Math. Z. 206, 589-604 (1991) Mathematische Zeitschriftypandey/R.Sridharan/91.pdf · Z. 206, 589-604...

Math. Z. 206, 589-604 (1991) Mathematische Zeitschrift

�9 Springer-Verlag 1991

Pfaffians, central simple algebras and similitudes

M.-A. Knus 1, R. Par imala z and R. Sridharan 2 1 Mathematik, ETH-Zentrum, Rfimistrasse 101, CH-8092 Zfirich, Switzerland 2 Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Bombay 400005, India

Received June 9, 1989; in final form January 22, 1990

In [KPS], pfaffians were constructed for certain modules over Azumaya algebras and this construction was used to classify rank 6 quadratic spaces over arbitrary commutat ive rings. The aim of this paper is to indicate some applications of this idea.

We prove here in a general context of Azumaya algebras of rank 16 over commutat ive rings, classical results of Albert [A] and Jacobson [J] on central simple algebras over fields.

Further, as a consequence of our computations, we get a necessary and sufficient condition for an involution of a rank 16 Azumaya algebra to admit an invariant quaternion subalgebra. This result seems to be of independent interest, even for algebras over fields, since there are examples of Amitsur- Rowen-Tignol of involutions on central simple algebras of dimension 16 which do not have any invariant quaternion algebras (see [ART]) .

Finally, using the same techniques, we compute the group of special similitudes of rank 6 quadratic spaces over commutat ive rings. This puts in a general set up the classical computat ions of Dieudonn6 [D] for forms of dimension 6 over fields of characteristic not equal to 2.

We thank the referee for useful comments, in particular for his doubt which has led us to the present formulation of Theorem 5.2 and its Corollaries.

1 Quadratic spaces and Clifford algebras

In this section, we recall some properties of Clifford algebras, which, at least over fields of characteristic not equal to 2, are well-known. Let R be a commutative ring. We denote the group of units of R by R'. Tensor products over R are unadorned. A quadratic space is by definition a pair (M, q), where M is a finitely generated projective R-module and q: M ~ R is a quadratic form such that the associated polar form bq(x, y)= q(x + y ) -q (x ) -q (y ) is nonsingular, i.e. induces an isomorphism M-V* M* = HomR(M, R). Let C(q) = Co(q) G C1 (q)

590 M.-A. Knus et al.

be the Clifford algebra of (M, q). Any isometry q~: (M, q)-~ (M', q') induces an isomorphism C(q)): C(q)~ C(q'). The involution of C(q) which is ( - 1)-Identity) on M is called the standard involution of C(q). A similitude ~o: (M, q)-~(M',q') is an isomorphism ~o: M ~ M ' such that q'@(x))=pq(x) for some unit ~t~R'. The unit # is called the multiplier of the similitude. Any similitude q~ induces in a unique way an isomorphism of algebras with involution Co@): Co(q)-% Co(q') such that Co@)(xy)=~t-~(p(x)q~(y), x, y~M (see [K] or [KPS]). Let now (I, h) be a discriminant module, i.e. 1 is an invertible R-module and h: I | I ~ R is an isomorphism. We write (R, d) for the discriminant module h(x, y)=xyd, x, y in R, d~R'. For any discriminant module (I, h) and any quadratic space (M, q) we define a quadratic space (1, h) | (M, q) by putting

(h | q) (x | y) = h (x, x) q (y), x e I, y e M.

If (I, h) = (R, d), any isometry (I, h) | (M, q) -~ (M', q') can be viewed as a similarity with multiplier d. More generally we call an isometry (1, h) | (M, q) -~ (M', q') a similitude with multiplier (I, h). Any such similitude ~0 induces an isomorphism of algebras with involution Co@): Co (q) ~ Co(q') (localize !).

If (M, q) is a quadratic space of even rank n = 2 m, then C(q) is an Azumaya algebra of rank 2". The centre Z of Co(q) is a quadratic etale R-algebra and Co(q) is Azumaya over Z. The isomorphism class of Z is called the Arf invariant of (M, q). We say that (M, q) has trivial Arf invariant if Z -~ R x R. As a quadratic algebra, Z has a nontrivial involution a. Then l (q)={x~ZIx+a(x)=O} is a discriminant module. The map l(q)| I (q)~ R is induced by the multiplication in Z. We call its isomorphism class the signed discriminant 6(q) of (M, q). Locally 6(q) is the class of ( R , ( - 1 ) " 2 ~ ) | where disc(q)=(A"m, ~q) and ~'q is the determinant of b 4 with respect to a basis of M. We have 6(qllq2) = 6(q 0 | 6(q2). If (M~, q0 and (M2, q2) are quadratic spaces of even rank, then C(q, J-q2) -~ C(q,) | C(6(qx) q2).

