- Email: [email protected]
Automorphism group of a polynomial ring and algebraic group action on an affine space
Ac;tomsrphism
of a Polynomiat Action on an Afine
Group
Ring and AIgebraic Space*
Group
The main cbject of study in this paper is the gro:up of au*-VP---‘-‘c :v ..JA~r,Lms of as z-variable polynomial ring over a field. We examine this group 2nd i:s algebraic, subgroups 5&h a view, mainly, toward berte; r understanding of algebraic group actions on on an afline z-space. Let G&(k) denote the k-algebra automorphism group of t!:e polynomial ring iz[T] = k[T, ,-.., T,j over a field k. A Srst natural index of an automorphism of k[T] is its degree relative to (T) = (Ti ,..., T,) as will be define2 belo~x;. I3y means of this and following Shafarevich [lzj, -3-e $tyo&nce 02 GL- due to x:an der I<& [13: &&g an z_ req.-i: LL 52s amalgamated free-product decomposition of GA,(k). r~‘&s pO;i-ercu’ far-reaching consequences such as the absence of non;rii+al seApr$+ forms 3: the a&c plane [8] and the determination of the algebraic group aztion on &e &fine plane as treated in Section 4 of the present paper. Cnfortunateiy, hen-ever, ICOone SC fzr has been able to extend such free-product dccom~osiri~r;, to iigber >I aimensions. Indeed, one does not even know box to list the tec;es of generatcxs kr GA,(k). The second ‘basic result for GA,(K) is a theorelr_ of Frzer and RIadez 1”: -\x;h;,h +-es, for the “principai par:” of the a-~;o,mo:p>iis~L gro”p, 2.&esct=~:~i~~g ;;;rltracor. by normal subgroups with each successke r=,uotient isomorphic to wme -;ector group. It is rather straightforviard tc generalize this resuk m arbitrarv GL4,(k) to the extent of finding a descending central series Cc: the
principal part G&l(k). However, for the proof of the key part that the successive quotients of the filtration are vector groups, Fraser and Mader had to draw on the assumption that n = 2 and the characteristic of K is 0. Our generalization of their theorem (Theorem 2.2 below) is valid for any n and any characteristic as long as the field k is algebraically closed. A major question in algebraic geometry of the afhne space asks if a linearly reductive algebraic group action (or at any rate an algebraic torus action) on an aSine space may be linearized by correctly choosing a coordinate system in the atline space (see Conjecture 3.1). An a%rmative answer to this question, even in modest caseslike a one-dimensional torus G, acting on the a&e 3-space As, would be quite significant. While we are unable to answer the question except at the dimension 2 level, the results in Sections 1 and 2 of this paper provide a setting for a cohomological interpretation of the problem formulated as Proposition 3.2 below. As mentioned already, the case of the afBne plane is a special one due to the free-product decomposition of GA,(K). In this case we not only can answer the preceding question (Conjecture 3.1) in the affirmative, but also can essentially classify all possible algebraic group actions on the atfine plane As (see Theorem 4.3). The previous results dealing with various types of algebraic groups acting on Aa are then included in our result as special cases(cf. Gutwirth [5-J,BialynickiBirula [2], Rentschler [ll], Miyanishi [9], Wright [16J, and Igarashi [6]).
0. NOTATIOKS, COWEIKKIONS,ANI OTHER PRELIMIXARIES The letter K will abvays denote a fixed ground field of arbitrary characteristic. Unless otherwise stated, k is assumedto be algebraically closed, in which case we suppress the mention of k. Thus, for instance, GL, stands for the general linear group GL,(k) of rank n, and G-4, the fill algebra automorpkism group GA,(k) of W-1 ,.a., T,], both over the algebraically closed field k. Algebraic varieties and algebraic groups appearing inthis paper are all afhne, reduced, and defined over algebraically closed k, but are not necessarily irreducible. They are identified with their respective k-rational point sets. So, the a;8ine n-space An is identified with kn. The automorphism group Aut An of the algebraic variety An will be viewed the same as GA, . Each k-algebra endomorphism of k[T] = k[T, ,..., T,J will be represented by the image of the Tls. Thus
+ = (61,-*-,+,a):VI + WI denotes an endomorphism 4 such that T,# = & = +i( T) E k[ T] for all 1 < i < n. Tke degreeof + relative to (T) = (TI ,..., TJ is defined as the maximum of the total degrees of #r ,..., & .
AUTOMORPHISM
OF POLYSOML4~
RIYG
44:
Le: &V denote the set of all n-tuples of nmmegatice integers. For each r,-tuple (i) = (i(l),..., i(n)) EN denote by P the monomial T$‘i .*. Tic”) whose total degree \ve write as i i 1 = i(1) L . ..+i(nj.Let(u~).(ijE.N,l
=
fj(Tl
T,)
v---9
:=
Tj
_
x
I ; .,
uj”;T(‘!,
i-!
i:>1
a generic principal formal power series with coeffkients in R := k[z@: (i) 5 A-. i i > 1, 1 < j < n]. Let 31 denote the ideal generated by TI ,..., T,, ins& the form4 power series ring R[[T; ,..., Z’,]], and Z all integers. LEMhI.4
0.1.
There exists a family of polynomials in TI :..., T,; {HP’:1
sati$yiq
Ot
the following conditions for ail j and Cz:
(a) H!,@E,[z#: 1 < h < n, 1 < j i prime subriG of k (zZ or ~Z,lpZ!); [b)
The total degreedeg Hjd) Q d;
(c)
T, = Hj’)(f,
(d)
HJd’ = Hj:d-l’
,..., fJ
< d][T, ,..., T,$
zhere r is the
(mod Jid-i);
(mod Md). (Here, H:‘! is defined to be Ofor allj.)
Proof. We const,uct the Hjd)‘s inductively on d > 0. For d = 1, let Hj’) : = Tj for each 1 < j < n. Obviously conditions (a), (b), (c), and (d) are satisfied. Sow suppose Hi’),..., HAd! have been constructed for all d < e satisfying (a) through (d). Then, for each 1 $ j 6 nt T; G Hi’“-l’( ,1 f ,..., f,,) +
1 aIf’T(‘) ;1=e
(71 t-;
(mod W-l)
because of(c) with d = e - 1, where the a$‘s clear!y belong to zf@; ‘.:
1 < ! < n.
1, ajf’T!i! i =e
for each 1 < j < n. Conditions (a), (b), and (d) are obviousiy ftifilled. Hl”(f:
,.... fn) = Hje-l’(fl = H.F-“(f,
,...,f.)
A c ai!!fe’..fi;(l) ..a f ijcn’ il=r
,..., f,J -
E Tj (mod W-1)
As for (c),
x aj:‘T(” i!=e by (9).
(mod ~WdAi) QED.
T. KAMTHYASI-II
442
Let {z$) : (i) E X, 1 < j < n} be variables over K just as above, and consider generic power series without constant terms gj = g,(T, ),.., T,) :=
x up)T(i) ;I'>0
(3)
for 1 < j < n. Then, the rz-tuple (gr ,..., gn) defines an endomorpbism y of the formal power series ring R[[T]] g iven by Tf/ = gj for all j. For the same j’s let L, := xii~XI u:‘)T’~), and let D be the Jacobian matrix (I#): 1
For each 1 < j < n and each (i) EN with I i i # 0, there
exists a polynomial Pi’
E r[A-l,
&I:
where A = det D,
1 < h < n, 1e 1 < 1i i],
with the followitag properties: Let 4 = (& ,..., 4,J E GA,(h), and # = (I/~ ,..., #,,), its inverse. Assume that the 4,‘s are without constant terms (viz., +,(O) = 0 far all j) andwritefmeachl
dj = 1 $)T”’
ad
#* = c bi(“)T(‘).
ii’>0
Iipl
Then, f-w em-y j and ewry (i), by) = @)(a). The corollary is an easy consequence following that lemma.
of Lemma
0.1 and the observations
Remark 0.3. The essential contention of Corollary 0.2 is that the inverse of an origin-fixing automorphism 4 of a polynomial ring can be obtained through an a@rithm from the coefficients of 4. This is more than just saying that the coefficients of the inverse are polynomials in the coefficients of 4. -More importantly, Mitsuhiro Takeuchi has kindly pointed out to us that this result is no longer valid if 4 does not fix the origin, i.e., if 4,(O) + 0 for somej. His counterexample: For each natural number m, let y,,, = (TI - 1, T, - TIW) E GA,(h). Then, r;r = ( TI - 1, Tz - (TI L 1)“). Since the coefficient of TI in Tz + (TI T 1)” is m, it is absolutely impossible for thii coefficient to be expressed as a polynomial in the coefficients of TI - 1 and Tz - TIN, which are O’s, l’s, and (- l)‘s, and do not involve m.
