Questions of ra t ional i ty for solvable algebraic

groups over nonperfect fi+;lds (+)

±~¢Iemoria di ~¢~AXWELL ~:~OSE~LICI-IT (a B e r k e l e y , Cal i forn ia )

Sunto . - Date u n gruppo algebrico risolubile, ins ieme con u n sue campo di def iniz ione, si d i m o s t r a che ~ possibi lv trocar+ certi so t togrupp i di t i p i im~or tan t i , p . e. tor i m a s s i m i , s enza i n t r o d u r r e a l c u n a inscpa+'ab~lit4

If G is a connected l inear algebraic group defined over the field k it is quite natural to ask whether G possesses CARTA~ subgroups or maximal tori that are defined over k, or whether the center of G is defined over k, or whether G is rat ional ly parametrizable over k. If k is perfect all these questions are to be answered posit ively, the one concerning the center trivially, the others by no means so [4]. ]f k is not perfect there are counter- examples to all sorts of conjectures, and at least the questions concerning the center and rat ional parametr izat ion are to be answered negatively. In view of this and the fact that the basic tools available reduce essentially to questions of intersection theory, often involving components of excessive di- mension, it is rather astounding that something nontrivial can actually be asserted in case k is not perfect. One of our main results, for example, says that if our group G is solvable then there indeed exists a maximal terns defined over k ITheorem 4). The methods of proof we use here are refine- ments of those of our previous Aunali paper [4] and cry for improvement ; there are unnatura l complexities~ and it seems that something new that is quite general, and possibly quite subtle, must be brought to light before appreciable progress can be made.

The basic notion is that of a k-solvable group tpreviously called a con- nected solvable l inear algebraic group wLth k as a field of definit ion for its solvability}: a k-solvable group is an algebraic group that has a normal chain each of whose members is an algebraic subgroup defined over k and each of whose successive factor groups (chosen to be defined over k, to- gether with the respective homomorphisms) is k-isomorphic to the additive group G~ or the mult ipt icat ive group G,~. Among the basic properties of

(×} This r e s e a r c h w as s u p p o r t e d b y the A i r Fo rce Off ice of Sc ient i f ic R e s e a r c h .

Annali di Maternatlca 13

98 ).[. ROSENLICWr: Q~cstio~s of ~.(ttbm~dity /~;r .~ol~'(~ble (tIgebr(~h'. etc.

such groups are the facts tha t a t r ans fo rma t ion space def ined over k for such a group admi ts a ra t iona l cross sect ion that is def ined over k [3, Th. 10], that any homomorph ic image de f ined over k of such a group is also such a group [4, Prop. 6], and that such a group which is in ma t r ix form may be t r i angu l a t ed over k [4, Prop. 7]. I f k is separab ly a lgebra ica l ly closed then any torus that is def ined over k is k-so lvable ( immediate consequence of [4, Prop. 1]). I f k is perfect , any connected un ipo ten t a lgebraic group that is de f ined over k is k -so lvab le [~, Prop. 5, Cor. 2]. F i n a l l y we recal l that if a k-so lvable un ipo ten t a lgebraic group is commuta t ive (and f u r t h e r m o r e is an- nu l led by the endomorph i sm x ~ x~ if the field charac te r i s t i c is p::~= 0) then it is k - i somorph ic to a p roduc t of G~ ' s [5, Prop. I, 2].

~Iany of the subs id ia ry resu l t s which follow are qui te f ami l i a r in a less precise f o r m u l a t i o n ; needless to say, the present precis ion is essent ial . There also occur a n u m b e r of cases where the con ten t ion at hand is clear, or has an obvious analogu% for charac te r i s t i c zero and in such cases, for brevi ty, we often assume wi thout giving expl ic i t notice tha t we are in the case of charac te r i s t i c 19 =[= 0.

LE~MA 1. - I f V--~ W ~ Z is a sequence of rational maps of varieties defined over k, wi th V--~ W generically s~,~rjeclive and V ~ W and V ~ Z defined over k, then the map W ~ Z is defined over k.

W e can assume Z to be a line, i . e . that W ~ Z is a numer i ca l func t ion and we are reduced to proving that ~2(W) (~ k(V) C k ( W ) , ~2 being the uni- versa l domain . But if f e Q ( W ) A k ( V ) we can write

f - - Y, alf~ / ~ a~g~,

where the sums are f ini te , each fi , g i e k ( W ) , and the a i ' s are e lements of ,Q that are l i nea r ly i n d e p e n d e n t over k. F rom E a ~ ( f ~ - - f g i ) = 0 and the l inear d is jo in tness of ~Q and k ( V ) over k we deduce tha t each f~ - - fgi - - O, so

fek(W).

LE)[~I± 2. - Let V be a principal fiber space for the algebraic group G. Then the quotient space V /G exists, and the canonical map V ~ V /G is the canonical projection from V to its base space. I f H is a normal algebraic subgroup of G then V / H exists and the induced operation of G on this variety makes it a principal fiber space with group G / H and base space V / G. If , in addition, H is connected, solvable, and linear, then V is a principal fiber space with group H and base space V / H .

(( P r inc ipa l f iber space-> here assumes that the space in ques t ion is local ly a direct product of base space and group, so it is t r ivial to ver i fy that the base space for V is indeed a quo t ien t space ~//G (el., for example , the s t anda rd def in i t ions as reproduced in [5, § 3] and [6]).

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If UC V/G is an open subset such that a regular cross section U ~ V exists, and if /=7 is the inverse image of U on V, then U can be identified with U X G. Thus, if H is an algebraic subgroup of G, ~f/H exists and equals U X ( G / H } [6, Lemma 3]; taking a set I UI of such U ' s that cover V/G, the corresponding / ~/i will cover V and the various I~ / /H f can be pieced together to get V / H (the legit imacy of the piecing-together process following, e .g . , from [6, Lemma 3, ff]). If H is normal in G, V / H is clearly a principal fiber space with group G/H and base space V/G. If H is also connected, solvable, and l inear we know that G is an H-pr inc ipa l fiber space over G/H, so there exist local cross sections G / H ~ G, hence UM (G/H) U X G, i.e. V / H ~ V, so V is an H-pr inc ipa l fiber space over V/H.

THEOREM i. - Let the variety V be a principal fiber space for the k-sol. vable unipolent algebraic groulv G, V and the operation of G on V being de. fined over k, and suppose thai the base space V~ G is projectively embeddible. Then V/ G and the natural projection V ~ V~ G may be taken to be defined over k and, i f this is the case, a local cross seclion I? / G ~ V that is defined over k may be found that is defined at any preassigned point of V / G. I f there is a regular cross seclio~+, i.e. one defined on all of V~ G, which is true i f V~ G is affine, such a regular cross section exisls that is defined over k.

That the quotient variety V/G and the projection V.-+ V /G may be taken to be defined over k follows from the projective embeddibiii ty of V~ G [6, Prop. 1], so this assumption can be deleted and we can suppose instead that V / G and V--+ V~ G are defined over k. W,e now prove this modified theorem by induction on dim G, first supposing dim G > 1 and that the re- sult is true for all groups of smaller dimension. Let H C G be a normal k-solvable subgroup such that G / H is k-isomorphic to G<~. By Lemma 2 we have a sequence of morphisms

v - - V / H ~ V / e ,

V being an H-pr inc ipa l fiber space over V / H and V / H being a G/H-pr in . cipal fiber space over V~ G. If V / G is affine our induct ion assumption shows that V / H ( = ( V / G ) X ( G / H ) ) is also aff ine; as a result there exist G-tara- riant k-open subsets V that cover V, each of whose images on V / H is affine. Thus we can modify V / H so as to be able to suppose that V / H and the map V ~ V / H are defined over k. This done, Lemma 1 implies that all the maps V ~ V / H ~ V / G are defined over k, and the theorem f oliows immediately from a double application of our induction assumption. I t there- fore remains to prove the case dim G - - l , i.e. G - - Go, with V / G and V ~ V / G both being defined over k.