2 Pfaffians revisited

In this section, we recall some properties of the pfaffian as defined in [KPS]. Let A be an Azumaya algebra of rank 4n 2 over a commutative ring R such that its class in the Brauer group of R is of order 2. Let q): A @R A-* Endg(P) be an isomorphism of R-algebras, P denoting a finitely generated faithful projective R-module. The switch map ~o of A | given by ~(a@b)=b | a, is an inner automorphism Intu, ue(A| such that in any splitting ct: S Q R A ~ E n d s ( V ) , ( a | 1 7 4 is the switch map of VQsV. If =tp(u)eEndR(P), then ~2=1. We call ~b the module involution of P (induced by ~o). We call

Alt (P) = {x - ~O (x), xen}

the set of alternating elements of P with respect to ~. If A = EndR(V), P = V| V and ~o is the canonical isomorphism Can: EndR(V) | EndR(V) ~ EndR(V@ V), then Alt (V|174174 y~EV } can be identified with A2V through the map x | 1 7 4 y. To the triple (A, P, ~o) (which we call a 2-torsion datum in [KPS]) we can associate in a functorial way, an invertible R-module Pf(P), called the pfaffian module of V and a map P f : Alt (V)~ Pf(P),

Pfaffians, central simple algebras and similitudes 591

such that P f ( V | and Pf: A2V-.~I~2nv is the s tandard pfaffian, if (A, P, ~0)=(EndR(V), V| V, Can).

The m a p P f satisfies

Pf(q~(a | a)(x))= na(a ) Pf(x), a~A, x eAl t (P) ,

where nA is the reduced n o r m on A, and is homogeneous of degree n. Thus, if n = 2 , the m a p P f : A l t ( P ) ~ P f ( P ) is quadrat ic . If Pf(P) is free, a choice of a genera tor of Pf(P) yields a quadra t ic space of rank 6 over R. It was shown in [ K P S ] tha t any quadra t ic space of r ank 6 and trivial Arf invar iant arises in this way f rom a 2-torsion datum.

Suppose that A is an A z u m a y a a lgebra with an involut ion a, i.e. an R-l inear a u t o m o r p h i s m of A such that a(xy)=a(y) a(x) and a 2 = 1. There is a faithfully flat splitting ~0: S | A-~ Ends(V ) such that r u xtu - 1, where x t is the t ranspose of x and u is an i somorph i sm V *-z, V such that u'=eu for some e~pz(R)={rlER[rl2=l}. The element e is called the type of a. We say tha t

is of orthogonal type if e = 1, of symplectic type if e = - 1 and of even symplectic type if u is alternating, i.e. of the form u = v - v t for some v: V* --, V.

If A has an involut ion, its class in the Brauer g roup Br(R) is of order 2. M o r e precisely, the m a p q~: A | A ~ EndR(A) defined by ~o,(x | (a)=xaa(y) is an i somorph i sm of R-algebras. Then the cor responding module involut ion r is in fact ca, where ee#2(R) is the type of a. We denote by Alt~ the set of a l ternat ing elements of A with respect to ~ k , = e a and by Pf(A) the pfaffian of the da tum (A, A, q~,). There is non-s ingular pair ing ~: Pf(A) x Pf(A) ~ R induced by a and satisfying

~(Pf(a), Pf(a))= nA(a), a e A l f f ( A ) .

