algebraic geometry, needed to deal with the elementary theory of algebraic groups. More algebraic geometry will appear as we go along. More delicate results involving ground fields are deferred to Chapter 11.
1.1. The Zariski topology
1.1.1. Let k be an algebraically closed field and put V = kn. The elements of the polynomial algebra S = k[T1, ... , Tn] (abbreviated to k[T]) can be viewed as kvalued functions on V. We say that v E V is a zero of f E k[T] if f (v) = 0 and that
= v is a zero of an ideal I of S if f (v) 0 for all f E I. We denote by V (/) the set of
zeros of the ideal I. If X is any subset of V, let I(X) c S be the ideal of the f E S
= with f (v) 0 for all v E X.
Recall that the radical or nilradical ✓I of the ideal I (see [Jac5, p. 392]) is the ideal of the f E S with fn E I for some integer n > 1. A radical ideal is one that coincides with its radical. It is obvious that all I(X) are radical ideals.
We shall need Hilbert's Nullstellensatz in two equivalent formulations.
1.1.2. Proposition. (i) If I is a proper ideal in S then V(I) f= 0;
(ii) For any ideal I of S we have I(V(/)) = ✓I.
For a proof see for example [La2, Ch. X, §2] . The proposition can also be deduced from the results of 1.9 (see Exercise 1.9.6 (2)).
1.1.3. Zariski topology on V. The function I ~ V(I) on ideals has the following properties:
(a) V({O}) = V, V(S) = 0;
(b) If I C J then V(J) c V(I);
(c) V([ n J) = V(l) u V(J); (d) If Ua)aeA is a family of ideals and I = LaeA Ia is their sum, then V(I) =
naeA VUa).
The proof of these properties is left to the reader (Hint: for (c) use that I.J c I n J). It follows from (a), (c) and (d) that there is a topology on V whose closed
subsets are the V(I), I running through the ideals of S. This is the 7.ariski topology. The induced topology on a subset X of V is the Zariski topology of V. A closed set in V is called an algebraic set.
= 1.1.4. Exercises. (1) Let V k. The proper algebraic sets are the finite ones.

2

Chapter 1. Some Algebraic Geometry

(2) The Zariski closure of X c Vis V(I(X)). (3) The map I defines an order reversing bijection of the family of Zariski closed
subsets of V onto the family of radical ideals of S. Its inverse is V. (4) The Euclidean topology on en is finer than the Zariski topology.

1.1.5. Proposition. Let X C V be an algebraic set. (i) The Zariski topology of Xis T1, i.e., points are closed; (ii) Any family ofclosed subsets of X contains a minimal one; (iii) If X 1 :, X2 :, ... is a descending sequence of closed subsets of X, there is an h
such that X; = Xhfor i > h;
(iv) Any open covering of X has a finite subcovering.
If x = (x1, ... , Xn) E X then x is the unique zero of the ideal of S generated by
T1 - x1, ... , Tn - Xn. This implies (i). (ii) and (iii) follow from the fact that Sis a Noetherian ring [La2, Ch. VI, §1], using 1.1.4 (3).
To establish assertion (iv) we formulate it in terms of closed sets. We then have to
= show: if Ua)aeA is a family of ideals such that naeA VUa) 0, there is a finite subset = B of A such that naeB VUa) 0. Using properties (a), (d) of 1.1.3 and 1.1.4 (3) we see that LaeA Ia = S. There are finitely many of the Ia, say I1, ... , Ih, such that 1
n~ lies in their sum. It follows that I1+ ...+Ih = S, which implies that =1 V (I;) = 0. D

A topological space X with the property (ii) is called noetherian. Notice that (ii) and (iii) are equivalent properties (compare the corresponding properties in noetherian rings, cf. [La2, p. 142]. X is quasi-compact if it has the property of (iv).

1.1.6. Exercise. A closed subset of a noetherian space is noetherian for the induced topology.
1.2. Irreducibility of topological spaces

1.2.1. A topological space X (assumed to be non-empty) is reducible if it is the
union of two proper closed subsets. Otherwise Xis irreducible. A subset A c Xis
irreducible if it is irreducible for the induced topology. Notice that X is irreducible if and only if any two non-empty open subsets of X have a non-empty intersection.

1.2.2. Exercise. An irreducible Hausdorff space is reduced to a point.

1.2.3. Lemma. Let X be a topological space.
(i) A C X is irreducible ifand only if its closure A is irreducible; (ii) Let f: X ➔ Y be a continuous map to a topological space Y. If Xis irreducible
then so is the image f X.
Let A be irreducible. If A is the union of two closed subsets A1 and A2 then A is

1.2. Irreducibility oftopological spaces

3

the union of the closed subsets An A1 and An A . Because of the irreducibility of 2
A, we have (say) An A1 = A, and Ac A1, A c A1. So Ais irreducible.

Conversely, assume this to be the case. If A is the union of two closed subsets

An B1, A n B , where B1, B are closed in X, then A c B1 U B . It follows that

2

2

2

An B1 = A, whence An B1 =A.The irreducibility of A follows.

The proof of (ii) is easy and can be omitted.

□

1.2.4. Proposition. Let X be a noetherian topological space. Then X has finitely many maximal irreducible subsets. These are closed and cover X.

It is clear from 1.2.3 (i) that maximal irreducible subsets of X are closed.

Next we claim that Xis a union of finitely many irreducible closed subsets. As-

sume this to be false. Then the noetherian property 1.1.5 (ii) and 1.1.6 imply that

there is a minimal non-empty closed subset A of X which is not a finite union of irre-

ducible closed subsets. But A must be reducible, so it is a union of two proper closed

subsets. Because of the minimality of A these have the property in question, and a

contradiction emerges. This establishes the claim.

Let X = X I U ... U Xs, where the Xi are irreducible and closed. We may as-

sume that there are no inclusions among them. If Y is an irreducible subset of X then
Y = (Y n X 1) U ... U (Y n Xs) and by the definition of irreducibility we must have
Y c X; for some i, i.e., any irreducible subset of X is contained in one of the X;.

This implies that the X; are the maximal irreducible subsets of X. The proposition

follows.

D

The maximal irreducible subsets of X are called the (irreducible) components of
X.
We now return to the Zariski topology on V = kn.

1.2.S. Proposition. A closed subset X of V is irreducible if and only if I(X) is a prime ideal.

Let X be irreducible and let f, g e S be such that f g e I(X). Then

X = (X n V(f S)) u (X n V(gS))

and the irreducibility of X implies that (say) X c V(f S), which means that f e

I(X). It follows that I(X) is a prime ideal.

Conversely, assume this to be the case and let X =

V(Ii) U V(/ )

=

V(/1 n Ji).

2

If X =/:- V(/1), then there is f e / 1 with f (/. I(X). Since f g e I(X) for all g e h

it follows from the primeness of I(X} that / 2 c I(X), whence X = V(/2). So X is

irreducible.

□

1.2.6. Exercises. (1) Let X be a noetherian space. The components of X are its

4

Chapter 1. Some Algebraic Geometry

maximal irreducible closed subsets.
= (2) Any radical ideal / of Sis an intersection I P1 n ... n Ps of prime ideals. If
there are no inclusions among them, they are uniquely determined, up to order.
1.2.7. Recall that a topological space is connected if it is not the union of two disjoint proper closed subsets. An irreducible space is connected. The following exercises give some results on connectedness and the relation with the notion of irreducibility.
1.2.8. Exercises. (1) (a) A noetherian space X is a disjoint union of finitely many connected closed subsets, its connected components. They are uniquely determined.
(b) A connected component of X is a union of irreducible components.
(2) A closed subset X of V = kn is not connected if and only if there are two ideals 11, Ii in S with /1 + /2 = S, Ii n h = I(X).
(3) Let X = {(x, y) e k2 I xy = O}. Then Xis a closed subset of k2 which is con-
nected but not irreducible.
1.3. Affine algebras
1.3.1. We now tum to more intrinsic descriptions of algebraic sets. Let X c V be one.
The restriction to X of the polynomial functions of S form a k-algebra isomorphic to S /I(X), which we denote by k[X]. This algebra has the following properties: (a) k[X] is a k-algebra offinite type, i.e., there is a finite subset {f1, ... , fr} of k[X]
such that k[X] = k[f1, ... , fr];
(b) k[X] is reduced, i.e., 0 is the only nilpotent element of k[X].
A k-algebra with these two properties is called an affine k-algebra. If A is an affine k-algebra, then there is an algebraic subset X of some kr such that A ::::::: k[X]. For A ::::::: k[T1, ... , Tr]/ I, where I is the kernel of the homomorphism sending the T; to the generator f; of A (as in (a)), then A is reduced if and only if I is a radical ideal. We call k[X] the affine algebra of X.
1.3.2. We next show that the algebraic set X and its Zariski topology are determined by the algebra k[X].

If I is an ideal in k[X] let Vx(/) be the set of the x e X with f (x) = 0 for all f e /. If Y is a subset of X letix(Y) be the ideal in k[X] of the f such that f (y) = 0
for all y e Y. If A is any affine algebra, let Max(A) be the set of its maximal ideals. If
X is as before and x e X, then Mx = Ix ({x}) is a maximal ideal (because k[X]/ Mx
is isomorphic to the field k).
1.3.3. Proposition. (i) The map x H- Mx is a bijection of X onto Max(k[X]), moreover x E Vx(/) ifand only if I C Mx; (ii) The closed sets of X are the Vx([), I running through the ideals ofk[X].

1.3. Affine algebras

5

Since k[X] ~ S/I(X) the maximal ideals k[X] correspond to the maximal ideals

of S containing I(X). Let M be a maximal ideal of S. Then 1.1.4 (3) and 1.1.5 (ii)

imply that M is the set of all f e S vanishing at some point of kn. From this the first

point of (i) follows, and the second point is obvious. (ii) is a direct consequence of

the definition of the Zariski topology of X.

□

From 1.3.3 we see that the algebra k[X] completely determines X and its Zariski
topology.
1.3.4. Exercises. (1) For any ideal / of k[X] we have Ix(Vx(/)) = ✓/; for any
= subset Y of X we have Vx(Ix(Y)) Y.
(2) The map Ix defines a bijection of the family of Zariski-closed subsets of X onto the family of radical ideals of k[X], with inverse Vx. (3) Let A be an affine k-algebra. Define a bijection of Max(A) onto the set of homomorphisms of k-algebras A ➔ k. (4) Let X be an algebraic set.
(a) Xis irreducible if and only if k[X] is an integral domain (i.e., does not contain
zero divisors =I- 0).
= (b) X is connected if and only if the following holds: if f e k[X] and f 2
f, f =I- 0 then f = 1. (c) Let X 1, . . . , Xs be the irreducible components of X. If X; n Xi = 0 for
1 < i, j < s, i =I- j, then there is an isomorphism k[X] ➔ EBt~i:9 k[X;], defined
by the restriction maps k[X] ➔ k[X;].

1.3.S. We shall have to consider locally defined functions on X. For this we need special open subsets of X, which we now introduce.

If f E k[X] put
Dx(f) = D(f) = {x e X I f (x) =/- 0}.
This is an open set, namely the complement of V(fk[X]). We have
D(fg) = D(f) n D(g), D(fn) = D(f) (n > 1).
The D(f) are called principal open subsets of X.

1.3.6. Lemma. (i) If f, g E k[X] and D(f) C D(g) then fn e gk[X] for some n > 1; (ii) The principal open sets form a basis of the topology of X.

Using 1.1.4 (3) we see that D(f) C D(g) if and only if ✓(fk[X]) C ✓(gk[X]), which implies (i). (ii) is equivalent to the statement that every closed set in X is an intersection of sets of the form Vx(fk[X]). This is obvious from the definitions. D

6

Chapter 1. Some Algebraic Geometry

1.3.7. F-structures. Let F be a subfield of k. We say that Fis a.field of definition
of the closed subset X of V = kn if the ideal I(X) is generated by polynomials with coefficients in F. In this situation we put F[X] = F[T]/I(X) n F[T]. Then the
inclusion F[T] ➔ S induces an isomorphism of F[X] onto an F -subalgebra of S and an isomorphism of k-algebras k ®F F[X] ➔ k[X]. But this notion of field of definition is not intrinsic, as it depends on a particular choice of generators of k[X].
= The intrinsic way to proceed is as follows. Let A k[X] be an affine algebra.
An F-structure on X is an F-subalgebra Ao of A which is of finite type over F and which is such that the homomorphism induced by multiplication
k ®F Ao ➔ k[X]
is an isomorphism. We then write Ao = F[X]. The set X (F) of F-rational points for
our given F-structure is the set of F-homomorphisms F[X] ➔ F. More generally, if W is any vector space over k (not necessarily finite dimensional), an F -structure on W is an F-vector subspace W0 of W such that the canonical homomorphism

is an isomorphism. A subspace W' of W is de.fined over F if it is spanned by W' n W0. Then W' n W0 is an F-structure on W'.
1.3.8. A closed subset Y of X is F -closed (relative to our F-structure on X) if the ideal Ix(Y) is defined over F. A subset is F-open if its complement is F-closed. The F-open sets define a topology, the F-topology. An example of an F-open set is a principal open set D(f) with f e F[X]. These form a basis of the F-topology.
1.3.9. Exercise. Let k = C, F = R, let k[X] = C[T, U]/(T 2 + U2 - 1) and let
a, b be the images in k[X] of T and U. Show that R[a, b] and R[ia, ib] define two different R-structures on X. (Hint: consider the sets of rational points.)
1.4. Regular functions, ringed spaces
1.4.1. Notations are as in 1.3. Let x e X. A k-valued function f defined in a neighborhood U of x is called regular in x if there are g, h e k[X] and an open
neighborhood V C Un D(h) of x such that f (y) = g(y)h(y)- 1 for y e V.
A function f defined in a non-empty open subset U of X is regular if it is regular in all points of U. So for each x e U there exist gx, hx with the properties stated above. Denote by Ox(U) or O(U) the k-algebra of regular functions in U. The following properties are obvious: (A) If U and V are non-empty open subsets and U C V, restriction de.fines a kalgebra homomorphism O(U) ➔ O(V);
(B) Let U = UaeA Ua be an open covering of the open set U. Suppose that for each

1.4. Regular functions, ringed spaces

7

a EA we are given fa e O(Ua) such that if Ua n U/J is non-empty, fa and ftJ restrict to the same element of O(Ua n UfJ)· Then there is f e O(U) whose restriction to Ua
is fa, for all a E A.

1.4.2. Sheaves of functions. Now let X be an arbitrary topological space. Suppose that for each non-empty open subset U of X, a k-algebra of k-valued functions O(U) is given such that (A) and (B) hold. The function O is called a sheafofk-valuedfunctions on X. (We shall not need the general notion of a sheaf on a topological space.) A pair (X, 0) consisting of a topological space and a sheaf of functions is called a ringed space.
Let (X, 0) be a ringed space. If Y is a subset of X, we define an induced ringed space (f, Olr) as follows. Y is provided with the induced topology. If U is an open subset of Y then Olr(U) consist of the functions f on U with the following property:
LJ there exists an open covering U C Ua of U by open sets in X, and for each a,
an element fa e O(Ua) such that the restriction of fa to Un Ua coincides with the
restriction of f.

We leave it to the reader to show that Olr is a sheaf of functions. Notice that if Y
= is open we have Olr(U) O(U) for all open sets U of Y.

1.4.3. Affine algebraic varieties. The ringed spaces (X, Ox) of 1.4.1 are the affine algebraic varieties over k, which we also call affine k-varieties. In the sequel we shall usually drop the Ox and speak of an algebraic variety X.
We denote by Ox.x or Ox the k-algebra of functions regular in x e X. By definition these are functions defined and regular in some open neighborhood of x, two such functions being identified if they coincide on some neighborhood of x. A formal definition is
Ox = limind O(U),
where U runs through the open neighborhoods of x, ordered by inclusion and limind denotes inductive limit. We write An for the affine variety defined by kn. This is affine n-space.

1.4.4. Exercises. (1) Ox.xis a local ring, i.e., has only one maximal ideal (namely the
ideal of functions vanishing in x ).
(2) Let Mx C k[X] be the ideal of functions vanishing in x E X. Show that Ox,x
is isomorphic to the localization k[X]Mx. (If A is a commutative ring and S a sub-
s- set that is closed under multiplication, the ring of fractions 1A is the quotient of
S x A by the following equivalence relation: (s, a) ,.__, (s', a') if there is s1 e S with
s1(s'a - sa') = 0. The equivalence class of (s, a) is written as a fraction s- 1a and
these are added and multiplied in the usual way. If P is a prime ideal in A and Sis
s- the complement A - P, then 1A is written A p and is called the localization of A at
P. See [La2, Ch. II, §3].)

8

Chapter 1. Some Algebraic Geometry

Let (X, Ox) be an algebraic variety. It follows from the definitions that there is a homomorphism</> : k[X] ➔ O(X).
1.4.5. Theorem. </> is an isomorphism.
It is obvious that</> is injective. We have to prove surjectivity. Let f E O(X). For each x E X there exists an open neighborhood Ux of x and gx, hx E k[X] such that hx does not vanish at any point of Ux and that for y E Ux

= By 1.3.6 (ii) we may assume that there is ax E k[X] with Ux D(ax). Then D(ax) C
D(hx) and by 1.3.6 (i) there exist h: E k[X] and an integer nx > 1 with

QXnx

-
-

h

X

h'x

•

The restriction off to Ux equals gxh:(a;x)- 1. Observing that D(ax) = D(a;x) we
see that we may assume that hx = ax.

Since Xis quasi-compact (1.1.5 (iv)) there are finitely many of the hx, h 1, ... , hs,
such that the open sets D(h;) (1 < i =:; s) cover X. Let g; e k[X] be such that the
restriction off to D(h;) equals g;h;1(1 < i < s). Since g;h; 1 and gih°;-1 coincide on
= D(h;)nD(hj) whereas h;hi vanishes outside this set, we have h;h j(g;hi-gih;) 0.
Since the D(h;) cover X, the ideal generated by hf, ... , h; is k[X]. So there exist
b; e k[X] with

s
Lb;hf = 1.
i=l

Let x e D(hj). Then

s

s

hJ(x) Lb;(x)g;(x)h;(x) = Lb;(x)hf(x)hj(x)gj(x) = hJ(x)f(x).

It follows that f = </>(Lf=t b;g;h;), which proves surjectivity.

□

1.4.6. Exercise. Let D(f) be a principal open subset of X. Show that there is an
= isomorphism onto Ox(D(f)) of the algebra k[X]j k[X][T]/(1 - f T). (k[X]j is
s- = isomorphic to the ring of fractions 1k[X], where S (fn)n?.O; see 1.4.4 (2).)

1.4.7. Morphisms. Let (X, Ox) and (Y, Or) be two ringed spaces. Let</>: X ➔ Y
be a continuous map. If f is a function on an open set V c Y, denote by <l>v f the
function on the open subset <J,- 1V of X which is the composite off and the restric-
tion of </> to that set. We say that </> is a morphism of ringed spaces if, for each open

1.4. Regularfunctions, ringed spaces

9

V E Y, we have that </Ji, maps Or(V) into Ox(<J,-1V). If, moreover, (X, Ox) and
(Y, Or) are affine algebraic varieties, a morphism of ringed spaces X ➔ Y is called a morphism ofaffine algebraic varieties. It is clear how to define an isomorphism of affine varieties.
If Xis a subset of Y, q, the injection X ➔ Y and Ox = Orix (see 1.4.2) then
q, defines a morphism of ringed spaces in the previous sense. In this case we say that
(X, Ox) is a ringed subspace of (Y, Or).
A morphism q, : X ➔ Y of affine varieties defines an algebra homomorphism
Or(Y) ➔ Ox(X),
whence by 1.4.5 a homomorphism q,* : k[Y] ➔ k[X]. Conversely, an algebra homo-
morphism VI : k[Y] ➔ k[X] defines a continuous map VI : X ➔ Y with (VI )* = VI
(view the points of X and Y as homomorphisms, cf. 1.3.4 (3)). Then VI is a morphism
of affine varieties. If <I> is as before, we have (q,*) = q,. The upshot of this section is
that affine k-varieties and their morphisms can be described in algebraic terms.

1.4.8. Exercises. (1) Complete the proofs of the statements of the last paragraph. (2) Make affine k-algebras and affine algebraic varieties over k into categories. Show that these categories are anti-equivalent. (For categories see [La2, Ch. I, §7] or [Jac5, Ch. 1].)
(3) A morphism of affine varieties q, : X ➔ Y is an isomorphism if and only if the algebra homomorphism q,* is an isomorphism.
1.4.9. Affine F-varieties. Let F be a subfield of k and let (X, Ox) be an affine k-variety. An F-structure on this affine variety is given by the following data: (a) an F-structure on X, in the sense of 1.3.7; (b) for each F-open subset U of X we are given an F-subalgebra Ox(U)(F) of Ox(U) such that the homomorphism induced by multiplication
k ®F Ox(U)(F) ➔ Ox(U)
is an isomorphism and that properties like (A) and (B) of 1.4.1 hold. An affine variety over k with an F-structure will be called an affine F-variety. It is clear how to define morphisms of these (called F-morphisms).
The proof of 1.4.5 carries over to the case of F -varieties and gives that F[X] :::: Ox(X)(F). We conclude that affine F-varieties and their morphisms can be described in algebraic terms. An instance of an affine F-variety is affine n-space An (n > 0), whose algebra is k[TI , ... , Tn], If Xis an affine F-variety, its set X (F) of F-rational points (l.3.7) can be viewed as the set of F-morphisms A0 ➔ X.

1.4.10. Exercise. Complete details in 1.4.9.

10

Chapter 1. Some Algebraic Geometry

1.5. Products

1.S.1. Let X and Y be two affine algebraic varieties over k. In accordance with the general notion of product in a category [La2, Ch. I, §7] we say that a product of X and Y is an affine variety Z over k, together with morphisms p : Z ➔ X, q : Z ➔ Y such that the following holds: for any triple (Z', p', q') of an affine variety Z' together with morphisms p' : Z' ➔ X, q' : Z' ➔ Y, there exists a unique morphism r : Z' ➔ Z
such that p' = p o r, q' = q o r. Put A = k[X], B = k[Y], C = k[Z]. Using
1.4.7 we see that Chas the following property: there exist k-algebra homomorphisms
a : A ➔ C, b : B ➔ C such that for any triple (C', a', b') of an affine k-algebra and k-algebra homomorphisms a' : A ➔ C', b' : B ➔ C', there is a unique k-algebra
homomorphism c : C ➔ C' with a' = co a, b' = cob.
Working in the category of all k-algebras, i.e., forgetting the condition that k[Z] is an affine algebra, it follows from familiar properties (see e.g. [La2, Ch. XVI, §4])
that C = A ®kB and a(x) = x ® 1, b(y) = 1 ® y satisfy our requirements.

1.S.2. Lemma. Let A and B be k-algebras offinite type. If A and B are reduced

(respectively, integral domains) then the same holds for A ®kB.
I:7= Assume that A and B are reduced. Let 1a; ® b; be a nilpotent element of

A® B. We may assume the b; to be linearly independent over k. For any homo-

morphism h : A ➔ k we have that h ® id is a homomorphism A ® B ➔ B. Then
I:7=1 h(a;)b; is a nilpotent element of B, which must be zero since Bis reduced. As

the b; are linearly independent, all h(a;) are zero, for any h. This means that the a;

lie in all maximal ideals of A. It follows that a; = 0 for all i (apply 1.3.4 (1) with

k[X] = A, I= (a;)), which shows that A® Bis reduced. Next let A and B be integral domains. Let f, g e A ® B, f g = 0. Write
f = L; a;® b;, g = Lj ci ® dj, the sets {b;} and {dj} being linearly independent. An argument similar to the one just given then shows that a;ci = 0, from which it

follows that f or g equals 0.

D

1.S.3. Exercises. (1) Show that C ®RC is not an integral domain. (2) Show that in 1.5.2 the assumption that A and B are of finite type can be omitted.

1.S.4. Theorem. Let X and Y be two affine k-varieties. (i) A product variety X x Y exists. It is unique up to isomorphism; (ii) If X and Y are irreducible then so is X x Y.

From the discussion of 1.5.1 it is clear that it suffices to show that if A and B are

affine k-algebras (respectively, affine k-algebras that are integral domains), the same

is true for A ®k B. This follows from 1.5.2. The uniqueness statement of (i) follows

formally from the definition of products.

□

1.6. Prevarieties and varieties

11

1.5.S. Exercises. X and Y are affine varieties. (1) Show that the set underlying X x Y can be identified with the product of the sets underlying X and Y. (2) With the identification of (1), the Zariski topology on the X x Y is finer than the product topology. Give an example where these topologies do not coincide. (3) Let F be a subfield of k. A product of two affine F-varieties exists and is unique up to F -isomorphism.

1.6. Prevarieties and varieties

1.6.1. Prevarieties. A prevariety over k is a quasi-compact ringed space (X, Ox) (or simply X) such that any point of X has an open neighborhood U with the property that the ringed subspace (U, Olu) (see 1.4.7) is isomorphic to an affine k-variety. Such a U is called an affine open subset of X. A morphism of prevarieties is a morphism of the ringed spaces. A subprevariety of a prevariety is a ringed subspace which is isomorphic to a prevariety.

1.6.2. Exercises. (1) A prevariety is a noetherian topological space. (2) If X is an irreducible prevariety and U an affine open subset, then U is irreducible.
The notion of a product of prevarieties is defined in the categorical manner; see 1.5.1.

1.6.3. Proposition. A product of two prevarieties exists and is unique up to iso-

morphism.