ACTOMORPHISM i.
Following
OF POLYSOXIAL
GA,# BXD ITS ALGEBRUC
Shafarevich
w
R:SG
SUBGROUS
[12] a set union
of aigebraic \-arieties & (1 < d < CC) over an algebraically closed base feld k wili be called an infinite-dimensionaiolaal algebraic variety pro\-ided tht each -Xi . is a closed subvarrety of X,-, . A mapping j? X = l&q Xi + Y = iim I-; of infinite-dimensional wrieties is a motphism, by definition, if the restriction of-.? to each Xi is a morphism of ordinary algebraic varieties to Y; for some j determinable from i. From now on we will not distinguish between isomorphic infinite-dimensional algebraic varieties. Such a \-ariety X is said 10 be @rre, smooth, or irreducible, respectkely, if X is isomorphic to 3.7’ = lim Xi with a!1 Xi affine, smooth, or irreducible, respectively. Xn infinite-dimensional algebraic group is defined to be a group object in the categorv of infinite-dimensional algebraic varieties. Thlq if G = !&I Gi is such and G + G (inverse) a group; the group mappings G x G + G (multiplication) induce morphisms of algebraic I-arieties Gi x G, + Gr(r,j)
and
G -
for ail indices 4, j, and h, and for suitably corresponding For each positive integer d, put
Gsm r = r(i, j) and s = :(A).
where “deg” means the degree of an endomorphism of k[T] variable set (T) = (T, ,..., TJ as defined in Section 0.
relative to 2 fixed
THEOREM 1.1. The set Y, has a natural structure of an afine aigebraic wriety and Yd is a closed sukariet_y of YdB1for mery d > 0.
Proof. Let I$ = (& ,..., &), # = (4 r ,..., #,)) be w-0 n-tuples in k[Tj, each of degree
of poiynomia;s
Then, for qbto be an element of Yd and 4 be the inverse of i, it is necessary and sufficient that
T. KQNIBAYeASHI
444
for every 1
GA, = U Ya = l& Ya is an inJinite-dimezsional algebraic
PUP. Proof.
If a E Yd and r E Ye , then UT E Ya, and u-l E Ya , 7-l E Y, , clearly. Q.E.D.
Remark. A. Bialynicki-Birula has shown, in a personal communication of 1973 to us, that the group of automorphisms of any aRine algebraic varieq may be given a structure of an infinite-dimensional algebraic group. The proof is a straight generealization of that of Theorem 1.1 above. Now let G be an algebraic group acting on the affine n-space An. By definition, this means the datum of a morphism u: G x An -+ An such that u(grgs, a) = u(g, , u(gs , a)) and u(e, a) = a for anyg, , g, E G, any a E A”, and e the neutral point on G. Given this, there arises a natural group homomorphism G + G& . THEOREM 1.3. Let G be an algebraic group acting on An, and let 01:G + GA, = u Yd be the associatedgroup homomorphism. Then, there exists some d suck that the image a(G) of (Yis contained in Ye as a closedsubset,and 01:G + Ye is a morphism of algebraic varieties.
Proof. Let F(G) = k[x, ,..., xm] and I’(A”) = k[ Tl ,..., T,J be the respective af?ine coordinate rings of G and An. Then, a k-algebra homomorphism T,Xk[T] + k[x] 8 k[T] E (k[x])[T, ,..., T,] corresponds to u. Write T,q = P<(x, T) E k[x, T] for 1 < i Q n. Then, each g E G is regarded as a homomorphism r(G) + k, and if (.vr ,..., x,) + ([r ,..., [,,) under this mapping g then the automorphism of k[T] corresponding to g is given by Ti - P,(& T) for all 1 < i < n. Thus, the total degree in the Ti’s of a(g) = (PI@, T),..., P&, T)) for any g E G never exceeds the maximum total degree of the Pi(x, T)‘s, say d. Hence or(G) C Y, . It is obvious from the preceding analysis that CCG + Ya is a morphism. To show the closedness of a(G), let z: Yd x Ya -+ Ye (for some e $ d’) and w: Ya + Ya be restrictions of the group multiplication and the inverse map, respectively, of GA, . Then er maps or(G) x or(G) into oL(G) and u maps z(G) into itself. Therefore, the closure G” of O(G) in Ya (and also in YJ is stable under the group maps o and w. G’ is consequently an algebraic group. In the homomorphism CX:G + G’ of algebraic groups the image has to be a closed subgroup, hence or(G) = G’. Q.E.D. The theorem above motivates
the following
A~TO&IORPHIshI
OF POLYSO1IIAL
RIXG
445
DEFINITIOS 1.4. An algebraic subgroup of G-4, = IJ I:1 is an abstract subgroup of GrZ, which is also a closed subset of Ya for some d. Thus, the sttidy of algebraic subgroups of GA,. is equivalent to the stx$- cf a!gebraic group actions on the afhne n-space An.
2. A DESCEKDING
FILTRATIOS
OS
Again, k is a Fixed algebraically closed field. Let us define various subgroups of G/f, = G&(k) group of k[T, ,..., T,,] = Aut, A”.
G-4, = the R-auromor?hism
(i) G,” = k+n, consisting of 4 E GA4, with & = Tid = Ti -- ai for some a, E k and for each i. These are translations. (ii) G&O = (4 : Ti+ E ..l;I for all 1 < i < ~3, w-here &I denotes th nzaximal ideal in k[T] generated by the Ti’S. These a=ethe origin- (augmerztuticvz-1 Jixing autwnorphisms. (iii) GL, = GL,(k) = th e automorphisms arising from linear substitutions of the Ti’s (called lirzear automorphisms). (iv) For each positive integer Y, put GA,,:’ = (4 E G_i(,i : & = TzS;,:= Ti (mod JP1) for all 1 < i < n). An automorphism belonging to G-&Y v!ll be ca!led v-principal if v > 1, principal if I, = 1. LESINA 2.1. Let v be a positice integer. For + = ($1 i...I +,J onL z) = & ,.=., &J E G..4,O,+,F E GAnp if und ~n/y q+I = I$; (-mod W+l) J-N c/l I
Proof. Suppose + = w/ with 01E GA,,!‘. Then, 3; = (T’r 1 PI(T) ,..., T, -.P,(T)) with ali P,(T) E X”fl. For any i, +i = Ti+ = Tide = (Ti 7- P,(T))! == z& $ P,(T$) = & (mod JP+l). Converse!y, assume& = & (mod W-1) for a3 1’. Write C&E tji s Ti + Qi( T) (mcd W+r) and +-I = (pr ,..., pR),C-l = (7; :.... :.;,). Then: by Corolla? 0.2, p, = 7i (mod MV+l), which we write as ~2-i - &(Tj. Nom-, 0 = T,+,IW - Tt = (Ts I Qi(T))+-i - Ti ES Ri(Tj 7- Q$‘- - R(T)) (mod 3P1) obviously. But TZ&V = (Ti - Qi(T)) #-’ = Ti - Ri( 1”) + Q.E.D. QI( T i R(T)) SE Ti (mod .JP1). Therefore, +$--~ E G&v. THESEN 2.2. (a) G/f, = G,* - GA,0 = GA4,O-G,” {the identity automorphism E>.
and G,” n GA,0 --
(b) With respect to the inclusion map GA,’ G G,sl,,* arzd the map GA,,” -* GL, sending 4 E GA,O to its linear part (viz., itc I . JTac&m m&ix at Tl = . .. == T, = 0), the sequence l+GA,l+GA,o+GL,+l is exact, and is split by the natural inckuiorz map GL, G GA,,3.