In par t icular (Pf(A), ~) is a d iscr iminant module. We call its i sometry class the pfaffian discriminant of a. If A=EndR(V), a the involut ion given by a ( f ) =uf fu-1 , for f ~ E n d R ( V ) and u: V * ~ V an i somorph i sm with u t = e u , e~#2(R), then Pf (A) = A z" V, where rank V = 2 n and Pf: Alt (EndR (V)) ~ P f (EndR (V)) is the compos i te

2 2n

A l t ( V | I | 1 7 4 P/ , A V

(Note that under the identification EndR(V)=V| a is s imply the m a p ~0 (u- 1 | u)). Fur ther , the par ing if: A 2" V ~ A 2" V* is A z " u - 1. The following property of the pfaffian discr iminant is an immedia te consequence of its definition:

L e m m a 2.1 Let u be a unit of A such that a(u)=eu for some eep2(R). Then r = Int u o a is an involution of A such that (Pf(A), z')~_(Pf(A), nA(u)~).

We now describe two cases where the discr iminant module (Pf(A), ~) is trivial.

Proposition 2.2 The discriminant module (Pf (A), 6) is isometric to the trivial discriminant module (R, 1) in the following two cases: 1) A = AI | A2, a = a l Q a 2 , A1, A2 Azumaya algebras of rank 4 with involutions ~1, 02" 2) A is of rank 4 n 2 and a is of even symplectic type.


Proof 1)We first consider the split case. Let Ai=EndR(V~), V~ free of rank 4. There exist linear isomorphisms vi: Vi *~, Vii with vit-- ei vi, (el denoting the type of o-i), such that ai(x)=vfxtv:d 1, i=1,2. Let {el, e2} be a basis for V~, {f~,f2} a basis for Vz and u~, u z the matrices of v~, v2 with respect to these bases and the dual bases in VI*, V2* respectively. Let {gi}~__<i__<4 be the basis {ei | } of V= Va | I/2, lexicographically ordered. The element

4 e, = det u~ det u2 ga ^ g2 A g3 A g4EA V

is independent of the choices of the bases {el, e2} and {f~,f2}; hence e~ is a basis for/X4V. One checks that the map if: A 4 V ~ A 4 V* induced by a sends e, to the dual basis element G.* Thus (Pf(A),ff) is trivial if A = EndR(V0 | End~(V2). The general case follows by descent. 2) The proof for involutions of even symplectic type is similar to the proof of 1). In the split case, if the involution is given by a(x)= ux t u- ~, u an alternating matrix, one takes

G = pf(u) e 1 ̂ ... ^ e2,,

where pf(u) is the classical pfaffian of u. []

Remark. For any Azumaya algebra A, let nA(a) denote the reduced norm and trA(a) the reduced trace of aeA. If A is of rank 4, then aA(a)=trA(a)--a is an involution of A. If A = M2(R),

a a b d - b 0 - 1 a A(C d ) = ( - c a )= (1 0 ) (c by[ 0

Thus tr A is of even symplectic type. We call aA the standard involution of A. We have XeA(X)=nA(X) for all x6A. Observe that, for any isomorphism c~: A ~ B of Azumaya algebras of rank 4, we have ~ ffA = aB ~.

3 Algebras of rank 16

Let A be an Azumaya algebra of rank 16 over R with an involution a. We assume that the invertible module Pf(A) is free and fix an isometry 2: (Pf(A),~)-~(R, d), i.e. ~7(x, y)=2(x) 2(y)d. Let Pfa: AltO(A)~ R be the corresponding quadratic form Pf~= 2 o P f The form Pf~ has the following remarkable property:

Proposition 3.1 There exists an R-linear automorphism p of AltO(A) such that xp(x) = p(x) x = Pf~(x) for every xeAlt~(A) and p2 = d. I f p' is an R-automorphism of Alff(A) such that xp ' (x )eR for every xeAlt~ (or such that p'(x) x e R for every xeAlff(A)), then p '=pp for some unit peR'.

Proof. We begin with the case A=M4(R) , a (x )=x t and we choose 2 such that Pf~ is the classical pfaffian pf (x), i.e. p f (x)= a12 aa4-a l 3 a24 + a14 a2a for