Let X and Y be prevarieties and let X = LJ~1 U;, Y = Uj=1 Vi be finite cov-
erings by affine open sets. The underlying set of the product X x Y will be the set

theoretic product X x Y, which is covered by the sets U; x Vi. On these sets we have

a structure of affine variety (by 1.5.4 and 1.5.5 (1)). We declare a set U c X x Y to

be open if U n (U; x Vi) is an open subset of the algebraic set U; x Vi, for all i, j.
This defines a topology on X x Y. A function f, defined in an open neighborhood

U of x E U; x Vj, is defined to be regular in x if its restriction to Un (U; x Vj) is

regular in x, for the structure of affine variety on U; x Vj. This defines a structure of

ringed space on X x Y. One verifies that it has the required properties. The uniqueness

statement is proved in the standard manner.

D

1.6.4. Exercise. Fill in the details of the proof of 1.6.3.

1.6.S. Separation axiom. Let X be a prevariety, denote by ~x the diagonal subset of X x X, i.e.,
~x = {(x, x) Ix EX},
and denote by i X ➔ ~x the obvious map. We provide ~x with the induced topology.

12

Chapter 1. Some Algebraic Geometry

1.6.6. Example. Let X be an affine k-variety. Then ll.x is a closed subset of X x X, namely the set VxxxU) (see 1.3.2), where I is the kernel of the homomorphism
= k[X x X] k[X] ®k k[X] ➔ k[X]
defined by the product of the algebra k[X]. The ideal / is generated by the elements
f ® 1 - 1 ® f (f e k[X]). Since k[X x X]/ I '.:::'. k[X], we conclude that i now
defines a homeomorphism of topological spaces X ::::'. ll.x.

1.6.7. Exercise. Prove the assertions made in 1.6.6.

1.6.8. Lemma. i : X ➔ ll.x defines a homeomorphism of topological spaces for any prevariety X.

Cover ll.x by open sets of the form U x U, with U affine open in X, and use that

the result holds for affine varieties (1.6.6).

D

1.6.9. The prevariety X is defined to be a variety (or an algebraic variety over k or a k-variety) if the following holds:
Separation axiom. ll.x is closed in X x X.
By 1.6.6 this holds if Xis an affine variety. See 1.6.13 (1) for an example of a prevariety which is not a variety.
It is clear how to define morphisms of varieties.
1.6.10. Exercises. (1) Show that a topological space X is Hausdorff if and only if
the diagonal ll.x is closed in X x X for the product topology.
(2) The product of two varieties is a variety. (3) A subprevariety of a variety is a variety. (4) Let X be a variety. Define an induced variety structure on open and closed subsets of X.
One needs the separation axiom to establish the following results.

1.6.11. Proposition. Let X be a variety and Y a prevariety.
= (i) If <I> : f ➔ Xis a morphism, then its graph r t/J {(y, </>(y)) I y E f} is closed in
y XX;
= (ii) If</>, 1/f : f ➔ X are two morphisms which coincide on a dense set, then <I> 1/f.

In the situation of (i), consider the continuous map Y x X ➔ X x X sending (y, x)
to (</>(y), x). Then r ,pis the inverse image of the closed set ll.x, hence is closed. This = proves (i). In the situation of (ii) it follows similarly that {y e f I </J(y) 1/f(y)} is

closed, whence (ii).

D

1.6. Prevarieties and varieties

13

The following result gives a useful criterion for a prevariety to be a variety.

1.6.12. Proposition. (i) Let X be a variety and let U, V be affine open sets in X.
Then U n V is an affine open set and the images under restriction of Ox(U) and Ox(V) in Ox(U n V) generate the last algebra; (ii) Let X be a prevariety and let X = LJ~=l U; be a covering by affine open sets. Then
X is a variety if and only if the following holds: for each pair (i, j) the intersection
U; n Ui is an affine open set and the images under restriction of Ox(U;) and Ox(Uj) in Ox(U; n Uj) generate the ~ast algebra.
In the situation of (i), we have that L!lx n (U x V) is closed in U x V. Now i
induces an isomorphism of ringed spaces Un V :::: t!lx n (U x V). It follows that Un Vis affine and that regular functions on t!lx n (U x V) are restrictions of regular
functions on U x V, i.e., of elements of k[U] ® k[V]. Then (i) follows from the fact
that this algebra is generated by k[U] ® 1 and 1 ® k[V].

The necessity of the condition of (ii) follows from (i). If it is satisfied, then for

each pair (i, j) the intersection t!lx n (U; x Vj) is an affine algebraic variety whose

algebra is a quotient of Oxxx(U; x Uj). This implies that L!lx n (U; n Uj) is closed

in U; x Ui. Hence t!lx is closed in Xx X.

□

1.6.13. Exercises. (1) Define the 'line with doubled point 0' X as follows. As a
set X is the union of A1 and a point 0', moreover A1 is an affine open set, with its
= = usual variety structure. Define <P : A1 ➔ X by <P (x) x E A1 if x -f:. 0, <P (0) 0'.
Then Im <I> can be given a structure of affine variety isomorphic to A1. Define Im <I> to
be an affine open subset. Show that X is a prevariety which is not a variety.
(2) Define the projective line P1in a similar way: P1 = A1U {oo}, with <t,(x) = x- 1 = = for x -=I- 0, <P (0) oo. Show that P1 is a variety. Show that CJp1 (P1) k and deduce
that any morphism of P1 to an affine variety is constant.
(3) (a) Let U C An be open and non-empty. If f E OAn(U), there exist g, h E
= k[T1, ... , Tn] such that h does not vanish on U and that f (x) g(x)h(x)- 1 for
x E U. (Hint: cover U by principal open sets and use 1.4.6).
(b) Let X = An - {0}, with the induced structure of variety. Show that Xis not
an affine variety if n > 2.

1.6.14. F-structures. Let F be a subfield of k. We say that the k-variety X has an F -structure or that X is an F -variety if we are given a family of open subsets of X, called F -open subsets, with the following properties: (a) the F-open subsets form a topology; (b) the F-open subsets, which are also affine open, cover X (these sets will be called affine F-open sets); (c) an affine F-open set has a structure of affine F-variety;
(d) if U and V are two affine F-open sets with V c U, then the inclusion morphism
of affine varieties V ➔ U is defined over F.

14

Chapter 1. Some Algebraic Geometry

If X and Y are two F-varieties, a morphism </> : X ➔ Y is defined over F if for
any F -open set V c Y, the set U = <1>- 1U is F-open in X and if, moreover, the
induced morphism U ➔ V is defined over F. We also say that </> is an F -morphism. The notion of an F -isomorphism is the obvious one. Likewise the notion of an F subvariety.
If a k-variety X has an F -structure, we shall say that F is a ground field for X.
1.6.15. To define an F -structure on a k-variety X, the following data suffice: a covering (U;) of X by affine open subsets, together with F-structures on the U; and their
intersections, such that all inclusion morphisms U; n Uj ➔ U; are defined over F.
An F -open set is defined to be an open set whose intersection with U; is F -open in U;, for all i. Then the properties of 1.6.14 hold.
1.6.16. If X is an F-variety, the set X (F) of its F -rational points is the set of F morphisms A0 ➔ X (cf. 1.4.9).
If X and Y are two F-varieties, there is a unique structure of F-variety on the product X x Y of 1.5 such that the projection morphisms to X and Y are defined over F.
1.6.17. Exercises. (1) Check the statements made in 1.6.15 and 1.6.16. (2) A morphism of F-varieties </> : X ➔ Y is defined over F if and only if its graph (1.6.11 (i)) is a closed F-subvariety of the F-variety Xx Y of 1.6.16. (3) Let X beak-variety. There exists a subfield F of k which is an extension of finite type of the prime field in k such that X has an F -structure.
1.7. Projective varieties
The most important example of non-affine varieties, and practically the only ones that we shall encounter, are the projective spaces and their closed subvarieties, to be discussed in the present section.
1.7.1. pn. The underlying set of projective n-space pn is the set of all one dimensional subspaces of the vector space kn+I or, equivalently, kn+I - {O} modulo the
equivalence relation: x ,.._, y if there is a e k* = k - {O} such that y = ax. Write x* = for the equivalence class of x. If x (x0, x 1, ... , Xn), we call the x; homogeneous
coordinates of x*. For O < i < n, put
U; = {(xo, ... , Xn)* E pn I x; =I= O}.
Define a bijection </>; : U; ➔ An by

1. 7. Projective varieties

15

and transport the structure of the affine variety on An to Ui via <Pi. Then <Pi (Ui n Ui)
is a principal open set D(/) in An. In fact, identifying k[An] with k[T1, ... , Tn], we
may take f = Ti+ 1 if j < i, f = 1 if j = i and f = Ti if j > i.
We make P" into a prevariety in the following manner (cf. 1.6.15). A subset U
is defined to be open if U n Ui is open in the affine variety Ui, for O :::: i :::: n. Let x e pn and assume x e Ui. A function f, defined in a neighborhood of x, is declared
to be regular in x if the restriction of f to Ui is regular in x for the structure of affine
variety of Ui which was introduced above. As in 1.4.1 we obtain a sheaf Opn and a
ringed space (P", Opn), which is a prevariety (check the details).
This is, in fact, a variety. From the definitions we see that O(Ui n Uj) (0 < i, j <
n) is the k-algebra of functions whose value in (x0, ... , Xn)* is a polynomial function
1 1 of xi- Ixo, . . . , x;- 1Xn, x 1xo, . . . , x 1Xn. It is then clear that the condition of 1.6.12
(ii) is satisfied.
= The variety thus obtained is a projective n-space. For n 1 we recover the vari-
ety of 1.6.13 (2) (check this). A projective variety is a closed subvariety of some pn,
i.e., a closed subset with the induced structure of a variety. A quasi-projective variety
is an open subvariety of a projective variety.

1.7.2. Exercises. (1) An invertible linear map of kn+l induces an isomorphism of pn.
(2) Let V be a finite dimensional vector space over k. Define a variety P(V) whose underlying point set is the set of one-dimensional subspaces of V and which is isomorphic to pn- I, where n = dim V. (3) Let F be a subfield of k. Define an F-structure on P", inducing on each Ui the F -structure obtained by transporting the F -structure of An.

1.7.3. Closed sets in pn. We shall now give a concrete description of closed sets
in P". Let S = k[To, ... , Tn] be the polynomial algebra in n + 1 indeterminates.
An ideal / in S is homogeneous if it is generated by homogeneous polynomials or,
equivalently, if / 0 + •••+ fh e /, where f; is homogeneous of degree i, then all /;
lie in/. If/ is a proper homogeneous ideal in S, then if x e kn+l is a zero of/, the same
is true for all ax, a e k*. Hence we can define a set V* (I) e P" by
V*(l) = {x* E pn IX E Vkn+I (/)}.
1.7.4. Proposition. The closed sets in pn coincide with the sets V*(l), I running through the homogeneous ideals of S.
Let / be a homogeneous ideal. It is easy to see that V* (/) n U; is closed for all i,
from which it follows that V*(l) is closed. Let U be open in P". To prove 1.7.4 it suffices to show that U is the com-
plement of some V*(l), and by an analogue of 1.1.3 (d), it suffices to do this if

16

Chapter 1. Some Algebraic Geometry

U = </>;1(D(f)) C U; (for some i), where f e k[T1, ... , Tn]. There is a homo-
geneous polynomial f * e k[To, ... , Tn] which is divisible by T;, such that

Then U is the complement of V* (f *S).

□

1.7.5. Exercises. (1) Let f e S be homogeneous and# 0. Let Sj be the algebra of ra-
tional functions gf-h, where g e Sis homogeneous of degree deg g = hdeg f. Show that D* (f) = pn - V* (f S) is an affine open subset of pn and that Opn (D* (f)) ~ Sj.
(2) Let/ be a homogeneous ideal in S.
(a) Show that V*([) = 0 if and only if there exists N > 0 such that 1t e / for
0 < i < n.
(b) Show that V* ([) is irreducible if and only ✓/ is a prime ideal.
(3) Let F be a subfield of k. The F -closed subsets of pn, for the F -structure of 1.7.2
(3), are the V*([), where the homogeneous ideal / is generated by polynomials with
coefficients in F. (4) (a) Define a map of sets <J,: pm x pn ➔ pmn+m+n by

Show that the image of <J, is a closed subset ym.n of pmn+m+n and that <J, defines an isomorphism of varieties of pm x pn onto the projective variety defined by ym.n.
(b) The product of two projective F -varieties is isomorphic to a projective F variety.
1.8. Dimension
1.8.1. Let X be an irreducible variety. First assume that X is affine. Then k[X] is
an integral domain (1.2.5). Let k(X) be its quotient field [La2, p. 69]. If U = D(f) is a principal open subset of X, then k[U] = k[X] 1 (1.4.6) from which it follows that
the quotient field k(U) is isomorphic to k(X). Using 1.3.6 (ii) we conclude that the same holds for any affine open subset U.
Now let X be arbitrary. Using 1.6.12 (ii) the preceding remarks imply that for any two affine open sets U, V of X, the quotient fields k(U), k(V) can be canonically identified. It follows that we can speak of the quotient field k(X) of X. If X is an irreducible F-variety, we define similarly the F-quotient field F(X). The dimension dim X of X is the transcendence degree of k(X) over k (see [La2, Ch. X, §1]). If Xis reducible and if (X;)I!'::i!'::m is the set of its components, we define
dimX = max;(dimX;).
= If X is affine and k[X] k[x 1, ... , Xr] then dim X is the maximal number of
elements among the x; that are algebraically independent over k.

1.9. Some results on morphisms

17

1.8.2. Proposition. If X is irreducible and if Y is a proper irreducible closed subvariety of X, then dim Y < dim X.

We may assume X to be affine. Let k[X] = A = k[x1, ... , Xr ], k[Y] = A/ P,

where P is a non-zero prime ideal. Let y; be the image in k[Y] of x; and put d =

dim X, e = dim Y. We may assume that y 1, ... , Ye are algebraically independent.

Clearly, x 1, ... , Xe are also algebraically independent, whence e < d. Assume that

d = e. Let f be a non-zero element of P. There is a relation H(f, x 1, ... , Xe) = 0,

with H e k[To, ... , Tel We may assume that His not divisible by T0. But then we

would have a non-trivial relation H(0, YI, ... , Ye) = 0. It follows that we must have

d < e.

□

1.8.3. Proposition. Let X and Y be irreducible varieties. Then dim X x Y dimX +dimf.

We may assume X and Y to be affine. Let x1, ... , Xd and Y1, ... , Ye be maximal sets of algebraically independent elements in k[X], respectively k[Y]. Then

is such a set in k[X] ® k[Y] = k[X x Y].

□

1.8.4. Exercises. (1) dim An = dimP" = n. (2) A zero dimensional variety is finite. (3) Let f e k[T1, ... , Tn] be an irreducible polynomial. The set of its zeros is an (n - 1)-dimensional irreducible subvariety of An. (4) Let X be an irreducible F-variety. Then dimX equals the transcendence degree of F(X) over F.

1.9. Some results on morphisms

1.9.1. Lemma. Let <I> : X ➔ Y be a morphism of affine varieties and let </>*
k[Y] ➔ k[X] be the associated algebra homomorphism. (i) If</>* is surjective, then <I> maps X onto a closed subset of Y; (ii)</>* is injective if and only if</>Xis dense in Y;
(iii) If Xis irreducible, then so is the closure </>X and dim</>X < dim X; (iv) Let F be a sub.field ofk. If X and Y are F -varieties and <I> is de.fined over F, then
<I> X is an F -subvariety of Y.

Put I =Ker</>*. If</>* is surjective, then </>X = Vy(/), whence (i). Also,</>* is injective if and only if Iy(<J>X) = {0}, which implies (ii). The first point of (iii) follows from 1.2.3 and the last point from (ii), applied to the restriction morphism X ➔ <J>X, using 1.8.2. Notice that <l>X = Vy(/). In the situation of (iv), the ideal I is spanned

18

Chapter I. Some Algebraic Geometry

by the kernel of the homomorphism F[Y] ➔ F[X] induced by <J>. This implies (iv). □
Theimage<J,Xneednotbeclosed. Asanexample,letX = {(x, y) E A2 1 xy = 1}
and define</> : X ➔ A I by </>(x, y) = x. Then </>Xis the open set A 1 - {0}.
1.9.2. For the proof of the main result 1.9.5 of this section, we need some algebraic results.
Let B be a reduced ring and A a subring such that B is of finite type over A. Suppose that we are given a homomorphism of A to an algebraically closed field. We want to extend it to a homomorphism of B. We first assume that we have the special
case that B = A[b] is generated over A by one element. Then B ::::'. A[T]/I, where / is the ideal of the f E A[T] with f (b) = 0. It does not contain non-zero constant
polynomials. Denote by :J(l) the union of the set of leading coefficients of the nonzero polynomials in/ and {0}. This is an ideal in A.
1.9.3. Lemma. Let K be an algebraically closed field and let </> : A ➔ K be a
homomorphism such that <J>:1(1) # {O}. Then</> can be extended to a homomorphism
B ➔ K.
Let f = Jo + /1 T + ... + f mTm E / be such that </>fm # 0. We may assume
that m is minimal. We shall proceed by induction on m. First extend </> to the obvious homomorphism A[T] ➔ K[T], also denoted by</>. Assume that </>I does not contain a non-zero constant. Then it generates a proper ideal of K[T]. Let z E K be a zero of
that ideal. It is immediate that </>b = z then defines an extension of </> to B.
We claim that the assumption always holds. If not then / contains a polynomial
g = go+ ... + gnTn with </>(g0) # 0, </>(g;) = 0 (i > 0). The division algorithm shows that there exist q, r E A[T] and an integer d > 0 such that J:/ig = qf + r and that deg r < m. Then </> (f:/i)<I> (go) = </>q .</>f + </>r. Since </>f has degree m > 0,
we have that <J>r is also a non-zero constant. This means that we may assume n < m.
Then g cannot exist if m = 1, proving the claim in that case. Assume that m. > 1 and
that the assertion of the lemma is true for smaller values of m.
If h =ho+ ... + hsTs E A[T] and hs # 0, put h = Tsh(T- 1) = hs + ... + hoTs. Let i be the ideal in A[T] generated by the ii with h E /. If a E in A, there is
an integer s > 0 such that aTs E /, whence (a TY E /. Since B is reduced, we
= have aT E /. We conclude that in A is the ideal J I {a E A aT E /}. If
<J>J # {0} we have m = 1, contradicting the assumption m > 1. So <J>J = {0}. Put
A = A/J, B = A[T]/i = A[b]. Then B is reduced. In fact, if f E A[T] and ft -E i, then T" ft -E / for some u ::-: 0. Since B is reduced we can conclude that
T f E /, whence / E /, proving that B is reduced.
Now </> defines a homomorphism 4, : A ➔ K. Notice that i contains g = gn + ... + goTn, with </>(g0) # 0. Since n < m the induction assumption shows that 4, extends to a homomorphism B ➔ K. As <J>(g;) = 0 for i > 0, we have 4,(b) = 0. But i also contains j and 4,(j) # 0. This contradiction establishes our claim. D

1.9. Some results on morphisms

19

1.9.4. Proposition. Let B be an integral domain and let A a subring such that B

is offinite type over A. Given b =I- 0 in B there exists a =I- 0 in A such that any

homomorphism <I> of A to the algebraically closed field K with </>(a) =I- 0 can be

extended to a homomorphism <I> : B ➔ K with </>(b) =I- 0.

We have B = A[b1, ... , bn], By an easy induction we may assume that n = 1,

i.e., that B = A[bi] ~ A[T]/I, as before. First assume that I =I- {0}. Let f E / be

non-zero and of minimal degree. Denote by a1 its leading coefficient. The division
af algorithm shows that g E / if and only if for some d > 0 we have that g is divisible
by f. Leth e A[T] represent the given element b, then h ¢ /. Since f is irreducible

over the quotient field of A, it follows that f and h are coprime over that field. Hence
there exist u, v E A[T] and a2 e A - {O} such that uf + vh = a2. Then a = a,a2

is as required. For if <I> is as in the proposition, it follows from the preceding lemma

that <I> can be extended to B. Then </>(v(b1))</>(b) = </>a2 =I- O, whence </>(b) =I- 0. This

settles the case / =I- {O}. The easy case I = {O} is left to the reader.

□

1.9.5. Theorem. Let <I> : X ➔ Y be a morphism of varieties. Then cpX contains
a non-empty open subset of its closure cp X.
Using a covering of Y by affine open sets, we reduce the proof to the case that Y is affine. Using 1.3.6 (ii) and 1.4.6 we see that X may also be taken to be affine. If X 1, . . . , X s are the irreducible components of X, we have c/JX = LJi cp Xi, from which we see that we may also assume X to be irreducible. Replace Y by cpX. Then the assertion of the theorem is a consequence of 1.9.4, with A= k[Y], B = k[X], b = 1.
□

1.9.6. Exercises. (1) Let X be a variety. A subset of X is locally closed if it is the intersection of an open and a closed subset. A union of finitely many locally closed sets is a constructible set.
(a) The complement of a constructible subset of Xis constructible. (b) Let <I> : X ➔ Y be a morphism. Deduce from 1.9.5 that the image </>X is a constructible subset of Y. (Hint: proceed by induction on dim X, using 1.8.2.) (c) If C is a constructible subset of X then </>C is constructible. (2) (a) Let E be a field and let F be a field extension of F which is an F-algebra of finite type. Then Eis a finite algebraic extension of F. (Hint: use 1.9.4.) (b) Let F be a field. If M is a maximal ideal in the polynomial ring S = F[T1, ... , Tn], then S/ I is a finite algebraic extension of F.
(c) Let / be an ideal in S. Then the radical ✓/ is the intersection af the maxima!
ideals containing /. (Hint: Let f be a non-zero element in that intersection and let f be its image in S/ I. Show that (S/ /) J = {O}.)
(d) Prove the Nullstellensatz 1.1.2 (note that the proof of 1.9.4 does not use results from the previous sections).

20

Chapter 1. Some Algebraic Geometry

Notes
This first chapter contains standard material from algebraic geometry and needs few comments. We have also included the definition of algebraic varieties over a ground field that is not algebraically closed, as well as some simple results on such varieties. More delicate results will be taken up in Chapter 11. In another terminology our algebraic varieties are the schemes of finite type over a field which are absolutely reduced. For more about algebraic geometry we refer to Hartshome's book [Har] or Mumford's notes [Mu2].
In 1.4 we have introduced only sheaves of functions. We did not discuss more general sheaves, as they will not be needed in the sequel. Generalities about sheaf theory can be found in [God].
In the literature 1.9.4 is often proved by using valuations. We used an elementary approach that goes back to Chevalley and Weil [We3, p. 30-31]. The auxiliary result 1.9.3 will also be used in 5.2.

Chapter 2
Linear Algebraic Groups, First Properties
In this chapter algebraic groups are introduced. We establish a number of basic results, which can be handled with the limited amount of algebraic geometry dealt with in the first chapter. k is an algebraically closed field and F a subfield. All algebraic varieties are over k.
2.1. Algebraic groups
2.1.1. An algebraic group is an algebraic variety G which is also a group such that
= the maps defining the group structure µ : G x G ➔ G with µ(x, y) xy and
i : x i--+ x- 1 are morphisms of varieties. {We may view the set of points of the variety G x G as a product set, see 1.5.5 (i) and 1.6.3). If the underlying variety is affine, G is a linear algebraic group. These are the ones we shall be concerned with. It is usual to use the adjective 'linear' instead of 'affine' (this is explained by 2.3.7 (i)).
Let G and G' be algebraic groups. A homomorphism of algebraic groups </J : G ➔ G' is a group homomorphism is also a morphism of varieties. The notions of isomorphism and automorphism are clear.
The product variety G x G', provided with the direct product group structure is an algebraic group, the direct product of the algebraic groups G and G' (check this).
A closed subgroup H of the algebraic group G is a subgroup that is closed in the Zariski topology. Then there is a structure of algebraic group on H such that the inclusion map H ➔ G is a homomorphism of algebraic groups.
We say that the algebraic group G is an F -group if G is an F -variety, if the morphisms µ and i are defined over F, and if the identity element e is an F -rational point (see 1.6.14). The notions of an F -homomorphism of F -groups and of an F subgroup are clear.
If G is an F -group, the set G(F) of F -rational points (1.6.16) has a canonical group structure.
2.1.2. Let G be a linear algebraic group and put A = k[G]. By 1.4.7 the mor-
phisms µ and i are defined by an algebra homomorphism a : A ➔ A ®k A (called comultiplication) and an algebra isomorphism t : A ➔ A (called antipode). Moreover, the identity element is a homomorphism e: A ➔ k. Denote by m : A® A ➔ A
= the multiplication map (so m(f ® g) f g) and let E be the composite of e and the
inclusion map k ➔ A. The group axioms are expressed by the following properties:
(associativity) the homomorphisms a® id and id® a of A to A ®A® A coincide;
(existence of inverse) mo (t ® id) o a =mo (id® i) o a = E;
(existence of identity element) (e ® id) o a = (id® e) o a = id (we identify k ® A
and A® k with A).