B,at the set of equivalence classes of G~-principal fiber spaces over a

100 ~[. ROSENLICHT: Questions of rationality for solvable (dgebrc~ic, etc.

variety W is known to be in one-one correspondence with the elements of the cohomolo~y group H~(W. @wt. @w being the sheaf of local rings on W. and H~(W, Ow) -- 0 if W is affine. Hence if V / G is affine, the fiber space V is trivia], i .e. there exists a regular cross section V / G ~ V. Since V / G is covered by its k-open affine subsets, it follows that we need only prove that i f there exists a regular cross section ~: V/ G ~ V, then there exists a regular cross section defined over k. To fix notation, let G operate on V on the r~ght, let 0 be the morphism (defined over k) from the set

F - - { ( v , vg) l v e V, g e G I C V X V

into G such that O(v, vg) ~--g, and let 7:: V ~ V~ G be the natural projection. I f v e V, vr:v is defined and (a=v, v) e F. Hence we have a morphism V --~ G given by v--~ 0(~7:v, v}. Using the natural coordinate function on G(-- G~), we write O(ar:v, v)--Ec~,~(v), where each ~ i e k [ V ] and the c~'s are elements of P., l inearly independent over k. with cl ~ 1. Wri t ing G additively, we have, for g ~ G,

vg) = v g ) = v ) + g, S o

Z c~¢p~(vg) = ~. c~(v) + g.

By the l inear disjointness of • and k ( V X G) over k we get ~dvg) -- ~i(v) for i > 1, ~l(vg)----cpa(v)+g. In particular, if i > 1 then ~ e k [ V / G ] , i .e. there exists ~ i e k [ V / G ] such that ~ d v ) ~ ~d=v) for all v e V. Now define the cross section ~': V~ G ~ V by a'(x) -- ¢~(x).(E~>ic~(x)). Then

v ) = - v)=

Hence the map v ~ ~'uv is defined over k, so (again using Lemma 1), ~' is defined over k.

Here is an easy example in which the last part of the theorem breaks down for G not unipotent : Take V to be a connected nilpotent l inear alge- braic group defined over a separably algebraically closed field k whose unipotent part V~ is not defined over k (so k is not perfect). The maximal torus G of V is defined over k and indeed k-solvable, so V is a G-principal fiber space over V~ G, all defined over k. Over kP - ~ we have V - - V, X G, but there is no regular cross section ~: V] G ~ V that is defined over k. For if there were, lett ing u be the homomorphism V--. V / G we could take vr:e ~ e and the map ~: v--~ vI~v) -~ would be a morphism V ~ G sending e into e, hence a rat ional homomorphism, separable since ~ -- identi ty on G, and defined over k, implying that its kernel Vu is defined over k.

COROLLXR¥ 1. - Let G be a connected linear algebraic group defined over k, H a normal k-solvable unipotent algebraic subgroup of G. I f we take

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the natural rat ional homomorphism G ~ G / H to be defined over k, then there exists a regular cross seclion G / H ~ G, also deflated over k. In particular, G is k-isomorphic (as a variety) to ( G / H ) X H.

COROLLARY 2. - I f G is a k-solvable unipotent linear algebraic group then G is k-isomorphic (as a variety) to the affine space ~Q~, where n - - d i m G. More precisely, i f G ~ Go ~ G~ ~ ... ~ G~ -- l e I is a normal chain exhibiting the k-solvability of G, then there exist functions x~ , . . . , x ~ e k [ G ] such that g ~ { x ~ { g ) , . . . , x~(g)) is such a k-isomorphism, such that for i - - 1 , . . . , n the subgroup G~ is defined by x~ ~ - - x~ -- 0 (so x~ gives the natura l homomor. ph i sm G~_~ ~ G~_~/ Gi - - G~), and the group composition is given by

with x~(g~g~) - - f~(x~ig~), . . . , x, ,(gl) , x~ig2), . . . , x , ( g ~ ) ) ,

f ~ (X~ , . . . ,X , . L , .... Y ~ ) e k [ X , Y] and

i - - l , . . . , n,

f Io, o) = o;

furthermore i f each Gi is normal in G we can suppose that

f~(X, Y) - - X ~ Y ~ e k [ X ~ , . . . , X~_~, Y~, . . . , Y~_I],

for all i -- 1,. . . , n.

Use induction, applying Corollary 1 to H---- G~.

For any algebraic group G, call a rat ional homomorphism f: G---* G~ an additive function. If G----(Ga) ~ and x~, .... x~ are additive coordinate functions on G, i .e . l inear coordinate functions on G with respect to one of its admis . - sible vector group structures, then the additive functions on G are precisely the p-polynomials in x~, ..., x~ (cf. [5, § 1]).

LE~lVIA. - A n y finite subgroup P of IG~p, where n > 0 , is contained in a connected algebraic subgroup of (G~) ~ of dimension 1. I f n -- 1, there exists an additive function on G~ whose zeros are precisely the elements of ~, each to order o~e.

Prove the second part first. We may do this by taking a coordinate funct ion on G ~ / r , a group isomorphic to G~; or. on the other hand, we can show directly that l i v e r ( X - - y ) is a p-polynomial in X by noting that the finite commutative group 1: is of type (p , . . . , p} and 1]i=o ..... ~ _ l ( X - - i a ) - -- XP -- aP-IX. To prove the first part, let ~o C Ga be a subgroup isomoJphic to r and try ~o find a rat ional homomorphism G ~ ( G ~ ) ~ extending the map ]:o ~ £. Looking at coordinates on (G~} ", we reduce to the case n - ~ 1. I f :¢i,..., av is a minimal set of generators for Fo, the problem reduces to f inding a .p-polynomial in X that vanishes at u~; ..., a~_~ but not at av, which is possible by the part already proved.

102 5I. R o s n ~ c H ~ : Q~estion.~ of ratio~¢~iity for .~'olwtblc algebraic, etc.

PROPOSITION. - Let H be an algebraic subgroup of G = (G,}" that is de. fined over k. Then any additive fu~etion on H that is defi ,ed over k is lhe restriction to H of such a fu~ction on G. A basis for for the ideal of H in

[G] may be found consisting of additive functions in k[G]. and i f y~ , . . . , y~ is such a basis, then any additive fu~lelion on G that vcmishes on H a~d is defined over k is a p-polynomial with eoefficie~ts in k in y~ .... , y~ ; furll~er- more, the mini,mat possible v a l u e o f s is given by s -- n ~ dim H, in which case y~, ..., y, are algebraically independe~d over ~2.

W e firs t p rove the f i r s t two s t a t emen t s u n d e r the a s sumpt ion that k- - -Q. By [5, Prop . 1] the re ex is t addi t ive coord ina te func t ions x ~ , . . . , x,~ on G such tha t Ho is g iven by x ~ - - - - x , . = 0 . T h u s we can wr i te G - - H o : K G', whe re Ho, G' a re each d i rec t p ro d u c t s of G~'s. W e ma y suppose dim G ' > 0. Now H : Ho X F, wi th P a f in i te su b g ro u p of G'. App ly the L e m m a to get a connec t ed a lgebra ic subgroup of G' of d imens ion one tha t con ta ins P. A n o t h e r app l i ca t i on of [5, P rop . 1] shows that G - - H o X G~ X G", where H - - H o X ( H N G~), and G" aga in is a, p ro d u c t of G~'s. Not ing tha t any

add i t ive func t i on on the p r o d u c t of two g roups is the sum of addi t ive func- t ions on the fac tors , we r e d u c e our f i rs t two s t a t e me n t s to the ease n-----l. In this case our s t a t emen t s r e d u c e to t r iv ia l i t i es unless H is f ini te , in which case the second s t a t e m e n t is pa r t of the L e m m a ; the fac t tha t any addi t ive f u n c t i o n on H can be e x t e n d e d to G also follows f rom the L e m m a , if we take a m in ima l set of gene ra to r s ~ , . . . , ~., of H and note tha t an add i t ive

func t ion on G can be found tha t van i shes on ~ .... , ~_~ bu t not on ~ . Th i s

p roves the f i rs t two s t a t emen t s if k = ~2.

