JOURNAL OF 187, 474᎐492Ž. 1997 ARTICLE NO. JA966809

Composition over a of Fractions

S. Pumplun¨

Fakultat¨¨ fur Mathematik, Uni¨ersitat¨ Regensburg, Uni¨ersitatsstrasse¨ 31, 93040 Regensburg, Germany

View metadata, citation and similar papersCommunicated at core.ac.uk by Georgia Benkart brought to you by CORE

Received December 5, 1995 provided by Elsevier - Publisher Connector

Using some theorems on composition algebras over rings of genus zero and elementary results from algebraic geometry, all composition algebras over a ring of fractions related to the projective line over a are enumerated and partly classified. ᮊ 1997AcademicPress

INTRODUCTION

Composition algebras over rings have appeared in a of articles in recent years. These papers established general properties of composi- tion algebras over arbitraryŽ. commutative, associative, and unital rings ŽMcCrimmonwx M , Petersson w P1 x , and Knus, Parimala, and Sridharan wxKPS. , and some of them were also devoted to the investigation of composition algebras over special classes of rings. Peterssonwx P1 adapted Harder’s classical result on symmetric bilinear forms over polynomial ringsŽ Knebuschwx Kn, 13.4.3 and Lam w L1, VI.3.13 x. to prove, in particular, that for any field k which is perfect or of not two all composition algebras over ktwxof rank ) 2 are defined over k. It is also known that, over a principal ideal , every composition algebra containing zero divisors is split and isomorphic to Ž.Ž . R[R, Mat 2 R the 2 = 2 matrices over R , or to Zorn’s algebra of vector matrices Zor R Žcf.wx P1, 3.6. . The proof goes back to van der Blij and Springerwx BS . Knus, Parimala, and Sridharanwx KPS proved that an algebra without zero divisors over the polynomial ring in n variables over a field k of characteristic not two is defined over k if its is defined over k, and that any with isotropic norm form is defined over k. Moreover, they constructed octonion algebras over kxwx,y, char k / 2,

474

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 475 whose norms, restricted to trace zero elements, are indecomposable as quadratic modules and, in particular, not defined over k, provided that there exists a nonsplit octonion algebra over the base field k. In this paper the composition algebras over the ring

gtŽ. Rsgj ktŽ. jG0, gt Ž.gktwxof deg g F 2 j ½5ftŽ.

kx,x 22, (wx01Ž.x01yax

2 == with ftŽ.st yagktwxand a g k not a in k , with char k / 2, are partly classified using the results fromwx Pu1 about composition alge- bras over rings of genus zero. The basic facts and definitions that cannot be found inwxwxwx P1 , P2 , Pu1 , and Hartshornewx H are summarized in Section 1. Since every composition algebra with zero divisors over R splits and is isomorphic to R [ R, Ž.Ž . Ž . Mat 2 R the 2 = 2 matrices over R , End R R [ L for the nontrivial elements L g Pic R, or Zorn’s algebra of vector matrices Zor R s Zor k mRŽwxwxPu, 4.1, 4.3 and P1, 2.7. , this paper only deals with composition algebras having no zero divisors. Here is a summary of its main results. The Picard of a nonsplit torus defined over k contains at most two elements, one being the torus itself viewed as an R-moduleŽ. 2.6 . Every nonsplit algebra over R which is not defined over k can be realized as a nonclassical Cayley᎐Dickson doubling of a torus defined over k or, equivalently, as a classical Cayley᎐Dickson doubling of a torus not defined over k Ž.2.8 . This follows directly from 2.4, one of the main results ofwx Pu1 , and from investigating the Cayley᎐Dickson doublings of tori not defined over k Ž.2.7 . Likewise, octonion algebras not defined over k can either be realized as a nontrivial Cayley᎐Dickson doubling of some Ž. ŽŽ.. c,dk mRwith a uniquely determined element in Pic rkc, d m R or, equivalently, as a classical Cayley᎐Dickson doubling of a quaternion algebra not defined over k Ž.3.6 . The critical step in the proof here is to ŽŽ . . show that there is at most one element in Pic rkc, d m R that gives rise to a Cayley᎐Dickson doubling resulting in an algebra not defined over k Ž.3.3 . Here, some algebraic geometry is used. Results concerning the classification of the composition algebras over R are given in the last section. These are then used to classify composition algebras, up to isomorphism, for certain base fields k Ž.4.3 . The main results of this article first appeared in the author’s doctoral thesiswx Pu2 . 476 S. PUMPLUN¨

1. PRELIMINARIES 1.1. Let R be a commutative associative ring with a unit element. In this paper the term ‘‘R-algebra’’ refers to unital nonassociative algebras, which are finitely generated projective of constant rank ) 0as R-modules. An R-algebra C is called quadratic in case there exists a Ž. 2 Ž. Ž. N:CªRsuch that N 1CCCs 1 and u y N 1,uuqNu1 s0 for all ugC. The form N is uniquely determined and called the norm of C.An R-algebra is called alternati¨e if its associator wxu, ¨, w s Ž.u¨ w y u Ž¨w .is alternating. An R-algebra C is called a composition algebra if it admits a quadratic form N: C ª R satisfying the following two conditions:

Ž.i Its induced symmetric N: C = C ª R, Nu Ž,¨ .[ NuŽ.Ž.Ž.q¨ yNuyN¨ is nondegenerate, i.e., it determines an R-mod- ˇ Ž. ule isomorphism C ª˜ C s Hom R C, R . Ž.ii N permits composition, i.e., NuŽ.Ž.Ž.¨ sNuN¨ for all u, ¨ g C. Composition algebras over rings are quadratic alternative algebras. In particular, the quadratic form N on C satisfying conditionsŽ. i and Ž ii . above agrees with the norm of the quadratic algebra C and therefore is uniquely determined. N is called the norm of the composition algebra C, U U Ž. sometimes it is denoted by NCC. The map : C ª C, u [ N 1,Cu1Cyu, which is an algebra , is called the canonical in¨olution on C. Composition algebras over rings only exist in ranks 1, 2, 4, or 8. A composition algebra of rank 2Ž. resp. 4, 8 is called a torus Žresp. quaternion algebra, octonion algebra.. Composition algebras are invariant under base change. R [ R together with theŽ hyperbolic . norm Nx ŽŽ,y .. sxy is a torus. A composition algebra over R is called split if it contains a composition subalgebra isomorphic to R [ R. Moreover, two composition algebras of rank - 8 are isomorphic if and only if they have isometric normsŽ Knusw K, V.Ž.Ž. 2.2.3 , 4.3.2x. . It is not known whether this is also true for octonion algebras. 1.2. For the convenience of the reader, we recall the generalized Cayley᎐Dickson doubling process by Peterssonwx P1, 2.4, 2.5 for composi- tion algebras over rings which will repeatedly be used in this paper.