22

Chapter 2. Linear Algebraic Groups, First Properties

The properties can also be expressed by commutativity properties of the following diagrams:

-----A®A
!id®L\
A®A ----- A®A®A,

A®A ~ A®A,
id®1
e®id
A--A®A
;d®, I"';"" lA
A®A - - - A.
Let G be an F-group and put Ao = F[G]. Then 6, t and e come from F-
algebra homomorphisms Ao ➔ Ao ®F Ao, Ao ➔ Ao, Ao ➔ F, respectively, which we denote by the same symbols. They have properties similar to the ones listed above.
2.1.3. Exercises. (1) Check the translation of the group axioms stated in 2.1.2.
(2) Define the notion of a prealgebraic group, based on the notion of a prevariety (see 1.6.1 ). Show that a prealgebraic group is an algebraic group.
2.1.4. Examples. The notations are as in 2.1.1.
(1) G = A1 = k with addition as group operation. This defines a linear algebraic
= group, with A k[T]. The homomorphism 6 : k[T] ➔ k[T] ® k[T] ~ k[T, U] is
given by l::t.T = T + U and t : k[T] ➔ k[T] by tT = -T. Moreover, the algebra
homomorphism e : k[T] ➔ k sends T to 0. We denote this algebraic group by Ga; it is the additive group. It is obvious that F[T] defines an F-structure on Ga, for any subfield F of k.

2.1. Algebraic groups

23

r- (2) G = A1 - {O} = k*, with multiplication as group operation. Now A= k[T, 11 and 6 : k[T, r-11 ➔ k[T, r- 11® k[T, r- 11 '.::'. k[T, r-1, U, u-11is given by r- 6T = TU. Moreover, tT = 1 and e send T to 1. This algebraic group is denoted
by Gm or by GL1; it is the multiplicative group. It has the F-structure defined by
F[T, r-1].
If n is any non-zero integer then </)x = xn defines a homomorphism of algebraic groups Gm ➔ Gm. If k has characteristic p > 0 and n is a power of p then </) is an isomorphism of abstract groups but not of algebraic groups (since</)* : A ➔ A is not surjective, see 1.4.8 (3)). (3) View the set Mn of all n x n-matrices as kn2 in the obvious manner. For X E Mn let D(X) be its determinant. Then Dis a regular function on Mn. The general
linear group GLn is the principal open set {X E Mn I D(X) # O}, with matrix
multiplication as group operation.
v- We have A= k[T;j, 1h=:;i,j=:;n, with D = det(T;j). Now 6 is given by
L n
6Tij = T;h ® Thj·
h=l
and tT;i is the (i, j)-entry of the matrix (Tab)- 1. The identity e sends T;i to aii (Kro-
necker symbol). For n = 1 we recover the previous example. Since Mn is an ir-
reducible variety so is GLn (by 1.2.3 (i)). Its dimension is n2. Finally notice that
F[T;j, v-11defines an F-structure.
(4) Any subgroup of GLn which is closed in the Zariski topology of GLn defines a linear algebraic group. Here are a number of examples:
(a) a finite subgroup; (b) the group Dn of non-singular diagonal matrices; (c) the group Tn of upper triangular matrices X = (xij) E GLn with Xij = 0 for i > j; (d) the group Un of unipotent upper triangular matrices, i.e., the subgroup of the previous group whose elements have diagonal entries 1; (e) the special linear group SLn ={XE GLn I det(X) = l};
(f) the orthogonal group On = {X E GLn I 'X.X = I}, where' X denotes the
transpose of X;
(g) the special orthogonal group SOn = On n SLn;
(h) the symplectic group Sp2n ={XE GL2n I 'X J X = J}, where J is the matrix
( -~. ~· )
(5) As examples of non-linear algebraic groups (not needed in the sequel) we mention the elliptic curves. These are closed subsets of the projective plane P2. Assuming for convenience that the characteristic is not 2 or 3, such a group G can be defined to be the set of (x0 , x1, x2)* E P2 (notations of 1.7.1) such that
xox? = Xf + ax1x5 + bxJ,

24

Chapter 2. Linear Algebraic Groups, First Properties

where a, b Ek are such that the polynomial T 3 + aT + b has no multiple roots.
The neutral element e is (0, 0, 1)*. The group operation of G is commutative and is written as addition. It is such that if three points of Gare collinear in P2 (i.e., if their
homogeneous coordinates satisfy a non-trivial linear relation c0x0 + c1x1 + c2x2 = 0),
= then their sum is e. This defines addition. It is easy to check that, if x (x0 , x1, x2)* E
= G, we must have -x (xo, x1, -x2)*.
The addition can be described by explicit formulas which, however, are not very enlightening. A proof of the associativity of the group operation based on these formulas would be clumsy. There are better, more geometric, ways to deal with the group structure on such a curve. We refer to [Har, p. 321].

2.1.5. Exercises. (1) Let V be a finite dimensional vector space over k.
(a) Define a linear algebraic group GL(V) whose underlying abstract group is the group of all invertible linear maps of V and which is isomorphic to GLdim v.
(b) An F-structure Vo on V (l.3.7) defines a structure of F-group on GL(V). The
corresponding group of rational points GL (V) (F) is the group GL (V0 ) of invertible F -linear maps of V0.
(2) Check that the subgroups of GLn listed in 2.1.4 (4) are indeed closed.
(3) We have A= k[SL2] = k[T1, T2, T3, T4]/(T1T4 - T2T3 - 1) = k[t1, t2, t3, t4] (t;
denoting the image of T;). Let B be the subalgebra of A generated by the products
t;tj(l < i, j ~ 4).
= (a) Let fl. and t define the group structure of SL2. Show that !l.B C B ® B, tB
B and deduce that there is an algebraic group PSL2whose algebra is B. Show that the inclusion map B ➔ A defines a homomorphism of algebraic groups SL2 ➔ PSL2
with kernel of order at most two.
(b) If char(k) =j:. 2, then B is the algebra of functions f E A such that f (- X) =
f (X) for all X E SL2,
(c) If char(k) = 2, the homomorphism of (a) defines an isomorphism of underly-
ing abstract groups but is not an isomorphism of algebraic groups.
(4) Show that the group Tn of 2.1.4 (c) is solvable. (5) Show that the automorphisms of G0 (= k) are the multiplications by non-zero ele-
ments of k.

2.1.6. Generalizations of algebraic groups. (a) The description of the notion of an algebraic group in terms of algebra homomorphisms, given in 2.1.2, leads to the following generalization. Let R be a commutative ring and A a commutative R-algebra. Assume given homomorphisms of R-algebras fl. : A ➔ A ®R A, t : A ➔ A, e : A ➔ R, such that we have the properties of 2.1.2, with k replaced by R. We say that
= the set of data G (A, fl., t, e) defines a group scheme over R (more precisely, an
affine group scheme.) We shall occasionally encounter this notion. But we shall not go into the theory of group schemes. It is dealt with, for example, in [DG, SGA3]. (b) Let G be a group scheme over R. It follows from the axioms of 2.1.2, that for

2. 2. Some basic results

25

each R-algebra S, the set G(S) of R-algebra homomorphisms A ➔ S has a canonical group structure. In fact, S ~ G(S) defines a functor from the category of R-algebras to the category of groups. Such a functor is an R-group functor. Group functors generalize group schemes. For more about group functors we refer to [loc.cit.]. (c) A more recent generalization that has become quite important is obtained by admitting in (a) non-commutative R-algebras A. In this case I!:.. and e are as before, butt is required to be an anti-automorphism. We impose the same axioms as before. Moreover, we require that the opposite algebra Aopp (i.e., A with reversed multiplication)
= has the same properties, relative to I!:.., ,-1, e. Now the set of data G (A, I!:.., t, e)
defines a quantum group over R. We refer the reader to [Jan2, Kas] for more about the theory of quantum groups and for examples.

2.2. Some basic results

Let G be an algebraic group. If g e G, the maps x ~ gx and x ~ xg define isomorphisms of the variety G. We shall frequently use this observation.

2.2.1. Proposition. (i) There is a unique irreducible component G0 ofG that contains the identity element e. It is a closed normal subgroup offinite index; (ii) G0 is the unique connected component of G containing e; (iii) Any closed subgroup of G offinite index contains G0.

Let X and Y be irreducible components of G containing e. If µ, and i are as in

2.1.1, it follows from 1.2.3 that XY =µ,(Xx Y) and its closure XY are irreducible.

Since X and Y are contained in X Y, it follows from 1.2.6 (I) that X = Y = X Y.

It also follows that X is closed under multiplication. Since i is a homeomorphism,
we see that x-1 is an irreducible component of G containing e, so must coincide

with X. We conclude that X is a closed subgroup. Using that inner automorphisms
define homeomorphisms, one sees that for x e G, we have xxx-1 = X, so that X

is a normal subgroup. The cosets xX must be the components of G, and by 1.2.4 the
number of cosets is finite. We have proved that G0 = X has the properties of (i).

We also see that the irreducible components of G are mutually disjoint. It then

follows that the irreducible components must coincide with the connected components

(use 1.2.8 (i)). This implies (ii).

If H is a closed subgroup of G of finite index, then H0 is a closed subgroup of

finite index of G0. Then H 0 is both open and closed in G0. Since G0 is connected,

= we have H0 G0, which proves (iii).

□

The proposition shows that, for algebraic groups, the notions of irreducibility and connectedness coincide. In the sequel we shall, as is usual, speak of connected algebraic groups, and not of irreducible ones. We always denote by G0 the component of the algebraic group G containing the identity (briefly: the identity component).
Dimensions being defined as in 1.8.1, we see from 2.2.1 that all components of G have dimension dim G.

26

Chapter 2. Linear Algebraic Groups, First Properties

2.2.2. Exercises. (I) The groups Ga, Gm, GLn, Dn, Tn, Un, SLn of 2.1.4 are connected. (2) Assume that k has characteristic =I= 2.
(a) The group On is not connected.
(b) Let V be the set of skew symmetric n x n-matrices. Then x i-+ ( 1+x )-1( 1- x)
defines an isomorphism of a non-empty open subset of SOn onto an open subset of V. Show that SOn is the identity component of On. (3) The variety of 1.2.8 (3) cannot be the underlying variety of an algebraic group. (4) Let G be a connected algebraic group and let N be a finite normal subgroup. Then N lies in the center of G. (Hint: for n e N consider the map x i-+ xnx-1 of G to N .)

= 2.2.3. Lemma. Let U and V be dense open subsets of G. Then UV G.

Notice that a subset of G is open and dense if and only if it intersects any component of G in a non-empty open subset. The intersection of two such subsets is one with the same properties.

Let x e G. Then xv-1 and U are both dense open subsets. They have a non-

empty intersection (1.2.1 ), which means that x e UV.

□

Notice that if G is connected we need only require U and V to be open and non-empty.

2.2.4. Lemma. Let H be a subgroup of G.
'fi) The Closure H is a subgroup of G;
(ii) If H contains a non-empty open subset of H then His closed.
= Let x e H. Then H xH c xH. Since xH is closed we have H C xH and
x- 1H C H, whence H H C H. Now let x e H. Then Hx C H, and a similar
= = argument shows that H is closed under multiplication. Since (H)- 1 H- 1 H, we
conclude that H is a group. If U c H is open in H and non-empty, then H, being
a union of translates of U, is open in H. By 2.2.3 we conclude that H = H. H = H. □

2.2.5. Proposition. Let </J : G --+ G' be a homomorphism ofalgebraic groups.

(i) Ker </J is a closed normal subgroup of G;

(ii) </J(G) is a closed subgroup of G';

(iii) If G and G' are F-groups and </J is defined over F then </J(G) is an F-subgroup

ofG';
= (iv) </J(G0) (</JG)0.

= Ker </J </J-1e is closed in G, whence (i). By 1.9.5, </J(G) contains a non-empty

open subset of its closure. Then (ii) follows from 2.2.4 (ii) and (iii) is a consequence

of 1.9.1 (iv). </J (G0) is a closed subgroup of G' by (ii), which is connected (by 1.2.3

(ii)) and which is of finite index in </JG. Using 2.2.1 (iii) we obtain (iv).

□

2.2.6. Proposition. Let (X;, </J;);eJ be a family of irreducible varieties together with morphisms </J; : X; --+ G. Denote by H the smallest closed subgroup of G containing
the images Y; = </J; X; (i e /). Assume that all Y; contain the identity element e.

2.2. Some basic results

27

(i) H is connected;

(ii) There exist an integer n 2: 0, a = (a(l), ... , a(n)) E in and E(h) = ±1 (I <

= h :::: n) sueh that H

YaE((Il))

•••

y

t(n)
a(n);

(iii) Assume, moreover, that G is an F-group, that all X; are F-varieties and that the

morphisms </>; are defined over F. Then H is an F-subgroup.

= We may assume that the sets Y;- 1 occur among the Yi. For each element a
(a(l), ... , a(n)) of some in, we write Ya = Ya(l)··•Ya(n)• This is an irreducible sub-

set of G, and so is the closure Ya (see 1.2.3 (i)). With obvious notations we have
= Yb. Ye Y(b,e). An argument as in the proof of 2.2.4 then shows that Yb. Ye C Y(b,e).

Now take a such that dim Ya is maximal. For any b, we then have Ya C Ya. Yb C Y(a,b).
= From 1.8.2 we conclude that Y(a,b) Ya and Yb C Ya, for all b. It also follows that

Ya is closed under multiplication, which implies that Ya is a group. By 1.9.5 and 2.2.3
= = we have Ya Ya. Ya. It follows that H Ya has the properties stated in (i) and (ii).

Now (iii) is a consequence of 1.9.1 (iv).

D

2.2.7. Corollary. (i) Assume that (G;);el is a family of closed, connected, subgroups of G. Then the subgroup H generated by them is closed and connected. There is an integer n 2: 0 and a = (a(l), ... , a(n)) E 1n such that H = Gao>--•Ga(n); (ii) If, moreover, G is an F-group and all G; are F-subgroups then His an Fsubgroup.

If Hand Kare subgroups of G, we denote by (H, K) the subgroup generated by thecommutatorsxyx- 1y- 1 withx E H,y EK.

2.2.8. Corollary. (i) If H and K are closed subgroups of G one of which is con-

nected, then (H, K) is connected,·

(ii) If, moreover, G is an F-group and H, Kare F-subgroups then (H, K) is a con-

nected F-subgroup.

= Assume that H is connected. Then (i) follows by applying 2.2.6 with / K, all
X; being H, with </>;(x) = xix-1i-1(i EK). The statement of (ii) follows from 1.9.1

(iv), using that by 2.2.6 (H, K) is the image under an F-morphism (H x K)n ~ G.

In particular, the commutator subgroup (G, G) of a connected F -group is a connected

F-subgroup.

□

2.2.9. Exercises. (1) (a) Give another proof of the connectedness of SOn in characteristic =j:. 2 (see 2.2.2 (2)) using 2.2.7 and the fact that On is generated by 'symmetries' (see [Jac4, p. 353].
(b) Prove by a similar argument that Sp2n is connected for arbitrary k, using that Sp2n is generated by 'symplectic transvections', see [loc. cit., p. 3731). (2) (char(k) =j:. 2) The complement of SOn in On is irreducible and generates On, Deduce that in 2.2.6 the condition e E Y; cannot be omitted. (3) Let G be a connected F -group and let n be an integer 2: 2. The subgroup o<n> of

28

Chapter 2. Linear Algebraic Groups, First Properties

G generated by the n-th powers of its elements is a connected normal F -subgroup. (4) Show by a counterexample that 2.2.8 (i) is not true if neither H nor K is connected. (Hint: take Hand K finite.)
2.3. G-spaces
2.3.1. A G-variety, or a G - space, is a variety X on which G acts as a permutation group, the action being given by a morphism of varieties. More precisely,
= there is a morphism of varieties a : G x X ➔ X, written a (g, x) g .x, such that g.(h.x) = (g.h).x, e.x = x. If, moreover, G is an F-group and Xis an F-variety,
then X is a G-space over F if a is defined over F. A homogeneous space for G is a G-space on which G acts transitively.
Let X and Y be G-spaces. A morphism <I> : X ➔ Y is a G-morphism, or is said
= to be equivariant, if <J>(g.x) g.<J>(x)(g E G, x EX).
Let X be a G-space and let x EX. The orbit of xis the set G.x = {g.x I g E G}. The isotropy group of xis the closed subgroup Gx = {g E G I g.x = x} (check that
G x is closed).
2.3.2. Examples. (1) X = G and G acts by inner automorphisms: a(g, x) = gxg-1.
The orbits are the conjugacy classes of G and the isotropy groups are the centralizers of elements of G.
(2) X = G and G acts by left (right) translations: a(g, x) = gx (resp. xg- 1). This is
an example of a homogeneous space, even of a principal homogeneous space or torsor, where the isotropy groups are trivial (which means that the action of G is simply transitive). (3) Let V be a finite dimensional vector space over k. A rational representation of G in V is a homomorphism of algebraic groups r : G ➔ GL(V) (see 2.1.5 (1)). We also say that V is a G-module (it being understood that r is also given). In this case we can view V as an affine algebraic variety isomorphic to Adim v, with a G-action
defined by g.v = r(g)v (g E G, v E V). We also have a structure of G-variety on the
projective space P(V) of 1.7.2 (2).
If G is an F-group, a rational representation over Fis a homomorphism of algebraic groups G ➔ GL(V) which is defined over F, where now V is a finite dimensional vector space with an F-structure, the F-group G L(V) being as in 2.1.5 (1).
Another version of the definition of a rational representation of G is: a homomorphism of algebraic groups r : G ➔ GLn, for some n > 1.
Assume the situation of 2.3.1.
2.3.3. Lemma. (i) An orbit G.x is open in its closure; (ii) There exist closed orbits.

2.3. G-spaces

29

Application of 1.9.5 to the morphism g ~ g.x of G to X shows that G.x con-

tains a non-empty open subset U of its closure. Since G.x is the union of the open

sets g.U(g e G), the assertion (i) follows. It implies that for x e X, the set
= Sx G .x - G .x is closed. It is a union of orbits. By 1.1.5 (ii) there is a mini-

mal set Sx. Because of (i) it must be empty. Hence the orbit G.x is closed, proving

(ii).

□

The lemma implies that an orbit G.x is a locally closed subset of X (see 1.9.6 (1)), i.e., an open subset of a closed subset of X. It then has a structure of algebraic variety (use 1.6.10 (4)). It is immediate that this is a homogeneous space for G.

2.3.4. Exercises. (1) Let G be a closed subgroup of GLn. Then An has a struc-
ture of G-space. Determine the orbits for G = GLn, Dn, SLn (see 2.1.4 (4)). (2) There is an action of G = GL2 on the projective line P1 (see 2.3.2 (3)), which
makes P 1 into a homogeneous G-space. Describe the isotropy group of a point. The
diagonal action of G on P1 x P 1 is not homogeneous. In fact, there are two orbits.
(3) Generalize the results of the previous exercise to GLn, acting on pn-1.

2.3.S. From now on we assume that G is a linear algebraic group. Let X be an
affine G-space, with action a : G x X ➔ X. We have k[G x X] = k[G] ®k k[X] and
a is given by an algebra homomorphism a* : k[X] ➔ k[G] ® k[X] (see 1.4.7). For
g E G, x E X, f E k[X] define

= (s(g))f (x) f (g-1x).

Then s (g) is an invertible linear map of the (in general infinite dimensional) vector space k[X] ands is a representation of abstract groups G ➔ GL(k[X]). The next result will imply thats can be built up from rational representations (see 2.3.9 (1)).
2.3.6. Proposition. Let V be a.finite dimensional subspace ofk[X]. (i) There is a.finite dimensional subspace W of k[X] which contains Vandis stable
under all s(g) (g e G);
(ii) V is stable under all s (g) ifand only ifa* V C k [G] ® V. If this is so, s de.fines a ma,p sv : G x V ➔ V which is a rational representation of G ; (iii)Jf, moreover, G isan F-group, X isan F-variety, Vis de.fined over F (see 1.3.7) and a is an F-morphism then in (i) W can be taken to be de.fined over F.

It suffices to prove (i) in the case that V = kf is one dimensional. Let
n
a* f =Lui ® f; (u; e k[G], /; e k[X]).
i=l

30

Chapter 2. Linear Algebraic Groups, First Properties

Then
L n
(s(g)f)(x) = f(g- 1x) = u;(g- 1)/;(x),
i=l
and we see that all s(g)f lie in the subspace W' of k[X] spanned by the f;. The subspace W of W' spanned by the s(g)f then has the properties of (i).
By a similar argument we see that if a* V c k[ G] ® V the space V is s(G)-stable.
Now assume that V is s(G)-stable. Let(/;) be a basis of V and extend it to a basis (/;) U (gj) of k[X]. Let f E V and write

+ a* f =Lu;® f; Lvi ®gj,

where u;, vi E k[G]. Then

= L L s(g)f

+ u;(g-l)f;

vi(g-l)gj,

= Our assumption implies that vi(g- 1) 0 for all g, hence all vi vanish. This proves

the first point of (ii). The second point is now immediate. The proof of (iii) is a copy

of that of (i) and can be skipped.

□

We now consider the case that G acts by left or right translations on itself (see 2.3.2 (2)). For g, x E G, f E k[G] define
()..(g)f)(x) = f (g-1x), (p(g)f)(x) = f (xg).
Then).. and pare representations (of abstract groups) of G in G L(k[G]), even in the
group of algebra automorphisms of k[G]. If tis the automorphism of k[G] defined by
= inversion (see 2.1.2), then p io)..oi-1. The representations).. and pare faithful, i.e.,
have trivial kernel. If, for instance, )..(g) = id, then f (g-1) = f (e) for all f E k[G],
whence g = e.

2.3.7. Theorem. (i) There is an isomorphism of G onto a closed subgroup of some GLn;
(ii) If G is an F-group the isomorphism of( i) may be taken to be defined over F.

= By 2.3.6 (i) we may assume that k[G] k[f1, ... , fn], where (f;) is a basis of a
p(G)-stable subspace V of k[G].

By 2.3.6 (ii) there exist elements (mij)i';i,j';n in k[G] with
n
= p(g)f; Lmi;(g)fj (g E G).
j=l

2.4. Jordan decomposition

31

Then <J,(g) = (mij(g)) 1~;,j~n defines a group homomorphism <J, : G ➔ GLn. If <J, (g) = e then p (g) /; = /; for all i. Since p (g) is an algebra homomorphism and since the /; generate k[G] it follows that p(g)f = f for all / E k[X] and g = e. Hence <J, is injective. Also, <J, is a morphism of affine varieties. The corresponding algebra homomorphism

(notations of 2.1.4 (3)) is given by <J,*T;i = mii, <J,*(D-1) = det(mij )-1. From
L f;(g) = mj;(g)/j(e)
j

it follows that <J,* is surjective. By 2.2.5 (ii), <J,(G) is a closed subgroup. Its algebra is

isomorphic to k[GLnl/Ker <J,*, hence is isomorphic to k[G]. It follows that <J, defines

an isomorphism of algebraic groups G ~ <J,(G). This proves (i). The easy proof of

(ii) is omitted.

D

2.3.8. Lemma. Let H be a closed subgroup ofG. Then
H = {g E G I )..(g)Ia(H) = Ia(H)} = {g E G I p(g)Ia(H) = Ia(H)}.

The notations are as in 1.3.2. It suffices to prove this for )... If g, h E H, f E

I 0 (H) then ()..(g)f)(h) = f(g- 1h) = 0, whence )..(g)f E I 0 (H). Conversely, if

this is so then f (g- 1) = ()..(g)f)(e) = 0 for all/ E I 0 (H) and g E H.

□

2.3.9. Exercises. (1) In the situation of 2.3.5 there exists an increasing sequence of finite dimensional subspaces (Vi) of k[X] such that (a) each V; is stable under s(G) and s defines a rational representation of G in V;, and (b) k[X] = LJ; V;. (2) Let X be an affine G-variety. There is an isomorphism <J, of X onto a closed subvariety of some An and a rational representation r : G ➔ GLn such that <J,(g.x) = r(g)<J,(x) (g E G,x EX) (Hint: adapttheproofof2.3.7 (i)).

2.4. Jordan decomposition

2.4.1. We begin by recalling some results from linear algebra. Let V be a finite dimensional vector space over k. An endomorphism a of V is semi-simple if there is a basis of V consisting of eigenvectors of a. So with respect to this basis, a is represented by a diagonal matrix. We say that an endomorphism a is nilpotent if as = 0 for some integers > 1 and that a is unipotent if a - 1 is nilpotent. Notice that if the characteristic p of k is non-zero, a is unipotent if and only if aPs = 1 for some integer s::: 1.

32

Chapter 2. Linear Algebraic Groups, First Properties

We denote by End(V) the algebra of endomorphisms of V. The group of its invertible elements is G L(V). Choosing a basis of V we may identify End(V) with an algebra Mn of n x n-matrices with entries ink and G L(V) with GLn.

2.4.2. Lemma. Let S C Mn be a set ofpairwise commuting matrices.
(i) There exists x e GLn such that xsx-1 consists of upper triangular matrices;
(ii) If all matrices of Sare semi-simple there is x E GLn such that xsx- 1 consists of diagonal matrices.
The assertions are obvious if all elements of S are multiples of the identity. If
not, there is s e S with an eigenspace that is a non-trivial subspace W of V = kn.
Because of our assumptions, Wis S-stable. By induction on n, we may assume that an assertion like (i) holds for the endomorphisms induced by Sin Wand V / W. Then (i) follows. (ii) is proved similarly, writing V as a direct sum of eigenspaces of s. □

2.4.3. Lemma. (i) The product of two commuting semi-simple (nilpotent, unipotent) endomorphisms of V is semi-simple (respectively: nilpotent, unipotent); (ii) Ifa E End(V), b E End(W) are semi-simple (nilpotent, unipotent) then the same is true for a EB be End(V EB W), a® be End(V ® W);
(iii) If a e End(V), b e End(W) are semi-simple (nilpotent) then the same is true
fora® I+ 1 ®be End(V ® W).

The assertion about semi-simple endomorphisms of (i) follows from 2.4.2 (ii).