Now drop this a s sumpt ion , and let f be an addi t ive fu n c t i o n on H that is de f ined over k. T h e n the re exis ts an addi t ive func t ion /9" on G that

ex tends f. W r i t i n g F - - - ~, ~ f~ , where each f~ e/~ [G], each ~i e f~, ~ - - l, and the va r ious g i ' s a re l i nea r l y i n d e p e n d e n t over k, we see t h a t each f~ is a p - p o l y n o m i a l , hence an add i t ive fu n c t i o n on G. H is de f ined over k, so k[Hl and fl a re l i nea r ly d is jo in t over k, g iv ing F----f~ on H and p rov ing

the f i rs t par t .

W e a l r eady know that add i t ive func t ions in ~Q[G] gene ra t e the ideal of H. Bu t any add i t ive func t i o n F on G that van i shes on H can be wr i t t en F ~ E ¢ ~ F i , wi th each F~ a p - p o l y n o m i a l in k[G] and the ~.~'s a set of ele- men t s of 9, tha t are l i nea r l y i n d e p e n d e n t over k, and l i nea r d i s jo in tness

shows tha t each £~ van i shes on H. Th u s addi t ive func t ions y~ , . . . , Ys in k[G]

exis t that gene ra t e the ideal of It. L e t t i n g x~, . . . . ;v,~ be a set of add i t ive coo rd ina t e func t ions for G tha t

are de f ined over k, each Yi is a p - p o l y n o m i a l in k[x]. Assume for a m o m e n t tha t x~ ac tua l ly appea r s in bo th y~ and Y2, and to at least as high a degree in Y2 as in y~. T h e n the re is an e l e me n t c e k and an i n t e g e r v_~_0 such

3[. Ros~.x~I.wH~': O~t('stio~.~ o] r~ttiomdity for .s'olcabl,,; tdgcbreic., etc. 103

that y ~ - - c y S has smaller degree in x~ than does Yy, or does not involve x~ at all. Notice that to prove the various s ta tements made about y~,...~y~ it suffices to prove the same s ta tements about y~, y~--cy~p ~, ys~ ...,y~ twhich is also a set of generators for the ideal of H).

By changing' the order :of x~ , . . . , x , , , of y~, .... ys, applying the above process i 'epeatedly, as necessary, and e l iminat ing y~'s which are ident ical ly zero, we arr ive at the following s i tuat ion: x~ appears in y~ but not in y : , . . . , y , , x2 appears in Y2 but not in y~, . . . , y~ , .... x~ appears in y~. This being the case, the funct ions induced on H by xl, ..., x~, are integral ly dependent on the subring induced by elements of k[x~+~, ...,x,~], which maps isomorphical ly into funct ions on H, so we immedia te ly get dim H = n - - s ; that y~, . . . , y~ are a lgebraical ly independent over ~ is in our cur ren t case clear. It remains to show that any additive funct ion f~k[x~, . . . , vc , ] that vanishes on H is actual ly a p -po lynomia l with coefficients in k in y~, . . . , y~.

By subtract ing from f a p -po lynomia l in k[y~] we can assume that t h e degree of f in x~ is less titan that of y~, or that x~ is not involved in f. Proceeding thus, the falsity of our content ion would produce such an f which does not involve xl , ..., xi-1 (for some i ~ s ) , but does involve xi and to a degree less than that of xi in y,~ or which is a nonzero element of k[xs+~, ... ,x,,]; in e i ther case it is trivial to verify that f is not in the ideal generated by y~, . . . ,y~, so the proof is complete.

LE~f~fA 1. - The unipolenl par t G~, of a k-solvable algebraic group G is k-solvable.

Trivial if dim G ~ I , so suppose dim G > 1 and use induction. G contains a k-solvable normal algebraic subgroup H such that G/H. taken to be defined over k, is k- isomorphic to G~ or G,~. H,, is normal in G and, by induction, k-solvable. Passing to G/H,~ we see that it is permissible to assume their H ~ , = t e } , i .e . that H i s a torus. I~ G / H z G .... then G i s a torus and G = { e 1, so st~ppose G / H = G~. Then dim G~, = 1.

Taking a matr ix representa t ion of G, G may be put into t r iangular form ()ver k, so G,~, which is the kernel of the (separable) ho~:;omorphism consi- sting of mapping matr ices into their diagonal parts, is defined over k. H is central in G, so G is nilpotent, hence commutat ive (since G~, is), and the

raap (h. u) ~ hu of H X G~, ~ G is a k- isomorphism. Thus G~ is k- isomorphic to G / H - - G~.

LE~I~[A 2. - Let G be a k-solvable algebraic group,

r - -=d im G~ , s = d i m GIG,, .

Then there exist xl , ..., x,.~ Y l . . . , y 8 ~ k(G), algebraically independent over k, such that k[G] :=- k[x~, ..., x~., yl, ..., ys, y1-1, ..., ys-~].

104 M. ROSENLICHT: Questio+~s Of '~'atio~atity ]or solvable algebraic~ etc.

G~ is k-solvable, so by the previous Corollary 1 G z (G/G~} X G, . By the previous Corollary 2 we have k[G~,] -- k[~c~, ..., x~], for suitable ~cl, ..., x~. On the other hand, G/G~ is a k-solvable torus, hence k-isomorphic to a con. nected group of diagonal matrices, hence to (Gin) ~, whence the result.

TttEOI~EH 2 . - Let G be a connected solvable linear algebraic group defined over k. Then G is k-solvable (resp. k-solvable and unipolent) i f and only i f there exist quantities x~, ...,x~v, algebraically independent over k, su(;h that k[G] is k-isomorphic to a subring of

k[x, .... , ~c~, ~ - ~ , . . . , ~N -~] (resp. k [x~, . . . , xN]).

Half of this is contained in Lemma 2. As for the converse, the part in parentheses is a consequence of the other part, for if k[G] C k[xl , ..., xN] then the rat ional characters on G, all of which are defined over k, are units in k [x l , . . . , xN], hence constant, so G is unipotent

It therefore remains to show that if G is connected, solvable, l inear and defined over k and

k[G] C k[xl , ..., xlv, x~ -~, ..., x~-~].

with x~, .... x~v algebraically independent over k, then G is k-solvable. First suppose that G is a torus. The embedding

k [ G ] c k[xl , ..., xN, x~ -1 .... , xF~],

defines a generically surjective morphism, defined over k, ~: (G~n} N ~ G. E~ch rat ional character on G induces a unit in ~2[IG,n}zv], hence a constant multiple of a rat ional character on (Gml N. Changing ~ by a translation, so that ~((1, 1, ..., 1})-- e, we get ~ to be a homomorphism; hence ~ is surjective. As the image of a k-solvable group under a rat ional homomorphism that is defined over k, G is k-solvable. We now prove the general case by induction on dim G, the case dim G----0 being trivial. The proper normal algebraic subgroup [G, G] of G is the image of G X G X ... X G (enough times) under a morphism defined over k, hence satisfies a condition of the given sor~ (possibly with much larger N) so. by the induction assumption, is k-solvable.