Let D be a composition algebra of rank F 4 over R, and let Pic r D denote theŽ. pointed set of isomorphism classes of projective right D-mod- ules of rank one. Furthermore, let X s Spec R and view the group of = 1Ž.= units D of D as a group scheme. Then Pic r D s HXˇ ,D as pointed sets in the sense of noncommutative Cechˇ cohomologywx Mi, III.4.6 . The = NDm: D ª G canonically induces a homomorphism of pointed sets NDr: Pic D ª Pic R. Given a projective right D- Q of COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 477

Ž. rank one, it is said to have norm one if NQD (R. In case Q has norm one, there exists a nondegenerate quadratic form N: Q ª R satisfying Ž. Ž.Ž. NwиusNwND u for w g Q, u g D, where и denotes the right = D-module structure of Q. N is uniquely determined up to a factor ␮ g R and called a norm on Q. Furthermore, N determines a unique R-bilinear map Q = Q ª D written multiplicatively and satisfying Ž.Ž.w и uwи¨s U NwŽ.¨ ufor w g Q, u, ¨ g D. Now the R-module CayŽ.D, Q, N [ D [ Q becomes a composition algebra under the multiplication X X X X X XU Ž.Ž.Žu, wu,wsuu q ww , w и u q w и u ., Ž. with norm NCayŽ D, Q, N . s ND [yN. Conversely, given a composition algebra C with rank C s 2 rank D containing D as a subalgebra, there are Q, N as above such that C s CayŽ.D, Q, N .

Considering the free right D-module D g Pic r D itself, D has norm one and any norm on D is similar to ND. It turns out that in this case we get the classical Cayley᎐Dickson doubling CayŽ.D, ␮ [ Cay ŽD, D, ␮ND . Žcf., for instance,wx P1, 2.1, 2.2. due to Albertwx A . We also use the abbreviation CayŽ.ŽŽ..D, ␮, ␩ [ Cay Cay D, ␮ , ␩ for rank D F 2. X 1 1.3. Let X [ Pk s Proj kxwx01,x be the projective line over a field k X X Ž. X with structure sheaf OXPand function field K [ kt. For P g X , O ,X denotes the local ring of OXPX at P, m the maximal ideal of OP,XX and Ž. X ␬P[OP, XPrm the corresponding residue class field. F Pdenotes the X stalk of an OX X-module F at P. The isomorphism Pic X ( Z is given by ; XXŽ. Ž. Ž. the assignment m ª OXXm , where O m [ Sm with S [ kxwx01,x , mgZwxH, p. 117 . Every locally free OX X-module F splits into the direct XXXŽ. Ž. sum of invertible OXX-modules, F ( O m1[ иии [ OXsm , m ig Z.An OX X- M in a K- V is a locally free OX X-submodule of the X constant sheaf V of V over X of rank n s dim k V. 1.4. From now on ‘‘point’’ without further specification always refers to X 1 a closed point. For any point P g X s PkP, O ,XX is a discrete valuation ring which corresponds with a discrete valuation on K that is trivial on k.

KˆˆP denotes the completion of K with respect to this valuation and OP, X X the respective valuation ring of KˆP . A composition algebra C over K is called unramified at P if CˆˆPP[ C m K either splits or is an unramified composition over KˆP . C is called Ž.separably ramified at P if CˆP is aŽ. separably ramified composition division algebraŽ cf. Kuting¨ wxKu¨ or w P1, 6.2 x. . C is unramified at P if and only if CˆP contains a self-dual orderwx P1, 6.3 . For the definition of unramified as well as Ž.separably ramified composition division algebras over fields complete with respect to a discrete valuation, the reader is referred towx P1, 6.2 . 478 S. PUMPLUN¨

2. TORI AND QUATERNION ALGEBRAS

X 1 Let k be a field of char k / 2, X s Pk s Proj kxwx01,x the projective Ž. 1 line over k, and K [ kt its function field. Every point P0 g Pk of degree two is represented by the principal ideal generated by a monic irreducible 2 polynomial ftŽ.st yagktwxof degree two; the residue class field of Ž. Ž' . 1 P00is ␬ P s ka. Removing P0from Pkresults in the affine scheme 1Ä4 X[Pk yP0 sSpec R of a ring R, and a straightforward computation shows that

gtŽ. R kt j 0, gt kt of deg g 2 kx,x 22. sgj Ž.G Ž.gwx F j ( w01 xŽ.x01yax ½5ftŽ.

= = From now on let R be this ring of f-fractions. Obviously, R s k is the set of invertible elements of R. R is a with quotient field QuotŽ.R s K, since Spec R is an open dense subscheme of the curve 1 Pk . It is not a . The ideal class group of R, which is isomorphic to the Picard group here, contains two elements, that is, Ž. Cl R s Pic R ( Z2 bywx H, II.6.4, 6.5 . Let L denote the nontrivial ele- ment in Pic R.

2 2 2.1 LEMMA. As an R-module L is isomorphic to ŽŽ1r t y a.Ž, tr t y a.., an ideal in R which is not principal. Proof. Suppose

1 tgtgtŽ. Ž. , mm, R 22s 22g ž/tyatya ž/Ž.tyat Ž.ya is a principal idealŽ. it is necessary that deg g - 2m . Assume gtŽ.to be 2 2 n relatively prime to t y a. Then there exists an element htŽ.r Žt ya. g 2 R,htŽ.relatively prime to t y a, such that 1 htŽ. gt Ž. nm 2s 22 tyaŽ.Ž.tyatya

2 nm 2 2 s and therefore Žt y a.qs Žt y aht.Ž.Ž. gt implying gt Ž.s␣ Žt ya., 2 r htŽ.s␤ Žt ya. with ␣, ␤ g k, ␣␤ s 1. Since gtŽ.and ht Ž.are rela- 2 tively prime to t y a, it follows that r s s s 0. Without loss of generality, 2 let gtŽ.s1, then m s 1 and 1r Žt y a.generates the ideal. Thus tqtŽ. 1