The easy proofs of the other assertions are left to the reader.

□

2.4.4. Proposition. Let a e End(V).
(i) There are unique elements as, an E End(V) such that as is semi-simple, an is
= nilpotent, asan anas and a= as+ an (additive Jordan decomposition ofa); = (ii) There are polynomials P, Q E k[T] without constant term such that as P(a),
an= Q(a);
(iii) If W C V is an a-stable subspace of V, then Wis also stable under as and au
= and a Iw as Iw + au Iw is the additive Jordan decomposition of the restriction a Iw.
A similar result holds for the endomorphism of V/ W induced by a;
= (iv) Let <p : V ➔ W, b e End(W) be linear maps. If <p o a b o <p then
<p o as = bs o <p, <p o an = bn o <p.
Let det(T.1 - a) = n(T - a; tj be the characteristic polynomial of a, the a; being
the distinct eigenvalues of a. Put

The V; are non-zero a-stable subspaces. By the Chinese remainder theorem there exists P e k[T] with
P(T) = 0 (mod T), P(T) = a; (mod (T - a;ti) for all i.

2.4. Jordan decomposition

33

= = Put as P(a). Since P(a;) a; the eigenvalues of as are the same as those of
a. Also, as stabilizes all Vi and the restriction of as to Vi is scalar multiplication by
a;. It follows that the V; are the eigenspaces of as and that V is their direct sum. We
conclude that as is semi-simple and that a-as is nilpotent. The assertions of (i) and (ii)
now readily follow, except for the uniqueness statement. To prove this, let a = bs + bn
be a second decomposition with the properties of (i). From (ii) we infer that bs and bn
= commute with as and an. From 2.4.3 (i) we conclude that as - bs bn - an is both
semi-simple and nilpotent, hence must be zero, establishing the uniqueness.
If W is as in (iii) it is clear from (ii) that as and an stabilize W. Now observe
that the characteristic polynomial of alw divides the characteristic polynomial of a. It
follows that the polynomial P introduced above can also serve for a Iw. The assertions
of (iii) about V follow. For V / W the arguments are similar.
To prove (iv) we use that¢ has a factorization

V --+ V EB W --+ W.

The first map is v 1--+ (v, q,(v)); it is injective. The second map is projection, it is

surjective. An easy argument shows that it suffices to prove (iv) if¢ is either injective

or surjective. These cases are taken care of by (iii).

D

2.4.5. Corollary. Let a E GL(V). There are unique elements as,au E GL(V)
= = such that as is semi-simple, au is unipotent and a asau auas (multiplicative Jor-
dan decomposition of a). We have properties similar to those of 2.4.4 (iii) with au
instead ofan.

Let a = as + an be the additive Jordan decomposition of a. Since a is invertible

it has no eigenvalue 0. From the proof of 2.4.4 we see that as is invertible. It follows
from 2.4.4 that as and au = I + a_;- 1an have the required properties. Conversely, if as
and au are as in 2.4.5 then a = as + as(au - I) is the additive Jordan decomposition

of a. The corollary follows from these observations.

D

We call as, an(au) the semi-simple, nilpotent, (unipotent) parts of a E End(V) (resp. G L(V)).

2.4.6. Corollary. Let a = asau, b = bsbu be the Jordan decompositions of a E
= G L(V) and b E G L(W). Then a EB b (as EB bs) + (au EB bu) is the Jordan decompo-
= sition ofaEBb E GL(VEBW)anda®b (as®bs)(au®bu)thatofa®b E GL(V®W).

This follows from 2.4.3 (ii).

□

2.4.7. Let V be a not necessarily finite dimensional vector space over k. We denote again by End (V) and G L (V) the algebra of endomorphisms of V and the group

34

Chapter 2. Linear Algebraic Groups, First Properties

of its invertible endomorphisms. We say that a E End(V) is locally finite if V is a union of finite dimensional a-stable subspaces. We say that a is semi-simple (resp. locally nilpotent) if its restriction to any finite dimensional a-stable subspace is semisimple (resp. nilpotent). If a is semi-simple, it is also semi-simple according to the definition of 2.4.1 (check this). For a locally finite a E End(V) we again have an
= additive Jordan decomposition a as +an as in 2.4.4 (i), with locally finite as and an
such that as is semi-simple and an locally nilpotent. To define asx and aux for x E V, take a finite dimensional a-stable subspace W containing x and put
asx = (alw)sx, aux= (alw)ux,
It follows from 2.4.4 (iii) that this definition is independent of the choice of Wand that as and an are as required. Similarly, we have a multiplicative Jordan decomposition a = asau if a E G L(V) is locally finite. Here au is locally unipotent, i.e., au - 1 is locally nilpotent. We have obvious analogues of 2.4.4 (iii), (iv) and 2.4.6 (for locally finite b).
Now let G be a linear algebraic group and put A= k[G]. Let g E G. By 2.3.9 (1) the right translation p(g) is a locally finite element of GL(A). So we have a Jordan decomposition p(g) = p(g)sp(g)u,
2.4.8. Theorem. (i) (Jordan decomposition in G) There are unique elements gs, gu E G such that p(g)s = p(gs), p(g)u = p(gu) and g = gsgu = Cues;
= (ii) If <p : G ➔ G' is a homomorphism of algebraic groups, then <p(g)s ¢(gs),
</J(g)u = </J(gu); (iii) If G = GLn, then Ks and Cu are the semi-simple and unipotent parts of 2.4.5
= (with V kn).
The elements gs and gu of (i) are called the semi-simple part and the unipotent part of g E G.
As in 2.1.2 let m : A ® A ➔ A be the homomorphism defined by multiplication. Since p (g) is an algebra automorphism we have
mo (p(g) ® p(g)) = p(g) om.
From 2.4.4 (iv) (applied tom) and 2.4.6 it follows that
mo (p(g)s ® p(g)s) = p(g)s om.
This means that p(g)s is also an automorphism of A, hence/ H- (p(g)sf)(e) defines a homomorphism A ➔ k, i.e., a point gs of G. Since p(g) commutes with all left translations A(x), x E G (which are locally finite) we have by 2.4.4 (iv) for f E A
(p(g)sf)(x) = (A(X- 1)p(g)sf)(e) = (p(g)sA(x- 1)/)(e) = (A(x-1)/)(gs) = f (Xgs),
and we see that p(g)s = p(gs),

2.4. Jordan decomposition

35

One obtains in the same way an element Ku E G with p(g)u = p(gu). The remaining assertions of (i) now follow from the fact that pis a faithful representation ofG.
A homomorphism of algebraic groups </> : G ➔ G' can be factored:

G ➔ Im</> ➔ G'.

Using 2.2.5 (ii) it follows that it suffices to prove (ii) in two special cases: (a) G is a closed subgroup of G' and</> is the inclusion map. Let k[G] = k[G']/ I. By 2.3.8

G = {g e G' I p(g)l = /}.

The assertion (ii) now follows from 2.4.4 (iii). (b) </> is surjective. In this case k[G'] can be viewed as a subspace of k[G] (see 1.9.1 (ii)), which is stable under all p(g) (g e G). Again, the assertion of (ii) follows from 2.4.4 (iii). Let G = G L(V) with V = kn. Let f be a nonzero element of the dual space yv. For v E V define f (v) E k[G] by

f (v)(g) = f(gv).
Then f is an injective linear map V ➔ k[G] and it is immediate that for g e G, v e
V

f (gv) = p(g)f(v).

From 2.4.4 (iv) we conclude that
= - -
f (gsv) p(g)sf (v),

and similarly for Ku• This implies (iii).

□

2.4.9. Corollary. x e G is semi-simple (resp. unipotent) if and only iffor any isomorphism </> of G onto a closed subgroup of some GLn we have that </> (x) is semi-
simple (resp. unipotent).

2.4.10. Exercises. Notations of 2.4.8. (1) Show that A(X)s = A(Xs), )..(x)u = A(Xu). (2) The set Gu of unipotent elements of G is closed. (3) Show by an example that the set Gs of semi-simple elements of G is not necessarily open or closed.

2.4.11. Let F be a subfield of k and G an F-group. If x e G(F) then Xs and Xu need not lie in G(F). Here is an example.

36

Chapter 2. Linear Algebraic Groups, First Properties

Assume that char(k) = 2 and that F =,:- F 2, so Fis non-perfect. Let G = GL2. If
a e F- F 2 then

has the semi-simple part

( Ja 0

JOa ) '

which does not lie in G(F). If, however, Fis a perfect field, then the semi-simple and unipotent parts of an element of G(F) also lie in G(F). We postpone the proof (see 12.1.7 (c)).
The linear algebraic group G is unipotent if all its elements are unipotent. An example is the group Un of 2.1.4 (4). The next result implies that any unipotent group is isomorphic to a closed subgroup of some Un.

2.4.12. Proposition. Let G be a subgroup of GLn consisting of unipotent matri-
ces. There is x E GLn such that xGx-1 C Un,

= Put V kn. Then G is a group of unipotent linear maps of V. Use induction
on n. If there is a non-trivial subspace W of V with G. W = W, then the statement

follows by induction. We are left with the case that such a W does not exist, i.e., that

G acts irreducibly in V. In this case we know by Burnside's theorem [La2, Ch. XVII,

§3] that the elements of G span the vector space End(V) of endomorphisms.

If g e G we have Tr(g) = n. It follows that Tr((l - g)h) = 0 for g, h e G.

But then this equality holds for all h e End(V), which can only be if g = 1. Hence

G = {1} and the assertion is trivial.

D

Recall that a group H is nilpotent if there is an integer n such that all iterated
= commutators (xi, ( ... (Xn-t, Xn) ... )) equal e (as before, (x, y) xyx- 1 y- 1 ). Such a
group is solvable (see [Jac4, p. 243, ex. 6].
2.4.13. Corollary. A unipotent linear algebraic group is nilpotent, hence solvable.
Using 2.3.7 (i) the proposition reduces the proof to verifying that a group Un of unipotent upper triangular matrices is nilpotent. Verification is left to the reader. D
A consequence of 2.4.12 is the fact that if G is a unipotent linear algebraic group and G ➔ G L(V) a rational representation, there is a non-zero vector in V fixed by all of G. This fact is used to prove the following geometric result (theorem of Kostant-Rosenlicht).

2.5. Recovering a group from its representations

37

2.4.14. Proposition. Let G be a unipotent linear algebraic group and X an affine G-space. Then all orbits of Gin X are closed.

Let O be an orbit. Replacing X by the closure O we may assume that O is dense

in X. By 2.3.3 (i), 0 is also open. Let Y be its complement. The group G acts locally

finitely on the ideal Ix(Y) (by 2.3.6 (i)), and there is a non-zero function f in this

ideal fixed by the elements of G. But then f is constant on O. Since O is dense, f is

constant on all of X. Henceix(Y) = k[X] and O = X, as asserted.

D

2.4.15. Exercise. Let G be a subgroup of GLn which acts irreducibly in V = kn.
Show that the only normal unipotent subgroup of G is the trivial one.

2.5. Recovering a group from its representations
2.5.1. The results of this section, which will not be used in the sequel, illustrate the elementary theory of linear algebraic groups.
We keep the notations of the preceding sections. Let G be a linear algebraic group. Recall (see 2.3.2 (3)) that a rational representation of G is a finite dimensional vector space V (called a G-module) together with a homomorphism of algebraic groups
= rv : G ➔ GL(V). We denote by I the trivial G-module: I= k and r1(g) 1 for all
g E G. A homomorphism of G-modules V ➔ W is a linear map ¢ which is equivariant,
i.e., satisfies¢ o rv(g) = rw(g) o cp for all g e G.
If V is a G-module the dual vector space vv is a G-module: if ( , } (or ( , }v) is the duality pairing, then for g e G, x e V, u e vv
(rv(g)(x), rvv(g)(u)) = (x, u).
If V and W are G-modules the tensor product V ® W has a natural structure of G-
module, with rv®w = rv ® rw.
As in 2.4.7 we define locally finite G-modules and their G-homomorphisms. An
= example is A k[G], the representation of G being p, the action by right translations.
= 2.5.2. Lemma. Let V be a G-module. For v e V, u e vv, g e G define <l>u (v)(g)
(rv(g)v, u). Then c/Ju(v) e A and </Ju is a homomorphism of G-modules V ➔ A.

</Ju (v) can be viewed as a matrix element of a representation G ➔ GLdim v, so

lies in A. The equivariance of </Ju is obvious from the definitions.

D

2.5.3. Theorem. [Tannaka's theorem] Assume given for any finite dimensional Gmodule V an element av e G L(V) such that the following holds: (a) If V and Ware G-modules then av®w =av® aw,

38

Chapter 2. Linear Algebraic Groups, First Properties

(b) if <I>: V ➔ Wis a homomorphism of G-modules then <I> o av= aw o </>,
(c)a1=l.
Then there is x E G with av = rv(x)for all G-modules V.
We begin the proof by defining av for a locally finite G-module V. Let v E V and let W be a finite dimensional G-stable subspace of V containing v. Define av(v) = aw(v). Then av is well-defined. It is an invertible linear map of V. These maps have the properties (a) and (b) (check these facts).
Consider the locally finite G-modules A (for the representation p) and A® A (for p ® p ). The multiplication map m : A ® A ➔ A of 2.1.2 is G-equivariant. Put a = aA. By (a) and (b) we have mo (a® a) = a om. This means that a is an automorphism of the algebra A. It follows that there is an automorphism <I> of the affine variety G such that (af)(g) = f (</>g) for f EA, g E G.
Next consider the homomorphism D. : A ➔ A ® A of 2.1.2. Then (D.f)(g, h) = f (gh) (f E A, g, h E G). It follows that 6. o p(g) = (id® p(g)) o 6. (g E G). Let B be the locally finite G-module A ® A, with rs = id® p. Then 6. : A ➔ B is equivariant. By the properties (a), (b), (c) we have as =id® a. We conclude from (b) that 6. oa = (id®a)o 6., which means that for g, h E G we have </>(gh) = g</>(h). Put x = </>(e). Then </>(g) = gx and a = p(x).
Now let V be a finite dimensional G-module. For any u in the dual vv we have the G-homomorphism <l>u : V ➔ A of 2.5.2. By property (b) we know that

<l>u O av = a O <l>u = p(x) 0 <l>u• This means that for g E G, v E V we have

(rv(g)av(v), u) = (rv(gx)v, u).

Since u E vv is arbitrary we can conclude that av = rv(x).

□

2.5.4. The theorem implies that one can recover the algebra A from the rational representations of G. We now discuss the explicit description of A in terms of representations. Let V be a finite dimensional G-module. For v E V, u E vv define a linear map 1/lv : V ® vv ➔ A by 1/lv(v ® u) = <l>u(v), where <l>u is as in 2.5.2. We have the following properties. (a) If <I> : V ➔ W is a homomorphism of G-modules then the maps 1/lv o (id ® <I>v) and 1/fw o (<I> ® id) of V ® wv to A coincide.
Here <I>v : wv ➔ vv is the transpose of </>. This property is a reformulation of the equality (v, </>v (u)) = (</>(v), u), where v E V, u E wv.
Next let V and W be two G-modules. There are canonical isomorphisms (V ® W)V '.::::'. vv ® wv and C : (V ® W) ® (V ® W)v '.::::'. (V ® vv) ® (W ® wv). As before, m : A ® A ➔ A is the multiplication map.
(b) V'V®W =mo (1/lv ® 1/lw) o c.

2.5. Recovering a group from its representations

39

Property (b) means that for v E V, w E W, u E yv, t E wv we have (v ® w, u ® t}v@w = (v, u}v(w, t}w.
This relation describes the canonical isomorphism between (V ® W) v and yv ® wv.

2.5.5. Let F be the direct sum of the vector spaces V ® vv, where V runs through all finite dimensional G-modules V. Denote by iv : V ® vv ➔ F the canonical injections and by 'R the subspace of F spanned by the elements iv(id ® q,v)(z) iw(<I> ® id)(z). Here <I> : V ➔ Wis any homomorphism of G-modules and z runs
through V ® wv. Put A= F/R. For v EV, u E yv denote by av(v ® u) the image
in A of iv(v ® u). If¢ is as before, then for v E V, u E wv

(1)

We define a multiplication of these elements. Let v E V, w E W, u E yv, t E wv. Then

= av(v ® u)aw(w ® t) av@w((v ® w) ® (u ® t)).

(2)

This product is well-defined, and defines a structure of commutative, associative al-
gebra on A. The identity element is provided by the trivial representation /. We leave
it to the reader to check these statements. Notice that the properties of av reflect the
properties (a) and (b) of2.5.4 of the maps 1/tv.

2.5.6. The homomorphisms of algebraic groups G ➔ Gm form an abelian group X, the character group of G. If V is a one dimensional G-module, there is a unique x E X such that g.v = x(g)v for g E G, v E V. Then V ® vv has a canonical basis element ev@vv, namely v ® u, where v E V, u E yv and (v, u} = I. We write
= a(x) av(ev@vv ). Then a is a homomorphism of X onto a subgroup of the set of
invertible elements of A. If V is a finite dimensional G-module of dimension d we have the canonical
= homomorphism <I> : ®d V ➔ I\d V with ¢ (Vt ® ... ® vd) Vt I\ ... I\ vd. Identify
(/\d V)v with /\d(Vv) via the pairing
= (Vt/\ ... A vd, Ut I\ ... I\ ud} det((ui, Vj}).

Then

L </>v(u t /\ ... /\ ud) = (sgn s)Us(l) ® ... ® Us(d),
seSd

where sgn s is the sign of the permutation s. Let detv be the character defined by the one dimensional G-module /\d V. It follows from (1) and (2) that for vi E V, ui E yv (1 <, ij, ~ d) we have

det(av(vi ® ui)) = det((vi, ui})a(detv).

(3)

40

Chapter 2. Linear Algebraic Groups, First Properties

2.5.7. Theorem. (i) A is a k-algebra offinite type;
= (ii) There is a surjective algebra homomorphism <I> : A ➔ A such <I> o av 1/rv for
all G-modules V;
(iii) The kernel of <I> is the ideal of nilpotent elements of A.

Let V be a G-module. Fix a basis (ui)l~i~d of vv. It follows from 2.5.2 and
the local finiteness of A that there is a finite dimensional G-submodule W of A such that, for all i, the images of the maps <Pu; of 2.5.2 lie in W. These maps then define an injective G-homomorphism </> : V ➔ Wd. Since </>v is surjective, an element
I:?= av(v ® u) is a sum 1aw(wi ® ti). It follows that A is generated by the images
of the av, where V is a finite dimensional G-submodule of A. Fix such a V which generates A as an algebra and contains I. The product in A defines for any integer
n > 1 a G-equivariant surjective linear map <l>n of V®n onto a finite dimensional
G-submodule Vn of A. We have Vn C Vn+l and A= Un~l Vn. Now let W be any finite dimensional G-submodule of A. It is contained in a Vn,
whence

Im aw C Im avn C Im av@n

(the second inclusion follows from the surjectivity of <l>n). The multiplication rules in

A show that Im av®n lies in the subalgebra generated by Im av. It follows that this

subalgebra must be the full algebra A. This proves (i).

Using property (b) of 2.5.4 we see that there exists an algebra homomorphism

A ➔ A which maps av(v, u) to 1/rv(v, u), for all V. This implies (ii).

Fix an algebra homomorphism~ : A ➔ k. Let V be a G-module. Fix bases (vi)
= and (uj) of V resp. vv. There is an endomorphism av of V such that (avvi, uj}

~(av(vi ® Uj)). From (3) we see that det(av(v; ® Uj)) is an invertible element of

A. Hence av is an invertible linear map. It now follows that the av satisfy the as-

sumptions of Tannaka's theorem. The conclusion of that theorem shows that there
= is a homomorphism x : A ➔ k (i.e., a point x of G) such that ~ x o <I>. Let = N ✓{O} be the ideal of nilpotent elements of A. There is an affine variety X with k[X] = A/N. Then <I> induces a morphism of G onto a closed subvariety of X. But

we have seen that these varieties have the same points. This can only be if A/N is

isomorphic to A, which establishes (iii).

□

2.5.8. Remarks. (I) We can also recover the comultiplication 8 : A ➔ A ® A
and the antipode t : A ➔ A of 2.1.2 from similar homomorphisms for A.
Let V be a G-module. Let (v;) be a basis of V and denote by (u;) the dual basis of
vv (so (v;, Uj} = ~ij)- Denote by X the linear map A ➔ A® A sending an element
av(v ® u) to Li av(v ® u;) ® av(v; ® u). It can be checked that X is an algebra
homomorphism and that (<I> ® <I>) o X = 8 o <I>. = Define a homomorphism i : A ➔ A by i(av(v ® u)) avv(u ® v). Then i
induces the antipode t of A.

Notes

41

(2) In fact, Li and i have the properties of 2.1.2. This means that A and these homo-
morphisms define a group scheme over k (2.1.6 (a)). If char k = 0 the algebra A is
reduced by a theorem of Cartier [DG, p. 255]. In that case the homomorphism <I> of 2.5.7 is an isomorphism.
2.5.9. Exercises. (1) Let F be a subfield of k. If G is an F-group then the algebra A of 2.5.6 can be given an F-structure. (2) Check the statements of 2.5.8.
Notes
The name 'algebraic group' (or rather 'groupe algebrique') seems first to occur in the work of E. Picard on the Galois theory of linear differential equations (around 1880). The Galois groups occurring in that work are, indeed, linear algebraic groups over C. Kolchin's work on algebraic groups of 1948 (in [Koll, Kol2]) was also motivated by the Galois theory of linear differential equations. His results were taken up by Borel in his fundamental paper [Bot]. Here the emphasis is on the analogy between Lie groups and linear algebraic groups.
The results of Sections 2 and 3 are contained in [Bo1, Ch. 1]. Several of the results (for example 2.2.1) go back to Kolchin (see [Kol 1, Kol2]). The useful result 2.2.6 is due to Chevalley (see [Chl, Ch. II, §7]).
The existence of closed orbits in a G-space (see (2.2.3 (ii)) was proved in [Bot, 15.4]. This simple result is a cornerstone of the theory of linear algebraic groups. For example, it is needed in the proof of the crucial fixed point theorem 6.2.6. It should be noted that the algebraicity of the action is essential: a complex Lie group acting on a complex variety need not have closed orbits.
Theorem 2.4.8 about the Jordan decomposition in a linear algebraic group was proved in [Bot, Ch. 2]. It is also a consequence of 3.1.1, which was first proved in [Kol2, §3]. The proof given here shows that the theorem is, essentially, a formal consequence of the functorial property 2.4.4 (iii) of the Jordan decomposition of a linear map.
Tannaka's theorem 2.5.3 goes back to [Tan], where a similar result is proved for compact Lie groups. The aim of [Tan] was to establish an analogue for such groups of Pontryagin duality in abelian topological groups. The results of 2.5 should nowadays be viewed in the context of the theory of tensor categories, which we did not go into. See [DelM, Del].

Chapter3 Commutative Algebraic Groups

This chapter deals with results about commutative linear algebraic groups which are basic for the theory of the later chapters. The important tori are introduced in 3.2, and we prove the classification theorem 3.4.9 of connected one dimensional groups. The notations are as in the previous chapter.

3.1. Structure of commutative algebraic groups

3.1.1. Theorem. Let G be a commutative linear algebraic group. (i) The sets Gs and Gu ofsemi-simple and unipotent elements are closed subgroups; (ii) The product map 1C : Gs x Gu ➔ G is an isomorphism ofalgebraic groups.

We may assume that G is a closed subgroup of GLn, for some n. From 2.4.3 (i) we

see that Gs and Gu are subgroups and by 2.4.10 (2) Gu is closed. It follows from 2.4.2
EB (ii) that there exists a direct sum decomposition of V = kn, say V = V;, together
with homomorphisms</); : Gs ➔ k* such that g.v = </J;(g)v if g e G, v e V;. The

V; are G-stable. Applying 2.4.2 (i) to the groups induced by G in the V; we arrange
things such that G C Tn, Gs = G n Dn (notations of 2.1.4). This shows that Gs is

closed, whence (i).

The uniqueness of the Jordan decomposition in G implies that 1C is an isomor-

phism of abstract groups. It is also a morphism of algebraic varieties. The map

sending x e G to its semi-simple part Xs is also a morphism, since it maps x to a set
of its matrix elements. Hence 1C-1 : x ~ (xs, x; 1x) is a morphism, which proves

(ii).

□

3.1.2. Corollary. If moreover G is connected then the same holds for Gs and Gu.

Gs and Gu are images of the connected space G under continuous maps.

D

3.1.3. Proposition. Let G be a connected linear algebraic group ofdimension one.
(i) G is commutative;
(ii) Either G = Gs or G = Gu;
(iii) If G is unipotent and p = char k > 0 then the elements of G have order dividing
p.

Fix g e G and consider the morphism <P : x ~ xgx- 1 of G to itself. By 1.2.3
(ii) the closure </JG is an irreducible closed subset of G. Using 1.8.2 we conclude that this set is either {g} or G. Assume we have the latter case. By 1.9.5 the complement G - </JG is finite. View G as a closed subgroup of some GLn. Then there are only

3.2. Diagonalizable groups and tori

43

finitely many possibilities for the characteristic polynomial det(T.1 - x) of x E G.

The connectedness of G implies that this characteristic polynomial is constant, and
equal to (T - lt. This means that G is unipotent, hence solvable (see 2.4.13). But

the commutator subgroup G' is a connected, closed, subgroup (see 2.2.8 (i)), and can

only be {e}. Since g- 1</>G c G' we get a contradiction. It follows that G is commu-

tative.