Dividing by [G. G] we are reduced to the case where G is commutative. If k has characterist ic zero (more generally, if k is perfect), G~ is known to be k-solvable and we are reduced to the case G~ G~, a torus, which case is known ; hence we may suppose the characterist ic to be p =~-0. The image GP of G under the homomorphism g ~ gP satisfies the same kind of condition as G. If GP--- G then G is a torus, which case has been settled, and in the contrary case GP is k-solvable by the induct ion assumption. We are thus

M. ROSENLlCHT: QucstioJts of ratio~,(dity for solvable algebraic, etc. 105

reduced to the consideration of G~ Gp, which is commutative and has each element (except e) of order p. By [5, Prop. 2], G is k-isomorphic to a sub- group of (G~)", for some n. Passing to quotients again, if necessary, it clearly suffices to prove the following : if x~, ..., xN e f] are algebraically independent over k and

5 ( z ) , . . . , (z) e k . . . ,

are not all in k, then the smallest k-closed subgroup H of (G~) ~ that contains the point (f~lx), ..., f,(x)l contains an algebraic subgroup that is k-isomorphic to G, . If f~(x~, .... x~,) ( k, then fi(x~, ..., x,~v-l, xl '~) ~ k for suitable integral v so it is permissible to assume N ' - I , i. e. /'1, ... . f , ek[t , 1/t], with t e f~ t ranscendental over k and not all [~{t),.., f,,(t) 6k. Switching from t to 1 / t if necessary, we may assume that at least one of f~(t),...,f~(t} contains t to a strictly positive power.

Let y~ be the coordinate function on the i th factor of (G,) ", so that y~,. . . , y,, are additive coordinate functions on iGa) ~. H is the zero locus of a set of p-polynomials in k[y~, .... y,~]. If one such p-polynomial actual ly contains a Yi such that fi contains t to a strictly positive power then more than one such Yi must occur, and we deduce that there exist i, j ---- 1, .. , n, i:@j, such that fitt), f~(g) have degrees > 0 in t, one degree being a power of p times the other. Choosing i #=j suitably, and v ~ 0 and c e k suitably, fs will have strictly positive degree in t, strictly greater than that of f~- -c f jP ~. Replacing the additive coordinate functions y~, ..., y,~ by the same ones, except for the replacement of y~ by Y i - - c Y S , we get the same si tuation as before, with the exception that one of the fk~s in which t occurs to a positive power is replaced by an element of kit, 1/t] in which t either occurs to a smaller power, or perhaps not at all.

Proceeding thus, we arrive eventually at the si tuation in which certain of the f~(t)' s, say f l , . . . , f~, where 1 ~ r ~ n , actual ly contain / to positive powers~ but H is the zero locus of a set of polynomials involving only x~+~, ..., ~c~. Then H contains the subgroup G, X 0 X ... X 0 of (G~) ~, a sub- group k-isomorphic to G, . This completes the proof.

COROLLARY. - I f G is a solvable algebraic group defined over k and G~, Gz are k-solvable algebraic subgroups of G, tl~en the commutator group [G~, G2], the group generated by G~ and G2, and the group generated by all n th powers of elements of G~, for any fixed n, are k-solvable.

For each is connected, solvable, and linear, and is the image of

G I X G~X G1X G~X ... X G l X G~

(sufficiently many times) under a morphism defined over k.

Annali dl Matematlca 14

106 M. ROSENLICHT: Questio~'~s o/ ratio~ality fo~' solvable algebrale~ etc.

We remark that the assumption that G is solvable is necessary for the preceding theorem to hold, for any connected simple algebraic group G that is defined over an algebraically closed field k is generated by its subgroups that are k-isomorphic to Ga, hence satisfies the condition that

k[G]Ck[x~, ..., x~]. with xl, ..., xN

algebraically independent over k. We also remark that it is not sufficient for the k-solvabil i ty of G to assume that G is connected, linear, solvable, defined over k, and has k(G) a purely t ranscendental extension of k. One counterexample is on p. 35 of [4] (dimension 1, characterist ic 2). A more general counterexample may be obtained by letting C be the projective line with coordinate funct ion x, choosing ~ ~ ~ algebraic over k of degree n. let- ting 0 be the local ring in k(C) consisting of all functions defined at the point ~c - -a and taking values there that are in k, and taking G to be the generalized jacobian variety for C and o, chosen so that G and the canonical map ¢p: C ~ G are defined over k. In this ease, if 0 is the integral closure of 0 in k(C), the functions x, x,2,..., x ~'-1 provide a basis for the vector space (over k) -0/0, and the conductor of 0 in 0 is the ideal generated by the minimal polynomial of ~ over k.

By the general theory [2, § 4], ~ induces a bijection between divisors on C of the form (uo-~- u ~ - { - ... T u~-~x~-~t, where uo, .... u,~_.~e~2 and E ula~::t=0 for any conjugate a of ~ over k, and points Q e G, the correspon- dence being such that ktQ)--k(uo/u~ .... ,u,~_~/'u~) if u~=~=0. It follows, in part icular , that k(G) is a purely t ranscendental extension of k of dimension n - - 1 . G contains no G~ or G,~ defined over k, for otherwise we could find Vo, . . . , v ,_~ek[ t ] (t an indeterminate), not all in k and with no common factor, such that Vo~ v~g-~ ... ~-v,-~0¢ ~-~ has no finite nonzero root, and by l inear disjointness this is impossible. Had ~ been chosen separable over k, G would be a torus, as we see by making a ground field extension. On the other hand, if k is a nonperfect field of characterist ic p and ~P~k, ~ ~ k, we have OP CO, so G p - - l e l and G is isomorphic (over ~) to (Ga)v -~.

L E p t A . - Let x~, ..., x~ be coordinate functions on P,'~ and let F e ~[Q~]

be of the form

F -- ~ (a~x~ i -~- (lower degree polynomial in x~)),

where each a~e~. Then i f F vanishes on a k-closed subset of ~2" that is k-isomorphic to ~2 ~, there exist e~, ..., c~ ~ k, not all zero, such that

a, c~ ~ + . . . + a , c ~ = O.

For there exist f~(t), . . . , f~(t)ek[t] (t an indeterminate), not all in k,

~[. ROSENLICHT: Questions of r¢ttiouality for solvable algebraic~ etc. 107

such that F{f~, . . . , f , ,}--0, and we need only examine the highest degree terms in t.

TttEOREM 3. - Let G be a unipolent algebraic group defined over k. I f G is K-solvable, where K is a separable eJctension of k, then G is also k-solvable.

Use induct ion on dim G, the case dim G - - 0 being trivial. If dim G > 0, [G, G] is a proper subgroup of G, defined over k, K-solvable by the previous corollary, hence k-solvable by the induction assumption. Thus G/[G, G] is to be proved k-solvable, and we may therefore assume that G is commutative. We need only consider the case in which k has characterist ic p=~=0.

Then the argument we have applied above to [G, G] can be applied to G P - - t g P l g e G f , and we are reduced to the case G p - - - l e l . That is, by [5, Prop. 2], we may assume that G is an algebraic subgroup of {Ga) ~, for some n. There is nothing to prove if G = {G~)'*. Supposing for a moment the ease in which G has codimension one in (G~) '~, if dim G < n - - l we can find a k-solvable subgroup H of (G,)" such that d i m / - / G = n - - 1 (for example, take H to be the direct product of several of the direct factors of (G~)").

Then HG is defined over k, K-solvable, and of codimension one in {Ga) ~, so, by our assumption, k-solvable. Thus HG is k-isomorphic to (Ga) "-~. But now G C (G~) '~-~, and we may continue to reduce n unti l we are done. Hence all reduces to proving our theorem in the special case that G is a k-closed subgroup of (G~)" of codimension one. In this case, if x~ , . . . ,x~ are the coordinate functions on (G~) ", the Proposit ion tells us that G is the zero locus of a single p-polynomial Fek[x ] . If some x~ does not appear in F then one of the direct factors of (G~,)", isomorphic to G~ over the prime field, is con- tained in G, and we may divide both G and (G~) '~ by this subgroup to get the same problem for smaller n. Thus we may suppose that each xi is in. volved in F, so that

~ n l~ ~P';~ F -- ~=~ ~ ¢ . i + lower degree terms in x~},

where al, ..., a, are nonzero elements of k. Since G is K-solvable the Lemma implies the existence of c~, ..., e, e K , not all zero, such that

a~ c~ ~' = 0.

