JOURNAL OF 201, 151᎐166Ž. 1998 ARTICLE NO. JA977263

Composition over Rings of Fractions Revisited

S. Pumplun¨

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

Communicated by Georgia Benkart View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received January 3, 1997 provided by Elsevier - Publisher Connector

Let k be a of not two, let fxhŽ.01,xgkxwx 01,x be an irreducible homogeneous polynomial and denote the of elements of degree zero in the homogeneous localization kx,x by kx,x . For deg f 3it wx01fhh wx01Žf. hs is proved that the composition algebras over kx,x not containing zero wx01Žfh. divisors are defined over k and that there is at most oneŽ. split composition algebra not defined over k. For deg f 4 all composition algebras over kx,x are h s wx01Žfh. enumerated and partly classified. ᮊ 1998 Academic Press

INTRODUCTION

There are only a few examples of rings where all composition algebras can be classified or can at least be listed. Because there exist composition algebras which cannot be realized by a Cayley᎐Dickson doubling in the generalized sense described by H. P. Petersson inwx 4, 2.5 , and because there also exist algebras which cannot be constructed out of a torus as explained by M. L. Thakur inwx 8 , composition algebras over arbitrary rings are difficult to understand. This paper is a continuation of the work presented inwx 5, 6 . In wx 5 the general theory of composition algebras over rings of genus zero was developed, and inwx 6 it was applied to investigate these algebras over the ring kx,x [ÄgxŽ.Ž.,x fx,xjkxŽ.,xj<0, gxŽ.,x wx01Žfh. 01rh01g 01G 01g j kxwx01,x and degŽgrfh.s 04 for an irreducible homogeneous polynomial fh gkxwx01,x of degree two, where k is a field of characteristic not two. Here, the theory presented inwx 5 is used to obtain results on composi- tion algebras over the rings kx,x where fxŽ.,x kx,x is an wx01Žfh. h01gwx 01 arbitrary irreducible homogeneous polynomial of degree either 3 or 4. The most important feature of these rings is their close relation to the projec-

151

0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 152 S. PUMPLUN¨

1 1 tive line ސk s Proj kxwx01,x : Let P 0g ސkbe a closed point with P0/ ϱ, deg P00s d. P is represented by the principal ideal generated by an irreducible homogeneous polynomial fxhŽ.01,xgkxwx 01,x of degree d and ސ1 Ä4P Spec kx,x by 1, II.2.5Ž. b . k y 001( wxŽfh. w x For deg P0 s 3 the composition algebras over the corresponding ring turn out to be all defined over k with one possible exception given by a split algebra. This is the content of Section 2.

However, for deg P0 s 4 this is not the case. The composition algebras over the corresponding ring prove to behave similarly to those over the ring kx,x with fxŽ.,x kx,x of degree two which were wx01Žfh. h01gwx 01 investigated inwx 6 . This is shown in Section 3. In particular, every composi- tion algebra of rank ) 1 over this ring can again be realized by a general- ized Cayley᎐Dickson doubling of a composition subalgebra of half rank defined over k Ž.Theorems 3.2 and 3.4 . A special role is played by the quadratic field extensions of k which are isomorphic to a subfield of the residue class field ␬Ž.P00at P , and by those quaternion division algebras which are split over ␬Ž.P0 . Viewed as composition algebras over R defined over k, any Cayley᎐Dickson doubling of these algebras turns out to be an algebra which itself is defined over k, or a split quaternion algebraŽ. Proposition 3.7 . Hence it depends on the Galois of the polynomial f how the composition algebras over this ring look. Many of the arguments used are similar to those inwx 5, 6 and hence we are brief in our presentation. For the convenience of the reader, however, we have included the basic definitions and some statements of the main theorems on composition algebras over rings of genus zero fromwx 5 in Section 1, as well as some notations taken fromwx 5, 6 . We use the standard terminology of algebraic geometry from Hartshornewx 1 and the theory of composition algebras over locally ringed spaces as presented inwx 4 .

1. TERMINOLOGY

Let R be a commutative associative ring with a unit element. A unital nonassociative R-algebra C which is finitely generated projective of con- stant rank ) 0asan R- is called a composition algebra if there exists a N: C ª R satisfying the following two conditions:

Ž.i Its induced symmetric N: C = C ª R, NuŽ.,¨ [ NuŽ.Ž.Ž.q¨ yNuyN¨ is nondegenerate, i.e., it determines an R-mod- ; ule isomorphism C ª Hom RŽ.C, R . COMPOSITION ALGEBRAS 153