(ii) follows from 3.1.1 and 1.8.2. Assume that G is unipotent and p > 0. Consider
the subgroups c<P''> generated by the ph-th powers of elements of G. They are closed

and connected (2.2.9 (3)), so must be either G or {e}. Viewing G as a subgroup of

GLn we see that G(P''> = {e} if ph ::: n. This can only be if G(P> = {e}.

□

In the rest of this chapter we shall first discuss the abelian algebraic groups whose elements are semi-simple and then the abelian unipotent groups with the property of 3.1.3 (iii).
3.2. Diagonalizable groups and tori
3.2.1. Let G be a linear algebraic group. A homomorphism of algebraic groups x : G ➔ Gm is called a rational character (or simply a character). The set of rational characters is denoted by X*(G). It has a natural structure of abelian group, which we write additively. The characters are regular functions on G, so lie in k[G]. By Dedekind's theorem [La2, Ch. VIII, §4] the characters are linearly independent elements of k[G].
A homomorphism of algebraic groups :>., : Gm ➔ G is called a cocharacter (or multiplicative one-parameter subgroup) of G. We denote by X.(G) the set of cocharacters. If G is commutative X .(G) also has a structure of abelian group (written additively). If G is arbitrary we still have in X .(G) multiplication by integers, defined by (n.:>.,)(a) = :>.,(a)n (for:>., E X.(G), n E Z, a Ek*). We write-}.,= (-1).:>.,.
A linear algebraic group G is diagonalizable if it is isomorphic to a closed subgroup of some group Dn of diagonal matrices. G is an algebraic torus (or simply a torus) if it is isomorphic to some Dn.
3.2.2. Example. G = Dn. Write an elementx E Dn as diag(x, (x), ... , Xn(x)). Then
X; is a character of Dn and we have k[Dn] = k[x,, ... , Xn, x11, ••• , x;' ]. From
Dedekind's theorem we see that the monomials x~' ···X!" with (a,, ... , an) E zn
form a basis of k[Dn] and that any character of Dn is such a monomial. It follows that X*(Dn) '.::::: zn. A homomorphism Gm ➔ Dn is of the form X I-► diag(xa 1, ••• , Xa") (x E k*), where the a; are integers. It follows that X.(Dn) ::::::: zn.
3.2.3. Theorem. The following properties of a linear algebraic group G are equivalent: (a) G is diagonalizable;

44

Chapter 3. Commutative Algebraic Groups

(b) X*(G) is an abelian group offinite type. Its elements form a k-basis of k[G]; (c) Any rational representation of G is a direct sum of one dimensional such representations.

If G is a closed subgroup of Dn then k[G] is a quotient of k[Dn]• Since the restriction of a character of Dn to G is a character of G, we see that the restrictions to

G of the characters of Dn span k[G]. By Dedekind's theorem they form a basis and

any character of G must be a linear combination of these restrictions. So the restriction homomorphism of abelian groups X*(Dn) ➔ X*(G) is surjective. As X*(Dn) ~ zn,

the group X*(G) is of finite type. We have proved that (a) implies (b).
= Assume (b). Put X X*(G). Denote by <I>: G ➔ Gl(V) a rational representa-

tion in a finite dimensional vector space V. We define linear maps Ax of V (x E X)

by

L = </>(x)

x(x)Ax·

xeX

Then Ax = 0 for all but finitely many X· From <J>(xy) = <J>(x)<J>(y) (x, y E G) we infer (using Dedekind's theorem for G x G) that, for X, 1/f E X, we have AxAv, =
= 8x.v,Ax. We also have Lx Ax 1. Put Vx = Im Ax. The properties of the Ax ex-

press that V is the direct sum of the non-zero Vx and that x E G acts in Vx as scalar
multiplication by x(x). This implies (c).

That (c) implies (a) is immediate if one views G as a closed subgroup of some GL (V)

(see 2.3.7 (i)).

D

3.2.4. Corollary. If G is diagonalizable then X*(G) is an abelian group of finite
type, without p-torsion if p = char k > 0. The algebra k[G] is isomorphic to the
group algebra of X*(G).

The first point follows from (b), using that k does not contain p-th roots of unity

=/:- 1 if p > 0. The second point is implicit in the first part of the proof of 3.2.3. We

shall make it more explicit.

D

3.2.S. Let M be an abelian group of finite type. The group algebra k[M] is the algebra
= with basis (e(m))meM, the multiplication being defined by e(m)e(n) e(m + n). If
M1 and M2 are two such groups we have
(4)
Define homomorphisms Li: k[M] ➔ k[M]®k[M], t: k[M] ➔ k[M], e: k[M] ➔ k
= by Lie(m) e(m) ® e(m), te(m) = e(-m), e(e(m)) = 1. Assume that M has no p-torsion if p = char k > 0.

3.2.6. Proposition. (i) k[M] is an affine algebra, and there is a diagonalizable linear
= algebraic group Q(M) with k[Q(M)] k[M], such that Li, t and e are comultiplica-
tion, antipode and identity element of Q(M) (see 2.1.2);

3.2. Diagonalizable groups and tori

45

(ii) There is a canonical isomorphism M ~ X*(Q(M)); (iii) If G is a diagonalizable group there is a canonical isomorphism Q(X*(G)) ~ G.

It is well-known that M is a direct sum of cyclic groups. By (4) and 1.5.2 it

suffices to prove the first point of (i) in the cyclic case. If M is infinite cyclic then

k[M] ~ k[T, T-1], an integral domain. If M is finite of order d then p does not

divided (if p > 0) and k(M] ~ k[T]/(Td - 1). Since the polynomial Td - 1 does

not have multiple roots, this is a reduced algebra. To complete the proof of (i) we have

to verify the properties of 2.1.2. We leave the verification to the reader.

The map x ~ e(m)(x) (m E M, x E Q(M)) defines a character of Q(M), i.e.,

e(m) is a character of Q(M). By Dedekind's theorem these characters are distinct

and, since they form a basis, any character of Q(M) is an e(m). It follows that the

map m ~ e(m) is an isomorphism, whence (ii). A similar map induces an algebra

isomorphism k[G] ➔ k[X*(G)], whence (iii).

□

3.2.7. Corollary. let G be a diagonalizable group. (i) G is a direct product of a torus and a finite abelian group of order prime top, where p is the characteristic exponent ofk; (ii) G is a torus if and only if it is connected; (iii) G is a torus if and only if X*(G) is a free abelian group.

First observe that Q(Zn) ~ Dn, as follows from 3.2.2 and 3.2.6 (iii). Now X*(G)

is isomorphic to a direct sum zn EB M, where M is finite. By (4), G is isomorphic to

the product of Dn and Q(M). The latter group is finite, as follows for example from

the argument of the proof of 3.2.6 (i). We have proved (i). (ii) is a consequence of (i)

and (iii) also follows.

□

3.2.8. Proposition. [rigidity of diagonalizable groups] let G and H be diagonalizable groups and let V be a connected affine variety. Assume given a morphism of varieties </J : V x G ➔ H such that for any v E V the map x ~ </J (v, x) defines a homomorphism ofalgebraic groups G ➔ H. Then </J (v, x) is independent of v.

If 1/r E X*(H) then 1/r(</J(v, x)) can be written in the form

L 1/r(</J(v, x)) =

fx.v,(v)x (x),

xeX*(G)

with fx.1/t E k[V]. For fixed v the right-hand side is a character of G. By Dedekind's
theorem fx.v,(v) = 1 for one x and Ofor the others, whence f;,v, = fx.1/t· The con= nectedness of V implies that fx.1/t 1 for one x and Ofor the others (use 1.2.8 (2)). D

We give an application of the proposition. If G is an arbitrary linear algebraic group and Ha closed subgroup, we denote by Za(H) and Nc(H) the centralizer and

46

Chapter 3. Commutative Algebraic Groups

normalizer of H in G, i.e.
Zc(H) = {x e G I xyx- 1 = y for ally e H},

Na(H) = {x e GI xHx- 1 = H}.

These are closed subgroups of G (check this) and Zc(H) is a normal subgroup of Nc(H).

= 3.2.9. Corollary. If H is a diagonalizable subgroup of G then Nc(H)0 Zc(H)0

and Na(H)/Zc(H) is finite.

= The first point follows from the proposition, with V Nc(H)0, <I> being the mor-

phism (x, y) 1--+ xyx- 1 of V x H to H. The last point follows from 2.2.1 (i).

D

3.2.10. Exercises. pis the characteristic exponent of k. G is a diagonalizable group with character group X. (1) Make diagonalizable groups and abelian groups without p-torsion into categories and describe an anti-equivalence between these categories. (2) Let <I> : G --+ H be a homomorphism of diagonalizable groups. Denote by </>* the induced homomorphism X*(H) --+ X*(G). If <I> is injective (surjective) then </>* is surjective (respectively injective). (3) Describe a canonical isomorphism of abelian groups G ~ Hom(X, k*). (4) For a closed subgroup H of G and a subgroup Y of X define
H1. = {x ex I x(H) = {l}},

y1. = {x E G I x (x) = 1 for all x E Y}.
Then (HJ.).L = H, and (YJ.).L = Y if X /Y has no p-torsion. (5) For a positive integer n prime top, denote by Gn the subgroup of elements of G of order dividing n.
(a) (Gn).L = nX.
(b) The subgroup of elements of finite order is dense in G. (6) The group of automorphisms of an n-dimensional torus is isomorphic to the group GLn (Z) of integral n x n-matrices with an integral inverse.
We conclude this section with some material on tori. Let T be a torus. Put
= = X X*(T), Y X.(T) (3.2.1). For X EX,).. E Y, a Ek* the map a 1--+ X()..(a)) = defines a character of the multiplicative group. By 3.2.2 (with n 1) there is an
= integer (X, )..) such that x()..(a)) a(x,1.).
3.2.11. Lemma. (i) ( , ) defines a perfect pairing between X and Y, i.e. any ho-
x momorphism X --+ Z is of the form 1--+ (x, )..) for some ).. e Y, and similarly for Y.

3.2. Diagonalizable groups and tori

47

In particular, Y is a free abelian group;

(ii) The map a ®).. 1-► )..(a) defines a canonical isomorphism of abelian groups

k* ® Y ~ T.

= It suffices to prove this in the case T Dn. Then (i) follows from the results of

3.2.2 (check this). The proof of (ii) uses the freeness of Y.

□

Let F be a subfield of k. An F-torus is an F-group which is a torus. An F-torus T which is F-isomorphic to some Dn is F -split. The study of non-split F-tori, which
requires Galois theory, is deferred to Chapter 13.

3.2.12. Proposition. (i) An F-torus T is F-split if and only if all its characters are defined over F. If this is so the characters form a basis ofthe algebra F[T]; (ii) Any rational representation over F of an F-split torus is a direct sum of one dimensional representations over F.

The proof of (i) is straightforward. (ii) is proved as 3.2.3 (c).

□

3.2.13. Let T, X and Y be as before. Let V be an affine T -space. We have a lo-
cally finite representations of T in k[V], as in 2.3.5. For x E X put
= = k[V]x {f E k[V] I s(t).f x(t)f for all t ET}.
It follows from 3.2.3 (c) that the subspaces k[ V]x define an X -grading of the algebra k[V], i.e.,
k[V] = ffik[V]x, k[V]xk[V]v, C k[Vlx+v, (x, VJ' EX).
xeX
= = If T Gm then X Zand we have a structure of graded algebra on k[V] in the
usual sense.
= If cp is a morphism of varieties Gm --+ Z we shall write lima-+O cp(a) z if </J extends to a morphism ii, : A 1 --+ Z with ii,(O) = z. If cp'(a) = </J(a- 1) we define lima-+oo cp(a) = lima-+O </J'(a).
If V is a T-space and ).. E Y, we denote by V()..) the set of v E V such that
lima-+O )..(a).v exists. Then V(-)..) is the set of v such that lima-+oo )..(a).v exists.

3.2.14. Lemma. Assume that Vis affine.
n (i) V ()..) is a closed subset of V;
(ii) V ()..) V (-}..) is the set offixed points of Im ).. i.e. {V E V ' )..(k*). V = {V} }.

Let f = Lx fx with fx E k[V]x be an element of k[V]. Then
s()..(a)).f = La(x.>-) fx,

48

Chapter 3. Commutative Algebraic Groups

n from which we see that lima➔ o ).(a).v exists if and only if v annihilates all functions
of the Vx with (X, ).) < 0. This proves (i). Then V ().) V (-).) is the set v annihi-
lating all Vx with (x, ).) # 0, which is also the set of v with f ().(a).v) = f (v) for all

/ E k[G], a Ek*, i.e. the set of fixed points.

D

3.2.15. Example. Let G be an arbitrary linear algebraic group and ). : Gm ➔ G
= a cocharacter. We let Gm act on G by a.x ).(a)x).(a)-' (a Ek*, x E G). We de-
n note by P().) the set of x E G such that lima➔o a.x exists. It is immediate that this is a
subgroup, which by 3.2.14 (i) is closed. By 3.2.14 (ii) the intersection P().) P(-).)
is the centralizer of Im A.

3.2.16. Exercises. (1) The category of Gm-modules is equivalent to the category
of finite dimensional graded vector spaces.
(2) Let A = EBnezAn be a graded affine k-algebra without zero divisors. Assume A # Ao. Let dZ be the subgroup of Z generated by then with An # {O}. Choose
non-zero elements f, g and integers i, j with f E A;, g E Aj, i - j = d and let B = A Jg (notation of 1.4.6). The grading of A induces one of B. Define an isomor-
phism B0 ® k[fg-•, gf- 1] '.::::'.Band show that B0 is an affine algebra.
(3) Deduce from the previous exercise the following properties of a Gm-action on an
affine variety V: There is a decomposition V = LJ~=o V; into disjoint irreducible lo-
cally closed pieces with the following properties:
(a) Vo is the set of fixed points,
(b) For i > 0 there is an affine variety V/, an isomorphism </J; : V/ x k* ➔ V; and = an integer d; such that for x E V/, t, u E k* we have </J; (x, td; u) t .</J; (x, u),
(c) The closure of a V; is a union of some Vi.
(4) In the situation of 3.2.15 let G = GL (V). In that case there is the following de-
scription of a group P().): there is a flag in V, i.e. a sequence of distinct subspaces
V = Vo :::) Vi :J Vz :J ... of V such that P().) is the group of all invertible maps of V = = stabilizing all V;. (Hint: consider the case that G GLn, ).(a) diag(ah 1, ••• , ah")
with h, > h2 > ... ~ hn)·
(5) An affine embedding of T is an irreducible affine T -space V containing T as an
open subvariety such that the action T x V ➔ V extends the product map T x T ➔ T.
Then V is an affine toric variety.
(a) In that case there is a finitely generated sub-semigroup S of the group X which
generates X, such that k[V] is isomorphic to the semigroup algebra k[S]. (Hint: view
k[V] as a T-stable subspace of k[T].)
(b) Conversely, for every S with the properties of (a) there exists an equivariant
affine embedding V with k[V] '.::::'. k[S]. It is unique up to isomorphism of T-spaces.
(For more about toric varieties see [Od].)

3.3. Additive functions

49

3.3. Additive functions
3.3.1. Additive functions. An additive function on the linear algebraic group G is a homomorphism of algebraic groups f : G ~ G0 • The additive functions form
a subspace A = A(G) of the algebra k[G]. If F is a subfield of k and G is an F -
= group, we write A(F) A(G)(F) for the F-vector space of additive functions that
are defined over F. Notice that, if p = char k is non-zero, then the p-th power of
an additive function is again one. This fact is the reason for the introduction of a ring
over which A will be a module.
= First assume that p > 0. Then </>x xP defines an isomorphism of F onto a subfield FP. Recall that F is perfect if FP = F. We define a ring R = R(F) as
follows. The underlying additive group is the space of polynomials F[T] and the multiplication is defined by

Then R is an associative, non-commutative ring (this is the case for any isomorphism
</> of F onto a subfield). Notice that the subfield F of R does not lie in the center. The
degree function deg on R is as in the case of the polynomial ring F[T] and has the
= usual properties. They imply that R has no zero divisors. If p 0 then F is perfect (by convention). We now define R = R(F) = F.

3.3.2. Lemma. Assume that p > 0. let a, b E R and assume that deg a > 0.
(i) There exist unique elements c, d ER such that deg d < deg a and b =ca+ d;
(ii) If F is perfect there also exist unique elements c, d with deg d < deg a and
b = ac+d.

The proof is like that of the well-known division algorithm in the polyniomial

ring F[T]. The proof of (ii) requires that one can extract p-th roots in F, whence the

perfectness assumption.

D

3.3.3. Lemma. (i) left ideals in R are principal. If F is perfect the same holds
for right ideals; (ii) R is left noetherian. If Fis perfect R is also right noetherian; (iii) If F is perfect any finitely generated left R-module M is a direct sum of cyclic
modules. If, moreover, M has no torsion then it is free.
The assertions are trivial if p = 0, so assume p > 0. Then (i) follows from 3.3.2,
as in the case of F [T] and (ii) is a consequence of (i). In the case of (iii) let (mi) 1~i ~s be a set of generators of M. We have a surjective homomorphism Rs ~ M sending
the canonical basis element ei to mi. Let K be the kernel. It follows from (ii) that K is
= finitely generated, say by elements Li~i9 aiiej, By multiplying the matrix A (aij)
on the left and right by suitable invertible square matrices, one reduces to the case that
= aij 0 for i =f:. j, in which case (iii) is obvious. The argument is the same as that in
the case of F[T], which can be found in [Jac4, p.177-178]. Because in the case of R

50

Chapter 3. Commutative Algebraic Groups

both the left and right division algorithms are needed, we must require in (iii) F to be

perfect.

□

3.3.4. As in 3.3.1 let A(F) be the set of additive functions of the F -group G which are defined over F. If p > 0 we can define a structure of left R-module on A(F) by

= = If p 0 then R F and it is trivial that A(F) is an R-module. = = As an example, take G G:. Then F[G] F[T1, ••• , Tn]- An additive function
in F[G] is now an additive polynomial, i.e. an element f E F[T1, ... Tn] satisfying

where the T; and Ui are indeterminates. The set of additive polynomials is a left Rmodule A(G:)(F).

3.3.5. Lemma. A(G:)(F) is a free R-module with basis (T;),~;~n·

The assertion means that an additive polynomial only involves monomials of the

form T/i, where p is the characteristic exponent. Let D; be partial derivation in
F[T1, ••• , Tn] with respect to T;. If f is an additive polynomial, it follows from (5)
L that D; f is a constant c; for all i. Then / - c; T; is another additive polynomial
g and all derivatives D;g are zero. If p > 0 this means that g involves only the p-th
= powers of the variables, i.e there is a polynomial h with g h(Tt, ... , T/). But

then h is an additive polynomial of lower degree and we may assume by an induction

r/. that it is expressible in the

= Hence f is as asserted. The case p 0 is left to the

reader.

□

Now let G be an arbitrary F -group.

3.3.6. Lemma. (i) If G is connected the R-module A(G)(F) is torsion free; (ii) If f,, ... , fv are elements of A(G)(F) that are algebraically dependent over k, then they are linearly dependent over R.

If A(G)(F) had torsion there would be a non-constant f E k[G] satisfying a relation
= I P' + a, fP/1-I + ... +ah f 0,
with coefficients in k. Such an f would take only finitely many values, which is impossible if G is connected. This proves (i).
In the situation of (ii) there is a non-zero polynomial H E k[T1, ••• , Ts] with
= H (/1, ••• , Is) 0. Assume that H is such a polynomial with minimal degree. If

3.4. Elementary unipotent groups

51

x E G then the polynomial H(T1 + /1(x), ... , Ts+ fs(x)) H(T,, ... , Ts) also
gives an algebraic dependence between the /;, but has degree strictly smaller than that
of H. Hence it is zero. Let ii (T1, ••• Tn) be the coefficient of some monomial in the
indeterminates U; in

Then ii has degree smaller than H and ii (/1, ••• Is) = 0. It follows that ii = 0,
L which means that H is an additive polynomial. Write H = c; H;, where the H; are

additive polynomials with coefficients in F and the c; lie in k and are linearly inde-

= pendent over F. Then H; (/ 1, •.• ff) 0 for all i, and (ii) follows.

□

3.4. Elementary unipotent groups

3.4.1. We say that the unipotent linear algebraic group G is elementary if it is abelian and, moreover, if p > 0 its elements have order dividing p. G is a vector group if it is isomorphic to some G:. We first establish some auxiliary results, which will be needed in the discussion of the structural properties of elementary unipotent groups.
Assume that p > 0. If n E N is a natural number we denote by

its p-adic expansion, where the integers n; are uniquely determined by the requirement O :::: n; < p. They are Ofor almost all i E N. If m, n E N we write n ::::":p m if n; :::: m; for all i E N. If m, n E N we write (m, n) for the binomial coefficient m!(n!(m - n)!)- 1, with the convention that it is zero if n > m.

3.4.2. Lemma. (i) (m, n) = n; (m;, n;) (mod p);
(ii) (m,n) "I= 0 (mod p) ifandonlyifn ::::':pm.

Over a field of characteristic p > 0 we have (T + t r = n(TP; +tr;. Then (i)

follows by expanding the powers of T + 1 and equating coefficients of the powers of

T. Now (ii) follows from (i).

□

3.4.3. We next establish a result about polynomial 2-cocycles. If p > 0 we define

p-1
= c(T, U) LP-'(p,i) Tp-iui.
i=I

Notice that p- 1(p, i) is an integer for i =f:. 0, p. Let F be a perfect field and assume that / E F[T, U] satisfies

f (T + U, V) + f (T, U) = f (U + V, T) + f (U, V).

(6)

52

Chapter 3. Commutative Algebraic Groups

In (6) T, U, V are indeterminates.

= 3.4.4. Lemma. (i) If p 0 there is g e F[T] with = f (T, U) g(T + U) - g(T) - g(U);

(ii) If p > 0 there is g e F[T] such that f (T, U) - g(T + U) + g(T) + g(U) is a
linear combination l ofpolynomials ci;
= (iii) Iffor p > 0 we have, moreover, Li:::i:::p-l f (T, iT) 0, then the polynomial l
of (ii) is 0.
If f satisfies (6) the same is true for its homogeneous components. It follows that
= we may assume f to be homogeneous of degree d. We use induction on d. If d 0
the assertion is trivial. So assumed > 0. Putting T = U = 0 in (6) we find that
= f (V, 0) = 0. Putting U V = 0 we obtain f (O, T) = 0. Write

= L f(T, U)

d chrhud-h_

h=O

= = We have c0 cd 0. Equating coefficients of Th U; Vi in both sides of the equality
(6) we obtain

(7)
where h, i, j e N, h + i + j = d. For h = 0 or j = 0 we find from (7) that

(8)

Now assume O < h, j < d. Then (7) and (8) imply

(9)
If p = 0 we can rewrite this as

and (i) readily follows.
= From now on suppose that p > 0. From (9) with j 1 and (8) we obtain
(10) First assume that d is prime to p. Put
It follows from (10) that¥/: = 0. Similarly~ = 0. Hence f 1 is a polynomial in T P = and UP. Since d is prime to p we have f1 0, proving (ii) if p does not divide d.

3.4. Elementary unipotent groups

53

Now assume that p divides d and that ch =I= 0, with p not dividing h. If p divides

j we have by 3.4.2 (ii) that p divides (d - j, h) and (9) shows that p must divide

(d - h, j). If d - h > p we have by 3.4.2 that (d - h, p) ¢ 0 (mod p) and (9) with
= j p would lead to a contradiction. It follows that d - h < p and by (8) also h < p.
We conclude that d = p. Now (10) implies that f is a multiple of c.

There remains the case that all h with ch =I= 0 are divisible by p. In that case / is

the p-th power of a polynomial that also satisfies (6) (here we use the perfectness of

F). Then (ii) follows by induction. We have proved (ii).

(iii) follows from (ii), observing that Li~;~p-l c(T, iT) is a non-zero multiple of

TP.

□

3.4.5. We need a multi-variable generalization of 3.4.4. We now consider polynomials
= = in 2n variables T (T1, ... , Tn), U (U1, ... , Un), Write F[T, U] for the polyno-
mial algebra F[T1, ... , Tn, U1, ... , Un], Define ch e F[T, U] to be c(Th, Uh), where
c is as before. Assume now that f e F[T, U] satisfies

f(T + U, V) + /(T, U) = /(U + V, T) + /(U, V).

(11)

= 3.4.6. Lemma. (i) If p 0 there is g e F[T] such that

/(T, U) = g(T + U) - g(T) - g(U);

(ii) If p > 0 there is g e F[T] such that f (T, U) - g(T + U) + g(T) + g(U) equals
i
a linear combination l ofpowers cC ;
= (iii) I/for p > 0 we have, moreover, Li~;~p-l f (T, iT) 0 then the polynomial l of
(ii) equals 0.
(i) is proved as 3.4.4 (i), using a multi-variable binomial formula

We leave it to the reader to work out the details.
We deduce (ii) and (iii) from the corresponding assertions of 3.4.4 by a trick. If G e F[T] let d(G) be the maximum of the degrees of Gas a polynomial in one of the individual variables Th. Let q be an integer> d(G). It follows from the properties of q-adic expansions that Ti°' T;2 ... Tna,, ~ T 01 +aiq+...a,,qll-l defines a linear bijection of
the space of G e F[T] with d(G) < q onto the subspace of F[T] of polynomials of degree < qn. We denote this map by / ~ /. We have a similar map, sending poly-
nomials in two sets of variables T, U of degree< q in all the variables individually to
polynomials in two variables T, U, of degree < qn in both T and U.
Now let / be as in (11) and assume p > 0. Choose a p-power q that is strictly larger
= thandS/). IfG e ~[T] withd(~) < q an~ H(T, U) G(T+ U)-G(T)-G(U), = then H(T, U) G(T + U) - G(T) - G(U), as follows by using 3.4.2 (i). Also,
f = ch cq'•-•. The polynomial e F[T, U] satisfies (6). Apply 3.4.4 (ii) to f. The
polynomials whose existence is asserted in 3.4.4 (ii) have degree < qn and hence can

54

Chapter 3. Commutative Algebraic Groups

be written in the form g and l, with polynomials g e F[T], l e F[T, U]. These have

the properties of 3.4.6 (ii). In a similar manner, (iii) follows from 3.4.4 (iii).