0 a12 at3 a14 ~

a t2 0 a23 a24 I

\ - - a14 - -a24 - -a34

Pfaffians, central simple algebras and similitudes

We put Alt~(M4(R))=Alt4(R). Then P1: Alt4(R) ~ Alt4(R) given by

has the required

593

Thus we get

and

Alt4(R)~Alt4(R) be such that p',(x)xeR for every xeAlt4(R ). We denote the standard basis of M4(R) by {Eij}, l<i,j<4. Let {gi}, 1<i_<6, be the basis {His= Eii-Ej~} of Alt4(R ), ordered lexicographically. The condition P'i(gl)g, eR implies that p'~(gx)=pg6, p e r and p'~(g6)g6eR implies that p'~(g6)=p'g, for p'eR. The condition P'l(gl +g6) (gl +g6)eR then implies p=p' and p'~ =PPl follows by further similar computations. If A is not split, let e: S| be a faithfully flat splitting of A with V free over S such that or, =e(1 | a )e ' is induced by h: V* ~ V. Then Pf:Alt~(Ends(V))~ Pf(Ends(V)) is the composite

map Alt~(V| V*) ,| ,Alt(V| el ,A4V " Let {ei}, 1__<i__<4, be

an ordered basis of V such that (1 | 2) (e, ^ e2 ^ ea ^ e4) = 1. We identify Ends(V ) and V| with M4(S), Alt(V| with Alt4(S), and Pf(Ends(V)) with S through the above choice of a basis of V Let u be the matrix of h: V*-~ V with respect to the basis {ei} and the dual basis {e*}. We have u'=eu, where E is the type of ~, and o-~ (x)=ux'u-~. Since

X--F,,blX t/A-I=(Xu-(Xu) t) u-1 =U(U-'X--(U-'x)t),

the maps y~-~c~(l| u and y~-*u-la(l| induce isomorphisms S| The map l | V | 1 7 4 is given by z~-~zu, zem4(s). Thus we have Pfz(y)=pf(c~(l | We define p(y) = c~- 1 ( u p , (a(1 | y) u)). By construction, we get

~(yp(y)) = c((1 | y) up, (e(1 | y) u) = Pfx(y)

c~(p(y) y)= up1 (ct(1 | y) u) ct(1 | y) uu- '

= u p f ( ~ ( 1 @ y) u) u - 1 = pf~(y)

The fact that p(y)eAlt~(A) follows easily by faithfully flat descent. We now verify that pZ= d. It follows from the relation

pf (u x u') = det (u) pf (x)

for the classical pfaffian, that

P 1 (u x u t) = det (u) u - ' p, (x) (u t) - '.

(p 2 (y)) = u p, (u p, (c~(1 | y) u) u)

= e det (u) u u-1 p2 (e(1 | y) u) (u')-'

= de((1 @ y)

I 0 --a34 a24 --a23X~ a34 0 --a14 a13

\a2a --ala alz

property. In particular we get p 2 = l . Let now p'~"


since det(u)=d. The last claim, about p', follows easily by descent from the corresponding result for P'I, proved above. []

We call p the pfaffian adjoint of A (for the involution a and the choice 2: Pf(A) -~ R).

Corollary 3.2 Let al , 0- 2 be two involutions on A such that the corresponding pfaffians PfI(A), Pf2(A) are free and let ~.,: PfI(A)-~ R be isomorphisms. Then the quadratic spaces (Alt ' ' (A), P f J and (Alff2(A), Pfa2) are similar.

Proof. We first assume that R is local. The automorphism al a21 of A then is inner, say ffl(X)=VffE(X)V-1. We have ai(v)=#v where ]2=e 1 ~2,/~i being the type of ai. Further x~-~xv and x~-~v- 1 x are isomorphisms Alff' (A)-~ Alff~(A). Let p'2(x)=vpx(xV). By 3.1, there is peR" such that p'2=pp2 and x~-~xv is a similitude with multiplier p. In general, we put

I= {xEAlal a21(a)x=xa, VaeA}

so that I is locally generated by v as above. Since p (as above) is uniquely determined by v, we can define b: I | I ~ R locally by b(v, v) = p and the multiplication in A induces an isometry

(Alt"(A),Pfj| []

We now compute the Clifford algebra of (Alff(A), Pf~). We take on M2(A ) the checker-board gradation.

( 0 O) yields a n Proposition3.3 The map Alf f (A)~M2(A) given by x~-~ p(x)

isomorphism of graded algebras C(Alt'(A),Pfz)-~ M2(A). In particular Co(Alff(A),Pfa) is isomorphic to A x A and the centre of Co(Alff(A),Pf).) is isomorphic to R x R. Furthermore, the standard involution of C(AIt~(A), Pfa) corre- sponds to

z(a dJ=\ea( (a(d) ea(b)~,a,b,c,d~M2(A)

where e is the type of a.