We wish to deduce that such c~'s exist that are in k. First, since K is separable over k, a suitable k-specialization of {cl, ..., c,,) that is different from (0, ..., 0) will be separably algebraic over k, so we may suppose each e~ to be an element of the separable algebraic closure k~ of k. Now suppose c l , . . . , e,,ek~ chosen so that number of nonzero c i ' s is minimal. Suppose

cl .... ,c~:~=0, ci----0 if i > r (so that 1 < r < n } ,

108 M. ROSENLICH~£: Que.~'tionS of ratio~ality for .~olv(~ble (algebraic, etc.

and suppose v~: ..., vr_~ ~ v,.. Dividing the relation

v . . . . p.~,' ~ , ~ - - 0 by e~;'",

we see that we may assume c , . - -1 . This being so, for any k-automorphism of k, we have

Y~ a~(~(c~) --~-c~tp ~; = O,

a shorter relation of the desired type unless ~(ci)--ci, all i. Thus v(ci)--c~ for all i and all ~, so that each c~ek. This being so, we have a relation

where each b i e k and v~, . . . , v,._~_~v,.. :Now change additive coordinate funct ions on (G~} '~ by replacing (x~,..., x,,) by (x~', .... x,',), where

x~' -~- xi --[- b~xq pvr-5' •

for i----- 1 , . . . , r - - i , x i ' - -x~ for i - ~ r , . . . , n . If we write F ( x ) - - F ' ( ~ ' ) , then F ' is a p-polynomial in k[x l ' , . . . , ~c'n] having the same degree in x( as F has in x,~ for i =[= r, but of smaller degree in x',. than F is in xr (if x', actually appears in F ' ) . We now have a reduction process that terminates with the completed proof.

CO~O~LAa¥. - Let G be a connected solvable linear algebraic group defined over k. Let G~, G2 be algebraic subgroups of G that are deflated over k, one of which is either k-solvable or a torus, Then [G~, G2] is k-solvable.

[G~, G~] is conner~ted, unipotent, and defined over k. Supposing that G~ is k-solvable or a torus, it is the image under a morphism defined over k~ of a product of G, , ' s , hence the same is true for a subset containing a dense open subset of the closure of l a g a - ~ g - ~ [ g e G ~ l , a being a fixed point of G rational over k~. Therefore [G~, G2], which is the group generated by all these subsets as a ranges over the points of G~ that are rat ional over k~, is the image under a morphism defined over k~ of a product of G , / s , hence is k,-solvable, hence k-solvable.

It may reasonably be asked if, when G is a connected unipotent l inear algebraic group defined over k, there exists a unique smallest extension field k ' D k such that G is k'-solvable. The answer is aff irmative if dim G--- 1: I f G CG', where G' is a complete curve defined over k and nonsinguIar with reference to k and if G is k'-solvabl% where k ' D k , then G is biratio- nal ly equivalent to G a over k', so the one point of G'-- G is rat ional over k'; if we adjoin the coordinates of G ' - - O to k and continue this process,

M. ROSE~'LICHT: Qt~estio~ts of rationality ]or solt:(~ble algebraic~ etc. 109

and if we recall that there exists a a finite algebraic extension of k over which G is isomorphic to Ga, we see that there is a unique minimal extension of k over which G has a nonsingular projective model obtained by adding one rational point to G, and this extension of k is contained in k'. But if G has a complete nonsingular model G' over k and G ' - - G is a point rat ional over k, then since G' has genus zero there is a function in k(G) whose divisor is (e)-- {G'-- G}, and this funct ion gives an isomorphism of G with Ga.

However if G is a connected unipotent group defined over k and dim G ~ 1, there need not exist a unique minimal h ' D k such that G is k'-solva- ble. For example, let k - -ko (a , b) be a field of characterist ic p=~=0, where k0 is a subfield and a, b are algebraically independent over ko, and let G be the subgroup of ( G J defined by ~ + xi-~ a x e + b~P3~0. The equation can be writ ten [xl-~ all'x2) p ~ bx~ ~ - - x l , so G is k(a~/P)-solvable, and simi- larly it is k(blfP)-solvable; this also shows that G is connected and k-closed and its equation is irreducible, so G is defined over k. But k(aiJp)(~ k(b~tP):k and G is not k-solvable, for otherwise the lemma to the previous theorem would give 1, a, b l inearly dependent over kP.

The following lemma clearly calls for a generalization which, at the least, should give a better condition that a vector group operated on by another group possess a second vector group structure with respect to which the given operation is linear.

LE~IMA. - Let T be a k-solvable torus operating as a group of c~uto. morphisms on an algebraic group H that is isomorphic over ~ to a product of Ga'8 , H a n d the operation of T on H being defined over k. Then H - - H o X H~ X ... X H,., where each H~ is a connected algebraic subgroup of H that is defined over k and invar iant under T, T operating trivially on Ho and H~ . . . . , H,. each being k-isomorphic to G a .

For any additive function y on H and any t e T, the function kty (defined by (kty)(h)== y(t-~h)} is also an additive function on H, and the vector space spanned by all these as t varies on T is finite dimensional. Hence we can find an embe4ding H C ( G a ) ~ such that the operation of T on H induces a l inear operation, also defined over k, of T on (G~)". Since T is k-solvable additive coordinate functions x~, ..., ;Jc,, on (Ga)" may be found that are defined over k and such that the operation of T on H is given by x~,(th)-----7,,(t)x~lh), i ~---1, ..., n, where each X, is a rational character on T.

If E~, ~ a~jw~ '~' is a p-polynomial on IG~t" that vanishes on H, where each

a~j ~f~, then E a~jIXfl~)p'~x~ ~.i also vanishes on H for each t e T~ and by the

l inear independence of characters we get that Ea~ix~ '~ is a sum of p-poly- nomials vanishing on H for each of which Ixi(t))~"~ ' is constant for all the

110 M. Ros~N~,~c~,2: Ouestions o] ratio~ality fo~ ~ solve~ble algebraic, ctc.

terms that actual ly appear, for each t e T. By the Proposition, there is a

~, ,~P~J with the basis for the ideal of H consisting of p-polynomials ~ , j ~ i j ~ property that either each X~ involved is l, or at most one v~j is 4=0 for

each i. In the latter case H has as one of its equations Z'~=1¢~-~-P~, with the a~'s in gt, not all zero. Taking the p~-th root, for suitable v, and reordering

x l , ..., x , , we get a function of the form b~x~l - } - . . ~b,_~xP,~"_~ - ~ - x,~ that vanishes on H.

Since H is defined over k we can even assume that b~, ..., b , _ ~ e k , and we get another embedding for H in a product of G~'s on which T operates linearly, but this time with smaller n, namely h ~ (x~(h), . . . , ~¢,_~(h)) Thus we may assume that x ~ ..., x~ are such that x~, ..., x,~ occt~r in p-poly- nomials vanishing on H. while ~,~+~, ..., x,~ do not, and X1 . . . . . Xm-~ 1. We are done if we take Ho~-- H ( ~ (~m+~ . . . . ~ x , , - - - 0 ) , / / 1 ~ / x ~ : ~ - - -- xm -- Xm+2 -- -- x , - - O l , . . . , H,. - - I x ~ - - -- ~c ,_~- -Ol , r being

For any algebraic group G that is defined over k, each member of the sequence G ~ [ G , G]~[G, [G, G]J~. . . is algebraic, normal, and defined over k, and the smallest member of this sequence is the smallest normat algebraic subgroup G* of G such that G/G* is ni lpotent; in part icular G* is defined over k and is connected it G is. If G is connected and l inear and C is any CARTA~ subgroup of G, then in the homomorphism G--~ G/G* the image of C is a CAnTA~ subgroup of G/G*, i.e. G/G* itself, so CG*~-G; since any two CA~AN subgroups of G are conjugate, they are conjugate under G* and, in the same way, if we note that a maximal torus of C is central and chara- cteristic in C, we see that any two maximal tori of G are conjugate under G +.