□

After these preparations we come to one of the main results of this section. The ring Rand the R-module A(G) are as before.

3.4.7. Theorem. The following properties of a linear algebraic group G are equivalent: (a) G is elementary unipotent; (b) A(G) is an R-module offinite type. Its elements generate the algebra k[G];
(c) G is a vector group if p = 0 and a product ofa vector group and a finite elemen-
tary abelian p-group if p > 0.
Recall that an elementary abelian p-group is a product of cyclic groups of order p.
Assume that G is elementary unipotent and connected. By 2.4.12 we may assume that G is a closed subgroup of some group Um of upper triangular unipotent matrices. Denote by f;i E k[G] the (i, j)-th matrix element function (1 ~ i < j < m). These functions generate the algebra k[G]. We prove (b) for this case by induction on m.
= The case m 1 being trivial, we may assume that m > 1 and that (b) is known for
connected elementary subgroups of Um-I· There are two homomorphisms of algebraic groups <I>1, <l>z : Um -+ Um- I, the first
one being obtained by erasing the first row and column of a matrix and the second
one by erasing the last row and column. Then k[<J,1G] is generated by the f;i with i > 1 and k[<J,2G] by the /;j with j < m. Observing that an additive function for <J,1G or <l>zG is also one for G, we conclude from our induction assumption that there
are additive functions a1, •.. , an e k[G] such that the f;i with (i, j) =I= (1, m) all lie
in the subalgebra k[a 1, ... , an]. Using 3.3.6 (ii) and 3.3.3 (iii) we see that we may assume the ah to be algebraically independent. By the multiplication rule of matrices
(using the notations of 3.4.5) there is f e k[T, U] such that for x, y E G we have

It follows that f satisfies (11 ). If p > 0 the fact that the elements of G have or-
der dividing p implies the property of 3.4.6 (iii). We conclude that there exists h E k[a 1, ... , an] such that fim - h is an additive function. This shows that k[G] is generated by a finite number of additive functions, which can be taken to be algebraically independent. Then G is a vector group (check this). We have established the implications (a) ::::} (b) and (b) ::::} (c), if G is connected.
Now let G be an arbitrary elementary unipotent group. If p > 0 choose an element in each coset of the identity component G0. These representatives form an elementary abelian p-group A and it is immediate that G is the direct product of A and the vector group G0. This proves (c). The verification of (b) is left to the reader.
If p = 0, G does not contain any elements of finite order > 1. On the other hand,

3.4. Elementary unipotent groups

55

it follows from the fact that G0 is a vector group that each coset of G0 is represented by an element of finite order. This can only be if G is connected. We have proved (b) and (c) in this case.
Since the implication (c) => (a) is obvious the proof of 3.4.6 is complete. □

3.4.8. Corollary. let G be an F -group. Then G is elementary unipotent if and only ifone of the foilowing equivalent conditions holds : (a) A(G)(F) generates F[G]; (b) G is F -isomorphic to a closed F -subgroup ofsome G:.

Here A(G)(F) is as in 3.3.1. (a) follows from 3.4.7 (b), using that an additive

function in k[G] is a linear combination of additive functions in F[G]. To see this,
observe that f E k[G] is additive if and only if ~f = f ® 1 + 1 ® f, where ~

denotes comultiplication (2.1.2). This shows that A(G) is the kernel of a linear map

k[G] -+ k[G] ® k[G] which is defined over F. It then has a basis in F[G].

We skip the proof of the equivalence of (a) and (b).

D

When F is perfect a connected elementary unipotent F -group is F -isomorphic to some G:, but this is not generally true (see 3.4.10 (3), (4)). We shall return to these matters in 14.3.

We can now deal with the classification of connected one dimensional groups.

3.4.9. Theorem. let G be a connected linear algebraic group of dimension one. Then G is isomorphic to either Ga or Gm.

We have already seen in 3.1.3 that G is commutative and either consists of semi-

simple elements or is elementary unipotent. In the first case G is diagonalizable by

2.4.2 (ii), and then 3.2.7 (ii) gives that G :::: Gm. In the second case 3.4.7'(c) implies

that G :::'. Ga.

□

3.4.10. Exercises. F is a subfield of k.
(1) Let R = R(k) be as in 3.3.1. Elementary unipotent groups over k form a category,
which is anti-equivalent to the category of left R-modules of finite type. (For further results along these lines see 14.3.6). (2) (p > 0) Let c be as in 3.4.3. Define a structure of algebraic group on k2 with product
(x, x').(y, y') = (x + x', y + y' + c(x, x')) (x, x', y, y' Ek).

This group is connected, unipotent, commutative, of dimension two. Show that it is not isomorphic to G~. (3) Assume F to be perfect and let G be a connected elementary unipotent F -group

56

Chapter 3. Commutative Algebraic Groups

that is F-isomorphic to a closed subgroup of a triangular unipotent group Um. Then G is F -isomorphic to some G:. (Remark: the triangulizability condition is redundant, see 14.1.2).
= (4) Fis a non-perfect field of characteristic panda e F - FP. Then G {(x, y) e
G; I xP - x = ayP} is an F-group isomorphic to Ga which is not F-isomorphic to
Ga (Hint: use 2.1.5 (5)).
Notes
3.1.1 is due to Kolchin [Ko12, §3]. The name 'torus' for a connected diagonalizable group was coined by Borel in [Bo1]. He realized the important role played by these groups, similar to the role played by compact tori in the theory of compact Lie groups.
3.2 contains standard results on tori. The proof of the rigidity theorem 3.2.8 gives a stronger result: the affine variety V of the statement of that result may be replaced by any connected scheme over k. This implies that a diagonalizable group has no 'infinitesimal automorphisms.'
The theory of elementary unipotent groups bears some resemblance to the theory
of tori, the character group being replaced by the R(k)-module A of 3.3.1. The use
of the ring R(k) seems to go back to [DG, Ch. IV, 3.6]. In [loc. cit., Ch. V, 3.4] one finds more general results, for arbitrary commutative unipotent groups. These are described by 'Dieudonne modules.'
One of the main results of this chapter is the classification theorem 3.4.9. The first published proof seems to be the one given by Grothendieck in [Ch4, Exp. 7]. In [Bo3, Ch. III, §10] a proof is given that uses the fact that an irreducible smooth projective curve with an infinite group of automorphisms fixing a point is isomorphic to P 1. The proof given here is more elementary. We use the classification of elementary unipotent groups. We also need the result on polynomial cocycles of 3.4.4 (due to M. Lazard [Laz, lemme 31). Another proof of the classification theorem (also using additive polynomials) can be found in [Hut, no.20].

Chapter 4
Derivations, Differentials, Lie Algebras
We first discuss tangent spaces of algebraic varieties and related algebraic matters. In the second part of the chapter, Lie algebras of linear algebraic groups are introduced and their basic properties are established.
4.1. Derivations and tangent spaces
4.1.1. We recall the definition of a derivation. Let R be a commutative ring, A an R-algebra and Ma left A-module. An R-derivation of A in Mis an R-linear map
D : A ➔ M such that for a, b E A
D(ab) =a.Db+ b.Da.
It is immediate from the definitions that Dl = O, whence D(r.1) = 0 for all r E R. The set DerR(A, M) of these derivations is a left A-module, the module struc-
ture being defined by (D + D')a = Da + D'a and (b.D)(a) = b.Da, if D, D' E
DerR(A, M), a, b EA.
The elements ofDerR(A, A) are the derivations of the R-algebra A. If <I>: A ➔ B is a homomorphism of R-algebras and N is a B-module, then N is an A-module in an obvious way. If D E DerR(B, N) then Do <I> is a derivation of A in N and the map DH- Do <I> defines a homomorphism of A-modules

whose kernel is DerA(B, N). Thus we obtain an exact sequence of A-modules

(12)

4.1.2. Tangent spaces, heuristic introduction. We use the notations of the first chap-
ter. Let X be a closed subvariety of An. We identify its algebra of regular functions
k[X] with k[T1, ... , Tn]/ I, where/ is the ideal of polynomial functions vanishing on
X. Assume that / is generated by the polynomials / 1, . . . , fs. Let x E X and let L be
a line in An through x. Its points can be written as x + tv, where v = (v1, ... , Vn) is
a direction vector, t running through k. The t-values of the points of the intersection
L n X are found by solving the set of equations

f;(x + tv) = 0, 1 ~ i < s.

(13)

Clearly, t = 0 is a solution.

58

Chapter 4. Derivations, Differentials, Lie Algebras

Let D; be partial derivation in k[T] with respect to T;. Then
n
= f;(x + tv) t L vi(Djf;)(x) + t2(...),
j=l
and we see that t = 0 is a 'multiple root' of the set of equations (13) if and only if
n
LVj(Djf;)(x) = 0, 1 < i::: s.
j=l
If this is so we call L a tangent line and v a tangent vector of X in x.
Write D' = Ej=1 viDj, this is a k-derivation of k[T]. The last set of equations then says that D' f;(x) = 0 for 1 < i ::: s. Denoting by Mx the maximal ideal in k[T]
of functions vanishing at x, it follows that D' I c Mx. The linear map f i-+ ( D' f) (x)
= factors through/ and gives a linear map D : k[X] ➔ k k[X]/ Mx, Viewing k as a
k[X]-module kx via the homomorphism f i-+ f (x), we see that Dis a k-derivation
of k[X] in kx, Conversely, any element of Derk(k[X], kx) can be obtained in this manner from a
derivation D' of k[T] with D' I c Mx, We conclude that there is a bijection of the set
= of tangent vectors v, such that (13) has a 'multiple root' t 0, onto Derk(k[X], kx),
4.1.3. Tangent spaces. The heuristic description of tangent vectors of 4.1.2 suggests a formal definition of the tangent spaces of an algebraic variety. First let X be an affine variety. If x e X we define the tangent space TxX of X at x to be the k-vector space Derk (k [X], kx ), where kx is as in 4.1.2. Let <I> : X ➔ Y be a morphism of affine varieties with corresponding algebra homomorphism</>* : k[Y] ➔ k[X] (1.4.7). The
induced linear map ¢0(see 4.1.1) is a linear map of tangent spaces
d</>x : Tx X ➔ Tq,x Y,
the differential of <I> at x, or the tangent map at x.
If l/1 : Y ➔ Z is another morphism of affine varieties then we have the chain rule

If</> is an isomorphism then so is d</>x and the differential of an identity morphism is an identity map.
We give two alternative descriptions of the tangent space TxX. Let Mx C k[X] be the maximal ideal of functions vanishing in x. If D e Tx X then D maps the elements of M} to 0. Hence D defines a linear function )..(D): Mx/ M} ➔ k.

4.1.4. Lemma.).. is an isomorphism of TxX onto the dual of Mx/ M}.

Let l be a linear function on Mx/ M}. Define a linear map µ,(l) : k[X] ➔ k by
µ,(l)f = l(f - f (x) + M}). Then µ(l) E TxX, andµ is the inverse of)... We skip the

easy proof.

□

4.1. Derivations and tangent spaces

59

Another description of the tangent space uses the ring Ox of functions regular in x (see 1.4.3). It is a k-algebra which has a unique maximal ideal Mx, consisting of the functions vanishing in x (1.4.4 (1)) and Ox/Mx '.::'. k. We view k as an Ox-module. There is an obvious algebra homomorphism a : k[X] ➔ Ox, inducing a linear map ao : Derk(Ox, k) ➔ Derk(k[X], kx),

4.1.5. Lemma. a0 is bijective.
We have a linear map /3 : Derk(k[X], kx) ➔ Derk(Ox, k), which comes from
the formula for differentiating a quotient. Let f E Ox and let g, h E k[X] be such that h(x) =I- 0 and that hf - g vanishes in a neighborhood of x (see 1.4.1). If D E Derk(k[X], kx) then
= ({3D)f h(x)-2(h(x)Dg - g(x)Dh)
defines an element of Der(Ox, k) and it is immediate that a0 and f3 are inverses. D

4.1.6. Lemma. Let <J, be an isomorphism of X onto an affine open subvariety of Y. Then d<J>x is an isomorphism ofTxX onto T,t,xY,

<J, induces an isomorphism Or,,t,x '.::'. Ox,x• The assertion follows from the previ-

ous lemma (check the details).

D

4.1.7. We can now define the tangent space in a point x of an arbitrary algebraic variety X (see 1.6). It follows from 4.1.6 that if U and V are open affine neighborhoods of x in X with V C U there is a canonical identification Tx U '.: : '. Tx V. This allows us to define the tangent space Tx X to be Tx U, for U as above. The formal definition is
Tx X = limproj Tx U,
a projective limit relative to the set of open affine neigborhoods of x, ordered by inclusion. It is clear how to define for a morphism of varieties <J, : X ➔ Y the tangent map
d</>x : Tx X ➔ T,t,x Y.
We say that x is a simple point of X, or that X is smooth in x or that X is non-singular
in x if dim Tx X = dim X (the dimension of X, defined in 1.8.1 ). X is smooth or
non-singular if all its points are simple.

4.1.8. Let X be an F-variety, where Fis a subfield of k and let x E X(F) (1.6.14). First assume that X is an affine F-variety. The point x defines an algebra homomorphism F[X] ➔ F, which makes F into an F[X]-module Fx. Define
TxX(F) = Derp(F[X], Fx);

60

Chapter 4. Derivations, Differentials, Lie Algebras

this is a vector space over F. We have a canonical isomorphism

We call TxX(F) the space of F-rational points of TxX and we view it as an Fsubspace of Tx. If cp is a morphism of affine F -varieties then for x E X ( F) the tangent map d</Jx maps TxX(F) to Tq,xY(F). If Xis an arbitrary F-variety, the definition of Tx X ( F) is similar to that of Tx X, given in 4.1.7.
4.1.9. Exercises. (1) Using 4.1.4, describe Tx X in the following cases: (a) X is a point,
(b) X = An,
= = (c) X {(a, b) E A 2 I ab= 0}, x (0, 0), = = = (d) (char k =/:. 2, 3) X {(a, b) E A 2 I a 2 b3}, x (0, 0).
(2) Let X and Y be algebraic varieties, let x E X, y E Y. Then T<x.y)(X x Y) ~
TxX EB TyY,
(3) Let X be an affine algebraic variety. Let k[r] = k[T]/(T2) be the k-algebra of dual numbers (r being the image of T). Show that there is a bijection of Tx X onto the set of k-homomorphisms cp : k[X] ➔ k[r] such that cp(f) - f (x) E kr:. (These are the 'k[r]-valued points of X lying over x' .) (4) If X is a closed subvariety of Y and cp : X ➔ Y is the injection morphism then d</>x is injective for all x E X. (5) Complete the details in 4.1.8.
4.2. Differentials, separability
We shall need a number of results about derivations, in particular about derivations of fields. To deal with them we introduce differentials.
4.2.1. Let R be a commutative ring and A a commutative R-algebra. Denote by
m: A®RA ➔ A
the product homomorphism (so m(a ® b) = ab) and let/ = Ker m. This ideal of
A® A is generated by the elements a® 1-1®a (a E A). The quotient algebra A® A/ I is isomorphic to A. We define the module ofdifferentials nA/R of the R-algebra A by
g_A/R = ///2•
This is an A® A-module, but since it is annihilated by/ we may and shall view it as an A-module.
Denote by da or dA/Ra the image of a® 1 - 1 ® a in nA/R· One checks that d is an R-derivation of A in nA/R and that the da generate the A-module nA/R· The following result shows that nA/R is a 'universal module for R-derivations of A.'

4.2. Differentials, separability

61

4.2.2. Theorem. (i) For every A-module M the map <l> : HomA (QA/R• M) ➔
DerR(A, M) sending</> to</> o dis an isomorphism of A-modules; (ii) A pair (QA/R, d) ofan A-module together with an R-derivation of A in nA/R with the property of (i) is unique up to isomorphism.

<l> is a homomorphism of A-modules, which is injective since the da generate
nA/R· Now let D E Der(A, M). Define an R-linear map 1/1 : A ® A ➔ M by
= 1/l(a ® b) bDa. Then

= + 1/l(xy) m(x)1/l(y) m(y)1/l(x).

It follows that 1/1 vanishes on / 2, hence defines an R-linear map</> : nA/R ➔ M,
= which in fact is A-linear. Since 1/l(a ® 1 - 1 ® a) =Dawe have <l>(</>) D. Hence

<l> is surjective, proving (i). The proof of (ii) is standard.

D

4.2.3. If </> : A ➔ B is a homomorphism of R-algebras, there is a unique homomorphism of A-modules
<pO : QA/R ➔ QB/R,
= with ¢0 o dA/R dB/Ro <J>. If N is a B-module and if </>o is as in 4.1.1, there is a
commutative diagram of A-modules:

HomB(nB/R, N) ~ DerR(B, N)

l

l¢o

HomA(Q• A/R, N) ---:- DerR•(A, N).

The horizontal arrows are isomorphisms, they are as in 4.2.2. The left-hand vertical
arrow is induced by ¢ 0.
Now let A be an R-algebra of the form A = R[T1, ... , Tm]/(f1, ... , fn)- Lett;
= be the image of 1'; in A and putt (t1, ... , tm). Denote by D; partial derivation in
R[T1, ... , Tm] with respect to T; (1 ::: i :::: m).

4.2.4. Lemma. The dt; (1 < i < m) generate the A-module nA/R· The kernel
= of the A-homomorphism</> : Am ➔ nA/R with <J>e; dt; is the submodule generated "£7': by the elements 1(D; Jj(t))e; (1 ::: j ::: n).

Here (e;) is the standard basis of An. Let D be an R-derivation of A in an A-module M. If f E R[T1, ... , Tm] then

m

= D(f(t)) L(D;f)(t).Dt;,

(14)

i=l

62

Chapter 4. Derivations, Differentials, Lie Algebras

whence

m
L(D;/j)(t).Dt; = 0 (1 < j < n).
i=l

Let K be the submodule of An described in the lemma. A straightforward check

shows that the A-module An/K (together with a derivation of A given by (14)) has

the universal property of 4.2.2, hence is isomorphic to QA/R by 4.2.2 (ii).

D

= 4.2.5. Exercises. (1) If A R[T1, ... , Tm] then nA/R is a free A-module with
basis (dT;)1~i~m•
(2) In the case of 4.2.4 with m = n = 1, give a necessary and sufficient condition on
/ 1 under which nA/R = 0. Consider the case when Risa field. (3) Let A be an R-algebra which is an integral domain and let F be the quotient field of A. Then nF/R ~ F ®A nA/R·
= (4) Let F be a field and let E F(x1, ... , Xm) be an extension field of finite type.
Then nE/F is a finite dimensional vector space over E spanned by the dx;. (5) Let A= k[T, U]/(T2 - U3). Show that nA/k is not a free A-module. (6) Let A and B be R-algebras. There is an isomorphism of A ®RB-modules

4.2.6. We next discuss the case of fields. Let F be a field and let E, E' be two
extensions of F of finite type, with E' c E. By (12) we have an exact sequence of
groups
which is also an exact sequence of vector spaces over E (the vector space structures coming from the second arguments). Using 4.2.2 (i) we obtain an exact sequence of E-vector spaces
n n Since the map u 1--+ I ® u of E'1F to E ®E' E'1F induces an isomorphism of
E-vector spaces
we get an exact sequence of £-vector spaces

4.2. Differentials, separability

63

These vector spaces are finite dimensional by 4.2.5 (4). Dualizing we obtain an exact sequence of finite dimensional £-vector spaces
(15)
= which is basic in what follows. Notice that a(l ® dE'/Fx) dE/FX (x E E').
Recall (see for example [La2, Ch. VII, §4]) that Eis a separable algebraic extension of F or is separably algebraic over F if for each x E E there is a polynomial
= / E F[T] without multiple roots such that f (x) 0. We may assume f to be ir-
reducible, in which case the derivative f' is a non-zero polynomial. If char F = 0
every algebraic extension of F is separable.
4.2.7. Lemma. If E is separably algebraic over E' then a is injective.
From the discussion in 4.2.6 we see that the injectivity of a is equivalent to the
surjectivity of the homomorphism of 4.1.1

An equivalent property is: any F-derivation of E' in E can be extended to an F -

derivation of E in E. To prove this it suffices to deal with the case of a simple
= extension E E'(x) '.'.:::: E'[T]/(/), where f is an irreducible polynomial with
= f'(x) =I= 0. Let D E DerF(E', E). If g L;>0 a;Ti E E'[T] define Dg E E[T]
= by Dg I:,(Da;)Ti. Then Dis extendible to-an F-derivation D' of E in E with

D'x = a if and only if f'(x)a + (Df)(x) = 0. Since f'(x) =I= 0 this equation has a

unique solution and the lemma follows.

□

= = 4.2.8. Lemma. Let E F(x). Then dimEQE/F ::::: I. We have QE/F 0 if and
only if Eis separably algebraic over F.

If x is transcendental over F this follows from 4.2.5 (1), (3). If x is algebraic we

are in the situation of 4.2.5 (2).

□

= We denote by trdegFE the transcendence degree of E over F. If E F(x1, ... , Xm)
this is the maximal number of x; that are algebraically independent over F. Recall that E is purely transcendental over F if the x; can be taken such that the transcendence degree equals m. We say that E is separably generated over F if there is a purely transcendental extension E' of F, contained in E, such that E is separably algebraic
= over E'. Let p char F. If p = 0 then E is always separably generated.
We can now state the main result about fields.

4.2.9. Theorem. (i) dimEQE/F ~ trdegFE; (ii) Equality holds in (i) ifand only if Eis separably generated over F.

64

Chapter 4. Derivations, Differentials, Lie Algebras

= We prove (i) and (ii) together, by induction on d dimEQE/ F·
= Let E F(x1, ... , Xm).

First let d = 0. If m = 1 we have (i) and (ii) by 4.2.8. If m > 1 then (15)

with E' = F(x1) shows that QE/F(xi> = 0. By induction on m we may assume
that E is separably algebraic over F(x1). Using 4.2.7 we conclude from (15) that
= QF<xi)/F 0, and x1 is separable over F by 4.2.8. It follows that E is separably = algebraic over F, proving (i) and (ii) in the case d 0. By induction on m one also = shows that QE/F 0 if Eis separably algebraic over F.

Now let d > 0 and assume that (i) and (ii) are known for smaller values. By
what we have already proved there is x e E with dE/ FX =/:- 0. We use (15) with
= = E' F(x). Since a(l ® dF(x)/Fx) dE/FX =/:- 0 we have QF(x)/F =/:- 0. Using
= 4.2.8 we conclude that dimF(x)QF(x)/F I and that a is injective. Consequently,
= + + dimEQE/F dimEQE/F(x) 1. By induction we have dimEQE/F 2: trdegF(x)E 1. = Since trdegFE trdegF(x)E +trdegFF(x) (see [La2, Ch. X, 8.51) and trdegFF(x) <
1, (i) follows. If we have equality in (i) then x is transcendental over F, and by

induction Eis separably generated over F(x), hence also over F.

To finish the proof we have to show that if E is separably generated over F we

have equality in (i). Now apply (15) for E' a purely transcendental extension over
= which Eis separably algebraic. We have already seen that QE/E' 0. Using 4.2.7

we find, using 4.2.5 (1),(3), that dimEQE/F = dimE,QE'/F = trdegFE' = trdegFE,

finishing the proof.

D

= We say that E is separable over F if either p 0 or if p > 0 and the following
holds: Let x 1, ... , Xs be elements of E which are linearly independent over F. Then
so are xf, ... , xf. It is immediate that if F is perfect any extension E is separable.
Recall that F is perfect if either p = 0 or if p > 0 and every element of F is a lh
power. In particular, an algebraically closed field is perfect.

4.2.10. Proposition. Assume that Eis separable over F. Then Eis separably gener-
ated over F.
= Let E F(x1, ... , Xm) and assume that x1, ... , x1 are algebraically independent
over F, with t = trdegFE. Let Ebe separable over F and let p > 0 (for p = 0 the result is trivial). We may also assume that 0 :::: t < m. If t < m - 1 then, by induction on m, there are algebraically independent elements Y1, ... , y1 in F(x1, ... , Xm-1) such that F(x1, ... , Xm- 1) is separably algebraic over F(y1, ... , y1). Likewise, there are algebraically independent elements z1, ... , Zr in F(y1, ... , y1, Xm) such that this field is separably algebraic over F(z 1, ... , z1). Using a transitivity property of separably algebraic extensions (see [La2, Ch. VII, §4)) we conclude that E is separably algebraic over F(z1, ... , z1).
= This reduces the proof to the case that t m - 1. Since E is separable over F

4.2. Differentials, separability

65

there is a non-zero polynomial f E F[T1, ... , Tm] such that f(x1, ... , Xm) = 0 and

that not all exponents of the powers of the indeterminates occurring in f are divisible
by p. Using 4.2.4 (with R = F, A = F[x1, ... , Xm]) and 4.2.5 (3) it follows that

dimEOE; F < m - I (notice that the kernel of the homomorphism (/J of 4.2.4 is non-

zero). Now the assertion follows from 4.2.9.