Proof. It is immediate from 3.1, by the universal property of the Clifford algebra, that the given map induces a homomorphism of algebras C(Alff(A),Pfz) --* M2(A). Since both are Azumaya algebras of the same rank this map is an isomorphism. The last claim follows from the fact that z is an involution and its restriction to the image of Alff(A) in M2 (A) is minus the identity. []

Let Q(A) be the similitude class of (Alff(A), Pfz). Obviously Q(A) does not depend on 2 and, in view of 3.2, Q(A) is independent of a. It follows that Q(A)= Q(B) if A and B are isomorphic Azumaya algebras of rank 16. We now prove the converse, so that:

Theorem 3.4 Two Azumaya algebras of rank 16 with involutions are isomorphic (as algebras) if and only if Q(A)= Q(B).


Proof As already remarked, A "~ B implies Q (A)= Q (B). Conversely, if we have (AIt'(A), Pfa,) -~(Alt"(B), Pfa~) | (I, d), then (see Sect. 1)

Co (Alt ~ (A), Pike) ~- Co (Alt" (B), Pfa~)

and, by 3.3, A x A ~- B x B. This implies that A -~ B. []

Remark. The explicit description of the Clifford algebra of the pfaffian associated to a rank 16 algebra with involution, given in 3.3, will be used in Sect. 6 to compute the group of special similitudes of a rank 6 quadratic space.

4 The Albert form, a theorem of Jacobson

In this section, we assume that 2 ~ R'. Let A = A x | A 2, A i being rank 4 Azumaya algebras over R. Let o=c r lQcr 2 denote the tensor product of the standard involutions on Ai (i.e. such that cri(x)xeR is the reduced norm). By 2.2 the discriminant module (Pf(A), #) is trivial. Let 2: (Pf(A), ~)"~ (R, 1) be the isometry given in 2.2. We shall describe the space (Alt(A), Pf2) as the Albert form of the pair (A~, A2) defined as follows. Let A'i denote the orthogonal complement of 1 in A i with respect to the reduced norm, i.e. A'i={xeAiltrA(x)=O}, tra denoting the reduced trace. Let hi: A'~--,R be the restriction of the reduced norm to A'~. The Albert form of the pair (A1, A2) is the quadratic space

Q(A1, A2)---(A'I, nx)_L (A'2, --n2).

Proposition 4.1 (AltO(A), Pfx) ~- Q(A1, a2).

Proof We obviously have A'I | Thus it clearly suffices to prove the proposition in a faithfully fiat splitting. Let ~: SQAI-~M2(S), i= 1, 2, be a faithfully flat splitting of Ai such that el(1 @ al)e~-1 is the standard involution

tr' of M2(S), i.e. ~'(x)=uxtu -~, x~M2(S ), u=(_Ul 10). We put o ' (x)=~. Then /

~| S|174 transports a l | into the involution

The type of a' is 1 since the type of ai is - 1. Thus,

Alt" (M4 (S)) = { x - o' (x)[x s M 4 (S)}

={(ya 1 x'lgt),x,y~S,a~M2(S) }.

The basis e, of P f A constructed in 2.2 is

e. = (det u) 2 gl A g2 ^ g3 A g# = gl ^ gz ^ g3 A g4


where {g~}, 1 <i__<4, is the standard basis of S 4. In view of the computation of Pfa given in Sect. 2,

\ du yu

= - x y - d e t ( a ) .

Let a=z. 1 +a' , zeS, a'~M2(S) with tr(a')=O. Then,

Pfx(yal x_' ~)= det (~ Xz) - det (a')

.(; :z).,o, n' denoting the restriction of the determinant form to M2(S)'. Thus,

(Alt ~ (M4 (S)), Pfx) ~- (M 2 (S)', n') A_ (M2 (S)', - n')

Q (M2 (S), M2 (S)).

This proves the proposition. []

The following theorem includes a result of Jacobson for fields (see [J]).

Theorem 4.2 Let A1, A2, B1, B2 be Azumaya algebras of rank 4 over a ring R in which 2 is invertible. Then AI |174 if and only if the Albert forms Q(A1, A2) and Q(BI, B2) are similar.

