automorphisms of the polynomial ring in two variables (dicks)

8/12/2019 Automorphisms of the Polynomial Ring in Two Variables (Dicks)

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Pub Mat UABVol 27 Ne

AUTOMORPHSMS OF THE POLYNOMAL RING I N TWOVARIABLESWarren Dcks

Let k be a fi el d, k[x, y] the pol ynomal ri ng i n two vari abl es, andAut k[x, y] the group of al l i ts k-al gebra automorphi sm Such an auto-morph smwl l be denoted by the ordered pai r (p,q) where p,q E k[x, y] arethe respecti ve images of x,y THEOREM The group Aut k[x, y] i s generated by (y,x), (x, y- uxn u E k n 0Moreover Aut k[x, y] = A UB where

A = {(X11x+12y+1 X21x+a22y+2) IX11x22 ~21>~12},B = {(a11x+Xl , a22y+(x)) I Xl l a22 Of(x) E k[x]}0 =AnB = { X11x+a1 , 21 x+a22ya2 I a 11 a 22~0}

The el ements of Aare cal l ed aff i ne automorphi sm, the el ements of Bde J onqui l res automorphi sm, and the el ements of the subgroup generated byAUBare cal l ed tame automorphi sm The fact that al l k-al gebra automorphi sm of k[x,y] are tame was proved by J ung [2] f or char k = 0, and thenby Van der Kul k [8] i n the general case Fromthei r work the coproductdecomposi ti on fol l ows fai rl y easi l y, but i t i s not clear who fi rst made theobservati on (Kambayashi [3] gi ves the credi t to Shafarevi tch [7] .

Rentschl er [5] gave a very simpl e proof of tameness for char k = 0, andthen al ong sl i ghtl y di f ferent l i nes Makar-L manov [4] gave a fai rl y simpl eproof for arbi trary character sti c (News of Van der Kul k s resul t seem notto have reached Moscow at that time, for Makar-Li manov refers to the resul t as

Semnar gi ven at Uni versi tat Autnoma de Barcel ona, J ul y 1981 155


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unpubl i shed work of Shafarevi tch I n Che spi ri t of Serre [6l , RogerAl peri n El ] gave an expl i ci t exampl e of a tree acted on by Aut k[x, yl fromwhi ch Che coproduct decomposi ti on can be read off

I n l bel owwe gi ve a modi f i ed versi on of Makar-Li manov s proof, andi n recal l A peri n s exampl e

I amvery grateful t o P . M Cohn f or provi d ng me wth hi s translati onof Makar-L manov s thes s

l The support of a pri mti ve el ementLet f,g) be an automorphismof k[x,yl I de can wite f =I x i j xl yj

ai j e k and defi ne supp f) _ {x yl I a i j 0} _ where i s Che free

abel an group generated by x, y Let m= x-deg f), n = y-deg f), that i s,mi s Che hi ghest exponent of x occurr ng i n supp f), and si ml arl y f or nSet o = {xl y l ni m < mn, i ? 0, j 3 0} S Geometri cal l y, supp f) l i esi n Che rectangl e determned by l , xmxmn, yn and A occupi es Che tri angl edetermned by l , x~yn

The obj ecti ve of Chi s secti on i s to show xmyn e supp f)and mn ornl m

I f mn = 0 Chi s i s cl ear Thus we may assume mn >O Let m = m mn), n = n/ mn) These are

copri me natural numbers, so we can choose natural numbers s, t such thatvsm-tn = 1 Let u = xm/ yn v = ys / xt i n so x = usvn y = ut vm

156

ys / xt =vn

m nx/y


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Thus k[x, y] c k[u, v] and we can wite =j u1v so supp f) _ {ul v]I pi j 0} We def i ne the l eadi ng v-component of f to be I f l = ~u . U )V e k[u] > x i 1Jwhere j = v-deg f ) I f then u-deg I f l ) = i we defi ne I f l l = ul v E cal l edthe l eadi ng termof f Thi s extends to a group homomorphi sm1 1 I I : k u, v) x - > Noti ce the superscr pt x i s bei ng used to denote theset of nonzero el ements . The fol l owng statement i nd cates the steps i nMakar-Li manov s argument TREOREM 1 There exi st a, s E k u) x x c k u, v)x such that f = aaaak x a EIN and x, y e k[a+, o] i i ) There then exi st wz E such that _ or and+x,y e semgp

i i ) Then xmyn E supp f ) c p and f = xmand = v) I f = Kx x extendi ng the correspond ng maps on k[u, v] Wevi ewk x as a subgroup of Kx x cK v1 x Si nce v-deg f ) > 0 thereexi sts a E Kx x such that the mage of a i n Kx x /k x generates a

xmaxi mal cycl i c subgroup contai ni ng the mage of I f l , say I f = aaa a E k a c ]N . By nduct on on a we shal l show that for any f,g e K v-1)) wthI f l = aaakx a E I N there exi sts R E Kx x such thatI k [ f ~g]I c k[a, o] The case a = 0 i s vacuous

Let us nowdef i ne a possi bl y f i ni te) sequence i nducti vel y Letan9, = g Suppose we have g for some i 1 I f I g i l = a l a f or somenia E k , ni E 7i we set g +1 = gi -x f ; f g = 0 or gi 0 and I g i l i s not


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of th i s formwe l et the sequence end att he sequence g1 92 , has a l imt g i n K v- 1 ,

I f g = 0 then k[f,g] c k f - l so I k[f 1c k[Ifl tl ] c1] and we can take R arbi trary .