446
T. KAYBAYASHI
(c) There is a chain of normal subgroups of GA,O : GA,’ 3 GA,’ 3 mm3 GA:‘3 GA,“3 m-nin which n,“=, GA,” = {E}. (d) For any v < ,u the factor group GA,,“/GA,p has a natural structure of a unipotent algebraic group, and GA,v/GALT1 is isomorphic to a oector group GE(‘) = k!$) of a certain dimension N(v). Proof. (a) It is clear that each # E GA, is written uniquely as a translation followed by an origin-fixing automorphism. (Sote, however, that neither God nor G-4,0 is normal in GA, .) (b)
Obvious.
(c) The normality of GA, in GA,O stems from the fact that each+ E GA,O induces a k-module automorphism +o) of ( Tl ,. .., T,#( Tl ,..., T,)“+’ in a natural fashion and G&v is the kernel of the reduction homomorphism 4 + #“J. The rest of the assertions in part (c) are clearly true. (d) Xe shah first show that GA’,-‘/GAn’ has a structure of a finitedimensional vector space over k. Let + = ($r ,..., +,J E GA;;‘. For each i one may write cJ~= Ti L C #‘T(j) (mod JP1) where the sum is taken over all = Y. When $ = (#r ,..., &J E GA’,- is (~)EN with ij 1 =j(l) + --a ‘j(n) similarly expressed as y$ = Ti + C vt’)T(n, one can see by Lemma 2.1 that + = # (mod G&u) if and only if (I#)) = (@) as ordered tuples on (j) and for every i. So, (~1~))may, and will, be viewed as coordinates of [+] E GA~M1/GA,v. Moreover, one can readily check the identity
for all 1 < i < n. Therefore, the mapping + ti (UP)) gives a one-to-one group homomorphism of GAk-‘,/GA,’ into the additive group of kv(“) for a certain N(V). Our proof will be complete if the image under this homomorphism is actually a k-vector subspace of k-v(U).To see that, we avail ourselves of “scalar operations on group functors” developed by Bass et al. [ 1, Sect. 23. As 4 preserves the filtration {My; 0 < Y < Loo} on k[T, ,..., T,], the scalar multiple a+ is well defined and automorphic on k[T, ,..., T,] for any s E k; if 4 H (z@), then %j ++ (s’4rj’)) according to [l, p. 2873. Now let c E k be given, and pick s E k with SY-~= c. Then, c(u!‘)) = (CUE’))= (sV-%#)) is the image of [“+I E GAi-l/ GAny, and we thus conclude that GA’,$/GAnv as embedded in 79(“) is closed under scalar multiplication. The fact that GA,y/GA,u (V < p) is naturally endowed with an algebraic group structure is easily shown by induction on ,o - Y. One proves that [#I E GA,u/GA,u is naturally coordinatized by the coefficients of the terms of total degree between Y + 1 and CL;the group operation is clearly polynomial on these Q.E.D. coefficients. Further details will be omitted.
IL‘TOMORPHISY
OF POLYKOMAL
RIKG
447
Rem& 2.3. The preceding theorem constitutes a generalization of a resuh of Fraser and Mader [4, Theorem 2.11, p. 28-J.Their proof of an equivalent cf part (d) abo\-e is by means of dimension counting and does not appear tc be generalizab!e beyond the case of n = 2, char(k) = 0 that they treat. On the other hand, note that the closure of k under extraction of radicals is of the essence in our proof abox-e. We record here a group-theoretical fact without proof, as it map be veriiied by elementary calculations. PROPOSTIOX
2.4.
The commutator [G&l,
3. -1 CONJECTURE ABOUT iiLGEBR4IC
GA’:;lj is contained in GA;.
GROW
&X10X
OS bin
One of the ultimate goals of these investigations is to determine and class@; all algebraic group actions on the afine n-space An. In the first nontrividi case of n = 2, there ha\-e been manr contributions bv \-arious authors. In Section 4 below, we ;d discuss and conclude this case.-When the dimension A > 2: how-ever, very little is known about such actions. One knows, thanks to Shafarecich, that any algebraic torus or any finite group whose order is a power of prime + char(k) acting on An must ha\-e a fixed point (cf. i2, I, Theorem 2I). Bialynicki-Birula has shown that a faithful action of either an ?z-dimensional or an (n - I)-dimensional torus on An may be linearized [Z;. 1r our language of Definition 1.4, this means that an algebraic torus subgroup of GA, of dimension R or r? - 1 may be conjugated to a subgroup of GL,, . This seems to be the best result so far known when n > 2. Con..ecture3.1. Let G be a linearly reductive algebraic group acting regularly on A”. Then, G has a fixed point, say P, and the action of G is iinear with respect to a suitzble coordinate system of A” having P as its origin. Supposing the existence of a fixed point, we may assume G to be an algebraic subgroup of GA, contained in GA, 0. Since G C Yd for some d as a closed set by Theorem 1.3 and Ya n GA,,” = {c} for all sufiiciently large 7: G n G.&’ =
448
T. KAhIE4AYASIil
When one infinite-dimensional algebraic group r acts on another such group A, we define H’J(r, A) and Hl(I’, A) in the usual way (cf. [17j, for instance), except that all the maps and cocycles involved are required to be morphisms of infinite-dimensional algebraic varieties. PROPOSITIOX 3.2.
The ttuth
of Conjecture
P(G,
G&l)
3.1 for G C GA,0 is epuiw&zt to
= (I),
wherethe action of G on GAt is dejimd by conjugation.
Proof. Assume first that H1(G, G&l) = {l}, and consider GL, 4 GA,0 through the natural inclusion which splits (6). So both G and L(G) are contained in GA,O. For each o E G, put c(u) := L(u) u-l. One verifies that c(uT) = L(a) L(7) 7-k-1 = L(u) u-1 - uL(7) 747-l = c(u) . “C(7). Therefore, c: G + GA,l is a I-cocycle. By assumption, there exists u E GA,1 such that c(u) = u-l * Du for all a E G. Then, L(u) u-1 = u%w+, hence L(u) = U-%U for all u E G, and L(G) = u-lGu as desired. Conversely, assume the conjecture for G C GA,O, and take any I-cocycle c: G + GA,l. Put G, := (c( u)u : u E G). One can see at once that a ++ c(u)” gives an isomorphism G N G, , and that L(G) = L(G,). Therefore, by assumption, G and G, are conjugate: There is u E GAf such that c(u)u = U%U for all u E G. It follow-s that c(u) = u-l . cu. Q.E.D. It is easy to see that GA,” is stable under the conjugation action by G for every Y large enough. Therefore, W(G, GA,l/GA,u) is well defined, and this cohomology set is trivial because GA,,l/GA,” is &potent. Let us remark also that the conjecture above is wide open for n > 2: One does not even know whether or not HI(G,, , GA,l) = (1) for G, = kX = onedimensional torus. Yet, an affirmative solution to this last question would have important consequences such as the resolution of the cancellation probleml: If X x A1 N As for a variety X, is then X N As ?
4. THE &SE OF THE AFFIX-E PLANS Let K be an arbitrary field and consider GA,(K) = Aut, K[X, Yj. Let A, I3 be subgroups of thii defined as follows:
A = (CY E GA,(h) : XCY= aX f bY + c, Ya = U’X + b’Y + c’), B=#IEGA,(K):X~=~X+~,Y~?=CY+P(X)}, 1 (Added March 8, 1979) This problem was recently solved in the affirmative by M. Miyanishi and T. Fujita. Cf. Fujita, Proc. Japan Acad. Ser. A. 55 (1979), 106-110.