T~EOaE~ 4. - Let G be a connected solvable l inear algebraic group defined over k. Then G has a m a x i m a l torus that is defined over k, and a~y two such m a x i m a l torg are conjugate by an element rat ional over k of the smallest normal algebraic subgroup G* of G such that G/G* is nilpole~t. I f G is k-solvable, so is any m a x i m a l lotus of G that is defined over k.

If G is k-solvabl% G~ is defined over k and any maximal torus of G that is defined over k is k-isomorphic to the k-solvable gro~(p G/G, , hence the last statement. Note next that the rest of the theorem is known if G is nilpotent. In the following we shall have occasion to use normal algebraic subgroups H of G that are connected, unipotent, defined over k~ =]={et, and minimal with respect to these properties. I f (; is not commutative such groups exist, since [G, G] satisfies all the desired properties but minimality. I f H is such a group, [H, HI < H and therefore [H, H]----{ei; s imilarly HP-- - {e } , so H is a vector group [over P-). Also, again by the minimal i ty property, [G, HI : H or {e}. We now prove, under the assumption that k - - k ~ , that G ires a maximal torus defined over k, and we use induction

~J[. ROSENLICHT: Questions of ratio~b(dity for solvable algebraic~ etc. 111

on dim G. Assume, as we may, ~hat G is not nilpotent, and let H be as above. By assumption G/H has a maximal torus defined over k, and the inverse image in G of the maximal torus of G/H contains one of G. Thus we may suppose that G/H is a torus. [G, H]----H, for otherwise [G, H l - - t e! and G is nilpotent. The operation of G on H by inner automorphisms induces an operation of G/H on H, and this lat ter operation is also defined over k. Since k - - k s , the torus G/H is k-solvable and we may apply the Lemma. [G, H] = H implies that H ~ {el and the minimali ty of H implies that H is k-isomorphic to Ga. If ~: G/H ~ G is a regular cross section defined over k (such exist, by Theorem 1), then for teG/H, heH, we have ~z~t})-~hz(t)--X(t)h, where X is a rat ional character on G/H (defined over k, as are all rational characters), and where we use the na tura l structure of H as a vector group. We also know that G is the semidirect product of any maximal torus and H, so there exists a regular cross section z': G / H ~ G that is also a group homomorphism.

Wri t ing ~'(t)--~(t)flt), where f : G / H ~ H is a morphism, we get from ~'~tl)z'(t~)- z'(tJ~} the equivalent relat ion

(So that, conversely, any morphism f : G / H ~ H that satisfies the last equation makes the morphism z': G / H ~ G defined by z'ttJ=~(tlf(t ) a homomorphism). Wri te f_ - -Ec i f i , where each fi: G / H ~ H is a morphism defined over k and where the c , ' s are elements of ~ that are l inearly independent over k, c~ being 1. Since the r ight -hand side of the above equation is a function defined ow~r k, we deduce, by l inear disjointness, that /'1 satisfies this equation. Hence the regular cross section ~: G/H--. G (defined over k) could have been taken to be a group homomorphism. This being so, z(G/H) is a maximal torus of G that is defined over k : indeed, if we note that the map ( G / H ) X H ~ G given by (t, h)~-~(t)h is biregular and defined over k, the group law on G is given by

our maximal torus defined over k being given by h - - 0 . This completes the proof that G has a maximal torus defined over k if k - - k ~ . We now prove by induction on dim G that two maximal tori of G that are defined over k, say T1, T2, are conjugate under (G*~k, still under the assumption that k - - k~ . Again we may suppose G not nilpotent. I f G1 is a connected algebraic subgroup of G that is defined over k and contains b o t h Tt and T2 we may restrict our at tent ion to G,, since Gt*C G*.

Thus, since TtG*--T2G*, we may suppose G * ' - G u. Now G contains normal algebraic subgroups that are connected, unipotent, k-solvable, and

112 M. ROSE~LICHT: Qtw~'tiom~ of ratio~lality for .~.olsabIc algcbra.i% etc.

# : { e } , for example [1'1, G], so G contains such a subgroup H of minimal dimension, and the same argument as above shows that H is a vector group (this time over k) and [G, H ] - - H or t e}.

H C G u ' - G*, and by our induction assumption T~H/H and T2H/H are conjugate in G / H by an element of ((G/H)*)k--(G*/H)k. But since H is k-solvable, each element of (G*/H)~ comes from an element of (G*)~, so it is permissible to assume that T~H/H--T~H/H, that is, we are again reduced to the case G / H - - t o rus . Precisely as above, [G, H ] - - I t , etc., and we get H k-isomorphic to G~, with G given by

and we may also suppose that T1 is the terns h - - 0 . Any other maximal terus T' of G is then of the form T ' : ( 1 , a) T~(1, a) -~ for some a e Q , i . e .

T ' - - (1, a)I'1(1, -- a) ---- l(t, a(x( t ) - 1) ) [ t~G/H l .

Here K ~ 1 since G is not nilpotent, so T' is defined over k only if a ~ k, completing the proof when k = k~. We now prove for arbi t rary k, again by induction on dim G, that G has a maximal torus defined over k.

Supposing G not nilpotent, we let H be as above and we get, as above, an immediate reduction to the case G / H ~ torus, [G, H ] - - H , H a vector group over k (but now, since we cannot suppose G/H k-solvable, we may have dim H > 1). Let T be a maxinaal torus of G thai is defined over a finite separable normal extension field K of k, and let I? be the galois group of K / k . For any ~ e P the group T ~ is also a maximal torus of G that is defined over K, so T'~--a:T(ct~) -~. for some a ~ e H k , by the special case proved above.

a~ is unique to within multiplication by an element of the normalizer of T in H, which is the centralizer of T in It, which is {e}, since this centralizer is connected and H - - I T , H], so a~ is unique; in part icular, a ~ H K . For z, ~ e l ~ ~e have

T:= = ( T~) ~ --(a,pTo((ct~)-~V --(a~Va~T(a~)-~((a&) -~ ,

so a~,.~-- (a~)°ao or, wri t ing additively, c t~- - (a~) ~ d- ao for all ~, xe F. Re- calling that H is a vector group over k and the vanishing of the cohomology of ]: in K, we get ~ e Hic such that a~--so--s¢~ for all ~ e 1 ~.

Thus (:¢T~-1) °--" ~Ts¢ -1 for all ~ e F , and therefore ccT~ -1 is defined over k. It remains only to show that if /'1, T2 are maximal tori of G that are defined over k, then they are con juga te under (G*)k. As before, use induction on dim G, taking G not nilpotent, and use the above reduction process to arrive at the case G---- TH, where T is a torus de[ined over k, H a vector

~[. ROSENLICHT: Q, uestio'J~S O] rationality ]or solvable algebrtdc~ etc. 113

group over k, and H - - [T, H]. Here again the normalizer of T in H (-- een. tralizer of T in H) is t e} , so the element he(G*)k - - H G such that hTJe- l=-T~ is unique. Thus h is left fixed by each k-automorphism of f~, hence is rat ional over k, which completes the proof.

It is not known if the above theorem remains true with <<maximal torus >> replaced by << Caftan subgroup >>. The problem seems difficult.

COROLLARY 1. - I f G is a connected solvable linear algebraic group defined over k then any torus of G that is defined over k is contained in a maximal torus of G that is defined over k.