2 s 2 j 2 tyatŽ.tyaya COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 479

2 j 2 for some qtŽ.r Žt ya.gR,qtŽ.relatively prime to t y a and qtŽ.s 2 j ttŽ ya.. Therefore j s 0 and qtŽ.st, which implies deg q G 2 j,a contradiction. So

1 t L ( 22, ž/tyatya as R-modules. The goal of this paper is to classify the composition algebras over R. Since WkŽ.ªWK Ž .is injective Ž see, for instance,wx S, 4.3.3. and for composition algebras over fields isometric norms imply isomorphic compo- sition algebrasŽ Jacobsonwx J. , obviously the composition algebras over R which are already defined over the base field k are classified whenever those over k are: C01( C is equivalent to C0mkR ( C1mkR for compo- sition algebras C01, C over k. Every composition algebra with zero divisors over R splitswx Pu1, 4.3 and Ž. Ž . therefore is isomorphic to R [ R, to Mat 2 R , to End R R [ L by 2.1 and wxP1, 2.7 , or to the algebra of Zorn’s vector matrices Zor R ( Zor k m R wxPu1, 4.1 . The question whether the split quaternion algebras Mat 2Ž.R Ž. and End R R [ L are isomorphic remains open. Regarding composition algebras without zero divisors, the following is known.

2.2 PROPOSITION wxPu1, 2.7, 2.8 . Let C be a composition algebra without zero di¨isors o¨er R. Ž.a The following are equi¨alent. Ž.i C is defined o¨er k. Ž. ii C mR K is defined o¨er k. Ž.iii C is unramified at P0. Ž. b C is not defined o¨er k if and only if C mR K ramifies exactly at P0. To determine whether or not a composition algebra over K ramifies at ŽŽ.. P0 s ft , the next lemma by Kuting¨ turns out to be helpful. Recall that any nondegenerate quadratic form ␾ over KˆP with ␲ a fixed prime ˆ ␾ ²:²:␲ element of K P can be written as s u1,...,urrH u q1,...,unwith X ²: uiPgO,Xfor 1 F i F n. Then u1,...,uris the first residue form of ␾ over the residue class field KˆˆPPof K . Ž. 2.3 LEMMA wxKu,¨ 2.2.3 . Let C11s Cay K, h ,...,hr be a composition Ž. Ä4 algebra o¨er K s k t with r g 1, 2, 3 and square free polynomials h1,...,hr Ä4 1 Ä4 ŽŽ.. Ž. gktwxy0. For P g Pk y ϱ , P s gt ,where g t g kwx t is a monic irreducible polynomial, and with OˆˆP, XPXthe ¨aluation ring of K the following holds. 480 S. PUMPLUN¨

Ž. a If g does not di¨ide any h1,...,hr , then C1 is unramified at P. In particular, C1 m KˆP splits if and only if the first residue form of the norm of C1 m KˆˆPP is isotropic o¨er the residue class field K . = Ž. Ä4 X b If g di¨ides hiP, i g 1,...,r , then there exists an ␣ g Oˆ,Xl ktwx =1 ˆX and a diagonalized form q with entries in OP, X l kwx t and dim q s 2 dim C1, such that the norm N11 of C can be written as N1( q H g␣ q. C1 ramifies at P if and only if the first residue form of q is anisotropic o¨er KˆˆP ;otherwise, C1 m KP splits and C1 is unramified at P. Using a result ofwx Pu1 , the composition algebras without zero divisors over R will be characterized.

2.4 THEOREM wxPu1, 3.10 . Let C be a composition algebra without zero di¨isors o¨er R of rank r G 2. If C is not defined o¨er k, then it can be realized by a generalized Cayley᎐Dickson doubling of a composition algebra defined o¨er k. The latter is uniquely determined up to isomorphism. Ž. Since the above theorem states that C ( Cay D0 mk R, Q, N for every composition algebra C without zero divisors of rank G 2, we have to find Ž. the elements Q g Pic r D0 mk R of norm one, for all composition alge- bras D0 over k of rank 1, 2, or 4, and calculate the respective norm N: Q ª R on each Q. Note that any classical Cayley᎐Dickson doubling Ž.Ž. Cay D0 mk R, ␮ ( Cay D0 , ␮ mk R is itself an algebra defined over k. Toriᎏthe Cayley᎐Dickson doublings of Rᎏare considered first.

2.5 PROPOSITION.iŽ.The nontri¨ial element L of Pic R has norm one. Ž. ii There exists a unique nondegenerate quadratic form N0: L ª R satisfying

11 N022s, ž/tyatya tt2 N022s, ž/tyatya and

1 t 2t N0 22, s 2. ž/tyatyatya

²2 : N00 is a norm on L and N m K ( t y a K . Ž.iii E¨ery torus o¨er R which is not defined o¨er k is isomorphic to Ž. = Cay R, L, ␮N0 for a suitable ␮ g k . COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 481

Proof. Ž.i L has order two and thus norm onewx P1, 2.10 . Ž.ii Consider the nondegenerate form N: K ª K given by N Ž.1 s 2 Ž. tya. Then NL;R, so it induces a quadratic form N0: L ª R with 11 N022s, ž/tyatya tt2 N022s, ž/tyatya and 1 t 2t N0 22, s 2. ž/tyatyatya Since

gtŽ. L m ktŽ. m)0, gt Ž. kt, and deg g 2m 1 sg2 g wx F y ½5Ž.tya is a fractional ideal,

1 Lˇ ( Ly s Ä4htŽ.gkt Ž. ht Ž. L;R gtŽ. m ktŽ. m)0, gt Ž. kt, and deg g 2m 1, sg2 g wx F q ½5Ž.tya and for any gtŽ. gtŽ. mmL,N , 22g 0y Ž.tyatž/ Ž.ya Ž. Ž2 .my1 is just the multiplication by the element 2 gtrt ya ,so N0 induces an isomorphism L ª Lˇ. Hence, N0 is nondegenerate on L as required. The uniqueness is clear. Ž. Ž . Ž Ž2 .. iii Cay R, L, ␮N00m K ( Cay K, ␮ t y a ramifies at P s Ž2.=Ž. Ž. tyafor all ␮ g k 2.3 and therefore Cay R, L, ␮N0 is without zero divisorsŽ otherwise it would be unramified at P0; cf.wx Pu1, 4.3. and not defined over k ŽŽ..2.2 b .