Ž.ii Nu Ž¨ .sNuN Ž. Ž.¨ for all u, ¨ g C. Composition algebras over rings are quadratic alternative algebras. The quadratic form N on C satisfying the previous conditions agrees with the of the quadratic algebra C and therefore is uniquely determined. N U is called the norm of C and is denoted by NC . The map : C ª C, U u[NCCŽ.1,u1 Cyu, which is an algebra , is called the canoni- cal in¨olution on C. Composition algebras over rings only exist in ranks 1, 2, 4, or 8, where they are called torus, quaternion,oroctonion algebra, respectively. They are invariant under base change. R [ R together with the hyperbolic norm NxŽŽ,y .. sxy is a torus. A composition algebra over R is called split if it contains a composition subalgebra isomorphic to R [ R. 1 Let k be a field of characteristic not two and let P0g ސks Proj kxwx01,x be a closed point of degree d with P0 / ϱ, represented by the principal ideal generated by an irreducible homogeneous polynomial fxŽ.,x kx,x of degree d. Then ސ1Ä4P Spec kx,x and h 01gwx 01 ky 001s wxŽfh. an easy calculation shows that kx,x ÄgtŽ.ft Ž.j ktŽ.< j 0 and wx01Žfh.( r g G gtŽ.gktwxof deg g F dj4\ R for t [ x01rx , where we may assume that ftŽ.[fth Ž,1 .gktwxis a monic irreducible polynomial of degree d. The ring R is a with quotient field QuotŽ.R s kt Ž. == \K. The invertible elements of R are R s k and Pic R ( ClŽ. R ( ޚ d bywxwx 1, II.6.4, 6.5 . By 5, 4.3 every composition algebra over R containing zero divisors is split and isomorphic to R [ R, to End RŽ.R [ L for LgPic R or to Zorn’s algebra of vector matrices ZorŽ.R . Now let C be a composition algebra over R of rank r which is not defined over k and has no zero divisors. Using the terminology fromwx 5 let C [ C˜ be the composition algebra over X [ Spec R determined by C, X 1 X X and define X [ ސkX. For the maximal O X-order C in C [ C m RK X extending C, i.e., C

d 0 XX X r)hXŽ.Ž.,CG␹Csr1y , Ž. ž/4 )

X 0 XX 1XX X where ␹ ŽC .[ hXŽ,C.yhXŽ,C.is the Euler characteristic of C XX Žwx5, 3.2, 3.4.Ž . ⌫ X , C .mk R is a composition subalgebra of C which is, up to isomorphism, the only composition subalgebra of C of rank s [ XX hX0Ž ,C.Ždefined over k wx5, 3.8. . It is easy to verify that C contains a composition subalgebra of rank s X s XXŽ.иии which is defined over k if and only if C ( OXXs[ O m q1[ [ 0 XX OXrXŽ.mfor m s1,...,mrigޚ, m Fy1. In particular, s s hXŽ,C.g q X Ä41, 2, 4 follows immediately. Moreover, C is a quadratic alternative OX X- X algebra whose norm NC X is degenerate exactly at P0 g X Žseewx 4 for the 154 S. PUMPLUN¨ definition of quadratic alternative OX X-algebras or composition algebras X X 1 X over X .. Define the structure morphism of X s Pk by ␴ : X ª Spec k. The notations fixed here will be used throughout the next sections. For the concept of the generalized Cayley᎐Dickson doubling of a composition algebra over a ring the reader is referred towxwx 4, 2.3,2.4, 2.5 or to 6, 1.2 . The definition of ramified and unramified composition algebras over KsktŽ.can be found inwxwx 4, 6.2 or 6, 1.4 . In this paper we will investigate the cases where d s deg P0 equals 3 or 4. It is clear that many of the arguments are also applicable to higher degrees, but achieving a classification seems to become more and more tedious.

2. THE CASE d s 3

Let ftŽ.gktwxbe a monic irreducible polynomial of deg f s 3. We will see that all composition algebras over the ring,

gtŽ. Rsgj ktŽ. jG0, gt Ž.gktwxof deg g F 3 j ½5ftŽ. are defined over k, with the possible exception of one .

Define Pic R [ Ä4R, L12, L . Every composition algebra which contains zero divisors is split and isomorphic to R [ R, to Mat 2Ž.R , to End Ri ŽR [ L .for i s 1, 2, or to Zorn’s algebra of vector matrices. Usingwx 4, 2.8 we observe that

RL12 RL End RŽ.R [ L1( ª ( End RŽ.R [ L2 ž/ž/LR21 LR tU au au b ¬ s y¨ ž/ž/¨b ¨b ž/yua is an algebra isomorphism. It remains an open question whether End RŽR [L12.Žis also isomorphic to Mat R., or whether it is not defined over k. In the latter case it will turn out to be the only such composition algebra over R.

Both L12and L are not self-dual because of Pic R ( ޚ3and therefore do not have norm one. Hence each torus over R is a classical Cayley᎐Dickson doubling of R itself and isomorphic to CayŽ.R, ␮ ( == CayŽ.k, ␮ mk R for a suitable ␮ g k s R , i.e., defined over k. Before we can prove that this is also true for nonsplit quaternion algebras over R,we need the following. COMPOSITION ALGEBRAS 155

2.1 LEMMA. Any Cayley᎐Dickson doubling of a nonsplit torus o¨er R is defined o¨er k. X Proof. Let T be such a torus. Then T s k mk R for a quadratic field extension kX of k. Because ftŽ.remains irreducible over kX we again get