First suppose that k - - k ~ and use induction on dim G. Supposing, as we may, that G is not nilpotent, and let t ing H be a normal connected unipotent proper algebraic subgroup of G that is defined over k and minimal wi th respect to these properties, the a rgument of the first part of the above proof gives an immediate reduction to the case where G is the semidirect product of a k-solvable torus T and a group H that is k-isomorphic to G~, the group law being given by

Z being a rational character on T. Any torus S of G is contained in a maximal torus which, again by the proof above, must have the form

1 (t, a{x(t) - - 1))It e T / ,

for some aeO~. If S i s defined over k and S ~ : ker X, a point of S rat ional over k and not in ker 7~ may be found, giving a ~k (and proving our result), while if S C k e r X then S is central in G and hence contained in any maximal torus of G.

To prove the result for an arbi t rary k, let S be a torus of G defined over k and T a maximal torus of G containing S and defined over k~. Then S is central in T, hence in any conjugate of T over k, hence in the group generated by all the conjugates of T over k, an algebraic subgroup of G that is defined over k. Thus S may be supposed central in G, and the corollary is got by applying the theorem to G/S.

COROLLARY 2. - I f G is defined over k then the smallest G/G* is nilpotent is k-solvable.

a connected solvable linear algebraic group normal algebraic subgroup G* of G such that

Let T be a maximal torus of G that is defined over k. Then [G, T] is a normal algebraic subgroup of G. G/[G, T] has a maximal torus that is central, hence is nilpotent, so G*C[G, T]. Hence G*, which is the smallest group in the chain G ~ [G, G] D [G, [G, G]] ~ ..., is also the smallest group in the chain [G, T]D [G, [G, T]]D .... Bat each group in the latter chain is k-solvable, by repeated application of the Corollary to Theorem 3.

Anna l i d i Matemat iea 15

l l 4 M. ROSENLICHT: Questio.t~s of rationality for solvable algebraic, etc.

COROLLARY 3. - I f G is a connected solvable linear algebraic group de- fined over k, there there exists a k-solvable unipotent algebraic subgroup G' of G that contains all other k-solvable unipotent algebraic subgroups of G, and G' is normal in G with G~ (7' nilpotent.

The existence of (7' is implied by the Corollary to Theorem 2, its nor- mality by the Corollary to Theorem 3; ~nd the nilpotence of G~ G' by the previous Corollary.

COROLLARY 4. - I f G is a corenected solvable linear algebraic group defined over the infinite field k, then the tori of G that are defined over k generdte the algebraic subgroup TG* of G, T being any maximal torus of G.

All maximal tori of G are conjugate over G* and these generate a sub- group F of G that is equal ly well generated by the conjugates of a given maximal torus by the elements of a dense subset of G*. Since G* is k-sol- vable and k is infini~.e, (G*)k is dense in G*; since G has at least one maximal torus defined over k, F is the group generated by all such. F is clearly ~normal in G and is the smallest normal algebraic subgroup of G such that G/F is unipotent. Hence I~DG* and since the semisimple part of the nilpotent group G/G ~" is TG*/G*, we have r _ - - T G * .

Apropos of Corollary 3, we give several examples of connected solvable algebraic groups defined over 'k, not tori, that contain no algebraic subgroup k- isomorphic to Ga.

Such a group G is necessari ly nilpotent, and if its maximal torus is T then G / T is also such a group, for otherwise G would contain an algebraic subgroup defined over k that is k , -solvable and larger than T, and the uni- potent par t o.f this would be k-solvable. The simplest example of such a group is I (x, .y) e (G~) 2 i Y~ - - Y -- awV f, where a ~ k, a ~ k v , and other examples may be found above, preceding Theorem 3. An example of such a group G, commutat ive and of dimension 2, which however possesses a connected algebraic subgroup defined over k giving rise to a factor group that is k- isomorphic to Ga is given by

G -- ~ (x, y~ z) e ( G J I x ~ -[- ay p + bz ~ + z -- O } ,

where a, b e k and 1, a, b are l inearly independent over k~; G is defined over k, since x ~ ~ ayP ~ bz ~ -~ z, x~ and al/Py ~ bl/Pz are additive coordinate functions for-(Ga) ~, G possesses no subgroup that is k- isomorphic to G a by the lemma to Theorem 3, and if H is the subgroup H - - G A tx -~ -0} then H is connected and defined over k and the factor group G / H is k- isomophic to Ga, having the funct ion x as an additive parameter , tAs a matter of fact any connected algebraic subgroup G of dimension > l, defined over k, of

.~[. ROSENLICHT: Q~testions of r~tiot~ality for solvable algebraic~ etc. 115

(Ga} '~ has a connected algebraic subgroup defined over k producing a factor group that is k-isomorphic to Ga. For, assuming, as we may, that k is infinite, if x,~, ...~ ~c,, are the coordinate functions on (Gal '~ then for suitable cl, . . . . c n ~ k tlle map ~ (~c~ . . . . , ~)---c~x~--}-.. . z~ c~x** induces a separable homomorphism of G (for separabil i ty we need only E c i d x ~ - O) whose kernel is connected (since the generic hyperplane section of G is irreduciblet).

Here is a more complicated, noncommutative, example of a unipotent algebraic group defined over k that has no algebraic subgroup that is k-iso- morphic to Ga: Let 1, a, a ~ k be l inearly independent over k ~ and let F, G~, G~ be the algebraic groups

r -- l(Vl, C2)~(Ga}:[Cl ~'Ji- aC2 p "Ji- ac~- - Of

For (c~ , c~) ~ F, (x, y ) ~ GI , define

~(~,, ~) (x,, y) - - (c~x - - c2y, c~x, - - c~y).

Then ¢~ induces a homomorphism from l: into Horn (G~, G~) and we get a group law of the desired type on the variety I~ X G~ X G~ by put t ing

(y, g l , g~) (7', gl' , g2') - - (Y -{- Y', gl --[- g~', g2 -t- g2' -[- % (g~'} ).

For some counterexamples to a number of guesses that might be made on the basis of the previous theorem and other known results, consider the composition law on ~2 3 given by

(x, y, z)(x', y', z') = (x "t- Ix', y "b Y', z "t- z' --]- f(y)g(x') ),

where f, g are p-polynomials with coefficients in k. This is easily seen to be a group law and the algebraic group structure we obtain on ~2 3 is k-sol- vable and unipotent. If we choose

a e k, a ~ k p, f(y) -- y~2_~ aye, g(x') - - ~c',

we find that the centralizer of ~][, 0, 0), which is also both the normalizer and centralizer of the subgroup y -- z - - 0, is the subgroup yS2.{_ aY r _ O, which is nei ther connected nor defined over k: fur thermore the center of the whole group is given by y~2 ~ ay p _. ~ - - 0 and is therefore also not connected and not defined over k. A connected k-solvable unipotent group

116 3[. ROSENLICHT: Question,s of ratio~ality lot .~'oh'~(ebIe algebraic, etc.

whose center is connected and not defined over k is given b~"

(x, y, z , u)(x', y', z', u') -- (~c ~ x', y + y', z -~ z', u -~ u ' + (x p -~ ay ~) z'),

where a is before; here the center is x ~ ~ ay ~ - - z - - O . The following lemma, which is slightly more detailed than necessar.y

for our purposes, is an easy ref inement of wel l -known facts.

LEMMA 1. - Let G -- Go D G~ D ... D G,. -- t e t be connected solvable linear algebraic groups defined over k, wi th each group normal in its predecessor. Then there exists a normal chain of connected algebraic subgroups of G that are defined over k which is a refinement of the given normal chain and such that each factor group is either a torus or isomorphic (over ~2 ) to a product o f G~,'s ; furthermore, i f each Gi i s n o r m a l in G then the refinement can also be so chosen, and i f all the groups in the given chain are k-solvable the refi- nement can be so chosen, with each subgroup normal in G i f each given Gi is, and these various refinements can be so chosen that each group is of codi. mension one in its predecessor in case G is either unipotent or k-solvable.