2.6 THEOREM. Let T0 be a torus o¨er k without zero di¨isors, i.e., let Ž.' Ž. Ä4 T0sk c be a quadratic extension of k. Then Pic T00m R s T m Rif Ž.'' Ž . Ž. T00(k a and Pic T m R ( Z2if T0\ ka.In the latter case let Ec denote the nontri¨ial element in this Picard group, then as a T0 m R-module ŽŽ2 .Ž2 .. Ec is isomorphic to the ideal 1r t y a , tr t y ainT0mR. 482 S. PUMPLUN¨

Proof. Let kX [ kcŽ.' and X X R [ R mk k

gtŽ. XX m ktmŽ.0, gt Ž. kt, and deg g 2m . sg2G g wx F ½5Ž.tya X Ž.'' Ž. 2 Ž .Ž' . In case k ( kas␬P0 the polynomial t y a s t y atqa X splits in ktwx, and '' XXtqatya Rsk , ty''atqa XXŽ. is a ring of Laurent polynomials over k . This implies Pic R s Pic T0 m R Ä4 sT0mR. 2 X In case kcŽ.''\ka Ž.the polynomial t y a is irreducible in ktwx, too, X Ž. which implies Pic R s Pic T02m R ( Z by the same argument which proves Pic R ( Z2 . 2.1 concludes the proof.

If there is no danger of confusion, we will abbreviate Ec s E. Next the classical Cayley᎐Dickson doublings of tori not defined over k are considered. They can be used to determine the right T0 m R-module structure of Eccand the norms on E . In particular, it turns out that Echas norm one. Ž.= 2.7 PROPOSITION. Let T [ Cay R, L, ␮N0 , ␮ g k , be an arbitrary torus not defined o¨er k. = Ž.a Cay ŽT, c . for c g k is not defined o¨er k and without zero di¨isors = if and only if c is not a square in k and kŽ.'' c \ ka Ž.. Ž. Ž . Ž Ž' . Ž Ž . .. b Cay T, c ( Cay kcmR,Ec,␮N00[yc N for e¨ery c g ==2 k with c f k and with kŽ.'' c \ ka Ž.. Ž. Ž.' i Ec sL[LasanR-module and the right k c m R-module structure of Ec is gi¨en by

Ž.Ž.Žw12, w и u 12, u s wu 11ycw 2221 u , wuywu 12. Ž. for w12, w g L, u 12, u g R, where u12, u g R [ R is identified with u1 ''Ž. qcu2gkcmR. Ž. Ž. Ž. Ž . ii Nc:Ec ªR,Nc [N00[yc N is a norm on Ec. Ž. Ž . ² : Ž .Ž Proof. a D [ Cay T, c has the norm NDs 1, yc [y␮ N0[ Ž.. ² Ž2 .Ž2 .: Ž . ycN0 . Since ND, K ( 1, yc, y␮ t y a , ␮ct ya K by 2.5 ii , the ŽŽ2.. algebra D m K ( Cay K, c, ␮ t y a ramifies at P0 if and only if c is = not a square in k and kcŽ.''\ka Ž.Ž.2.3 . The assertion follows from 2.2 and the fact that every composition algebra with zero divisors is unramified at P0 Žcf.wx Pu1, 4.3. . COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 483

Ž.b D[Cay ŽT, c .is not defined over k and without zero divisors for ==2 all c g k , c f k with kcŽ.''\ka Ž.by Ž. a . Moreover, D s T H T s ²: Ž . ² : RHLHRHLhas norm ND s 1 Hy␮N00HycHc␮Nand a straightforward calculation shows that T1 [ R H 0 H R H 0 is a composi- tion subalgebra of D canonically isomorphic to the torus kcŽ.' mR.It Ž. ŽŽ..' follows that D s Cay T1, P, N for a P g Pic kcmRof norm one H with P s T1 s 0 H L H 0 H L, P ( L [ L as an R-module. Any classical Cayley᎐Dickson doubling of T1 is defined over k, therefore P s Ec g Pic T1. This implies that EcDPhas norm one. Moreover, N syN < s0H Ž.ŽŽ.. Ž. ␮N0000H0Hyc␮N (␮N[ycN implies that Nc [N0[ Ž. ycN0 is a norm on Ec. A straightforward calculation proves that the right T1-module structure of P s Ec is

Ž.Ž.Žw12, w и u 12, u s wu 11ycw 2221 u , wuywu 12. for w12, w g L, u 12, u g R. Ž.'' Ž . Alternatively, Ec s L m kcsL[y cL(L[Lalso follows from 2.6. The multiplication in 2.7Ž.Ž. b i then corresponds to '' ' Ž.Ž.w121211222112ycw u y cu sŽ.Ž.wu ycw u y cwuywu .

2.8 COROLLARY. A quaternion algebra o¨er R without zero di¨isors which is not defined o¨er k is isomorphic to ' CayŽ.kcŽ.mkcR,E,␮Ž.N00[yŽ.cN ,

= where kŽ.'' c is a torus such that kŽ. c \ ka Ž.',and ␮ g k . It can also be realized as a classical Cayley᎐Dickson doubling CayŽ.T, cof Ž. = T[Cay R, L, ␮N0 , a torus not defined o¨er k, for suitable ␮, c g k . Proof. 2.4 and 2.7.

3. OCTONION ALGEBRAS

We will prove the first two theorems in this section for rings S satisfying 1 Ä4 1X Spec S s Pk y P0 ,...,Pn with closed points P0 ,...,PnkgP \X such Ýn that is0deg Pi s 2. Therefore they also hold when two k-rational points are removed, that is, for the ring of Laurent polynomials over k. Recall the following fromwx Pu1, 1.10 and Sect. 3 : Let C be a composi- X X tion algebra over S, C [ C mSXK. Then an O X-lattice C is called a X X X maximal OXXX-order in C if C is an O X-subalgebra of the constant sheaf C X X X X X of C over X such that CPPis a maximal O ,XX-order in C for all P g X . 484 S. PUMPLUN¨

XX X Choose a maximal OX X-order C in C satisfying C

3.2 THEOREM. Let C be a composition algebra o¨er S of rank r ) 2, X without zero di¨isors and not defined o¨er k, and let C be the maximal XX Xr2 r2 X rXXŽ.r X OXXXX-order in C with C

Proof. For C not defined over k, the proof ofw Pu1, 3.3Ž. ix together with 0XX X wxPu1, 3.4 shows that rr2 G hXŽ,C.ŽG␹C.srr2, implying 1 XX 0XX X hXŽ ,C.s0 and hXŽ,C.Žsrr2s␹C.for the Euler characteristic X X ␹ ŽC .. Moreover, C contains a composition subalgebra of rank rr2 which X r 2 r 2 Ž. ޏ rXXÝ r Ž. is defined over k wPu1, 3.3 ix and so C ( Xi[ s1OXim with X 1XX X Ž . miXgZfor the locally free O -module C . Now hX,Cs0 implies 0ŽXX. miiGy1 and hX,Csrr2 implies m F 0, 1 F i F rr2wx H, p. 225 , proving the assertion.