Pic T ( CLŽ. T ( ޚ3 bywx 1, II.6.4, 6.5 and because R is Dedekind, it is easy to verify that the canonical map Pic R ª Pic T is injective. In particular, it follows that Pic T [ Ä4T, Q12, Q where we may assume that QiiRsLmTand that Qiii( L [ L is an R-module, for i s 1, 2. How- ever, Qi does not have norm one: Otherwise there exists a quaternion 22 42 algebra over R with R-module structure T [ Qii( R [ L , and ⌳ ŽR [ 2 Lij.(L\Rfor i, j g Ä41, 2 , j / i. This cannot beŽw 2, V.Ž. 4.3.1x. . Thus there only is the classical Cayley᎐Dickson doubling of T which is defined == over k because of k s R . 2.2 THEOREM. Any nonsplit quaternion algebra o¨er R is defined o¨er k. Proof. Let C be a quaternion algebra over R which is nonsplit, i.e., without zero divisors. Assume that C is not defined over k. Then XX 0XX ⌫Ž X,C.mkRis a composition subalgebra of C and hXŽ,C.s XX X dim k ⌫ŽX , C .g Ä41, 2 . By Ž.) from Section 1 we know that ␹ ŽC .s 0 XX 1XX hXŽ ,C.yhXŽ,C.s1. 0 XX In case hXŽ ,C.s2 the algebra C contains a torus defined over k and can be realized as a generalized Cayley᎐Dickson doubling of it. This 0 XX 1XX contradicts Lemma 2.1. Thus hXŽ ,C.s1. It follows that hXŽ,C.s0 X 3 X and that C ( OXXXX[ O Ž.y1asanOXX-module. For the norm NCXof C X this implies that NC XXs ²:1 X H N with an OX -quadratic form N: 3 OXXXXŽ.y1ªOwhich is degenerate exactly at P0. N induces an OXX- 33 3 XXˇXX linear map N: OXXŽ.y1 ª ŽŽ..O y1 , that is N g Hom XŽŽ.OXy1, 3 U OX XXŽ.1.ŽŽ..sMat 3 OX 2 ( ␴ Mat 3012Žkxwx,x ., where kxwx 012,x are the homogeneous elements of degree two in kxwx01,x . Hence N is represented by a symmetric matrix,

axŽ.,x , Ž.ij 011Fi,jF3 with axijŽ.01,xgkxwx 012,x , and

; gtijŽ. N

X with gtijŽ.[at ij Ž,1 .gktwxof deg g ij F 2. Because C

We conclude that there is no nonsplit quaternion algebra over R which is not defined over k.

2.3 LEMMA. Let C be an o¨er R which is not defined o¨er k. Then C does not contain a quaternion algebra which is defined o¨er k. Proof. C does not contain zero divisors. The inequality Ž.) becomes 0 XX X 0XX 8)hXŽ ,C.ŽG␹C.s2 implying hXŽ,C.gÄ42, 4 . C contains a 0 XX 1XX torus defined over k if and only if hXŽ ,C.Žs2 and hXŽ,C..s0.C 0 XX contains a quaternion algebra defined over k if and only if hXŽ ,C.s4 1 XX 0 XX Žand hXŽ ,C..s2 . Assume the latter. Then hXŽ,C.s4 and 1 XX X 4 2 2 X 4 hXŽ ,C.s2 yield C ( OXXXX[ O Ž.y2 [ O XXŽ.y1orC(OXX[ 3 OXXXXŽ.y3[O Ž.y1. XX Let D00[ ⌫ŽX , C ., then D m R is the quaternion subalgebra of C X X X defined over k, and with D [ D0 m OX X , we get C s D H P where X 2 2 X 3 PsOXXXXŽ.y2[OŽ.y1 , respectively, P s OXXXXŽ.y3 [ O Ž.y1is XXXX canonically a right D -module via и: P = D ª P , Ž.w, u ¬ w и u, the multiplication in CX. X 2 2 In case P s OXXXXŽ.y2 [ O Ž.y1 this module structure induces an OX X-algebra ,

Xop X D ª EndXX Ž.P , which, by passing to global sections, gives rise to a k-algebra homomor- phism,

Mat 220Ž.k Mat Ž.kxwx,x11 Dop . 0 ª 0 Mat k ž/2Ž.

Following this homomorphism with the projection to the upper left-hand op corner of the block matrices on the right yields a homomorphism D0 ª Mat 20Ž.k . Because D is a , this is a contradiction. X 3 In case P s OXXXXŽ.y3 [ O Ž.y1 we analogously get a k-algebra ho- momorphism,

kkxwxwxwx01,xkx222 01,xkx 01,x op 0 D0 ª , 0 Mat 3Ž.k 00 and following it with the projection to the upper left-hand corner of the op block matrices on the right yields a homomorphism ␸: D0 ª k which is a contradiction. COMPOSITION ALGEBRAS 157