NJTO3IORF’HISM
OF POLSXOMIAL
RISG
449
where a, b, c, a’, b’, c’ E k, and P(X) E K[XJ L et -cls ca:i A the afins subgroup and l3 the Jonquit%e subgroup. We know that G&(K) = A kc B with C = A n B, i.e., GA,(k) is a free product of 3 and B amalgamated by their intersection. This theorem is due in essence to van der Kulk [13] &though not in this form, is announced in this formulation and without proof by Shafarevich [!2], and is acmally proved in Kagata [lo, Theorem 3.3, p.31j and Kambayashi [g, Theorem 2, p. 4541 by someMrhat different methods. The rewit goes back ro Jung [7] w-ho showed that GA&) is generated by A and B. This free-product decomposition of GA,(k) is what sets the dimension 2 case apart from the higher-dimensional case where no satisfactory counterparts of such a decomposition are available as yet. For example, one knows that all separable k-forms of k[X, Y] are trivial (cf. [& 3]j. Also, one can answer Conjecture 3.1 in the aflirmative and in a much strcnger vssy-: as will be seen belovs (see Theorem 4.3). Let o E GAq(k), o $ C = A n B. Define the length of o, denoted i(u), to be the num!ber of factors when (I is expressed as a product of members of d - C and B - C appearing alternatingly. Put Z(T) = 0 if T E C. As always, for any u E GA,(k), the notation a = (a 1, u2) means (T: = Xu and ~a = Yz, both belonging to k[X, Y]. Again, the degreeof u means the larger of the total degrees of o1 and a, and is written as deg u. The following lemma is fcund in Wright’s thesis [14, 5.31, p. 651. LEMWA 4.1. Let u = (q , ua) E GA,(k). If c begins with an r E 3 - Cj i.e., if u = a;Ba’.** with a!, a’,... in A - Cared /3,... in B - C, then degq > dgo,.
If~be&switha~~B-C,i.e.,o=@@‘.**el.ith~,,T ,... itzB-Ca,zdc ,... +z A - C, then deg u1 < deg uy . In either case, deg u = (deg p) * (deg ,5’).~. ;j’ ejeenzents of B - C occur in the free-product ex-cression$or u; wd cieg (r = I zjf otherwise. = 1. I’rooJ Induction on Z(uj. The assertions are obvious for any c with I Assume, therefore, that Z(u) > 1 and thzt the iemma is vaZd for all automorphisms of smaller lengths. First, treat the case when G = c&‘.~. = 37 v-ith q- = pa’... , Z(rj < Z(u). We write Xa = aX + bY + c: YLI = a’X A 6’Y G c’ with b + 0. So, Xu = url 7 bre $ c, Yu = a’rl L Fr, + c’ ~4th 3eg rl < deg ~a . Eence deg u = deg u1 = deg ~a > deg a, and ieg LT= (deg jSj*-* by the induction hypothesis. r\‘ext, suppose (I = ,&+I’... = /I? xvith p = &I’.*. Write X/3 = aX i b, I[L3 = cY + P(X), where P(Xj E k:X] is of degree >!. Then, Xu = up, - 1, Yu = cp%+ P(pl), but deg p1 2 deg pr so that deg uI < Q.3.D. degG* = degpndegp, = degflmdegp. COROLLARY 4.2. If a set of elements of GA,(k) z’sbomzdedin degree, thm it is bounded in length (but not vice versa).
Let us return to geometry and to the assumption that k is algebraical& closed.
450
T. KAMFiAYASHI
Let G be an algebraic subgroup of GA, . By Corollary 4.2, there is a maximum in (Z(G) : u E G}, which we may call & length of G. The following theorem asserts that G is conjugate to a subgroup of the afline group A or of the Jonqui&e group B. It was originally proved by induction on the length of G. Here, however, we wiIl give an alternative proof observed by M. Pavaman Mm-thy, which shows the theorem to be a special case of one of our earlier results [8, Theorem 1, p. 4511. THEO-I 4.3. Let G be an algebraic group, not necessarily connected, acting on the aBne plane A2 as a group of automorphisms. Then, with respect to a suitable coordikate of a2, the action of G is either aflne or of the Jonquit!re type. Proof. Let G act on GA, trivial~, and consider H1(G, GA,). According to (8, Theorem 1, p. 4511, iF(G, GA,) is the coproduct of Hr(G, -4) and HI(G, B) over Hr(G, A n B). (In the cited paper G is assumed to be finite. But, as one can readily check, the proof works without change for any G as long as the values E GA, of the cocycles are bounded in length.) So, any 1-cocycle G + GA, is cohomologous to either a I-cocycle G + 9 or one G + B. In our situation, this reads exactly as follows: Any homomorphism G -+ G-4, whose image is bounded in length is conjugate to either a homomorphism G + A or one G + B of bounded length. ;“;ow let f: G -+ G-4, be the homomorphism associated to the given action of G on A2. Then, by Theorem 1.3 and Corollary 4.2, f(G) is bounded Q.E.D. in length, so the foregoing argument applies. COROLLARY 4.4. Let G be an a&ebraic group which is not a closed subgroup of any vector group GaN. Then, any action of G on the afine pknae A2 is a$‘ine with respect to suitable coordinates of A=.
Indeed, no such group G can be embedded in the Jonquiere subgroup B. (Sate that, if char(k) = 0, the assumption on G is equivalent to saying simply that G is not a vector group.) Theorem 4.3 includes as special cases all of the previous results on algebraic group action on the afline plane: Gutxirth [5], G, action; Bialynicki-Birula [2], torus action; Rentschler [l 11, G, action, char(k) = 0; Miyanishi [9], G, action, char(k) > 0; and Igarashi [6], finite group action. Also compare Wright [16] for Abelian group action on Aa.
1. H. BBS, E. H. COXKZLL, .tim D. WRIGHT, Locally polynomial algebras are symmetric algebras, invent. Math. 38 (1977), 279-299. 2. A. BIALWICKI-BIRULA, Remarks on the action of an algebraic torus on k”, I, Bull. Acad. Poh. Sci. SC+. Sci. Ahtk. Astronom. Pkys. 14 (1966), 177-181; II, 15 (1967), 123-125.
MJ’TOMORPHISM
OF POLYSOMIAL
RIKG
qj!
3. bI. BRY%~GI, Forms of the rings R[X] and R[X, Yj, Sk~sgcx Zzirrth. ,;- 13 (1972), 91-97. 4. hL. FRW,ES A&?) X. >hDER, Xutomorphisms of the poiyncmiai ring, J. dtgebra 25 (19733, 25-39. 5. A. GCTWIRTH, The action of an algebraic rorus on an afine plane, TWW. =Iww. MatI:. Sot. 105 (1962). 407-414. 6. T ic.;.aasm, “Finite Subgroups of the Xutomorphism Group of the ASine Plane,’ 31. A. thesis, Osaka University, 1977. 7. H. W. E. Just, cber ganze birationale Transformmatioten der Zene, J. Rhe Angen. Ma& l&4 (1942). 161-174. 8. T. KAN~AYASHI, On the absence of nontriviai separable forms of the affine piane, J. Algebra 35 (1975), 44H56. 9. Ai. M:SASISHI, G.-action of the affine plane, ATago:va ,“.a&. J. 41 (;97!), 97-iOO. 10. 31. ~AGAC.4, “On Automorphism Group of k[x, y], -r wmres in Mathematics, Ryoto LYniv.: \-cl. 5, Kinokuniya Book-Store Co.. Tokyo, 1972. :I. R. Rzs:sca~~~, OpCrations du groupe additif sur le plan ethne, C. I?. &ad. SC!. Parts Ser. r: 267 (1968). 384-387. 12. I. R. SH.WARE~-ICH, On some infinite d&nensional algebraic groups, in “.ItdSimposio Inrernaz. Geom. Algebrica, Roma, 1965,” Edizioni Cremonese, Rome, 1967; Rerrd. Ma:. e Appl. (5) 25 (1966), 208-212. 13. W. v.4~ nna Kcrx, On po!ynomial rings in two veriebles, AVAw A& ‘B’Zz. (3) 1 (I953), 33-41. 14. D. \TkIGHT, “Algebras w-hich Resemble Symmetric -%lgebra,” Ph.D. dissertation, Columbia University, 1975. IS. D. ITRIGHT, The amalgamated free product structure of GL? @LX, ,..., X,,]) anl the weak Jacobian Theorem for two variables, Bd. Amer. Xath. SW-. 82 (!3751, 724--726. 16. D. \~R:GHT, Abelian subgroups of amalgamated free products, and applicaticns tc actions on the aSine plane, Illinois J. Math., in press. in Lecture Sotes in Mathematics I%o. 5, 17. J.-P. SERRB, “Cohomologie Galoisienne,” Sprlnge>\ver!ag, Berlin;Heidelberg:Kew York, 1965.