3.3 THEOREM. Let D0 be a composition di¨ision algebra o¨er k of rank Ä4 Ž. lg2, 4 . There exists at most one element Q g Pic r D0 mk S of norm one Ž. such that Cay D0 mk S, Q, N has no zero di¨isors and is not defined o¨er k for a suitable norm N on Q. Ž. Proof. Let Q12, Q g Pic rD0mkS be of norm one such that the Ž. Ž composition algebras C10[ Cay D mkS, Q11, N and C 2[ Cay D 0mk S,Q22,N.are not defined over k and without zero divisors, for suitable norms N121respectively N . Extend C˜to the uniquely determined maximal XX X Ž. OX -order C11in C m K with C 1

XX l XXŽ. X Ž Then C120( C ( D m OXXH O y1asO X-modules see 3.2 andw Pu1, XXl X . XXŽ. 3.8x . Define D [ D0 m OXX. Then Q [ O y1 is a right D -module XX XŽ. both via и11: Q = D ª Q , w, u ª w и u [ uw, the multiplication in the XXXX X Ž. quadratic alternative OX -algebra C12, and via и : Q = D ª Q , w, u ª wи2 u[uw, the multiplication in the quadratic alternative OX X-algebra XXX C2 . This is a consequence of the fact that Q␰␰is a right D -moduleŽ with ␰ X. X the generic point of X via the multiplication in C1, ␰ ( C1 m K. More- X Ž .Ž. over, the trace T of C1 cf., for instance,wx P1, 1.3 satisfies T ¨, uw s 0 X 0 X TŽ.¨u,ws0 for w g HUŽ,Q.,u,¨gHUŽ,D.and any open set U ; X Ž X . 0Ž X . X X implying w и10u s uw g HU,Q. So due to HU,Q;Q␰and 0ŽX.XX HU,D;D␰ the above map и1 induces a right D -module structure on X Q . The same applies for и2 . X Now the D -module structure и1 induces an OX X-algebra morphism Xop X XŽ .Ž . f:DªEndX Q , u ª w ª w и12u , and, analogously, the map и an Xop X X XŽ .Ž . OXX-algebra morphism g: D ª E nd Q , u ª w ª w и2u . X X Let ␴ : X ª Spec k denote the structure morphism of X . Then X op U op X l XXŽ .ŽŽ.X. D(␴D0 .Furthermore,End XXXQ s E nd O y1 ( UŽ. U op U Ž. op Ž. ␴Mat l k ,so f,g:␴ D0 ª␴ Mat l k . Since D0 and Mat l k are op finite-dimensional k-vector spaces, Hom kŽ D0 , Mat lŽ..k can be identified U op U with Hom X XŽ␴ D0 , ␴ Mat lŽ..k . This is a straightforward calculation U involving the fact that the natural map ␳ V : V ª ␴#␴ V is an isomor- phism for any finite-dimensional k-vector space V. The identification then U is possible, because ␳WV( f s ␴#␴ f ( ␳ for every homomorphism f g Hom kŽ.V, W , with V and Wk-vector spaces. Therefore f and g can be viewed as k-algebra Ž. op f, g: D0 ª Mat l k , since also D00( D . By the theorem of Skolem and Ž .ŽŽ.. Noether cf., for instance,wx S, 8.4.2 , there exists a ⌽ g Autkl Mat k , Ž. y1 Ž.= ⌽xsbxbwith b g Mat l k such that f s ⌽( g. Again we may U X Ž Ž ..XX Ž Ž.. ŽX Ž .. identify Autkl Mat k s Aut X␴ Mat lk s Aut XE nd XQ . Then 1 U X Ž. y XŽ . ⌽hs⌿ (h(⌿for ⌯ [ ␴ b g Aut X Q and

1 fuŽ.s⌿y(gu Ž.(⌿

m⌿(fuŽ.sgu Ž.(⌿ m⌿Ž.Ž.fuŽ.Ž. w sgu Ž.⌿ Žw .

m⌿Ž.Ž.wи12us⌿wиu

XX ŽX.ŽX . for all u in D and w in Q . This implies that ⌿: Q , и12ª˜ Q , и is an X ˜˜ isomorphism of right D -modules and that ⌿

For l s 2 and R the ring of f-fractions, the above theorem is a weaker version of 2.6. Nonetheless, the result for l s 4 is useful to gain insight on which Cayley᎐Dickson doublings are possible for those quaternion alge- bras over R that are defined over k. Since any classical Cayley᎐Dickson Ž. doubling of D0 m R is a composition algebra defined over k 2.4 , the element Q satisfying the conditions in 3.3 has to be nontrivial.

3.4 THEOREM. Let D be an arbitrary quaternion algebra without zero ŽŽ' . Ž.. di¨isors and not defined o¨er k, that is, D s Cay kcmR,Ec,␮Nc , = with ␮ g k and kŽ.'' c a torus o¨er k with kŽ. c \ ka Ž.'. Ž. Ž . = Ž. a Cay D, d , d g k , is not defined o¨er k if and only if c, disak di¨ision algebra which does not ha¨e kŽ.' a as a splitting field. Ž. Ž . ŽŽ . . = b Cay D, d ( Cay c, d k m R, Q, N for each d g k such that Ž. Ž. c,disadik ¨ision algebra which does not ha¨e k' a as a splitting field. Moreo¨er, Ž. ŽŽ . . i QgPic rkc, d m R is a nontri¨ial element of norm one with Ž. QsEcc[EsL[L[L[LasanR-module. The right c, d km R- module structure of Q is gi¨en by

U U Ž.Ž.Žw12, w и u 12, u s w 11и u y dw 2221и u , w и u y w 12и u .