2.4 THEOREM. Any octonion algebra o¨er R is defined o¨er k. Proof. Assume that there exists an octonion algebra C which is not defined over k. We may choose C to be without zero divisors, since otherwise C ( ZorŽ.R . 0 XX By the Proof of Lemma 2.3 we know that hXŽ ,C.s2 and that 1 XX X 2 6 hXŽ ,C.s0. Thus C ( OXXXX[ O Ž.y1 . Denoting the, up to isomor- phism, uniquely determined subtorus of C which is defined over k by X keŽ.' mk Rwe observe that NC XXs ²1, ye :X H N with an OX -quadratic 6 form N: OXXXXŽ.y1 ª O . N induces an OXX-bilinear map N g 6 6 U Hom XXXXŽŽ.O y1,O XXŽ.1.(␴Mat6012Žkxwx,x .and is thus represented by a symmetric matrix,

bxŽ.,x , Ž.ij 011Fi,jF6

with bxijŽ.01,xgkxwx 012,x . Therefore,

; stijŽ. N

where stijŽ.[bt ij Ž,1 .gktwxhave deg sij F 2. Again NC s ²:1, ye H 6 6 = ²Žstij Ž .rft Ž ..: and det NC syŽe .det ŽŽstij Ž ...rft Ž . \htŽ.rft Ž. fk, because deg h F 12. This yields a contradiction.

Note that for an arbitrary monic irreducible ftŽ.gktwxof odd degree d j it is also true that any torus over the ring Ä gtŽ.rft Ž.gktŽ.< jG0, gtŽ.g ktwxof deg g F jd4is defined over k, and that any Cayley᎐Dickson dou- bling of a nonsplit torus only results in which again are defined over k. The proofs previously given can be used without change.

3. THE CASE d s 4

Let ftŽ.gktwxbe a monic irreducible polynomial of degree 4. We will investigate the composition algebras over the ring,

gtŽ. Rsgj ktŽ. jG0, gt Ž.gktwxof deg g F 4 j . ½5ftŽ. 158 S. PUMPLUN¨

Define Pic R [ Ä4R, L123, L , L with L 2denoting the element of order two. Every composition algebra containing zero divisors is split and is isomorphic to R [ R, to Mat 2Ž.R , to End Ri ŽR [ L .for i s 1, 2, 3, or to ZorŽ.R . Again bywx 4, 2.8 we observe that End RŽ.R [ L1 ( End R Ž.R [ L3 . It remains an open question whether End RŽ.R [ L1 and End R Ž.R [ L2 are also isomorphic, and whether one or both are defined over k. Because only L2 is self-dual it is the only nontrivial element in Pic R of norm one. Put L [ L2 . 3.1 PROPOSITION.iŽ. As an R-module L is isomorphic to ŽŽ.1rft, 2 trftŽ.,trftŽ..,an ideal in R which is not principal.

Ž.ii There exists a unique nondegenerate quadratic form N0: L ª R 2 2 4 satisfying N00Ž1rft Ž ..s1rft Ž .,Nt Žrft Ž .. strftŽ.,Nt 0 Ž rftŽ..strftŽ., 2 2 N00ŽŽ.Ž..1rft,trft s2trft Ž.ŽŽ., N 1rft,trftŽ..s2trftŽ., and 2 3 Nt000ŽŽ.rft,trftŽ..s2trftŽ..N is a norm on L and N m K ( ²ft Ž.:K. Ž.iii E¨ery torus o¨er R which is not defined o¨er k is isomorphic to = CayŽ.R, L, ␮N0 for a suitable ␮ g k . j Proof. Ž.i Because OXXŽ.y2

Let S [ kxwx01,x and denote by Snthe set of homogeneous elements of degree n in S.

3.2 THEOREM. Let C be a nonsplit quaternion algebra o¨er R not defined o¨er k. Then C contains a torus which is defined o¨er k, and which is uniquely determined up to isomorphism. 0 XX X Proof. The inequality Ž.) here becomes 4 ) hXŽ ,C.ŽG␹C.s0. XX Because ⌫Ž X , C .m R is a composition subalgebra of C we know that 0 XX hXŽ ,C.gÄ41, 2 . 0 XX 1XX X Assume first that hXŽ ,C.s1. Then hXŽ,C.s1 and C ( OX X[ 2 X OXXXXŽ.y2[O Ž.y1 . Furthermore, NCXs ²:1 X H N with N s OXXŽ.y2 COMPOSITION ALGEBRAS 159

2 [OXXXXXXŽ.y1ªOis an O -quadratic form inducing

22 NgHom XXXXŽ.O Ž.y2 [ OXXXX Ž.y1,OXX Ž.2[O Ž.1

SSS433

US3 (␴ . Mat 22Ž.S S 03 ŽŽ .. Thus N is represented by a symmetric matrix axij 011,x Fi,jF3where axijŽ.01,xgS 2for 2 F i, j F 3, axijŽ.01,xgS 3for i s 2, 3 and j s 1, and ax11Ž. 0,x 1gS 4. ²: ²Ž Ž. Ž.. : Ž. Ž . It follows that NCis 1 H gtjrft 1Fi,jF3with gtij[at ij ,1 gktwxof deg gij F 2 for 2 F i, j F 3, deg gij F 3 for i s 2, 3 and j s 1, and deg g11 F 4Ž. similar to the Proof of Theorem 2.2 . This implies that ŽŽ Ž.. . Ž.3 Ž. Ž.3 = det NCis det gtj1Fi,jF3rft \htrft fk, because deg h F 8, and therefore is a contradiction. 0 XX 1XX We conclude that hXŽ ,C.shXŽ,C.s2 and that C contains a torus defined over k which is uniquely determined up to isomorphism by wx5, 3.8 .