of a Polynomiat Action on an Afine
Group
Ring and AIgebraic Space*
Group
The main cbject of study in this paper is the gro:up of au*-VP---‘-‘c :v ..JA~r,Lms of as z-variable polynomial ring over a field. We examine this group 2nd i:s algebraic, subgroups 5&h a view, mainly, toward berte; r understanding of algebraic group actions on on an afline z-space. Let G&(k) denote the k-algebra automorphism group of t!:e polynomial ring iz[T] = k[T, ,-.., T,j over a field k. A Srst natural index of an automorphism of k[T] is its degree relative to (T) = (Ti ,..., T,) as will be define2 belo~x;. I3y means of this and following Shafarevich [lzj, -3-e $tyo&nce 02 GL
principal part G&l(k). However, for the proof of the key part that the successive quotients of the filtration are vector groups, Fraser and Mader had to draw on the assumption that n = 2 and the characteristic of K is 0. Our generalization of their theorem (Theorem 2.2 below) is valid for any n and any characteristic as long as the field k is algebraically closed. A major question in algebraic geometry of the afhne space asks if a linearly reductive algebraic group action (or at any rate an algebraic torus action) on an aSine space may be linearized by correctly choosing a coordinate system in the atline space (see Conjecture 3.1). An a%rmative answer to this question, even in modest caseslike a one-dimensional torus G, acting on the a&e 3-space As, would be quite significant. While we are unable to answer the question except at the dimension 2 level, the results in Sections 1 and 2 of this paper provide a setting for a cohomological interpretation of the problem formulated as Proposition 3.2 below. As mentioned already, the case of the afBne plane is a special one due to the free-product decomposition of GA,(K). In this case we not only can answer the preceding question (Conjecture 3.1) in the affirmative, but also can essentially classify all possible algebraic group actions on the atfine plane As (see Theorem 4.3). The previous results dealing with various types of algebraic groups acting on Aa are then included in our result as special cases(cf. Gutwirth [5-J,BialynickiBirula [2], Rentschler [ll], Miyanishi [9], Wright [16J, and Igarashi [6]).
0. NOTATIOKS, COWEIKKIONS,ANI OTHER PRELIMIXARIES The letter K will abvays denote a fixed ground field of arbitrary characteristic. Unless otherwise stated, k is assumedto be algebraically closed, in which case we suppress the mention of k. Thus, for instance, GL, stands for the general linear group GL,(k) of rank n, and G-4, the fill algebra automorpkism group GA,(k) of W-1 ,.a., T,], both over the algebraically closed field k. Algebraic varieties and algebraic groups appearing inthis paper are all afhne, reduced, and defined over algebraically closed k, but are not necessarily irreducible. They are identified with their respective k-rational point sets. So, the a;8ine n-space An is identified with kn. The automorphism group Aut An of the algebraic variety An will be viewed the same as GA, . Each k-algebra endomorphism of k[T] = k[T, ,..., T,J will be represented by the image of the Tls. Thus
+ = (61,-*-,+,a):VI + WI denotes an endomorphism 4 such that T,# = & = +i( T) E k[ T] for all 1 < i < n. Tke degreeof + relative to (T) = (TI ,..., TJ is defined as the maximum of the total degrees of #r ,..., & .
AUTOMORPHISM
OF POLYSOML4~
RIYG
44:
Le: &V denote the set of all n-tuples of nmmegatice integers. For each r,-tuple (i) = (i(l),..., i(n)) EN denote by P the monomial T$‘i .*. Tic”) whose total degree \ve write as i i 1 = i(1) L . ..+i(nj.Let(u~).(ijE.N,l
=
fj(Tl
T,)
v---9
:=
Tj
_
x
I ; .,
uj”;T(‘!,
i-!
i:>1
a generic principal formal power series with coeffkients in R := k[z@: (i) 5 A-. i i > 1, 1 < j < n]. Let 31 denote the ideal generated by TI ,..., T,, ins& the form4 power series ring R[[T; ,..., Z’,]], and Z all integers. LEMhI.4
0.1.
There exists a family of polynomials in TI :..., T,; {HP’:1
sati$yiq
the following conditions for ail j and Cz:
(a) H!,@E,[z#: 1 < h < n, 1 < j i prime subriG of k (zZ or ~Z,lpZ!); [b)
The total degreedeg Hjd) Q d;
(c)
T, = Hj’)(f,
(d)
HJd’ = Hj:d-l’
,..., fJ
< d][T, ,..., T,$
zhere r is the
(mod Jid-i);
(mod Md). (Here, H:‘! is defined to be Ofor allj.)
Proof. We const,uct the Hjd)‘s inductively on d > 0. For d = 1, let Hj’) : = Tj for each 1 < j < n. Obviously conditions (a), (b), (c), and (d) are satisfied. Sow suppose Hi’),..., HAd! have been constructed for all d < e satisfying (a) through (d). Then, for each 1 $ j 6 nt T; G Hi’“-l’( ,1 f ,..., f,,) +
1 aIf’T(‘) ;1=e
(71 t-;
(mod W-l)
because of(c) with d = e - 1, where the a$‘s clear!y belong to zf@; ‘.:
1 < ! < n.
1, ajf’T!i! i =e
for each 1 < j < n. Conditions (a), (b), and (d) are obviousiy ftifilled. Hl”(f:
,.... fn) = Hje-l’(fl = H.F-“(f,
,...,f.)
A c ai!!fe’..fi;(l) ..a f ijcn’ il=r
,..., f,J -
E Tj (mod W-1)
As for (c),
x aj:‘T(” i!=e by (9).
(mod ~WdAi) QED.
T. KAMTHYASI-II
442
Let {z$) : (i) E X, 1 < j < n} be variables over K just as above, and consider generic power series without constant terms gj = g,(T, ),.., T,) :=
x up)T(i) ;I'>0
(3)
for 1 < j < n. Then, the rz-tuple (gr ,..., gn) defines an endomorpbism y of the formal power series ring R[[T]] g iven by Tf/ = gj for all j. For the same j’s let L, := xii~XI u:‘)T’~), and let D be the Jacobian matrix (I#): 1
For each 1 < j < n and each (i) EN with I i i # 0, there
exists a polynomial Pi’
E r[A-l,
&I:
where A = det D,
1 < h < n, 1e 1 < 1i i],
with the followitag properties: Let 4 = (& ,..., 4,J E GA,(h), and # = (I/~ ,..., #,,), its inverse. Assume that the 4,‘s are without constant terms (viz., +,(O) = 0 far all j) andwritefmeachl
dj = 1 $)T”’
ad
#* = c bi(“)T(‘).
ii’>0
Iipl
Then, f-w em-y j and ewry (i), by) = @)(a). The corollary is an easy consequence following that lemma.
of Lemma
0.1 and the observations
Remark 0.3. The essential contention of Corollary 0.2 is that the inverse of an origin-fixing automorphism 4 of a polynomial ring can be obtained through an a@rithm from the coefficients of 4. This is more than just saying that the coefficients of the inverse are polynomials in the coefficients of 4. -More importantly, Mitsuhiro Takeuchi has kindly pointed out to us that this result is no longer valid if 4 does not fix the origin, i.e., if 4,(O) + 0 for somej. His counterexample: For each natural number m, let y,,, = (TI - 1, T, - TIW) E GA,(h). Then, r;r = ( TI - 1, Tz - (TI L 1)“). Since the coefficient of TI in Tz + (TI T 1)” is m, it is absolutely impossible for thii coefficient to be expressed as a polynomial in the coefficients of TI - 1 and Tz - TIN, which are O’s, l’s, and (- l)‘s, and do not involve m.