Ž.' Ž. for w12, w g Ec, u12, u g kcmR\T.Here u 12, u g T [ T is canon- Ž. Ž . ically ¨iewed as an element of c, d k m R s Cay T, d . Ž.ii Nc Ž.[y ŽdNc . Ž.:QªR is a norm on Q. Ž. Ž . ² : Proof. a C [ Cay D, d has the norm NC s 1, yc, yd, cd [ Ž.ŽŽ.Ž.Ž.. ² : Ž.Ž2 y␮Nc [ydNc . Since NC,K( 1, yc, yd, cd K[y␮ t y 2 a.²1, yc, yd, cd:Ž.K by 2.5 ii , the algebra C m K ( CayŽŽK, c, d, ␮ t y a.. Ž 2 .Ž. ramifies at P0 s t y a if and only if c, d k is a division algebra and kaŽ.' not a splitting field of Ž.c, d k ; cf. 2.3. The proof follows with 2.2 and from the fact that every composition algebra with zero divisors over R ramifies at P0. Ž.b According to the assumption C [ CayŽ.D, d is not defined over k. Ž.' Define T [ kcmR,E[Ec. Then C s D H D s T H E H T H E has ²:Ž.Ž.² : Ž. norm NC s 1, yc Hy␮Nc Hyd,dc H d␮Nc. A straightfor- ward calculation shows that D1 [ T H 0 H T H 0 ( T [ T as an R-mod- ule is a composition subalgebra of C canonically isomorphic to the Ž. ŽŽ. . quaternion algebra c, d k m R. It follows that C s Cay c, d m R, Q, N ŽŽ . . H for a nontrivial Q g Pic rkc, d m R of norm one with Q s D1s 0 H E H0HE(E[Eas an R-module. Furthermore, N syNC

A straightforward calculation shows that the right D1-module structure of Q is

U U Ž.Ž.Žw12, wu 12,usw 11иuydw 2221и u , w и u y w 12и u . for w12, w g E, u 12, u g T.

We write FŽc, d. [ Q for the above nontrivial element. If there is no danger of confusion, we will abbreviate FŽc, d. s F.

3.5 PROPOSITION. Let D0 be a quaternion di¨ision algebra o¨er k which Ž.' has k a as a splitting field. Then any Cayley᎐Dickson doubling of D0 m R is defined o¨er k. Ž. Proof. We may assume D0 s a, c k . For an arbitrary Cayley᎐Dickson Ž.X doubling C [ Cay D0 m R, Q, N obviously C [ C m K ( CayŽŽ..Ž.K, a, c, gt for a gt gktwxyÄ40 , which can be assumed to be 2 Ž. X square free. If t y a does not divide gt, then C is unramified at P0 by 2 2 Ž. Ž .Ž. Ž. X Ž . 2.3. If gt s t yaht with ht gktwx, write NC ( q H t y a ␣ q with q [ ²:Ž1, a, c, ac and ␣ [ ht.Ž.RK kt. The first residue y y g P0 l wx form of q over Kˆ kaŽ.' is isotropic and so again CX is unramified at P0 ( P0 by 2.3. The assertion follows from 2.2. 3.6 COROLLARY. An octonion algebra o¨er R not defined o¨er k is isomorphic to

CayŽ.Ž.c, d km R, FŽc,d., ␮Ž.NcŽ.[y ŽdNc . Ž.

Ž. Ž. for some di¨ision algebra c, dk which does not ha¨e k' a as a splitting = field, and a suitable ␮ g k . It can also be realized as the classical Cayley᎐Dickson doubling CayŽ.D, d ŽŽ' . Ž.. of a quaternion algebra D [ Cay kcmR,Ec,␮N c not defined o¨er k. = Ž. In this case d is an element of k such that c, disadik ¨ision algebra not ha¨ing kŽ.' a as a splitting field. Proof. Every octonion algebra over R not defined over k is the Cayley᎐Dickson doubling of a quaternion algebra defined over k Ž.2.4 . Since the classical Cayley᎐Dickson doublings of these quaternion algebras are defined over k, 3.3 together with 3.4Ž. b and 3.5 implies that only the ŽŽ . Ž Ž . nontrivial Cayley᎐Dickson doublings Cay c, d km R, FŽc,d., ␮ Nc [ Ž . Ž ... = Ž. Ž . ydNc , with ␮ g k and c, d kk\ a, y a division algebra, yield octonion algebras not defined over k.

3.7 Remark. Note that Eccis a free R-module, since E s L [ L ( R[LmLand L has order twoŽ cf., for instance, Balcercyk and Jozefiak´ wxBJ, p. 136.Ž. . The norm Nc is not defined over k, because this assump- 488 S. PUMPLUN¨ tion would imply NcŽ.( ²1, yc :R by a simple computation and ŽŽ' . Ž.. Ž. Cay kcmR,EcR,␮Nc m K(Cay K, c, ␮ would be defined over k = ŽŽ' . Ž.. for any ␮ g k . Hence D [ Cay kcmR,Ec,␮Nc would be defined over k by 2.2, which contradicts 3.4Ž. b . X However, the maximal OX X-order C in D m K constructed at the X 2 2 XXŽ.Ž. beginning of Section 3 satisfies C ( ޏXX[ O y1 cf. 3.2 , and N Nis the norm of this quadratic alternative O X-algebra with kŽ 'c .mOXX [1 X 2 XXŽ. N1:OXXy1ªOdegenerate exactly at P01. Therefore N is repre- ŽŽ␣ .. ␣ Ž. sented by a symmetric matrix i, j x011, x Fi,jF2with the i,jx01, x g kxwx01,x homogeneous polynomials of degree two, and so