3.3 LEMMA. Let C be as in Theorem 3.2. Then

X 2 2 C(OXXXX[OŽ.y2.

0XX 1XX X Proof. From hXŽ,C.shXŽ,C.s2 it follows that either C ( 2 X 2 2 OXXXX[OŽ.y3[O X X Ž.y1orC(OXXXX[O Ž.y2 . Assume the first case, XX XXX and let T00[ ⌫ŽX , C ., T [ T m OXX. We know that C ( T H X OXXXXŽ.y3[O Ž.y1 , and we know that Q [ OXXXXŽ.y3 [ O Ž.y1isa XXXX right T -module via Q = T ª Q , Ž.w, u ¬ uw, the multiplication in the X quadratic alternative OX X-algebra C Žcf.wx 6, 3.3. . This module structure Xop X induces a homomorphism T ª E ndXXXXŽQ . of O -algebras and passing op Ž.kS2 to global section a k-algebra homomorphism T0 ª 0 k . However, it can be easily seen that there is no such homomorphism. We get a similar result for octonion algebras. The arguments used in the following proofs are almost exactly as the ones used before in Theorems 2.2 and 3.2, andwx 6 , hence we will be brief in our presentation. 3.4 THEOREM. Let C be an octonion algebra o¨er R not defined o¨er k. Then C contains a quaternion algebra which is defined o¨er k and uniquely determined up to isomorphism. 160 S. PUMPLUN¨

0 XX X Proof. Using Ž.) we know 8 ) hXŽ ,C.ŽG␹C.s0 and so 0 XX 1XX 0XX 1XX hXŽ ,C.shXŽ,C.gÄ41, 2, 4 . Assume that hXŽ,C.shXŽ,C. X 6 s1. Then C ( OXXXX[ O Ž.y2 [ O X X Ž.y1 and NCX( ²:1 H N with N: 6 OXXXXŽ.y2[O Ž.y1ªO X Xbeing represented by a symmetric matrix,

axŽ.,x , Ž.ij 011Fi,jF7 where axijŽ.01,xgS 2for 2 F i, j F 7, axijŽ.01,xgS 3for i s 1, 2 F jF7, and ax11Ž. 0,x 1gS 4. This implies that NCijs ²:²ŽŽ.1 H gtr Ž.. : Ž. Ž . ft 1F i, jF 7 with gtij[at ij,1 gktwxof deg g ij F 2 for 2 F i, j F 7, deg gij F 3 for i s 1, 2 F i, j F 7, and with deg g11 F 4. However, then ŽŽ Ž.. . Ž.7 Ž. Ž.7 = det NCis det gtj1Fi,jF7rft \htrft fk because deg h F 0XX 1XX 16, a contradiction. Assume now that hXŽ,C.shXŽ,C.s2. Then X 2 5 X 2 2 4 C(OXXXX[OŽ.y3[O X X Ž.y1orC(OXXXX[OŽ.y2[O XXŽ.y1.In = the first case NC Xs ²:1, ye H N for a suitable e g k which is not a 5 and N: OXXXXŽ.y3 [ O Ž.y1 ª O X Xis represented by a symmetric matrix,

bxŽ.,x , Ž.ij 011Fi,jF6 with bxijŽ.01,xgS 2for 2 F i, j F 6, bxijŽ.01,xgS 4for i s 1, 2 F jF6, and with bx11Ž. 0,x 1gS 6. This yields NCijs ²:²ŽŽ.1, ye H htr Ž.. : Ž. Ž . ft 1F i, jF 6 with htij[bt ij,1 gktwxof deg h ij F 2 for 2 F i, j F 6, deg hij F 4 for 2 s 1, 2 F j F 6, and with deg h11 F 6. Again, det NC s Ž . ŽŽ Ž.. . Ž.6 Ž. Ž.6 = yedet htij 1Fi, jF6 rft \htrft fk because deg h F 16, a contradiction.

Now consider the second case. Then NC Xs ²:1, ye H N and N s 2 4 OXXXXŽ.y2[OŽ.y1ªO X Xis represented by a symmetric matrix,

mxŽ.,x , Ž.ij 011Fi,jF6 with mxijŽ.01,xgS 4for 1 F i, j F 2, mxijŽ.01,xgS 3for i s 2, 3 and 3FjF6, and mxijŽ.01,xgS 2for 3 F i, j F 6 implying NCs ²:1, ye H ²ŽŽ. Ž.. : Ž. Ž . ltij rft 1Fi, jF6 , where ltij[mt ij,1 gktwxhas deg l ij F 3 for is2, 3 and 3 F j F 6, deg lij F 2 for 3 F i, j F 6 and deg lij F 4 for 1 F i, Ž . ŽŽŽ.. . Ž.6 Ž. Ž.6 = jF2. Because det NCisyedet ltj1Fi,jF6rft \htrft fk because of deg h F 20 this is a contradiction. 0 XX 1XX It remains the case that hXŽ ,C.shXŽ,C.s4. Usingwx 5, 3.8 the assertion is thus proved.