□

The converse of 4.2.10 is also true (as a consequence of 4.2.12 (5)).

Let E, E', F be as in 4.2.6.

4.2.11. Corollary. Assume F to be perfect. Either of the following conditions is necessary and sufficient for E to be separably generated over E': (a) a : E ®E' QE'/F ➔ QE/F is injective; (b) DerF(E, E) ➔ DerF(E', E) is surjective.

The equivalence of (a) and (b) follows from 4.2.6. To obtain the criterion (a) ob-
= = serve that by 4.2.9 (ii) and 4.2.10 we have dimE,QE'/F trdegFE', dimEQE/F
trdegFE. It follows that dimEQE/E' = trdegE,E if and only if the map a of (15) is

injective. Then 4.2.11 follows from 4.2.9 (ii).

□

4.2.12. Exercises. (I) If in the case of 4.2.11 E is separably algebraic over E', then a is an isomorphism.
(2) Let p = 0. If x EE, dE;Fx = 0 then xis algebraic over F.
(3) Let p > 0. Show that QE/F = 0 if and only if E = F(EP). (Hint: use (15) and 4.2.8.) (4) Assume that E has the following property: for any algebraic extension K of F the algebra K ®Eis reduced. Then Eis separably generated over F. (5) (a) If Eis separable over E' and E' is separable over F then E is separable over F.
(b) If E is separably algebraic over F then E is separable over F.

4.2.13. In applying the preceding results to geometric questions we need some auxiliary results, pertaining to linear algebra.
Let R be an integral domain with quotient field F. If f E R, f =I= 0 denote by R1 the ring R[T]/(1 - fT) (see 1.4.6). We may and shall view it as the subring R[f-1] of F, i.e., the ring of elements of F of the form f-na (a E R, n > 0). If Mis an R-module we denote by MI the Rrmodule R1 ®R M.
= Let A (aij) be an m x n-matrix with entries in R. Denote by r the rank of A,
viewed as a matrix with entries in F. Define the R-module M(A) or MR(A) to be the
quotient of Rn by the submodule generated by the elements I:j=1 a;iei (I < i :'.Sm),
where (ei) is the canonical basis. We denote by GLm (R) the group of those m x mmatrices with entries in R that have an inverse with entries in R.
= 4.2.14. Lemma. (i) If B E GLm(R) then M(BA) M(A),; if C E GLn(R)

66

Chapter 4. Derivations, Differentials, Lie Algebras

then M(AC) ~ M(A),· (ii) There exist f ER, f =/:- Oand BE GLm(R), CE GLn(R) such that
~ A= B ( ~) C.

Here Ir is an identity matrix. (i) is easy. The assertion of (ii) is true if R is a field, by linear algebra. Take B E GLm(F), CE GLn(F) with the required property and
c- choose f ER such that B, C, B-1, 1 have entries in R1. Then we have (ii). □
4.2.15. Lemma. There is f E R, f =I- 0 such that M(A) 1 is a free Rrmodule
of rank n - r. We may choose f such that n -r ofthe images e; of the elements 1® e; of R1 ® Rn form a basis ofM(A)1-

The first point follows from 4.2.14 (ii). Let (f1, ... , fn-r) be a basis of M(A) 1 and assume that the elements e~+l • ... , e~ are linearly independent over F. Write

= Ln-r

e;+i

C;j/j (1 < i ~ n - r),

j=l

with cij E R1 , det(cij) =I- 0. By modifying f we may assume that the inverse matrix

(cii )-1 has entries in R1. Then e~+1, . . . , e~ are as required.

D

4.3. Simple points

4.3.1. Let X be an irreducible affine variety over k. If x E X let again Mx be the maximal ideal in k[X] of functions vanishing in x. If M is an R-module, put
M(x) = M/ MxM; this is a vector space over k.
Let A be as in 4.2.13, with R = k[X]. Since the entries of A are functions on X
the matrix A(x) with entries in k is defined. It is clear that

(notations of 4.2.13). As before r is the rank of A, as a matrix with entries in the field k(X).
4.3.2. Lemma. (i) dimk(X> Mk(X)(A) = n - r;
= (ii) The set of x E X such that dimk M(A)(x) n - r is open and non-empty;
(iii) If x E X and dimk M(A)(x) = n - r there is f E k[X] with f (x) =I- 0 such that
M(A)1 is a free k[X]rmodule of rank n - r.
= One knows from linear algebra that dimk M(A)(x) n - r if and only if r is the
maximal size of a square submatrix of A(x) with non-zero determinant. By the same result, r equals this maximal size for the matrix A. (i) and (ii) follow from these facts.

4.3. Simple points

67

Let x be as in (iii). We may assume that det(aij(X)i:::i.j:::r) =I= 0. Put f = e; det(aij)l::,i,j:::r• If E M(A) I is as in 4.2.15 then

n
LaiieJ = 0 (1 ~ i < m).
j=I

Our assumption implies that we can express the e; with 1 ~ i < r as linear combina-

tions with coefficients in k[X] 1 of e~+I' ... , e~. Using (i) we conclude that the latter

elements form a basis of M(A) 1, whence (iii).

□

= We put nx Qk[XJ/k· If x E X the tangent space TxX is isomorphic to
Hom(S'lx, kx), by 4.2.2 (i). Since for any k[X]-module M we have

Homkcx1(M, kx) ~ Homk(M(x), k)

it follows that

The dual vector space of the tangent space TxX is the cotangent space (TxX)*. It can be identified with S'lx(x).

We come now to the basic results on simple points (defined in 4.1.7).

4.3.3. Theorem. Let X be an irreducible variety ofdimension e. (i) If x is a simple point of X there is an affine open neighborhood U of x such that
nu is a free k[U]-module with a basis (dg 1, ... , dge), for suitable g; E k[U];
(ii) The simple points of X form a non-empty open subset of X; (iii) For any x E X we have dimk TxX > e.

We may assume that Xis affine and that k[X] = k[T1, ... , Tmll(fi, ... fn)- With

the notations of 4.2.4 we have nx ~ M(A), where A is them x n-matrix (Djf;(t)).
= From 4.2.9 (ii) and 4.2.10 we see that dimk(X) Qk(X)/k e. If r is as before it follows from 4.3.2 (i) that e = n - r. Now (i) follows from 4.3.2 (iii) and 4.2. 15, and (ii) is a

consequence of 4.3.2 (ii). For any x E X the dimension of TxX is n - s, wheres is

the rank of (Difi(x)). It is clear that r > s, whence (iii).

D

4.3.4. Exercises. (1) The functions g1, ... , 8e of 4.3.3 (i) are algebraically independent. (2) Let X be an affine variety. If nx is a free k[X]-module then Xis smooth.

4.3.5. A morphism </> : X ~ Y of irreducible varieties is called dominant if </> X is dense in Y. It follows from 1.9.1 (ii) that, if</> is dominant, there is an injection of quotient fields k(Y) ~ k(X). So we can view k(X) as an extension of k(Y). We say that </> is separable if this extension is separably generated.

68

Chapter 4. Derivations, Differentials, Lie Algebras

Assume, moreover, that X and Y are affine. The homomorphism </>* : k[f] ➔ k[X]
defining</> induces by 4.2.3 a homomorphism of k[f]-modules

Let x e X and let kx be as in 4.1.2. Viewed as a k[f]-module (via</>*) it is kq,x• Consider the linear map of 4.1.3
d</>x : Tx X ➔ Tq,x Y.
From the preceding remarks, using the diagram of 4.2.3, we see that we can view d</>x as the homomorphism

deduced from (</>*)0. It can also be viewed as a linear map
Hom(Ox(x), k) ➔ Hom(Or(</>x), k).
4.3.6. Theorem. Let </> : X ➔ Y be a morphism of irreducible varieties.
(i) Assume that x is a simple point of X such that </>xis a simple point of Y and that d</>x is surjective. Then</> is dominant and separable;
(ii) Assume that</> is dominant and separable. Then the points x e X with the prop-
erty of (i) form a non-empty open subset of X.
It follows from 4.3.3 that it suffices to consider the case that X and Y are affine
and smooth, and that nx and Or are free modules over k[X] resp. k[Y] of rank
d = dim X and e = dim Y.
The homomorphism (q,*)0 of 4.3.5 leads to a homomorphism of free k[X]-modules
V, : k[X] ®k[Y] Qy ➔ nx.
Fixing bases of these modules, v, is described by a d x e-matrix A with entries in
k[X]. Let x e X be such that d</>x is surjective. Then by the remarks of 4.3.5, the matrix A(x) has rank e. An argument involving determinants, as in the proof of 4.3.3, shows that the rank of A (as a matrix with entries in k(X)) is at least e. Since this
rank is at most e it must equal e. It follows that v, is injective. Then the same holds
for (q,*)0. As nx and Qy are free modules, the homomorphism¢* : k[Y] ➔ k[X]
is also injective, which means that¢ is dominant. By 4.2.5 (3) it also follows that the
= = = homomorphism a of (15) in 4.2.6, with E k(X), E' k(Y), F k, is injective
(on suitable bases it is given by the matrix A). The separability of q, now follows from
4.2.11. We have proved (i).

4.4. The Lie algebra ofa linear algebraic group

69

If</> is dominant and separable, the rank of A (as a matrix with entries in k(X))

equals e. The set of x E X such that A(x) has rank e is then non-empty and open,

whence (ii).

□

To conclude this section we give some consequences of the preceding results for homogeneous spaces (defined in 2.3.1).

4.3.7. Theorem. Let G be a connected algebraic group.
(i) Let X be a homogeneous space for G. Then Xis irreducible and smooth. In par-
ticular, G is smooth;
(ii) Let</> : X ➔ Y be a G-morphism of homogeneous spaces. Then </> is separable if and only if the tangent map d<f>x is surjective for some x E X. If this is so then d</>x is
surjectiveforall x EX; (iii) Let</> : G ➔ G' be a surjective homomorphism of algebraic groups. Then</> is
separable if and only if d</>e is surjective.

If X is as in (i) and x E X, the morphism G ➔ X sending g to g .x is surjective.

Hence X is irreducible by 1.2.3 (ii). Also, for fixed g, the map x H- g .x is an iso-

morphism of X. Hence x is simple if and only if g .x is simple. Now (i) follows from

4.3.3 (ii) and (ii) from 4.3.6. Finally, (iii) is a consequence of (ii) (view G and G' as

homogeneous spaces for G).

□

4.4. The Lie algebra of a linear algebraic group

4.4.1. Let G be a linear algebraic group. Denote by ).. and p the representation of
= G by left and right translations in A k[G] (2.3). We view A ®k A as the algebra
of regular functions k[G x G]. If m : A® A ➔ A is the multiplication map, then
= for F E k[G x G] we have (mF)(x) F(x, x). So I = Ker m is the ideal of
functions in k[G x G] vanishing on the diagonal (see 1.6.5). It is clear that for x E G the automorphisms )..(x) ® )..(x) and p(x) ® p(x) of k[G x G] stabilize / and / 2,
= hence induce automorphisms of !la I/ 12, also denoted by )..(x) and p (x). We thus
have representations).. and p of G in !la, which are locally finite (by 2.3.6(i)). The derivation d: A ➔ !la of 4.2.1 commutes with all )..(x) and p(x).
For x E G the map Int(x) : y H- xyx-• is an automorphism of the algebraic group G, fixing e. It induces linear automorphisms Ad x of the tangent space Te G and (Ad x)* of the cotangent space (TeG)*. Thus, for u E (TeG)* we have
= ((Ad x)*u)X u(Ad(x-1)X) (x E G, XE TeG)
Let Me C A be the maximal ideal of functions vanishing in e. By 4.1.4, (TeG)* can
be identified with Me/ M;. If/ E A we denote by l,f the element f - f (e) + M; of
(TeG)*. For X E TeG = Derk(A, ke) we have (l,/)(X) = X/, as follows from the proof of 4.1.4.

70

Chapter 4. Derivations, Differentials, Lie Algebras

4.4.2. Proposition. There is an isomorphism ofk[G]-modules

(the module structure ofthe right-hand side coming from the firstfactor), such that (a) <I> o ).(x) o <1>- 1 = ).(x) ® id, <I> o p(x) o <1>-1 = p(x) ® (Ad x)* (x e G);

(b) if f E k[G] and !if = L; /; ® g;, then

L /; <l>(df) = -

® ~g;.

i

In (b) fl : A ➔ A® A is the comultiplication of 2.1.2 (so (llf)(x, y) = f (xy)).

The map sending (x, y) to (x, xy) is an automorphism of the algebraic variety
G x G. The corresponding algebra automorphism VF of A ® A is given by
(VF F)(x, y) = F(x, xy) (x, y e G).
It follows that VF/ is the ideal of functions vanishing on G x {e}, which is k[G] ® Me. Then VF / 2 = k[G] ® M; and it follows that VF induces a bijection of Oa onto k[G] ®
Me/ M;. Let <I> be the composite of this bijection and the isomorphism coming from
4.1.4. From the definition of VF it follows that for x e G
().(x) ® id) o VF = VF o ().(x) ® ).(x)),

(p(x) ® Int(x)) o VF = VF o (p(x) ® p(x)).

These formulas imply that <I> satisfies (a). With the notations of (b) we have
L VF(f ® 1 - 1 ® f)(x, y) = /;(x)(g;(e) - g;(y)),
i

from which we see that <I> satisfies (b).

D

4.4.3. We assume that the reader is familiar with the basic facts about Lie algebras (which can be found in [Bou2, Ch. 1] or [Jacl]).
If A is an arbitrary commutative k-algebra, the space V = Derk(A, A) has a Lie algebra structure, the Lie product being given by [D, D'] = Do D' - D' o D (D, D' e V).
If p = char k > 0 then by Leibniz's formula
p
+ DP(ab) = L(P, i)(Dia)(DP-ib) = a(DPb) (DPa)b (a, be A, DEV)
i=O
(where (p, i) is a binomial coefficient). So DP is again a derivation. In this case V is an example of a restricted Lie algebra (or p-Lie algebra) . This means that

4.4. The Lie algebra ofa linear algebraic group

71

the Lie algebra has a p-operation D 1--+ D[PJ (which in the case of derivations is the ordinary p th power) such that the following holds for a E k, D, D' E V (with (ad D) D' = [D, D'])
= (a) (aD)[PJ aP D[P1, (b) ad(D[P1) = (ad D)P, (c) (Jacobson's formula) (D + D')lP1 = D[PJ + D'[PJ + °Ef:/ i-1s;(D, D'),
where s;(D, D') is the coefficient of a; in ad(aD + D')P-1(D'). For a further discussion see [Bou2, Ch. I, p. 105-106]. Now let G and A be as in 4.4.1. If necessary we write V = Va. Then ). and p
define representations of Gin V, denoted by the same symbols. So
= ).(x)D ).(x) o Do ).(x)-1 (x E G, D EV),
and similarly for p. The Lie algebra L(G) of G is the set of D EV commuting with all ).(x) (x E G). It is immediate that L(G) is a subalgebra of V, stable under the p-operation if p > 0. Since left and right translations commute, all p(x) stabilize L(G). We denote the induced linear maps also by p(x).
4.4.4. Corollary. There is an isomorphism ofk[G]-modules

(the module structure of the right-hand side coming from the first factor), such that
w- = w- = (a) \If O ).(x) 0 1 ).(x)@ id, \If O p(x) 0 1 p(x)@ Ad X (x E G);
(b) (notations of4.4.2 (b))

This is a consequence of 4.4.2 and 4.2.2 (i). These results give an isomorphism of Ve

onto Homk[GJ(k[G] ®k (TeG)*, k[G]), which is a module isomorphic tok[G] ®k TeG,

There is an isomorphism of the latter module onto the former which sends f ® X to the
= homomorphism</> with </>(g ® u) u(X)fg (f, g E k[G], X E TeG, u E (TeG)*).

The assertions of the corollary are readily checked. For the last one observe that

Xg; = (Sg; )(X).

D

Let ac =a: Va-+ TeG be the linear map with (acD)f = (Df)(e).

4.4.5. Proposition. (i) a induces an isomorphism of vector spaces L(G) :::::'. TeG, We have for x E G
= a o p(x) o a-1 Ad x;

(ii) Ad is a rational representation ofGin TeG (the adjoint representation).

72

Chapter 4. Derivations, Differentials, Lie Algebras

Let \II be as in 4.4.4. It follows from 4.4.4 that \ll(L(G)) = 1 ® TeG. Moreover,

4.4.4 (b) implies that (with the previous notations)

L (a O w- 1)(1 ® X)(f) = - f;(e)(Xg;) = -Xf,

i

L; since f = f;(e)g;. Now (i) and (ii) readily follow.

□

4.4.6. Corollary. dimk L(G) = dim G.

This is a consequence of (i) and 4.3.7 (i) (applied to the identity component G0).
□
Next let H be a closed subgroup of G. Denote by J c k[G] the ideal of functions
vanishing on H, so k[H] = k[G]/J. Put
Va.H = {D e Va I DJ c J}.
Then Va,H is a subalgebra of the Lie algebra Va and there is an obvious homomorphism of Lie algebras <I>: Va,H ➔ Vff. Notice that

4.4.7. Lemma. <I> de.fines an isomorphism ofVa,H n L(G) onto L(H).
It follows from the definitions that aH o <I> is the restriction of aa to Va,H- The injectivity of <I> on L(G) then follows from 4.4.5 (i). To finish the proof of the lemma
we show that, if X e TeH, we have D = w-10 ® X) e Va,H, where \II is as in
L 4.4.4. If f e J and !if = f; ® g; then we may assume that for each i one of the
elements f; or g; lies in J. Then 4.4.4 (b) shows that DJ e J, whence the lemma. D
4.4.8. Henceforth we identify the Lie algebra L(G) and the tangent space TeG via aa. We thus obtain a Lie algebra structure on the latter space. We shall denote the Lie algebra of linear algebraic groups G, H, ... either by L(G), L(H), ... or by the corresponding gothic letters g, ~ ....
If <I> : G ➔ G' is a homomorphism of linear algebraic groups, we write dq> for the tangent map d</>e : g ➔ g'. We call d</> the differential of q>. If Fis a subfield of k and G is an F-group, we denote the F-vector space TeG(F) of 4.1.8 by L(G)(F) or g(F). This is the set of F-rational points of g. It is a Lie algebra over F. If <I> : G ➔ G' is a homomorphism of F-groups then d<J> induces an F-linear map g(F) ➔ g'(F).
4.4.9. Proposition. Let <I> : G ➔ H be a homomorphism of linear algebraic groups. Then dq> is a homomorphism of Lie algebras, which is compatible with the p-operation if p > 0.

4.4. The Lie algebra ofa linear algebraic group

73

Using the factorization of cp G ➔p G xH ➔a H,

where px = (x, cpx) (x E G) and a is a projection, we see that it suffices to prove the proposition in the cases that ct, is an injection of a closed subgroup or a projection like
a. We leave the second case to the reader.
Let cxa be as in 4.4.5. If cp is an injection we have for X E g, f E k[H]

(a·c/(X))(f o cp) = (cx1/(dcp(X)))(f).

This formula implies the assertions.

D

4.4.10. Examples. (1) G = Ga. Then k[G] = k[T]. The derivations of k[G]
commuting with all translations T 1---+ T + a (a E k) are the multiples of X = /T. If
p > 0 we have XP = 0. So g is the one dimensional Lie algebra kX with [X, X] = 0 and XP = 0 (if p > 0).
(2) G = Gm. We have k[G] = k[T, T-11. The derivations of k[G] commuting with
the translations T 1---+ aT (a E k*) are the multiples of T /T. If p > 0 we have X P = X. g is as in the previous example, but the p-operation is different (if p > 0).
(3) G = GLn. Now k[G] = k[T;j, v-111~;.j~n• where D = det(T;j) (see 2.1.4 (3)).
Denote by g[n the Lie algebra of all n x n-matrices over k, with product [X, f] = X Y - YX, and the usual pth power as p-operation if p > 0. If X = (xij) E g[n then
L n
DxT;j = - T;hxhi
h=l
defines a derivation of k[G] commuting with all left translations, hence lies in L(G). Since the map X 1---+ Dx is injective, it follows from 4.4.6 (since dim G = n2) that L(G) consists of the Dx. We conclude that we can identify g and g[n (with the p-
= power p-operation). For x E GLn, X E g[n we have Ad(x)X xxx- 1. Also, if H
is a closed subgroup of GLn we can view ~ as a subalgebra of g[n.

4.4.11. Exercises. (1) The Lie algebra of SLn is the subalgebra s[n of g[n of matrices with trace zero. (Hint: use 4.4.7.)
(2) Determine the Lie algebras of the groups Dn, Tn, Un of 2.1.4.
(3) Let cp : SL2 ➔ PSL2 be the homomorphism of 2.1.5 (3). Show that dcp is bijective if and only if p -:fa 2. Describe dcp if p = 2. (4) Let T be a torus. There is a canonical isomorphism L(T) ➔ k ®z X.(T) (where X.(T) is as in 3.2.1). (5) L(G) = L(G0). (6) Show that Ad xis an automorphism of the Lie algebra g (x E G). (7) Let cp : G ➔ H be a homomorphism of linear algebraic groups. Show that (dcp)((Adx)(X)) = Ad(cp(x))(dcp(X)) (x E G, X E g).

74

Chapter 4. Derivations, Differentials, Lie Algebras

Next we give some differentiation formulas, to be used in the sequel. As before, G is a linear algebraic group. We denote byµ : G x G ➔ G and i : G ➔ G multiplication and inversion (2.1.1). We identify the vector spaces L(G x G) and 9 EB 9 (see 4.1.9 (2)). In fact, the Lie algebras L(G x G) and 9 EB 9 are isomorphic (we leave it to the reader to check this).
+ = 4.4.12. Lemma. (dµ)(e,e) : 9 EB 9 ➔ 9 is the map (X, Y) 1-+ X Y and (di)e -id.

µ defines a k-linear map µ = (µ*)° : Oa ➔ Oaxa = (Oa ® k[G]) EB (k[G] ® Oa)
L = (see 4.2.3 and 4.2.5 (6)). If f e k[G], t:,.f f; ® g; then µ(df) = L(df; ® g; + f; ® dg;).
L Since f = f;(e)g; = }:g;(e)f; we have

µ(df) - df ® 1 - 1 ® df e Mce,e)nGxG•

= Hence the linear map of Oa(e) to Oaxa(e, e) Oa(e) EB Oa(e) induced byµ sends

u to (u, u). As (dµ)(e,e> is the dual of this map, the first assertion follows. The second

one follows from the fact thatµ o (id, i) is the trivial map G ➔ {e}.

D

4.4.13. Lemma. (i) Let a : G ➔ G be a morphism of varieties and put </>(x) =
= (ax)x- 1. Then d</>e dae - 1; = (ii) Let a E G. [f v,(x) axa-1x- 1 then d1/le =Ada - 1.

The morphism <I> of (i) is the composite of the morphism x 1-+ (a(x), x) of G to

G x G and µ o (id, i). To prove the formula of (i) use the chain rule of 4.1.3 and

4.4.12 (and observe that the tangent map of the first morphism at xis (dax, id)). The

proof of (ii) is similar.

D

If V is a finite dimensional vector space over k, we write 9C(V) for the Lie algebra of endomorphisms of V. Then 9C(V) ~ 9[dimv· If <I> : G ➔ GL(V) is a rational representation (2.3.2 (3)) its differential d</> is a Lie algebra homomorphism 9 ➔
9C(V), i.e., a representation of 9 in V. Now let G 1 and G2 be two linear algebraic groups and let <I> : G; ➔ GL(V;) be
= a rational representation (i 1, 2). Let </>1 EB </>2 and </>1 ® </>2 be the direct sum and
tensor product representations of G1 x G2 in Vi EB V2 and Vi ® V2, respectively. We
identify the vector spaces L(G1 x G2) and 91 EB 92-

4.4. The Lie algebra ofa linear algebraic group

75

4.4.14. Lemma. (i) d(<J,1 EB <J,2) = d<J,1 EB d<J,2: (ii) (d(¢1 ® <P2HX1, X2))(v1, v2) = ((d<J,1)(X1))v1 ® v2 +vi® ((d<J,2)(X2))v2.

We only prove (ii); the proof of (i) is similar. It suffices to deal with the case that

X2 = 0. Then observe that the left hand side of (ii) is the tangent map ate of the

morphism x i-+ q,1(x )v1 ® v2 of G to Vi ® Vi, evaluated at X 1.

□

4.4.15. Exercises. (1) Let q, : G ➔ GLn be a rational representation and write
</J(x) = (fij(x)), where fii E k[G]. For XE g we have d¢(X) = (X/ij)-
(2) Let V c k[G] be a finite dimensional subspace of k[G] which is stable under all
leftmultiplications}..(x), x e G. Let¢: G ➔ GL(V)betherationalrepresentation
defined by A. For X e g, f e V we have d(/J(X)(f) = Xf, where Xis viewed as
an element of L(G) c V 0 . Give a similar result for right translations. (Hint: use that
p(x) =to }..(x-1) o t, where tis the isomorphism of k[G] induced by inversion).
(3) The differential of the adjoint representation Ad (4.4.5) is given by

dAd(X)(Y) = [X, Y] (X, Y e g).