Thus we may assume g 0 so the sequence i s f i ni te andk[f g] l ~ k[f . g ]

I f I f 1 , I g * are al gebrai cal l y i ndependent over k then i t i s easy t osee I k [ f , g ] x c k[I fI1, I g*I ] and we can take S = I g*I .

Th s l eaves the case where I f j , I g * I are al gebrai cal l y dependent over k .I f c =v-deg f , d =v-deg g t hen I f i d, I g I , are al gebrai cal l y dependentover k and are v- homogeneous wth the same v-degree I t fol l ows thatI f l d/ i g*I c l i e s i n K and i s al gebrai c over k so l i e s i n k . ThusI g*I c - I f l d - aad mod kx . But Kx x /kx i s a torsi on- free abel i angroup, and the mage of a generates a maxi mal cycl i c subgroup, so ci ad andI g*I = ab mod kx where b = ad/ c Saylg1= pab , p e kx . By the defi ni ti onof g we know al b, say b = aq+r 0 and we can takew= a l , z =11011 . Thi s l eaves t h e , case where I I a I I , I I S I I a r e dependent . Letwbe a generator of 11 . 11, 1101 , say 11-11 =w . 11611 =w , w= I I a i r i l 0 I h Here

Si nce v-deg g1 > v-deg g2 >ng* =g-a l f1- a 2 f n2

g] xi I k f - 1 x


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I l a ~ i ~ I I Pl I l =wj so there i s a un que V e kx such that z =I l a - PPi j j w jBut z and wj have t he same v-degree so wz are i ndependent . Let a = acPd P = aj / P 1Then I I k [ a+ , P +17 x 1 l l k [ a + . P +u) + ] xl l c I i k [ a . P ] x I l


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PROOF . Let f , g be the i nverse of p, q and l et f be as i n l l ideg pm deg gn then deg f p, q = mx{deg pm, deg gn } . But f p, q = x sop or q i s a pol ynomal i n x of degree 1 and the desi red conclusi on fol l owseasi l y Thi s l eaves the case where deg pm = deg gn Here m> n so nl manddeg pr = deg q for r = m We my assum p, q i s not af f i ne so deg q > 1 nSi nce f p, q = x i t fol l ows that p0, g0 are al gebrai cal l y dependent over k Hence q0/ pr o i s al gebrai c over k so l i es i n k, say q00= u . Thendeg q- upr < deg q as dsi red

By i nducti on on deg q i t f ol l ows easi l y f romTheorem2 that al lk-al gebra automrphi sm of k[x, y] are tam I t i s even a si ml e mtter toobtai n the decomosi ti onTHEOREM3 Aut k[x, y] = A*CB PROOF Let P be the ori ented graph whose verti ces are the k-subspaces ofk[x, y] and whose edges are the i ncl usi on mps Then Aut k[x, y] acts i n anatural _ way on r_he .r phP_ Let T be the orbi t of k+kx 3 1 1 1 , . L - - el ai mthat T i s a tree .

Any vertex of T i s of the formk+kp or k+kp+kq where p, q s somautomrph sm We def i ne deg k+kp = deg p and deg k+kp+kq _mx{deg p , deg q } - } . I t i s easy to see these are wel l -def i ned .

Consi der a vertex of the formk+kp . We can f i nd an automrph smp, q wth deg q mni ml , so deg q < deg p or p, q i s af f i ne . A l thenei ghbours of k+kp are of the f ormk+kp+k q+h where h e k[p] . The onl ynei ghbour of k+kp wth sml l er degree i s k+kp+kq al l the others havegreater degree

Consi der a vertex of the formk+kp+kq where deg q < deg p Thenei ghbours are of the f ormk+k ap+Sq where a,R e kX are not both zero ; onl yk+kq has sml l er degree, al l the others have greater degree


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Fi nal l y, the vertex k kx ky has smal l er degree than al l i ts nei ghbours Thus every path fromk kx ky i s stri ctl y i ncreasi ng so T has no

ci rcui ts and f romeach vertex there i s a str ctl y decreasing path whi chmust necessari l y arri ve at k kx ky so T i s connected Hence T i s a tree

Nowk kx k kx ky i s a transversal i n T for the acti on of Aut k[x, y]and the stabi l i zer of k kx s B whi l e the stabi l i ze_ of k kx ky s A Tb simpl i es G = A*c B c C6]


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REFERENCES R. C ALPER N Homol ogy of the group of automorph sm of k[x, y], J Pure

and Appl Al gebra 15 1979) 109-1152 H W E J UNG Uber ganze bi rati onal e transformati onen der Ebene, J rei ne

angew Math 184 1942) 161-174 3 T KAMBAYASH , On the absence of nontri vi al separabl e form on the aff i ne

pl ane, J Al gebra 35 1975) 449-456 4 L G MAKAR-LI MANOV, On automorphi sm of certai n al gebras Russi an),

Ph D Thes s, Moscow 1970 5 R RENTSCHLER, Oprati ons du groupe addi ti f sur l e pl an aff ne, C. R

Acad Sc Pari s, Ser A 267 1968) 384-387 6 J . -P SERR, Arbres, amal games et SL Astr sque No. 46, Soci t Math de France, 1977

7 I . R SHAFAREVI TCH On some i nf n te di mensi onal groups, pp. 208-212,Att -Si mpos o I ntecnaz di Geom Al g Roma, 1965

8 W VAN DER KULK, On pol ynomal ri ngs i n two vari abl es, N euwArchi efvoor Wsk 3 I 1953) 33-41

Rebut el 15j ul i ol 1982Revi sat el 20 abri l 1982Bedford Col l egeRegent s ParkLondon, NW 4NS

automorphisms of the polynomial ring in two variables (dicks)

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