Proof The claim is an immediate consequence.of 4.1 and 3.4. []

5 Splitting of rank 16 algebras into tensor products of rank 4 algebras, a theorem of Albert and an application

In this section, we give some conditions under which rank 16 Azumaya algebras, whose classes in the Brauer group Br(R) are of order 2, split into a tensor product of rank 4 algebras. All these results are generalizations of a result of Albert for fields (see [A] ). We begin with the following

Theorem 5.1 Let A be an Azumaya algebra of rank 16 over a semilocal ring R. I f the class of A in the Brauer group of R is of order 2, then A is a tensor product of rank 4 Azumaya algebras.


Proof. Since the class of A in the Brauer g roup of R is of order 2 and R is semilocal, there exists an involut ion o on A (see IS]). Let 2: Pf(A)--*R be chosen such that (AltO(A), Pfx) represents 1. Any quadra t ic space over a semilocal ring is the o r thogona l sum of binary spaces. So we can decompose Pfz ~-q~-l-qzlq3, qi binary forms, and we can assume that q~ represents 1. If q =q2_l_q3, and if 6 denotes the signed discriminant , we have 6(q)=6(qO, 6(Pf~) being 1, further C(Pf~)~-C(qO| as graded algebras and C(qO

Mz(R). It follows that Co(Pfz) e ~- C(6(q I ) q), where e is an idempoten t generating the centre R x R of Co(Pf;.). In view of 3.3 we get A~-C(6(ql)q) ~- C(6(qx) q2) | C(6(q~) 6(q2) q3). [ ]

Observe that we do not assume 2 invertible in 5.1. Fo r the next result we need that 2 is a unit. We have seen in Sect. 2 that if A = A a | is a tensor p roduc t of two A z u m a y a algebras of rank 4, then A has an o r thogona l involut ion a, namely trl| t7 i denot ing the s tandard involut ions of Ai, with (Pf~,| 6) trivial as a discr iminant module . The following is a converse to this fact.

Theorem 5.2 Let A be an Azumaya algebra of rank 16 over a ring R in which 2 is invertible. Suppose A has an orthogonal involution a such that (Pf(A), ~) is trivial as a discriminant module. Then A is isomorphic as an algebra with involution to a tensor product of rank 4 Azumaya algebras with involution the tensor product of the standard involutions.

Proof. Let 2: (Pf(A), ~)=~(R, 1) be an isometry. By 3.1, there exists a m a p p: Alt ~ (A) - Alt ~ (A) such that x p (x) = p (x) x = Pfz (x) and p2 = 1. Let

AltO(A) + = {x + p(x) txeAlt~(A)}

AltO(A) - = { x - p ( x ) lxeAlt~(A)}.

It m a y be verified, using the identities xp (x )=p(x )x=Pfx (x ) and p2(x) = x , xeAlt~(A), that AltO(A) + and AltO(A) - are mutua l ly o r thogona l for the form Pfx and AltO(A)=Alt~(A)+| -. Let n + and n - denote the restic- t ions of Pf~. to AltO(A) + and AltO(A) - respectively. Then,

(Alt ~ (A), Pfa) ~- ( Alt~ (A) +, n +) l (Alt ~ (A) - , n -).

We verify tha t Al t ' (A) + and Al t ' (A) - are rank 3 modules by going over to a splitting. Since 2 e R ' , there exists a faithfully flat splitting S | A ~-M4(S) for A such that 1 | t r anspor t s to x~--~x t on M4(S ). Clearly, P=Pl (see Sect. 3) and S | AltO(A) + = {x + pl (x), x ~Alt4(S)}, S | AltO(A)- = {x - p~ (x), x~Alt4(S)} are r ank 3 free modules over S. Since the Ar f invar iant of (Alt ' (A), Pfx) is trivial, disc(Alt ' (A), Pfz)~-(R, - 1 ) , disc denot ing the unsigned discriminant. In part icular , disc(Al t~(A)+,n+)~-(R,-1) | In view of ( [ K O S ] ) , any rank 3 quadra t ic space over R is similar to (B', n), where B is a rank 4 A z u m a y a algebra, B' its submodu le of trace zero elements and n the reduced norm. Thus there exist A z u m a y a algebras B~, Bz and a d iscr iminant module (I, d) such that

(Alt ~ (A) +, n +) -~ (I, d) | (B], n)

(Alt ~ ( a ) - , n - ) _ (I, d) | (B~, - n)


(noting that disc(B'i, n)-~(R, 1)). Thus,

(Alt ~ (A), Pfx)~-(I, d ) | Q (B~ , B2) ,

where Q(B~, B2) is the Albert form of the pair (B~, B2) and by 3.4 the algebras A and Ba @ B 2 are isomorphic. The fact that there exists an isomorphism of algebras with involution is a consequence of the following proposition. The notations are the same as in the proof of 5.2.