␣i,jŽ.t NcŽ.( ¦;ž/t2 a yi,j Ž. Ž . for ␣i, jit [ ␣ ,jt,1 gktwxof degree less than or equal to two. In general, these ␣i, jŽ.t seem to be difficult to calculate explicitly. 3.8 Remark. Let us briefly consider the situation that k is a field of 1 characteristic two. Then a separable point P0 g Pk of degree two is 2 represented by an irreducible polynomial ftŽ.st ytyagktwx, and removing P0 from the projective line again results in the affine scheme Spec R with

gtŽ. Rsgj ktŽ. jG0, gt Ž.gktwxof deg g F 2 j , ½5ftŽ. where Pic R s Ä4R, L and L ( ŽŽ.Ž..1rft,trft , since analogous proofs also hold in this case. Every composition algebra with zero divisors over R is isomorphic to Ž. Ž R[R, to Mat 2 R, to End R R [ L , or to Zor R wxPu1, 4.1 is also true in characteristic two for the projective line. . If k is perfect, then every composition algebra over R of rank ) 2 without zero divisors which is not Ž. defined over k is isomorphic to Cay D0 mk R, Q, N with D0 a composi- tion algebra over k of half rank which is uniquely determined up to isomorphismwx Pu1, 3.10 . Since the proofs of 3.1, 3.2, and 3.3 hold in this Ž. situation, there is at most one element Q g Pic r D0 mk R that allows such a Cayley᎐Dickson doubling. Moreover, similar as in 2.6, it can be Ž. shown that for any nonsplit torus D00over k the Picard group Pic D mkR Ž. is trivial if D00( ␬ P , and that it contains exactly one nontrivial element ŽŽ.Ž.. Ewhich is isomorphic to the ideal 1rft,trft in D0 mk R if D0 \ ␬Ž.P0 . We can also prove that any Cayley᎐Dickson doubling of a quater- nion algebra without zero divisors which contains ␬Ž.P0 as a subfield is always defined over k. To get more information, however, some additional work is required. COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 489

4. SOME CLASSIFICATION RESULTS AND EXAMPLES

Consider the Cayley᎐Dickson doublings of two composition algebras

D1 mk R, D2 mk R of rank - 8 and assume that these Cayley᎐Dickson Ž.Ž. doublings Cay D1 mk R, Q11, N , Cay D 2mkR, Q22, N are algebras with- Ž. out zero divisors and not defined over k. Then Cay D1 mk R, Q11, N and Ž. Ž. Cay D2 mk R, Q22, N are not isomorphic if D 1\ D 22.4 . It remains to investigate when two Cayley᎐Dickson doublings of the same composition algebra D1 m R are isomorphic.

4.1 PROPOSITION.aŽ.Let T10[ CayŽ.Ž.R, L, ␮N , T20[ Cay R, L, ␩N , = with ␮, ␩ g k , be two tori not defined o¨er k. ==2 =2 Ž.iLet c, e g k with c, e f k and c, e k akŽ.. Then for c k =2 eŽ k.Ž the Cayley᎐Dickson doublings Cay T12, c. and CayŽT , e. both are not defined o¨er k and not isomorphic. Ž. Ž . Ž. = ii Cay T11, a is defined o¨er k, Cay T ,1 splits, and for c, e g k =2 with c, e k 1, akŽ., =2 CayŽ.T11, c ( Cay Ž.T , e if and only if c ' ekŽ.. Ž. ŽŽ.''Ž .. ŽŽ. b Let D1[ Cay kcmR,Ec,␮N c and D2[ Cay kem Ž.. = R,Ee,␩N e with ␮, ␩ g k be two quaternion algebras without zero di¨i- sors not defined o¨er k. Ž. = Ž.Ž. iLet d, s g k such that c, d kk, e, s are di¨ision algebras which Ž.' Ž.Ž.Ž. do not ha¨e k a as a splitting field. If c, d kk\ e, s then Cay D1, d Ž. and Cay D2 , s both are not defined o¨er k and not isomorphic. Ž. = Ž. Ž. ii For d g k with c, d k ( Mat 2 k , the Cayley᎐Dickson doubling Ž. = Ž. Ž . Cay D1, d is isomorphic to Zor R. For d g k such that c, d kk( a, y , Ž. = the Cayley᎐Dickson doubling Cay D1, d is defined o¨er k. For s, d g k Ž.Ž. Ž. such that c, d kk, c, s are di¨ision algebras which do not ha¨ek' a asa splitting field,

CayŽ.D11, d ( Cay Ž.D , s implies Ž.Ž. c, d kk( c, s . Ž.Ž. Ž . Ž . ŽŽ.' Proof. a i Cay T12, c ( Cay T , e if and only if Cay kcm Ž.. ŽŽ' . Ž.. Ž. R,Ece,␮Nc (Cay kemR,E,␩Ne by 2.7 b . These are quater- nion algebras not defined over k ŽŽ..2.7 a , which implies kcŽ.'mR( =2 keŽ.'mR Ž.2.4 and thus c ' ekŽ.. Ž.ii follows from a straightforward calculation and 4.1Ž.Ž. a i . Ž.Ž. Ž . Ž . ŽŽ . bi CayD12,d(Cay D , s if and only if Cay c, d km Ž Ž . Ž . Ž ... ŽŽ . ŽŽ.Ž. R, FŽc,d., ␮ Nc [ydNc (Cay e, s km R, FŽe,s., ␩ Ne[ys NeŽ ...by 3.4 Ž b . . These algebras are not defined over k Ž3.4 Ž a .. , which Ž. Ž. Ž. Ž. Ž. implies c, d kkm R ( e, s m R 2.4 and thus c, d kk( e, s by the comments following Lemma 2.1. Ž.ii again follows from a straightforward calculation and 4.1Ž.Ž. b i . 490 S. PUMPLUN¨

4.2 PROPOSITION. Let kŽ.'' c be a torus o¨er k with kŽ. c \ ka Ž.', Ž. Ž. e,dk a quaternion di¨ision algebra not ha¨ing k' a as a splitting field, and = ␮, ␩ g k . Ž. Ž . Ž . Ž =2. a Cay R, L, ␮N00( Cay R, L, ␩N if and only if ␮ ' ␩ k . Ž.ŽŽ.b Cay kc''mR,E,␮NcŽ ..\Cay Žkc Ž . mR,E,␩NŽ c .. whene¨er ŽŽ' ..= ␮k␩mod NkckŽ'c. . Ž . ŽŽ . Ž Ž . Ž . Ž ... ŽŽ . c Cay e, d k m R, F, ␮ Ne [ydNe \Cay e, d k m R, F, ␩ŽNe Ž . Žd . N Ž e ... whene er ␮ ␩ mod NeŽŽ,d .=.. [y ¨ k Že,d.k k Ž. Ž . Ž . ² Ž2 .: Proof. a Cay R, L, ␮N00( Cay R, L, ␩N implies 1, y␮ t y a K 2 ( ²Ž1, y␩ t y a.:K for the respective norms over K. Therefore ␮ ' ␩Ž K=2 .Žand thus ␮ ' ␩ k=2., since k is algebraically closed in K. Con- versely, ␮ ' ␩Žk=2 . trivially implies the above assertion.