(Hint: first deal with the case of GLn)-
(4) (a) The Lie algebra of the commutator group (G, G) (see 2.2.8) is a subalgebra g
containing all elements (Ad(x) - l)X and all [X, Y] (x E G, X, Y E g).
(b) If G is commutative (solvable) the same holds for g.
(5) Let q, : G ➔ G L(V) be a rational representation. Define the representation/\h ¢
of G in the exterior power /\h V by (/\h ¢ )(x)(v1A ... A vh) = ¢ (x )vi I\ ... I\¢ (x )vh.
Then/\h q, is a rational representation and

:I: h

h

d(j\ ¢)(X)(v1 A ... A vh) = vi I\ ... I\ (d(/J)(X)v; I\ ... I\ vh.

i=l

(6) Lets E Mn and let G = {g E GLn I gs(' g) = s }. Then G is a closed subgroup of
GLn. Its Lie algebra is contained in {X E g[n I Xs + s(' X) = O}.

4.4.16. As an application of the results of this chapter we shall prove a fundamental result about algebraic groups over finite fields. Let F = Fq be a finite field with q elements and assume k to be an algebraic closure of F. Notice that
F = {a E k I aq = a}.

If Xis an affine F-variety then f i-+ fq defines an algebra endomorphism of F[X], whence an F-morphism a : X ➔ X, depending functorially on X. This is the Frobenius morphism of X. If Xis a closed F-subvariety of An and x = (x1, ... , Xn) E X then ax = (xf, ... , x%). It is immediate from the definitions that the homomorphism (a*)0 (see 4.3.5) is the zero homomorphism, from which it follows that dax = 0 for

76

Chapter 4. Derivations, Differentials, Lie Algebras

all x E X. It is also clear that the fixed point set xu = {x E X I ax = x} is finite.
Similar results hold for arbitrary F -varieties. Now let G be an F -group, not necessarily affine. Its Frobenius morphism is an
endomorphism of the algebraic group G.

4.4.17. Theorem. [Lang's theorem] Assume G to be connected. Then Ax= (ax)x- 1 defines a surjective morphism of G.

It follows from 4.4.13 (i) that dAe is bijective. Let a E G. Denote by p(x) :

y H- yx right translation by x in G. It is an isomorphism of the variety G. Put
a' = p(A(a)) o a and put A'(x) = a'(x)x-1. Then A' = A o p(a) and (dA')e =

dAa o dp(a)e- As before, (dA')e is bijective. It follows that dAa is bijective for all a.

Let X be the closure of AG; this is an irreducible closed subvariety of G. It

follows from 1.9.5 and 4.3.3 (ii) that there is a E G such that Aa is a simple point of
X. We conclude that dim X = dim TAa X = dim Ta G = dim G, whence X = G. By

1.9.5, AG contains a non-empty open subset U of G.

A similar argument shows that for a E G there is a non-empty open subset V of G

consisting of elements of the form a (x )ax-1. Since G is irreducible, the intersection

Un Vis non-empty. This means that a E AG, proving Lang's theorem.

D

If G is as in the theorem, the fixed point set Gu is a finite group. Many interesting finite groups are of this kind. Lang's theorem is a basic tool in the study of such groups, see e.g. [Ca]. We give an instance of an application in the next exercise.

4.4.18. Exercises. (1) Let G and Gu be as before. Let a E Gu and let
Z(a) = {x E G I xa = ax}

be its centralizer. This is a closed subgroup, which is a-stable. Assume that Z(a) is connected. If b E au is conjugate to a in G then b is conjugate to a in the finite group au (more details about such matters can be found in [Bo2, Part E]).
(2) Let G be a connected linear algebraic group and r : G ➔ G an endomorphism of
algebraic groups such that dr : g ➔ g is a nilpotent linear map. Then x H- (rx)x- 1
is surjective.

4.4.19. Jordan decomposition in the Lie algebra. Let k be algebraically closed and let G be a linear algebraic group over k. We view the elements of the Lie algebra gas derivations of k[G], in particular they are linear maps of the vector space k[G]. We have a Jordan decomposition for g, which is similar to that of 2.4.8
Let X E g. It follows from 2.3.6 and 4.4.15 (2) that X is a locally finite linear map
of k[G]. Let X = Xs + Xn be its additive Jordan decomposition (2.4.7).
4.4.20. Theorem. (i) Xs and Xn lie in g and [Xs, Xn] = 0;

Notes

77

(ii)/f</>: G-+ G' isahomomorphismofalgebraicgroups, then ((d</>)X)s = d</>(Xs), ((d</>)X)n = d</>(Xn): (iii) If G = GLn then Xs and Xn are the semi-simple and nilpotent parts of the matrix
X E Mn (see 4.4.10 (3)).
The proof is similar to the proof of 2.4.8 and is left to the reader. Xs and Xn are the semi-simple and nilpotent parts of X.
4.4.21. Exercises. (1) If G is a torus then all elements of g are semi-simple. If G is unipotent then all elements of g are nilpotent. (2) (p = char k > 0). If X is as above then x!P1 and X~Pl are the semi-simple and nilpotent parts of X£Pl.
Notes
4.1 and 4.3 contain standard material on tangent spaces and simple points of algebraic varieties. The basic geometric results are 4.3.3 and 4.3.6.
The treatment given here makes use of modules of differentials. The discussion in 4.2 of their formal properties has been kept brief. A more extensive discussion can be found in [EGA, Ch. 0, §20] or in [Ma, Ch. 10]. We have also included the relevant algebraic results about separably generated field extensions.
In the discussion of the Lie algebra of a linear algebraic group G and its properties, use is also made of differentials. The basic properties of the Lie algebra of G (such as 4.4.5) are deduced from the properties of the differential module 0 0 given in 4.4.2.
Another approach to the Lie algebras uses the description of tangent spaces via dual numbers (see 4.1.9 (3)). This is the approach followed in [Bo3].
Lang's theorem, basic in the study of 'finite groups of Lie type', was first proved in [Lal], with a view to applications to abelian varieties over finite fields.

Chapter 5 Topological Properties of Morphisms, Applications

The first part of the chapter deals with general results about morphisms of algebraic varieties. Then these results are applied in the theory of algebraic groups. One of the main items of the chapter is the construction in 5.5 of the quotient of a linear algebraic group by a closed subgroup.

5.1. Topological properties of morphisms

5.1.1. X and Y are two irreducible algebraic varieties over the algebraically closed field k and <I> : X ➔ Y is a dominant morphism (4.3.5). We shall establish a number of general facts about the topological behavior of</>.
View the quotient field k(X) as an extension of the field k(Y) (see 1.8.1 and 1.9.1 (ii)). The transcendence degree trdegk(Y)k(X) equals dim X - dim Y. In 5.1.6 (ii) we shall give a geometric interpretation of this integer.
Let F be a field and E a finite algebraic extension. We denote by [E : F] its degree. The elements of E that are separable over F form a subfield Es, which is a separable algebraic extension of F. Its degree [E : F]s is the separable degree of the
= extension E/ F. Let p be the characteristic. If p 0 we have E =Es.If p > 0 then
E is a purely inseparable extension of Es, i.e., for any x e E a power xPe lies in Es
= (see [La2, Ch. VII, §4] for these facts). If <I>: X ➔ Y is as before and dim X dim Y
then k(X) is an algebraic extension of k(Y). In 5.1.6 (iii) we shall give a geometric
interpretation of [k(X) : k(Y)]s.
If k(X) = k(Y) then <I> is said to be birational.

5.1.2. Lemma. <I> is birational if and only if there is a non-empty open s.ubset U of X such that </JU is open and <I> induces an isomorphism ofvarieties U ::::: </JU.

It follows from the definition of quotient fields (l.8.1) that <I> is birational if the

condition of the lemma is satisfied. Assume that <I> is birational. We may assume X
and Y to be affine. Then k[X] = k[Y][f1, ... , fr], where all /i lie in k(Y). Take

f e k[Y] such that f =I- 0 and the f Ji lie in k[Y]. Then <I> induces an isomorphism

= k[Ylt :::::: k[X]f, and U Dx(f) (see 1.3.5) is as required.

□

The main result of this section is 5.1.6. We first deal with some special cases.
Assume now, moreover, that X and Y are affine and that there is f e k[X] with
k[X] = k[Y][f] (</> being defined by the inclusion map k[Y] ➔ k[X]).

5.1.3. Lemma. Assume that f is transcendental over k(Y). (i) <I> is an open morphism;

5.1. Topological properties ofmorphisms

79

(ii) If Y' is an irreducible closed subvariety of Y then q,-1Y' is an irreducible closed
subvariety of X, ofdimension dim Y' + 1.

Recall that an open (closed) map of topological spaces is a continuous map such that the image of an open set is open (respectively the image of a closed set is closed). We say that the morphism <J, is open (closed) if it defines an open map (respectively a closed map).
We may assume that X = Y x A 1 and that <J, is projection on the first factor. Let g = "'E,~=o g; Ti E k[X] = k[Y][T]. Then

LJr
<J,(Dx(g)) = Dr(g;),
i=O

whence (i).

If Q is the prime ideal in k[Y] defined by the irreducible closed subvariety Y',
then q,- 1Y' is the set of points of X in which the functions of the ideal P = Qk[X] =

{L;>o f; Ti I /; E Q} vanish. Then k[X]/ P ~ (k[Y]/ Q)[T]. Since the last ring is an

integral domain, P is a prime ideal and q,-1Y' is irreducible. The last point of (ii) is

clear.

D

5.1.4. Lemma. Assume that f is separably algebraic over k(Y). There is a nonempty open subset U of X with the following properties: (i) The restriction of <J, to Ude.fines an open morphism U ~ Y; (ii) If Y' is an irreducible closed subvariety of Y and X' is an irreducible component
ofq,- 1Y' that intersects U, then dim X' = dim Y';
(iii) For x EU the.fiber<J,-1(</)(x)) is a.finite set with [k(X): k(Y)] elements.

We have k[X] = k[Y][T]/ I where / is the ideal of polynomials vanishing in
f. Let F be the minimum polynomial off over k(Y) (the irreducible polynomial in k(Y)[T] with leading coefficient one, with root f). Choose a E k[Y] such that all coefficients of F lie in k[Y]a.
Let f 1, ... , fn be the roots of F, in some extension field of k(Y). Since f is
separable over k(Y) the roots are distinct and the discriminant d = ni<j(/; - Jj)2 is
a non-zero element of k(Y), which can be expressed polynomially in the coefficients of F (see [Jac4, p. 250-251]). It follows that there is b E k[Y] and m > 0 such that
amd = b.
We may replace X and Y by Dx(ab), respectively Dr(ab). We are then reduced to proving the lemma when, moreover, the following holds: (a) / contains the minimum polynomial F. From this it follows, using the division algorithm, that/ is the ideal generated by F. It also follows that k[X] is a free k[Y]module.

80

Chapter 5. Topological properties ofmorphisms, applications

(b) If F(T) = 'E7=o hi Ti then for ally e Y the polynomial
L n
F(y)(T) = hi(y)Ti
i=O
has distinct roots. We shall show that in this situation the statements of the lemma
hold, with U = X. We may assume that

X = {(y, t) E y X A 1 I F(y)(t) = O},

<I> being the first projection. Let G e k[Y][T] and denote by g its image in k[X]. Then

Dx(g) = {(y, t) e X I G(y)(t) -I- O}.
Write G = Q F + R, where R = 'E7:-~ ri Ti is a polynomial in T of degree < n =
degF. Then <J>Dx(g) is the set of y e Y such that not all roots of F(y)(T) are roots of R(y)(T). Since the first polynomial has n distinct roots, this implies that

LJn-l
</>Dx(g) = Dr(ri),
i=O

whence (i).

Next let Y' be as in (ii) and let Q be the corresponding prime ideal in k[Y]. Then

<F-1>-t1hYe'

is the closed set defined by the ideal image of F in A[T]. We claim that

Qk[X]. Let A = k[Y]/ Q and Qk[X] is a radical ideal, i.e.,

denote
A[T]/(

Fb-y)

is reduced (1.3.1). Let H e A[T] and assume that Hm is divisible by F. We may

assume that degH < n. It follows from property (b) that F has distinct roots and that

H is divisible by F, as polynomials with coefficients in the quotient field of A. But
since H has lower degree than F, this can only be if H = O, which implies the claim.

By 1.2.6 (2) we know that Qk[X] is an intersection of prime ideals of k[X], say
Qk[X] = n~=l Pi. We may assume that there are no inclusions among the Pi, The

irreducible components of the <1>- 1Y' are the sets Vx(Pi) (notation of 1.3.2). We show
= that P; n k[Y] Q (1 ~ i ~ r). If this is not so we have, say, P1 n k[Y] -1- Q. Take
x1 e P1 n k[Y] - Q and xi e Pi - P1 (2 < i ~ r). Then x 1x2, ... , x, e Qk[X]. Since
k[X] is free over k[Y], it follows that x2, ... , x, e Qk[X] c P1, which is impossible
= if r > 1. If r = 1 we have a contradiction, since Qk[X] n k[Y] Q.

It follows that the quotient field of k[X]/ Pi is an algebraic extension of the quo-

tient field of A, which proves (ii).

If Y' is a point then Q is a maximal ideal of k[Y] and A= k. The preceding anal-

kys[iTs]s/(hFo-w).s

SthinactenoF-wis<1a>-p1oYl'yinsotmheialzeorfoddeigmreeennsiowniathl

variety defined by distinct roots, (iii)

the k-algebra follows. D

5.1.S. Lemma. Let p = char k > 0 and assume that f P E k(Y). There is a
non-empty open subset U of X with the following properties:

5.1. Topological properties ofmorphisms

81

(i) The restriction of <I> to U is an open morphism U ➔ Y which induces a homeo-
morphism U ~ <J>U; (ii) If Y' is an irreducible closed subvariety of Y, there is at most one irreducible com-
ponent X' of<1>-1Y' that intersects U. If X' exists, we have dim X' = dim Y'.

Let f P = g. Replacing X and Y by Dx(a) and Dy(a) with suitable a E k[Y],

we may assume that g E k[Y] and that k[X] is free over k[Y]. We shall prove that we
may then take U = X.

We view X as the set of points (y, g(y) 1f P) in Y x A1, <J> being the first projection.
Then k[X] = k[Y][T]/(TP - g). The proof that <I> is open is like that of 5.1.4 (i).

Since <I> is bijective (i) follows.

Let Y' be as in (ii) and let Q and A be as in the proof of 5.1.4. Let g be the image of gin A. Now k[X]/Qk[X] ~ A[T]/(TP - g). If TP - g is irreducible over the quotient field of A, then Qk[X] is a prime ideal and (ii) follows. If T P- g is reducible
= then in A[T]/(TP - g) the elements b with bP 0 form a prime ideal. It follows
that P = {h E k[X] I hP E Qk[X]} is a prime ideal, and is the radical of Qk[X]. If
= = h E P n k[Y] then hP E Qk[X] n k[Y] Q. Hence P n k[Y] Q and (ii) follows,

as before.

□

The following theorem is the main result of this section. Its content is that a morphism <I> as in 5.1.1 behaves well on an open set. The exercises (3) and (4) of 5.1.8 show that one cannot expect good behavior everywhere.

5.1.6. Theorem. Let X and Y be irreducible varieties and let <I> : X ➔ Y be a
= dominant morphism. Put r dim X - dim Y. There is a non-empty open subset U of
X with the following properties: (i) The restriction of<I> to U is an open morphism U ➔ Y; (ii) If Y' is an irreducible closed subvariety of Y and X' an irreducible component
= of <1>-1Y' that intersects U, then dim X' dim Y' + r. In particular, if y E Y, any
irreducible component of<1>-1y that intersects U has dimension r; (iii) If k(X) is algebraic over k(Y), then for all x E U the number of points of the fiber <1>-1(<J>x) equals [k(X) : k(Y)] 5.
= Assume we have a factorization <I> <I>' o 1/1, where v, : X ➔ Z, <I>' : Z ➔ Y are
dominant morphisms, Z being irreducible. If (i) and (ii) hold for <I>' and v, they also
hold for <J>. To prove the theorem we may assume that X and Y are affine. Since k[X] is a k[Y]-algebra of finite type, we can find a factorization of <I>

X

=

Xr

➔ ,Pr

Xr-1

,P➔r-I

...

➔ "'2

X
1

➔ ,PJ

Xo =

f ,

where the </>; are cases covered by one of the three preceding lemmas. (i) and (ii) then follow by application of these lemmas.

82

Chapter 5. Topological properties ofmorphisms, applications

= In the case of (iii) let k[X] k[Y] [/1, ... , Is] and let Z be the variety with

e being such that all J( are separable over k(Y). We obtain a factorization as in the

beginning. Refining it we obtain a factorization of <I> to which we can apply the last

two lemmas.

D

The proof gives the following useful corollary.

5.1.7. Corollary. In the case of 5.1.6, we may replace (i) by the following stronger property, (i)' For any variety Z the restriction of <I> to U defines an open morphism U x Z ➔ f X Z.

It suffices to prove this for Z affine. Observe that if (i)' holds for Z, and if Z' is a

closed subvariety of Z, then (i)' also holds for Z'. Hence it suffices to establish (i)' for
= Z Am. This will follow if we prove the corresponding result in the cases of 5.1.3,

5.1.4, 5.1.5. The first case is trivial. In the others (i)' follows by observing that the

minimum polynomial of an element of k(X) over k(Y)(T1, ... , Tm) coincides with

the minimum polynomial over k(Y).

□

5.1.8. Exercises. (1) Using 5.1.6 (iii) show that an isomorphism <I> : A1 ➔ A1 is
of the form q,t =at+ b (a E k*, b Ek). Deduce that an isomorphism of the projec-
tive line P1 (see 1.6.13 (2) or 1.7.1) is induced by an element of GL2.
= = = (2) Let X {(x, y) E A2 I x 2 y3}. Define <I> : X ➔ A1 by q,(x, y) x- 1y if = (x, y) -:/= (0, 0) and q,(0, 0) 0. Show that q, is a morphism of irreducible varieties
that is birational and bijective, but is not an isomorphism of varieties.
= (3) Consider the morphism q, : A2 ➔ A2 with q,(x, y) (x, xy). Show that it is
birational, but not open. Determine the components of the fibers q,-1z, z E A2.
= = = (4) Define <I> : A3 ➔ A3 by q,(x, y, z) (x, xy, z). Let X {(x, y, z) E A3 I y2 = 1 + x}. Show that X and Y q,X are irreducible closed subvarieties of A3 of dimen-
sion two. Put

Y' = {(x,y,z) E YI y =xz, z2 = 1 +x}.

Show that Y' is irreducible, closed of dimension one and that if char k -:/= 2, q,-1Y' n X
has a component of dimension zero.

5.2. Finite morphisms, normality

5.2.1. Let A be a ring and B an A-algebra. We say that B is finite over A if B is an A-module of finite type. We say that b E B is integral over A if it satisfies an

5.2. Finite morphisms, normality

83

equation

with coefficients in A.

S.2.2. Lemma. Let B be an A-algebra of finite type. Then B is finite over A if and only ifevery element of B is integral over A.

Assume that Bis finite over A. There are bi E B such that B = Ab1 + ... + Abm,
Let b E B. There exist aii in A such that

L m

= bbi

aij bj, l < i < m.

j=l

It follows that det(~ijb - aii) = 0, showing that bis integral over A. The proof of the

converse statement is straightforward.

D

It follows from 5.2.2 that if, moreover, C is a B-algebra that is finite over B, then C is finite over A. It also follows from 5.2.2 that if Bis any A-algebra, the elements of B that are integral over A form a subalgebra.
Now let <I> : X ➔ Y be a morphism of affine varieties. The algebra homomorphism <I>* : k[Y] ➔ k[X] makes k[X] into a k[Y]-algebra. We say that <I> is finite if k[X] is finite over k[Y]. Then <t,-1(y) is finite for ally E Y.

S.2.3. Lemma. A finite morphism is closed.

Asume that</> is finite. Let B = k[X], A = k[Y]. We have B = (</>* A)[b1, ... , bh] and an easy argument (see the proof of 5.1.6) shows that we may assume h = l.

Applying 1.9.3 to B and its subring <I>* A, we see that Im <I> is closed. Applying this

to a closed subvariety X' of X and the induced morphism X' ➔ Y, we conclude that

<t,(X') is closed. Hence <I> is closed.

D

We say that <I> is locally finite in a point x E X if there exists a finite morphism µ, : Y' ➔ Y and an isomorphism v of an open neighborhood U of x onto an open set in Y' such thatµ, o v is the restriction of <I> to U.
Let VI : Y ➔ Z be another morphism of affine varieties.
S.2.4. Lemma. If <I> is locally finite in x and VI in <I> (x) then VI o <I> is locally fi-
nite in x.
= We may assume that Y Dz,(f), where Z' is finite over Z, with f E k[Z']. If Y' = is finite over Y then k[Y'] BJ, where B is integral over k[Z']. Hence Bis integral

84

Chapter 5. Topological properties ofmorphisms, applications

over k[Z]. It follows that Y' '.:::'. Dv(g), where Vis finite over Z, with g E k[V]. □

From now on we assume that X and Y are irreducible and that q, is dominant. We view A = k[Y] as a subring of B = k[X].

5.2.5. Lemma. Assume that there is b E B with B - A[b]. Let x E X. We have the following alternatives: (a) q,-1(q,x) is.finite and q, is locally.finite in x, (b) q,-1(q,x) '.:::'. A 1.
= We have B A[T]/1, where I is the ideal of the polynomials f E A[T] with
f (b) = 0. Let E : A ➔ k be the homomorphism defining q,x. Extend E to a homomorphism A[T] ➔ k[T] in the obvious manner. If El = {0} then k[q,-1(q,x)] '.:::'. k[T],
whence q,-1(q,x) '.:::'. A 1.

If El # {0} the polynomials in El vanish in b(x); hence El contains non-constant
polynomials and no non-zero constants. This implies that 4>-1(q,x) is finite. It also follows that there is f E I of the form

fnTn + ... + fmTm + ... + fo,
where E(fn) = ... = E(fm+d = 0, Efm # 0, m > 0. Puts= fnbn-m + ... + fm, Then
s # 0 and

sbm + f m-1bm-l + ... +lo= 0.

We see that sb is integral over A[s] and that bis integral over the subring A[s-1] of

the quotient field of A. But since s E A[b], it follows thats is integral over A[s-1],

i.e., that s is integral over A. Now the assertion of (a) follows by observing that

= Bs A[sb, S]3 •

□

5.2.6. Proposition. Let x E X. If the fiber 4>-1(q,x) is finite then q, is locally .finite in x.

We have B = A[b1, ... , bh], If h = 1 the assertion is true by 5.2.5. We have a
.! factorization of q,: X X' ~ Y, where k[X'] = A[bi]. Clearly v,-1(v,x) is finite.
By induction on h we may assume that v, is locally finite in x. We may then assume
that there is a finite morphism of affine varieties v,' : X" ➔ X' such that X is an
affine open subset of X" and that v, is induced by v,'.
Put F = (t/>')-1(q,x). Assume that Fis infinite. By 5.2.5 it is isomorphic to A1.
Let C be a component of (v,')- 1( F) of dimension > 1 passing through x. Now X n C is an open subset of C containing x, hence must be infinite. But X n C lies in the
finite set 4>-1(q,x) and we get a contradiction. Hence the components of (v,')-1(F)
of dimension > 1 do not contain x. Replacing X by a suitable open neigborhood of x

5. 2. Finite morphisms, normality

85

we may assume that no such component exists. Then F is finite. The theorem follows

by using 5.2.5 and 5.2.4.

□

5.2.7. Corollary. In the situation of5.2.6 we have dim X = dim Y.

An integral domain A is normal if every element of its quotient field that is integral over A lies already in A. A point x of an irreducible variety X is normal if there exists an affine open neighborhood U of x such that k[U] is normal. X is normal if all its points are normal.
The next result is (a version of) Zariski's main theorem.

5.2.8. Theorem. Let </> : X ➔ Y be a morphism of irreducible varieties that is bijective and birational. Assume Y to be normal. Then </> is an isomorphism.

Let x E X. Replace X and Y by affine open neighborhoods U of x, respectively V

of </>x. We deduce from 5.2.6 that we may assume U is isomorphic to an affine open

subset of an affine variety V' which is finite over V. But our birationality assumption

implies that k(V') :::: k(V). Now the normality of Y implies that that finite morphism

V' ➔ V is in fact an isomorphism. This shows that</> is an isomorphism of ringed

spaces, hence an isomorphism of varieties (see 1.4.7).

D

5.2.9. Exercises. (1) Let </> : X ➔ Y be a finite morphism of affine varieties. Show that for any variety Z the morphism(</>, id) : X x Z ➔ Y x Z is finite and closed. (2) For any field F the polynomial algebra F[T1, ... , Tn] is normal. (3) An irreducible affine variety Xis normal if and only if k[X] is normal. (4) Let</> : X ➔ Y be a surjective morphism of irreducible affine varieties. If</> is lo-
cally finite in all points of X, then </> is finite. (Hint: For f E k[X] there are non-zero
F E k[Y][T] with F(f) = 0. Consider the leading coefficients of such F).
(5) In 5.2.8 the normality assumption cannot be omitted.

5.2.10. Lemma. Let A be a normal integral domain with quotient field F. Let B be an integral domain containing A, which is an A-algebra of.finite type. Assume that the quotient field E of B is a separable algebraic extension of F. There is a non-zero element a E A such that Ba is normal.

Let B = A[b1, ... , bh], Using that any non-zero element of B divides a non-

= zero element of A one sees that it suffices to deal with the case h 1. So assume

B = A[b]. We may assume that bis integral over A. Then (1, b, ... , bn- 1) is a basis

of E over F, where n = [E: F].

E7::~ Assume that x =

a;b; (a; E F) is integral over A. Then for O < j < n

n-l
Tr(xbj) = Z:,a;Tr(bi+j)
i=O