ACTOMORPHISM i.
Following
OF POLYSOXIAL
GA,# BXD ITS ALGEBRUC
Shafarevich
w
R:SG
SUBGROUS
[12] a set union
of aigebraic \-arieties & (1 < d < CC) over an algebraically closed base feld k wili be called an infinite-dimensionaiolaal algebraic variety pro\-ided tht each -Xi . is a closed subvarrety of X,-, . A mapping j? X = l&q Xi + Y = iim I-; of infinite-dimensional wrieties is a motphism, by definition, if the restriction of-.? to each Xi is a morphism of ordinary algebraic varieties to Y; for some j determinable from i. From now on we will not distinguish between isomorphic infinite-dimensional algebraic varieties. Such a \-ariety X is said 10 be @rre, smooth, or irreducible, respectkely, if X is isomorphic to 3.7’ = lim Xi with a!1 Xi affine, smooth, or irreducible, respectively. Xn infinite-dimensional algebraic group is defined to be a group object in the categorv of infinite-dimensional algebraic varieties. Thlq if G = !&I Gi is such and G + G (inverse) a group; the group mappings G x G + G (multiplication) induce morphisms of algebraic I-arieties Gi x G, + Gr(r,j)
and
G -
for ail indices 4, j, and h, and for suitably corresponding For each positive integer d, put
Gsm r = r(i, j) and s = :(A).
where “deg” means the degree of an endomorphism of k[T] variable set (T) = (T, ,..., TJ as defined in Section 0.
relative to 2 fixed
THEOREM 1.1. The set Y, has a natural structure of an afine aigebraic wriety and Yd is a closed sukariet_y of YdB1for mery d > 0.
Proof. Let I$ = (& ,..., &), # = (4 r ,..., #,)) be w-0 n-tuples in k[Tj, each of degree
of poiynomia;s
Then, for qbto be an element of Yd and 4 be the inverse of i, it is necessary and sufficient that
T. KQNIBAYeASHI
444
for every 1
GA, = U Ya = l& Ya is an inJinite-dimezsional algebraic
PUP. Proof.
If a E Yd and r E Ye , then UT E Ya, and u-l E Ya , 7-l E Y, , clearly. Q.E.D.
Remark. A. Bialynicki-Birula has shown, in a personal communication of 1973 to us, that the group of automorphisms of any aRine algebraic varieq may be given a structure of an infinite-dimensional algebraic group. The proof is a straight generealization of that of Theorem 1.1 above. Now let G be an algebraic group acting on the affine n-space An. By definition, this means the datum of a morphism u: G x An -+ An such that u(grgs, a) = u(g, , u(gs , a)) and u(e, a) = a for anyg, , g, E G, any a E A”, and e the neutral point on G. Given this, there arises a natural group homomorphism G + G& . THEOREM 1.3. Let G be an algebraic group acting on An, and let 01:G + GA, = u Yd be the associatedgroup homomorphism. Then, there exists some d suck that the image a(G) of (Yis contained in Ye as a closedsubset,and 01:G + Ye is a morphism of algebraic varieties.
Proof. Let F(G) = k[x, ,..., xm] and I’(A”) = k[ Tl ,..., T,J be the respective af?ine coordinate rings of G and An. Then, a k-algebra homomorphism T,Xk[T] + k[x] 8 k[T] E (k[x])[T, ,..., T,] corresponds to u. Write T,q = P<(x, T) E k[x, T] for 1 < i Q n. Then, each g E G is regarded as a homomorphism r(G) + k, and if (.vr ,..., x,) + ([r ,..., [,,) under this mapping g then the automorphism of k[T] corresponding to g is given by Ti - P,(& T) for all 1 < i < n. Thus, the total degree in the Ti’s of a(g) = (PI@, T),..., P&, T)) for any g E G never exceeds the maximum total degree of the Pi(x, T)‘s, say d. Hence or(G) C Y, . It is obvious from the preceding analysis that CCG + Ya is a morphism. To show the closedness of a(G), let z: Yd x Ya -+ Ye (for some e $ d’) and w: Ya + Ya be restrictions of the group multiplication and the inverse map, respectively, of GA, . Then er maps or(G) x or(G) into oL(G) and u maps z(G) into itself. Therefore, the closure G” of O(G) in Ya (and also in YJ is stable under the group maps o and w. G’ is consequently an algebraic group. In the homomorphism CX:G + G’ of algebraic groups the image has to be a closed subgroup, hence or(G) = G’. Q.E.D. The theorem above motivates
the following
A~TO&IORPHIshI
OF POLYSO1IIAL
RIXG
445
DEFINITIOS 1.4. An algebraic subgroup of G-4, = IJ I:1 is an abstract subgroup of GrZ, which is also a closed subset of Ya for some d. Thus, the sttidy of algebraic subgroups of GA,. is equivalent to the stx$- cf a!gebraic group actions on the afhne n-space An.
2. A DESCEKDING
FILTRATIOS
OS
Again, k is a Fixed algebraically closed field. Let us define various subgroups of G/f, = G&(k) group of k[T, ,..., T,,] = Aut, A”.
G-4, = the R-auromor?hism
(i) G,” = k+n, consisting of 4 E GA4, with & = Tid = Ti -- ai for some a, E k and for each i. These are translations. (ii) G&O = (4 : Ti+ E ..l;I for all 1 < i < ~3, w-here &I denotes th nzaximal ideal in k[T] generated by the Ti’S. These a=ethe origin- (augmerztuticvz-1 Jixing autwnorphisms. (iii) GL, = GL,(k) = th e automorphisms arising from linear substitutions of the Ti’s (called lirzear automorphisms). (iv) For each positive integer Y, put GA,,:’ = (4 E G_i(,i : & = TzS;,:= Ti (mod JP1) for all 1 < i < n). An automorphism belonging to G-&Y v!ll be ca!led v-principal if v > 1, principal if I, = 1. LESINA 2.1. Let v be a positice integer. For + = ($1 i...I +,J onL z) = & ,.=., &J E G..4,O,+,F E GAnp if und ~n/y q+I = I$; (-mod W+l) J-N c/l I
Proof. Suppose + = w/ with 01E GA,,!‘. Then, 3; = (T’r 1 PI(T) ,..., T, -.P,(T)) with ali P,(T) E X”fl. For any i, +i = Ti+ = Tide = (Ti 7- P,(T))! == z& $ P,(T$) = & (mod JP+l). Converse!y, assume& = & (mod W-1) for a3 1’. Write C&E tji s Ti + Qi( T) (mcd W+r) and +-I = (pr ,..., pR),C-l = (7; :.... :.;,). Then: by Corolla? 0.2, p, = 7i (mod MV+l), which we write as ~2-i - &(Tj. Nom-, 0 = T,+,IW - Tt = (Ts I Qi(T))+-i - Ti ES Ri(Tj 7- Q$‘- - R(T)) (mod 3P1) obviously. But TZ&V = (Ti - Qi(T)) #-’ = Ti - Ri( 1”) + Q.E.D. QI( T i R(T)) SE Ti (mod .JP1). Therefore, +$--~ E G&v. THESEN 2.2. (a) G/f, = G,* - GA,0 = GA4,O-G,” {the identity automorphism E>.
and G,” n GA,0 --
(b) With respect to the inclusion map GA,’ G G,sl,,* arzd the map GA,,” -* GL, sending 4 E GA,O to its linear part (viz., itc I . JTac&m m&ix at Tl = . .. == T, = 0), the sequence l+GA,l+GA,o+GL,+l is exact, and is split by the natural inckuiorz map GL, G GA,,3.