Ž.ŽŽ.b Cay kc''mR,E,␮NcŽ ..(Cay Žkc Ž . mR,E,␩NcŽ .. yields

22 22 ²1,yc,y␮Ž.Ž.t y a , c␮ t y a :²KK( 1,yc,y␩ Ž.Ž.t y a , c␩ t y a : for the respective norms over K. Therefore ␮²:1, yc KK( ␩ ²:1, yc , implying that ␮␩ is represented by²: 1, yc Kkand thus by ²: 1, yc wL2, ²:²: ŽŽ..'= IX.1.2x . So ␮ 1, yc kk( ␩ 1, yc and ␮ ' ␩ mod NkckŽ'c.. Ž.c is proved analogously. Using the above results, we consider some special base fields k.

==2 4.3 EXAMPLES.Ž. a Let k be a field with

gtŽ. R[ j jG0, gtŽ.gQpwxt of deg g F 2 j . ½5t2p Ž.y

Ž.' Ž . Ž . Ž. Then Q ppu m R, Q ''p m R, Q ppu m R, and Cay R, L, ␮N0 with ␮ g Ä41, p, u, pu are up to isomorphism the nonsplit tori over R ŽŽ..2.5 iii . Since Q Ž.'p is a splitting field of Ž.u, p , every octonion algebra splits p Q p and is isomorphic to Zor R Ž.3.5 . COMPOSITION ALGEBRAS OVER A RING OF FRACTIONS 491

Ž. =2 In the case that p ' 1 4 , that is, y1 g Q p , then the quaternion algebras without zero divisors can be determined up to isomorphism as follows: Ž. Ž Ž.. Ž Ž.. Obviously, N000000[yuN(uN[yuN and pN[yuN ( ŽŽ.. pu N00[yuN imply that ' D0[Cayž/Q pŽ.u m R, E, N00[yŽ.uN ' (Cayž/Q pŽ.u m R, E, uNŽ.00[yŽ.uN and ' D1[ Cayž/Q pŽ.u m R, E, pNŽ.00[yŽ.uN ' (Cayž/Q pŽ.u m R, E, puŽ. N00[yŽ.uN .

Moreover, D D because the assumption N N yields 01\ DD01( 2 2 2 2 ²Ž1, yu, y t y p.Ž, ut yp.:KK( ²1, yu, ypt Ž yp.Ž,pu t y p.: . Thus ²:² : Ž. 1, yu KK( p, ypu , which leads to the contradiction u, p K( Ž. Ž.ŽŽ..Ž Mat 200K . Likewise, N [ypu N ( pu N000[ypu N and pN[ Ž..Ž Ž.. ypu N00( uN[ypu N 0imply that

D2[Cayž/Q pŽ.'pu m R, E, N00[yŽ.pu N

(Cayž/Q pŽ.'pu m R, E, puŽ. N00[yŽ.pu N and

D3[ Cayž/Q pŽ.'pu m R, E, pNŽ.00[yŽ.pu N

(Cayž/Q pŽ.'pu m R, E, uNŽ.00[yŽ.pu N .

Also, D23\ D since NDD( N implies analogously as above the contra- Ž. Ž.23 diction u, p K ( Mat 20123K . Therefore D , D , D , and D are, up to isomorphism, the only nonsplit quaternion algebras over R Ž.2.4 . If p ' 3Ž. 4 a similar result can easily be obtained in the same way. Ž 2 .Ž2 . 1 Note that although removing the point t y u or t y pu from Pk results in two nonisomorphic rings of fractions both different from the ring 1 ÄŽ 2 .4 R considered above with Spec R s Pk y t y p , the composition alge- bras over each of these rings can analogously be computed up to isomor- phism. 492 S. PUMPLUN¨

REFERENCES wxA A. A. Albert, Quadratic forms permitting composition, Ann. of Math. 43Ž.Ž2 1942 . , 161᎐177. wxBJ S. Balcerzyk and T. Jozefiak,´ ‘‘Commutative Noetherian and Krull Rings,’’ Ellis Horwood, Chichester, 1989. wxBS F. van der Blij and T. A. Springer, The arithmetics of octaves and the group G2 , Nederl. Akad. Wetensch. Indag. Math. 21 Ž.1959 , 406᎐418. wxH R. Hartshorne, ‘‘Algebraic Geometry,’’ Springer-Verlag, BerlinrNew York, 1977. wxJ N. Jacobson, Composition algebras and their , Rend. Circ. Mat. Palermo 7Ž.1958 , 55᎐80. wxKn M. Knebusch, Grothendieck- und Wittringe von nicht-ausgearteten symmetrischen Bilinearformen, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., 1970. wxK M.-A. Knus, ‘‘Quadratic and Hermitian Forms over Rings,’’ Springer-Verlag, BerlinrNew York, 1991. wxKPS M.-A. Knus, R. Parimala, and R. Sridharan, On compositions and , J. Reine Angew. Math. 457 Ž.1994 , 45᎐70. wxKu¨¨ B. Kuting, ‘‘Kompositionsalgebren¨ uber rationalen Funktionenkorpern,’’¨ dissertation, Fernuniversitat,¨ Hagen, 1987. wxL1 T. Y. Lam, ‘‘Serre’s Conjecture,’’ Lecture Notes in Math., Vol. 635, Springer-Verlag, BerlinrNew York, 1978. wxL2 T. Y. Lam, ‘‘The Algebraic Theory of Quadratic Forms,’’ Math. Lecture Notes Series, W. A. Benjamin, Reading, MA, 1973. wxM K. McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 Ž.1985 , 85᎐108. wxMi J. S. Milne, ‘‘Etale´ Cohomology,’’ Princeton Univ. Press, 1980. wxP1 H. P. Petersson, Composition algebras over algebraic curves of genus zero,’’ Trans. Amer. Math. Soc. 337Ž.Ž1 1993 . , 473᎐491. wxP2 H. P. Petersson, Composition algebras over a field with a discrete valuation. J. Algebra 29 Ž.1974 , 414᎐426. wxPu1 S. Pumplun,¨ Composition algebras over rings of genus zero, Trans. Amer. Math. Soc., to appear. wxPu2 S. Pumplun,¨¨ ‘‘Kompositionsalgebren uber Ringen vom Geschlecht Null,’’ dissertation, Fernuniversitat,¨ Hagen, 1995. wxS W. Scharlau, ‘‘Quadratic and Hermitian Forms,’’ Springer-Verlag, BerlinrNew York, 1985.