Proposition 5.3 Let R be a ring in which 2 is invertible. Let (A, a), (B, z) be Azumaya algebras of rank 16 with involutions of orthogonal type. I f the diserimin- ant modules (Pf(A), ~) and (Pf(B), z') are trivial, the following properties are equivalent: i) A and B are isomorphic as algebras with involution. 2) There exists a similitude f : (Alt'(A), Pfa)~-(AW(B), P f J | (I, d) such that

f (Al t" (A)+)=Al f f (B) + | and f ( A l t ~ - @I.

Proof 1)=~2): Let ~0: (A, tr)~-(B, z) be an isomorphism of algebras with involution, Then q~ restricts to an isomorphism q~: Alt~(A)-~AW(B) as R-modules and gives rise to an isometry 3: (Pf(A), 6 ) ~ (Pf(B), ~). Let 21:(Pf(A), ~)-~ (R, 1) be an isometry. We set 2z=21o~ -1. Then ~0 is an isometry of (Alt'(A), Pfa,) with (AW(B), Pfa2)" Let p, , p~ respectively, be the map p defined in 3.1 for AltO(A), AW(B) respectively. After a scaling, pc= +_~opMo -~. Thus we get r +) = Alt r (B) +, q~ (Alt ~ (a ) - ) = AlP (B) -. 2 )~1 ) : The similitude f induces an isomorphism Co(f): Co(AltO(A), Pfz,) -~Co(AW(B), P f j which satisfies Co(f ) ( x y ) = i ( f ( x ) | x ,y~Alt~(A), where i: Co(AW(B), P f j | I @ Co(AW(B), Pfa~) | I ~ Co(AW(B), P f J is the map i ( xQ~ |17 4 We have Co(Al t~(A))=A• Co(AW(B)) = B x B. We claim that the composite

f * : A d Co~f) ~ A x A , B x B ,B,

where A is the diagonal map and g is one of the projections, is an isomorphism respecting the involutions. To prove this, we may assume (by localizing) that (1, d)=(R, #) so that f : Alt~(A)-~AW(B) is a. similitude with multiplier #~R'.

The imageo fA l t ' (A )~Mz(A) , x~ -*{ (pOo(x ) ; ) ( s ee3 .3 )genera te sM2(A) .S ince

Alt ~ (A) = Alt ~ (A) + @ Alt r (A)-

and p~(x) = x if x~Alt~(A) +, p~(x) = - x if xeAl t"(A)- , an easy verification shows that A(A) is generated by products x y with x, y both in Alt"(A) + or AltO(A) -. Since f (A l t " (A )+)=AW(B) + and f ( A l t " ( A ) - ) = A W ( B ) - and Co(f ) (xy) = # - l f ( x ) f ( y ) , it follows that Co ( f ) maps A (A) to A (B). The map Co (f) respects the standard involution on the even Clifford algebras and the involutions on Co(AltO(A)), Co(AW(B)) respectively, are given by (a, b)w-~(a(b), ~r(a)), (a, b)~, (z(b), z(a)) respectively. Thus f * respects the involutions. []


The next results are immediate consequences of 5.2. In view of the example given in [ART] (see the Introduction), they are of interest even for central simple algebras over fields.

Corollary 5.4 Let R be a ring in which 2 is invertible and let A be an Azumaya algebra of rank 16 which has an involution a of orthogonal type. There exists a a-invariant rank 4 Azumaya subalgebra of A if and only i f(Pf(A), ~) is a trivial discriminant module.