446
T. KAYBAYASHI
(c) There is a chain of normal subgroups of GA,O : GA,’ 3 GA,’ 3 mm3 GA:‘3 GA,“3 m-nin which n,“=, GA,” = {E}. (d) For any v < ,u the factor group GA,,“/GA,p has a natural structure of a unipotent algebraic group, and GA,v/GALT1 is isomorphic to a oector group GE(‘) = k!$) of a certain dimension N(v). Proof. (a) It is clear that each # E GA, is written uniquely as a translation followed by an origin-fixing automorphism. (Sote, however, that neither God nor G-4,0 is normal in GA, .) (b)
Obvious.
(c) The normality of GA, in GA,O stems from the fact that each+ E GA,O induces a k-module automorphism +o) of ( Tl ,. .., T,#( Tl ,..., T,)“+’ in a natural fashion and G&v is the kernel of the reduction homomorphism 4 + #“J. The rest of the assertions in part (c) are clearly true. (d) Xe shah first show that GA’,-‘/GAn’ has a structure of a finitedimensional vector space over k. Let + = ($r ,..., +,J E GA;;‘. For each i one may write cJ~= Ti L C #‘T(j) (mod JP1) where the sum is taken over all = Y. When $ = (#r ,..., &J E GA’,- is (~)EN with ij 1 =j(l) + --a ‘j(n) similarly expressed as y$ = Ti + C vt’)T(n, one can see by Lemma 2.1 that + = # (mod G&u) if and only if (I#)) = (@) as ordered tuples on (j) and for every i. So, (~1~))may, and will, be viewed as coordinates of [+] E GA~M1/GA,v. Moreover, one can readily check the identity
for all 1 < i < n. Therefore, the mapping + ti (UP)) gives a one-to-one group homomorphism of GAk-‘,/GA,’ into the additive group of kv(“) for a certain N(V). Our proof will be complete if the image under this homomorphism is actually a k-vector subspace of k-v(U).To see that, we avail ourselves of “scalar operations on group functors” developed by Bass et al. [ 1, Sect. 23. As 4 preserves the filtration {My; 0 < Y < Loo} on k[T, ,..., T,], the scalar multiple a+ is well defined and automorphic on k[T, ,..., T,] for any s E k; if 4 H (z@), then %j ++ (s’4rj’)) according to [l, p. 2873. Now let c E k be given, and pick s E k with SY-~= c. Then, c(u!‘)) = (CUE’))= (sV-%#)) is the image of [“+I E GAi-l/ GAny, and we thus conclude that GA’,$/GAnv as embedded in 79(“) is closed under scalar multiplication. The fact that GA,y/GA,u (V < p) is naturally endowed with an algebraic group structure is easily shown by induction on ,o - Y. One proves that [#I E GA,u/GA,u is naturally coordinatized by the coefficients of the terms of total degree between Y + 1 and CL;the group operation is clearly polynomial on these Q.E.D. coefficients. Further details will be omitted.
IL‘TOMORPHISY
OF POLYKOMAL
RIKG
447
Rem& 2.3. The preceding theorem constitutes a generalization of a resuh of Fraser and Mader [4, Theorem 2.11, p. 28-J.Their proof of an equivalent cf part (d) abo\-e is by means of dimension counting and does not appear tc be generalizab!e beyond the case of n = 2, char(k) = 0 that they treat. On the other hand, note that the closure of k under extraction of radicals is of the essence in our proof abox-e. We record here a group-theoretical fact without proof, as it map be veriiied by elementary calculations. PROPOSTIOX
2.4.
The commutator [G&l,
3. -1 CONJECTURE ABOUT iiLGEBR4IC
GA’:;lj is contained in GA;.
GROW
&X10X
OS bin
One of the ultimate goals of these investigations is to determine and class@; all algebraic group actions on the afine n-space An. In the first nontrividi case of n = 2, there ha\-e been manr contributions bv \-arious authors. In Section 4 below, we ;d discuss and conclude this case.-When the dimension A > 2: how-ever, very little is known about such actions. One knows, thanks to Shafarecich, that any algebraic torus or any finite group whose order is a power of prime + char(k) acting on An must ha\-e a fixed point (cf. i2, I, Theorem 2I). Bialynicki-Birula has shown that a faithful action of either an ?z-dimensional or an (n - I)-dimensional torus on An may be linearized [Z;. 1r our language of Definition 1.4, this means that an algebraic torus subgroup of GA, of dimension R or r? - 1 may be conjugated to a subgroup of GL,, . This seems to be the best result so far known when n > 2. Con..ecture3.1. Let G be a linearly reductive algebraic group acting regularly on A”. Then, G has a fixed point, say P, and the action of G is iinear with respect to a suitzble coordinate system of A” having P as its origin. Supposing the existence of a fixed point, we may assume G to be an algebraic subgroup of GA, contained in GA, 0. Since G C Yd for some d as a closed set by Theorem 1.3 and Ya n GA,,” = {c} for all sufiiciently large 7: G n G.&’ =
448
T. KAhIE4AYASIil
When one infinite-dimensional algebraic group r acts on another such group A, we define H’J(r, A) and Hl(I’, A) in the usual way (cf. [17j, for instance), except that all the maps and cocycles involved are required to be morphisms of infinite-dimensional algebraic varieties. PROPOSITIOX 3.2.
The ttuth
of Conjecture
P(G,
G&l)
3.1 for G C GA,0 is epuiw&zt to
= (I),
wherethe action of G on GAt is dejimd by conjugation.
Proof. Assume first that H1(G, G&l) = {l}, and consider GL, 4 GA,0 through the natural inclusion which splits (6). So both G and L(G) are contained in GA,O. For each o E G, put c(u) := L(u) u-l. One verifies that c(uT) = L(a) L(7) 7-k-1 = L(u) u-1 - uL(7) 747-l = c(u) . “C(7). Therefore, c: G + GA,l is a I-cocycle. By assumption, there exists u E GA,1 such that c(u) = u-l * Du for all a E G. Then, L(u) u-1 = u%w+, hence L(u) = U-%U for all u E G, and L(G) = u-lGu as desired. Conversely, assume the conjecture for G C GA,O, and take any I-cocycle c: G + GA,l. Put G, := (c( u)u : u E G). One can see at once that a ++ c(u)” gives an isomorphism G N G, , and that L(G) = L(G,). Therefore, by assumption, G and G, are conjugate: There is u E GAf such that c(u)u = U%U for all u E G. It follow-s that c(u) = u-l . cu. Q.E.D. It is easy to see that GA,” is stable under the conjugation action by G for every Y large enough. Therefore, W(G, GA,l/GA,u) is well defined, and this cohomology set is trivial because GA,,l/GA,” is &potent. Let us remark also that the conjecture above is wide open for n > 2: One does not even know whether or not HI(G,, , GA,l) = (1) for G, = kX = onedimensional torus. Yet, an affirmative solution to this last question would have important consequences such as the resolution of the cancellation probleml: If X x A1 N As for a variety X, is then X N As ?
4. THE &SE OF THE AFFIX-E PLANS Let K be an arbitrary field and consider GA,(K) = Aut, K[X, Yj. Let A, I3 be subgroups of thii defined as follows:
A = (CY E GA,(h) : XCY= aX f bY + c, Ya = U’X + b’Y + c’), B=#IEGA,(K):X~=~X+~,Y~?=CY+P(X)}, 1 (Added March 8, 1979) This problem was recently solved in the affirmative by M. Miyanishi and T. Fujita. Cf. Fujita, Proc. Japan Acad. Ser. A. 55 (1979), 106-110.