3.5 LEMMA. Let C be as in Theorem 3.4. Then

X 4 4 C(OXXXX[OŽ.y2. COMPOSITION ALGEBRAS 161

Proof. Let D0 m R be the quaternion algebra contained in C which is defined over k and is uniquely determined up to isomorphism by Theo- X X X X X rem 3.4. Define D [ D0 m OX X . Then again C ( D H P and P is a right DX-module with its module structure given by the multiplication in X Xop C . This structure induces an OX X-algebra homomorphism D ª X EndX X Ž P . and passing to global sections a k-algebra homomorphism ␸: op X X 0 XX 1XX D0 ª⌫Ž X,EndX XŽ P ... Using hXŽ ,C.shXŽ,C.s4 we get the following possibilities: X 3 Ž.iPsOXXXXŽ.y5[O Ž.y1 . Then

kSSS444

op 0 ␸ : D0 ª . 0 Mat 3Ž.k 00 Using an analogous argument as in the Proof of Lemma 2.3 we can show that such a homomorphism cannot exist. X 2 Ž.ii P s OXXXXXXŽ.y4 [ O Ž.y2 [ O Ž.y1 . Then

kS233 SS

op 0kSS11 ␸: D0 ª , 00 Mat Ž.k 000 2 and following ␸ with the projection to the lower right-hand corner of the op block matrices on the right yields a homomorphism D02ª Mat Ž.k . This cannot be, because D0 is a division algebra. X 2 Ž.iii P s OXXXXXXŽ.y3 [ O Ž.y1 [ O Ž.y2 . Then

kS211 SS 00 00 op ␸ : D0 ª , 0 S1 Mat Ž.k 0S 2 01 and following ␸ twice with the projection to the lower right-hand corner of op the respective block matrices yields a homomorphism ␸: D02ª Mat Ž.k which again cannot exist. X 2 2 Ž.iv P s OXXXXŽ.y3 [ O Ž.y1 . Then

Mat 22Ž.k Mat ŽS2 . op ␸: D0 00 , ª Mat Ž.k 000 2 op again yielding a homomorphism D02ª Mat Ž.k which does not exist. X 4 Thus P s OX XŽ.y2 is the only remaining possibility. 162 S. PUMPLUN¨

The next theorem is proved analogously aswx 6, 3.3 . We only need to s U know Lemmata 3.3 and 3.5 as well as that E ndXXXXŽŽ.O y2 .( ␴ Mat sŽ.k for s g Ä42, 4 .

3.6 THEOREM. Let D0 be a composition di¨ision algebra o¨er k of rank sgÄ42, 4 . There exists at most one element Q g Pic rŽ.D0 m R of norm one such that CayŽ.D0 mk R, Q, N has no zero di¨isors and is not defined o¨er k for a suitable norm N on Q.

Recall fromwxwx 4, 1.5, 6.2 or 6, 1.4 that KˆP denotes the completion of KsktŽ.with respect to the k-discrete valuation of K corresponding with X 1 a closed point P g X s ސkPand that ␬Ž.P ( Kˆ denotes the correspond- ing residue class field.

3.7 PROPOSITION. Let D00 be as in Theorem 3.6, and let D mk␬Ž.Pbe0 a split composition algebra. Then any Cayley᎐Dickson doubling of D0mk Ris a composition algebra which is defined o¨er k or a split quaternion algebra.

Proof. Let C [ CayŽ.D0 m R, P, N be an arbitrary Cayley᎐Dickson doubling of D0m R. For s s 2 there is a quadratic field extension keŽ.' X of k such that D0 s keŽ.'and we observe that w.l.o.g. C s C mR K ( CayŽŽ..K, e, gt for a square free gtŽ.gktwxyÄ40.If ftŽ.does not divide X gtŽ.then C is unramified at P0 bywx 3, 2.2.3Ž cf.wx 6, 2.3.Ž.Ž.Ž. . If gt sftht with htŽ.gktwxthen NCX( ²1, ye :H ftht Ž. Ž.²1, yc :K and ² 1, ye :is X hyperbolic over ␬Ž.P0 by assumption. Therefore again C is unramified at X P00bywx 3, 2.2.3 . For s s 4 write D s Ž.e, d kR. Then w.l.o.g. C s C m K s CayŽŽ..K, e, d, gt for a square free gtŽ.gktwxyÄ40 and similar to the X case l s 2 we find out that C is unramified at P0 bywx 3, 2.2.3 . The assertion now follows fromwx 5, 2.7 and 2.8 .

From now on let l [ ␬Ž.P0 . = 3.8 PROPOSITION. Let T [ CayŽ.R, L, ␮N0 , ␮ g k , be an arbitrary torus not defined o¨er k. = Ž.a Cay ŽT, e . for e g k is not defined o¨er k and without zero di¨isors if and only if e is not a square in k and kŽ.' e is not isomorphic to a subfield of l.