Between Gi and G~+I we may place the group [G~, G~]G~_~ or, if this equals G~+~, the group (G~)~,G~+~ in characterist ic zero or (G,)~G~+~ in charac- teristic p ~ 0 and we get a proper intermediate group unless G~/Gi+~ is either a torus or (~Q-)isomorphic to a product of Gffs. Note that repetition of this procedure will produce f inal ly a ref inement s~tisfying all our° de- mands, in all special cases, except possibly the condition of codimension one.

I f G is unipotent, then it is nilpotent and its commutator with any normal subgroup is a proper subgroup of the latter, so the question reduces to f inding a connected algebraic subgroup defined over k of codimension one of a central subgroup that is defined over k, hence to the question of f inding a connected algebraic subgroup defined over k of codimension one of an algebraic group that is defined over k and ~Q-isomorphic to a product of (Ga) ~ s, and a much stronger result has been shown above at the end of the paragraph after the last eorol}ary.

If G is k-solvable the only remaining problem is to show that if G~ is a normal k-solvable algebraic subgroup of G of dimension greater than one then G~ has a proper algebraic subgroup that is normal in G and k-solvable.

I f G~ is a torus it is central in G and there is no problem; in the con- t rary case we may suppose G1 unipotent, in fact a vector group over k that is central in Gu (otherwise take its commutator with Gu, etc.), so the tor~us G/Gu operates on G. by inner automorphisms, and the lemma to the pre- vious theorem settles the matter.

L E ~ A 2. - Let V be a homogeneous space for the connected commutative algebraic group G, all defined over k. Then there exists an algebraic group G',

M. Ros~NLIC~W: Question,s of ratio~ality for solvable algebraic, etc. 117

a surjective rational homomorphism ~,: G ~ G', and an operation of G' on V, all defined over k, such that g v = z ( g ) v for all g ~ G, v ~ V, and such that V is a principal homoger~eous space for G'.

Let g be a generic point of G over k, ~c a point such that k(x) is tile smallest overfield of k which is a field of definition for the birat ional map v ~ gv of V. By [7, Prop. 2], the locus of w over k is a pregroup of transfor- mations of V. Let t ing G' be the algebraic group that is birat iona]ly equivalent over k to this pregroup, we get a sur ject ive homomorphism ~: G ~ G' and an operation of G' on V (a priori only r~tional, not regular), all defined over k, such that gv -- ~,(g)v, at least generically, and such that k(~(g)) is the smallest overfield of k which is a field of definition for the automorphism of V given by v ~ g v , i .e. such that G' operates fai thful ly on V. By homo- geneity, G' operates regular ly on V and the equation gv -- ~(g)v holds every- where, so we may simplify by replacing G by G'.

Thus we may assume now that G operates fai thful ly on V, and we want to prove that in this case V is pr incipal for G. This is a problem that is independent of the field of definition k lfor example, by [7]) so we may, for simplicity, assume k algebraical ly closed. Since G is commutat ive there is only one isotropy group I / for all the points of V, and G / H operates ratio- nally on V. Since the operation of G on V is fai thful we must have H = te l . F ix a point v o e V t h a t is rational over k and define a morphism ~: G ~ V by ~(g)'--gVo; ~ must be proved birational.

is defined over k and one-one, hence purely inseparable, and if g~, g2 are independent generic points of G over k then

so 5,(g~g2) is rational over

k(g,, ~(g~) ) N k(g~, ~(g~) ) ~-- k4~(g~), ~ig2) )

(the last point is verified, for example, by working with bases for k(gi) over k(~(g~} ), i - - I, 2). Thus the morphism of V given by v ~ g~v, i .e.

is defined over k(~(g~)}, and since G operates fai thful ly on V we must have k t~(g~))- k(gQ. So '$ is birational.

LEMMA 3 . - Let V be a homogeneous space for the connected algebraic grottp G and let H C G be a normal algebraic subgroup of G that operates nontrivially on F, all defined over k, with H k-isomorphic to G~. Then there exists a quotient space V / H, an algebraic gro~tp H ' that is k-isomorphic to

118 ~ . I:~OSENLICHT: Questions Of rationality for solvable algebraicj etc.

Ga, a morphism ~: ( V / H ) X H ~ H' that is a homomorphism in the secottd variable, and a rational operalion of H ' on V. all defined over k, such that i f z: V ~ V / H is the natural map, lhen

(1) hv- -~ (xv , h)v for all h e l l , v e V

(2) V is a principal fiber space for H ' over V/ H.

The orbit space V / H is a priori define'd only up to a b'irational transfor- mation, but it is prehomogeneous for G and therefore we can find a genuine G-homogeneous orbit space V / H and a morphism ~: V ~ V /H, all defined over k, z being the map v ~ Hv, such that g(xv) = z(gv) (cf. [3, § 11t.

~, as usual, is separable and, since there exists a rat ional cross section V / H ~ V, there exist, by translation, local cross sections V / H ~ V that are regular a t p r e s c r i b e d points of V/H, so that V / H is actual ly a quotient variety in the stcict sense. For any v e V, let H , be the i so t ropy .g roup l h e H I h v - - v l ; for g~G, Hgv- 'gH,~g -~.

Since H operates nontr ivial ly on V, H~ is finite for all v. For any g e G, the automorphism of H given by h ~ g h g -~ is defined over k(g), so if we use the natural vector group s t ructure of H we have ghg - ~ - - x(g)h, with xek(G) ; X is a rational character on G that is trivial on H and we have Hg, - - x(g)Hv for all g e G, v e V.

Now let x be a f ixed additive coordinate function on H that is defined over k. For any v ~ F, fixed for the moment, consider the operation of H on the orbit Hv. By Lemma 2, f ir is a principal homogeneous space for a homomorphic image of H, all defined over k(v). But any homomorphic image of H with finite kernel is itself isomorphic to G~ and comes from a p -po ly . nomial in ~c, and any principal homogeneous space for a group is isomorphic to the group. Thus we obtain a unique monic p-po lynomia l lynx)~ k(v)[x] that gives the homomorphism from H into a group for which Hv is principal, and we have in fact k(v)[Itv] isomorphic to k(v)[fv(X)] if we use the map h ~ hv to embed ~2[Hv] in Q[H].

:Note that the degree of f~(x.) is the degree of the map h - ~ hv, while the roots of fv(~C) are points of Hv, and degree and roots together determine f~(x) since the latter is a p-polynomial . Still keeping v fixed, choose a f ixed g e G and consider fg,(x}. Hv is isomorphic (as a variety) to Hgv if we make correspond the points hv and gh~--(ghg-~)gv, implying that fg~(x} and f~(x) have equal degrees. But the roots of fay(X) are Hay--x(g)Hv and therefore

fg,,(x) - - ()~(g) )~ f~((~((gt)-~x).

v ta~power of p) being the degree of fv(x). ][n part icular, each coefficient of fv(x) is semi- invar iant for G, mult iplying by a power of ~({g) when we go from v to gv. Since ~( is trivial on H, each coefficient of fv(~) is H-invar iant .

120 M. R o s ~ x L I c t t ~ : O~testio*zs of ra t i ona l i t y for soh;ablc algebraic~ etc.

t 4 ] - -

[ 5 ] - -

[ 6 ] - -

[7] A. ~'*EIL~ 0J~ a~gebraic groups of tra~sformatio~s, vol. 77 (1955).

So~J,e ~'ationality questions on algebraic g~'oups. ~ Annal i di matemai ica laura ed applicata,~, Ser ies IV , vol. 43 (1957).

., Extension of vector groups by abelian varieties, ~, Amer i can Jo l l rna l of Mathema- tics *, -col. 80 {1958).

, OJ~ quotient va~'ieties and the affine embedding of ce~'tain homoge~eous spaces~ ¢ Transact ions of the A m e r i c a n Mathemat ica l S o c i e t y , . ~o]. I01 (1961).

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