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where a, b, c, a’, b’, c’ E k, and P(X) E K[XJ L et -cls ca:i A the afins subgroup and l3 the Jonquit%e subgroup. We know that G&(K) = A kc B with C = A n B, i.e., GA,(k) is a free product of 3 and B amalgamated by their intersection. This theorem is due in essence to van der Kulk [13] &though not in this form, is announced in this formulation and without proof by Shafarevich [!2], and is acmally proved in Kagata [lo, Theorem 3.3, p.31j and Kambayashi [g, Theorem 2, p. 4541 by someMrhat different methods. The rewit goes back ro Jung [7] w-ho showed that GA&) is generated by A and B. This free-product decomposition of GA,(k) is what sets the dimension 2 case apart from the higher-dimensional case where no satisfactory counterparts of such a decomposition are available as yet. For example, one knows that all separable k-forms of k[X, Y] are trivial (cf. [& 3]j. Also, one can answer Conjecture 3.1 in the aflirmative and in a much strcnger vssy-: as will be seen belovs (see Theorem 4.3). Let o E GAq(k), o $ C = A n B. Define the length of o, denoted i(u), to be the num!ber of factors when (I is expressed as a product of members of d - C and B - C appearing alternatingly. Put Z(T) = 0 if T E C. As always, for any u E GA,(k), the notation a = (a 1, u2) means (T: = Xu and ~a = Yz, both belonging to k[X, Y]. Again, the degreeof u means the larger of the total degrees of o1 and a, and is written as deg u. The following lemma is fcund in Wright’s thesis [14, 5.31, p. 651. LEMWA 4.1. Let u = (q , ua) E GA,(k). If c begins with an r E 3 - Cj i.e., if u = a;Ba’.** with a!, a’,... in A - Cared /3,... in B - C, then degq > dgo,.
If~be&switha~~B-C,i.e.,o=@@‘.**el.ith~,,T ,... itzB-Ca,zdc ,... +z A - C, then deg u1 < deg uy . In either case, deg u = (deg p) * (deg ,5’).~. ;j’ ejeenzents of B - C occur in the free-product ex-cression$or u; wd cieg (r = I zjf otherwise. = 1. I’rooJ Induction on Z(uj. The assertions are obvious for any c with I Assume, therefore, that Z(u) > 1 and thzt the iemma is vaZd for all automorphisms of smaller lengths. First, treat the case when G = c&‘.~. = 37 v-ith q- = pa’... , Z(rj < Z(u). We write Xa = aX + bY + c: YLI = a’X A 6’Y G c’ with b + 0. So, Xu = url 7 bre $ c, Yu = a’rl L Fr, + c’ ~4th 3eg rl < deg ~a . Eence deg u = deg u1 = deg ~a > deg a, and ieg LT= (deg jSj*-* by the induction hypothesis. r\‘ext, suppose (I = ,&+I’... = /I? xvith p = &I’.*. Write X/3 = aX i b, I[L3 = cY + P(X), where P(Xj E k:X] is of degree >!. Then, Xu = up, - 1, Yu = cp%+ P(pl), but deg p1 2 deg pr so that deg uI < Q.3.D. degG* = degpndegp, = degflmdegp. COROLLARY 4.2. If a set of elements of GA,(k) z’sbomzdedin degree, thm it is bounded in length (but not vice versa).
Let us return to geometry and to the assumption that k is algebraical& closed.
450
T. KAMFiAYASHI
Let G be an algebraic subgroup of GA, . By Corollary 4.2, there is a maximum in (Z(G) : u E G}, which we may call & length of G. The following theorem asserts that G is conjugate to a subgroup of the afline group A or of the Jonqui&e group B. It was originally proved by induction on the length of G. Here, however, we wiIl give an alternative proof observed by M. Pavaman Mm-thy, which shows the theorem to be a special case of one of our earlier results [8, Theorem 1, p. 4511. THEO-I 4.3. Let G be an algebraic group, not necessarily connected, acting on the aBne plane A2 as a group of automorphisms. Then, with respect to a suitable coordikate of a2, the action of G is either aflne or of the Jonquit!re type. Proof. Let G act on GA, trivial~, and consider H1(G, GA,). According to (8, Theorem 1, p. 4511, iF(G, GA,) is the coproduct of Hr(G, -4) and HI(G, B) over Hr(G, A n B). (In the cited paper G is assumed to be finite. But, as one can readily check, the proof works without change for any G as long as the values E GA, of the cocycles are bounded in length.) So, any 1-cocycle G + GA, is cohomologous to either a I-cocycle G + 9 or one G + B. In our situation, this reads exactly as follows: Any homomorphism G -+ G-4, whose image is bounded in length is conjugate to either a homomorphism G + A or one G + B of bounded length. ;“;ow let f: G -+ G-4, be the homomorphism associated to the given action of G on A2. Then, by Theorem 1.3 and Corollary 4.2, f(G) is bounded Q.E.D. in length, so the foregoing argument applies. COROLLARY 4.4. Let G be an a&ebraic group which is not a closed subgroup of any vector group GaN. Then, any action of G on the afine pknae A2 is a$‘ine with respect to suitable coordinates of A=.
Indeed, no such group G can be embedded in the Jonquiere subgroup B. (Sate that, if char(k) = 0, the assumption on G is equivalent to saying simply that G is not a vector group.) Theorem 4.3 includes as special cases all of the previous results on algebraic group action on the afline plane: Gutxirth [5], G, action; Bialynicki-Birula [2], torus action; Rentschler [l 11, G, action, char(k) = 0; Miyanishi [9], G, action, char(k) > 0; and Igarashi [6], finite group action. Also compare Wright [16] for Abelian group action on Aa.
1. H. BBS, E. H. COXKZLL, .tim D. WRIGHT, Locally polynomial algebras are symmetric algebras, invent. Math. 38 (1977), 279-299. 2. A. BIALWICKI-BIRULA, Remarks on the action of an algebraic torus on k”, I, Bull. Acad. Poh. Sci. SC+. Sci. Ahtk. Astronom. Pkys. 14 (1966), 177-181; II, 15 (1967), 123-125.
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3. bI. BRY%~GI, Forms of the rings R[X] and R[X, Yj, Sk~sgcx Zzirrth. ,;- 13 (1972), 91-97. 4. hL. FRW,ES A&?) X. >hDER, Xutomorphisms of the poiyncmiai ring, J. dtgebra 25 (19733, 25-39. 5. A. GCTWIRTH, The action of an algebraic rorus on an afine plane, TWW. =Iww. MatI:. Sot. 105 (1962). 407-414. 6. T ic.;.aasm, “Finite Subgroups of the Xutomorphism Group of the ASine Plane,’ 31. A. thesis, Osaka University, 1977. 7. H. W. E. Just, cber ganze birationale Transformmatioten der Zene, J. Rhe Angen. Ma& l&4 (1942). 161-174. 8. T. KAN~AYASHI, On the absence of nontriviai separable forms of the affine piane, J. Algebra 35 (1975), 44H56. 9. Ai. M:SASISHI, G.-action of the affine plane, ATago:va ,“.a&. J. 41 (;97!), 97-iOO. 10. 31. ~AGAC.4, “On Automorphism Group of k[x, y], -r wmres in Mathematics, Ryoto LYniv.: \-cl. 5, Kinokuniya Book-Store Co.. Tokyo, 1972. :I. R. Rzs:sca~~~, OpCrations du groupe additif sur le plan ethne, C. I?. &ad. SC!. Parts Ser. r: 267 (1968). 384-387. 12. I. R. SH.WARE~-ICH, On some infinite d&nensional algebraic groups, in “.ItdSimposio Inrernaz. Geom. Algebrica, Roma, 1965,” Edizioni Cremonese, Rome, 1967; Rerrd. Ma:. e Appl. (5) 25 (1966), 208-212. 13. W. v.4~ nna Kcrx, On po!ynomial rings in two veriebles, AVAw A& ‘B’Zz. (3) 1 (I953), 33-41. 14. D. \TkIGHT, “Algebras w-hich Resemble Symmetric -%lgebra,” Ph.D. dissertation, Columbia University, 1975. IS. D. ITRIGHT, The amalgamated free product structure of GL? @LX, ,..., X,,]) anl the weak Jacobian Theorem for two variables, Bd. Amer. Xath. SW-. 82 (!3751, 724--726. 16. D. \~R:GHT, Abelian subgroups of amalgamated free products, and applicaticns tc actions on the aSine plane, Illinois J. Math., in press. in Lecture Sotes in Mathematics I%o. 5, 17. J.-P. SERRB, “Cohomologie Galoisienne,” Sprlnge>\ver!ag, Berlin;Heidelberg:Kew York, 1965.