Ž.b Cay ŽT, e .( Cay Žke Ž' .mR,Ee,␮ ŽN00[y Že . N .. for e¨ery e g k= which is not a square in k, and where kŽ.' e is not a subfield of l. Moreo¨er,

Ž.i Ee sL[LisanR-module and the right kŽ.' e m R-module structure of Ee is gi¨en by

Ž.Ž.Žw12, w и u 12, u s wu 11yew 2221 u , wuywu 12., COMPOSITION ALGEBRAS 163

for w12, w g L, u 12, u g R, whereŽ. u12, u g R [ R is identified with u1 q''eu2gkeŽ.mR.

Ž.ii Ne Ž.:Ee ªR,Ne Ž.[N00[y Že . N is a norm on Ee. For the Proof of Proposition 3.8Ž. a we need Proposition 3.1, as well as wx3, 2.2.3Ž cf.wx 6, 2.3. andwx 5, 4.3 . It is then similar to the proof ofwx 6 , 2.7Ž. a and will be therefore omitted here. The proof of Proposition 3.8Ž. b is completely analogous to the one ofw 6, 2.7Ž. bx . 3.9 COROLLARY. A nonsplit quaternion algebra o¨er R which is not defined o¨er k is isomorphic to

CayŽ.keŽ.' mkeR,E,␮Ž.N00[yŽ.eN , where kŽ.' e is a torus o¨er k which is not isomorphic to a subfield of l, and = ␮ g k . It can also be realized as a classical Cayley᎐Dickson doubling of a torus not defined o¨er k. Proof. Theorem 3.2 and Proposition 3.8.

3.10 THEOREM. Let D be an arbitrary quaternion algebra without zero di¨isors o¨er R and not defined o¨er k, i.e., D s CayŽŽke' .mR,Ee,␮NeŽ.. = with ␮ g k and kŽ.' e a torus o¨er k not isomorphic to a subfield of l. = Ž.aFor d g k , CayŽ.D, d is not defined o¨er k if and only ifŽ. e, disk adi¨ision algebra which remains a di¨ision algebra o¨er l. = Ž.b Cay ŽD, d .( Cay ŽŽe, d .k m R, Q, N . for each d g k such that Ž.e,disadik ¨ision algebra which remains a di¨ision algebra o¨er l. Moreo¨er,

Ž.i QgPic rk ŽŽe, d .m R . is a nontri¨ial element of norm one with QsEee[EsL[L[L[LasanR-module. The rightŽ. e, 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 .,

for w12, w g Ee, u12, u g keŽ.'mR\T.HereŽ. u12, u g T [ T is canon- ically ¨iewed as an element ofŽ. e, d k m R s Cay ŽT, d .. Ž.ii Ne Ž.[y ŽdNe . Ž.:QªR is a norm on Q. For the proof of Theorem 3.10Ž. a we need Proposition 3.1, as well asw 3, 2.2.3xw . It is similar to the proof of 6, 3.4Ž. ax and will be omitted here, too. The proof of Theorem 3.10Ž. b again is completely analogous to the one of w6, 3.4Ž. bx . We write FŽe, d. [ Q for the preceding nontrivial element. 3.11 COROLLARY. An octonion algebra o¨er R not defined o¨er k is isomorphic to

CayŽ.Ž.e, d km R, FŽe,d., ␮Ž.NeŽ.Ž.Ž.[ydNe , 164 S. PUMPLUN¨

for some di¨ision algebraŽ. e, dk which does not split o¨er l, and a suitable = ␮ g k . It can also be realized as a classical Cayley᎐Dickson doubling of a quaternion algebra o¨er R not defined o¨er k. The proof is the same as the one ofwx 6, 3.6 , only here we use Theo- rems 3.4, 3.6, Proposition 3.7,and Theorem 3.10. We now have a list of all composition algebras over R. Going one step further we proceed to give some results concerning a classification of these algebras. Note that the composition algebras over R which are defined over k are classified whenever those over k are. Consider the Cayley᎐Dickson doublings of two composition algebras

D1 mk R,D2 mk Rof rank - 8 and suppose that CayŽ.D111m R, P , N and CayŽ.D222m R, P , N are nonsplit and are not defined over k. Then CayŽ.Ž.D111m R, P , N are Cay D 222m R, P , N are not isomorphic unless D12(Dby Theorems 3.2 and 3.4. It remains to investigate when two Cayley᎐Dickson doublings of the same algebra D1 m R are isomorphic. The arguments in the next two propositions are almost exactly as inw 6, 4.1 and 4.2x , thus we omit these proofs and refer the reader to the paper wx6 and the corresponding proofs there.

3.12 PROPOSITION.1Ž.Let T102[ Cay ŽR, L, ␮N . and let T [ = CayŽ.R, L, ␩N0 with ␮, ␩ g k . = Ž.i If c, e g k are no squares in k, kcŽ.''\ke Ž.,and k Ž.Ž. '' c , ke are not isomorphic to subfields of l, then the algebras CayŽ.T1, c and CayŽ.T2 , e both are not defined o¨er k and are not isomorphic. = Ž.ii For any e g k which is no square in k, and where kŽ.' e is isomorphic to a subfield of l, the algebra CayŽ.T1, e is defined o¨er k. = CayŽ.T1,1 splits. For c, e g k which are no squares in k, and where kcŽ.Ž.'',k e are not isomorphic to a subfield of l,

=2 CayŽ.T11, c ( Cay Ž.T , e if and only if c ' e mod k .

Ž.b Let D1[ CayŽŽkc'' .mR,Ec,␮NŽ.. c and D2[ CayŽŽke .m R,Ee,␩NŽ.. e be two nonsplit quaternion algebras not defined o¨er k. = Ž.iChoose d, s g k such thatŽ. c, dkk and Ž. e, s are di¨ision algebras which do not split o¨er l. If they are not isomorphic then CayŽ.D1, d and CayŽ.D2 , s are not defined o¨er k and are not isomorphic. = Ž.ii For d g k such thatŽ. c, d k ( Mat 2 Ž.k it follows that = CayŽ.D1, d ( Zor Ž.R . For d g k such thatŽ. c, d l ( Mat 2 Ž.l the algebra = CayŽ.D1, d is defined o¨er k. For s, d g k such thatŽ. c, dll and Ž. c, s are di¨ision algebras,

CayŽ.D11, d ( Cay Ž.D , s implies Ž.Ž. c, d kk( c, s . COMPOSITION ALGEBRAS 165

3.13 PROPOSITION. Let kŽ.' e be a torus o¨er k which is not isomorphic to a subfield of l, letŽ. e, dkl be a quaternion algebra such that Ž. e, disa = di¨ision algebra, and take ␮, ␩ g k . =2 Ž.a Cay ŽR, L, ␮N00 .( Cay ŽR, L, ␩N . if and only if ␮ ' ␩ mod k .

Ž.ŽŽ.b Cay ke''mR,Eee,␮NeŽ ..\Cay Žke Ž . mR,E,␩NŽ e .. when- = e¨er ␮ k ␩ mod NkekŽ'e.ŽŽ' ...

Žc . Cay ŽŽe, d .k m R, FŽe, d., ␮ŽNe Ž .[y ŽdNe . Ž ...\Cay ŽŽe, d .k m R, F , ␩ŽNe Ž . Žd . N Ž e ... whene er ␮ ␩ mod NeŽŽ,d .=.. Že,d.Ž[y ¨ k e,d.kk Hence it turns out that composition algebras over the ring R considered here behave similarly to those over the ring of fractions investigated inwx 6 , where a point of degree 2 was removed from the projective line, instead of one of degree 4 as assumed in this section. It still remains to investigate the conditions that set apart quadratic field extensions kX of k which are isomorphic to a subfield of l and quaternion algebras over k which become split over l. From now on let char k / 2, 3. Then we may assume that

4 2 ftŽ.stq␣tq␤tq␥.

X Let t14,...,t be the zeros of f, then l [ ktŽ.14,...,t is the splitting field X of f, and the Galois group G [ GalŽl rk.is isomorphic to a subgroup of 322 2 the symmetric group S4. Let gyŽ.[yy2␣yqŽ␣y4␥.yq␤ be 432 the cubic resolvent of f with zeros y123, y , y . Let ⌬Ž.f s 16␣␥y4␣␤ 2 2 2 4 3 y128␣ ␥ q 144␣␤ ␥ y 27␤ q 256␥ denote the discriminant of f. The following possibilities arise:

X Ž.1 wl: kxs24 and G ( S4 if and only if g is irreducible and =2 ⌬Ž.ffk . In this case there is exactly one quadratic field extension of k which is a subfield of lX. However, it is not contained in l. Therefore any nonsplit torus over k remains nonsplit over l. X Ž.2 wl: kxs12 and G ( A4 if and only if g is irreducible and =2 ⌬Ž.fgk . In this case there is no quadratic field extension of k which is contained in lX, and therefore also no quadratic field extension contained in l. Again, any nonsplit torus over k remains nonsplit over l. X =2 Ž.3 wl:kxs8 and G ( D4 if and only if g is reducible, ⌬Ž.f f k and f is irreducible over kŽ.'⌬Ž.f . Then there are three quadratic field extensions of k which are subfields of lX, and either exactly one or all three of them are also subfields of l. By Proposition 3.7 any Cayley᎐Dick- X X son doubling of the torus T [ k mk R, where k is such a subfield, is defined over k. 166 S. PUMPLUN¨

X =2 Ž.4 wl:kxs4 and G ( ޚ4 if and only if g is reducible, ⌬Ž.f f k and f is reducible over kŽ.''⌬Ž.f . Then k Ž.⌬Ž.f is the only quadratic X field extension of k contained in l s l, and by Proposition 3.7 any Cayley᎐Dickson doubling of T [ kŽ.'⌬Ž.f mkR is defined over k. X Ž.5 wl:kxs4 and G ( ޚ22= ޚ if and only if g decomposes into linear factors over k. In this case there are three quadratic field extensions X of k which are subfields of l s l. Again, a Cayley᎐Dickson doubling of any of these viewed as tori over R, results in a quaternion algebra already defined over k by Proposition 3.7. We conclude that depending on the Galois group of f there can be from one up to three ‘‘exceptional’’ tori which cannot be used to construct quaternion algebras not defined over k, or there is none, as it happens in casesŽ. 1 and Ž. 2 . A complete classification of the composition algebras over R can be obtained when choosing special base fields k as in the examples presented inwx 6, 4.3 .

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