Proof If (Pf(A), ~) is trivial, there exists a a-invariant rank 4 Azumaya subalgebra in view of 5.2. Conversely, if there exists such an invariant subalgebra A~, then A ~ - A I | a is the tensor product of involutions a l , a2 on A1, A2 and the claim follows from 2.2. []

Corollary 5.5 Let R be a ring in which 2 is invertible and let A be an Azumaya algebra of rank 16 which has an involution a of orthogonal type. Assume that a has an invariant Azumaya subalgebra of rank 4. Let u be a unit of A such that o(u)=u. The involution z = I n t u o a admits a z-invariant rank 4 Azumaya subalgebra if and only if nA (U) is a square in R.

Proof The claim follows from 5.4 and 2.1. []

Remark 1 Let K be a field of characteristic not 2, with separable closure /( and Galois group G. Let O, be the orthogonal group of the diagonal form (1, ..., 1) and let PO, be the corresponding projective group. Let n be even. It is known that H 1 (G, PO,) classifies central simple K-algebras of dimension n z with an involution of orthogonal type. Central simple K-algebras of dimension n 2 with an involution of orthogonal type of trivial pfaffian discriminant are classified by H I(G, PSO,) (see [KS]).

Remark 2 We give an example of a domain R, in which 2 is invertible, which admits an Azumaya algebra of rank 16 whose class is of order 2 in Br(R), but which is not a tensor product of rank 4 Azumaya algebras. Let R = K [X, Y], K being a field of characteristic ~ 2 which admits a rank 6 anisotropic quadratic space qo of discriminant - I (i.e. the Arf invariant of q0 is trivial). In view of [P], there exists an indecomposable quadratic space q of rank 6 over R whose reduction modulo (X, Y) is isometric to qo. Thus disc q = - 1. Let Co be the Clifford algebra of q and e an idempotent generating the centre of C 0. Then Co e is a rank 16 Azumaya algebra over R, which is of order 2 in Br(R). We claim that Co e is not isomorphic to a tensor product of rank 4 Azumaya algebras. Suppose, indeed, that Co e ~ A I | Ai rank 4 Azumaya algebras. By Theorem (9.4) of [KPS], q would be similar to the Albert form Q(A 1, A2) which is an orthogonal sum of spaces of rank 3, a contradiction to the indecom- posability of q. We observe that, in contrast, if R = K [X], char K 4= 2, every Azumaya algebra A over R whose class is of order 2 in Br(R), being extended from K, is a tensor product of rank 4 Azumaya algebras.

Remark 3 Let K be a field of characteristic 4:2 and let L = K ( ~ d ) be a quadratic field extension of K. Let D be a rank 16 central division algebra over L whose class in Br(L) is of order 2. To D we have associated the similarity class Q(D) of a 6-dimensional form. We claim that D is extended from a division algebra over K if and only if the similarity class Q(D) is extended from K. Let q be in the class Q(D) such that 2 q - ~ L | . . . . . as) for some 2~L and


ao, ..., aseK'. If Illa i - - - i in K'/K "a, then <ao . . . . . as> is similar to a form (AltO(H1 | , Pf), Hi, i= 1, 2 being quaternion algebras over K, and hence D =~ L | 1 | by our classification. If Hi ai @ -- 1 in L, then H i ai =- d in K'/K "a, since Ili ai = - --1 in E. Then (ao . . . . . a4 , -das> has trivial Arf invariant and its similarity class over L is also Q(D). The proof can be completed as in the case//~ a~- - 1. The converse is clear. We remark that the statement is analogous to the statement for rank 4 central division algebras that the algebra is extended if and only if the reduced norm is extended.

6 The group of similitudes

Let (M, q) be a quadratic space over R. We recall that a similitude of (M, q) with multiplier #~R" is an R-linear isomorphism f : M-~ M such that q(f(x)) = # q (x) for all x e M. A similitude f induces an automorphism Co ( f ) of Co (M, q) such that Co (f) (x y)= p - i f ( x ) f ( y ) for x, y ~ M (see Sect. 1). We call a similitude f special if C0(f) induces the identity map on the centre Z of Co(M, q). We denote by GO(M, q) the group of similitudes of (M, q) and by GO+(M, q) the subgroup of special similitudes. In this section, we shall compute the group GO+ (M, q) of a rank 6 quadratic space (M, q) over any commutative ring R with Pie(R) trivial. We also indicate the computation of this group if the rank of M is equal to 4.

Let (M, q) be a quadratic space over R of rank 6 and with trivial Arf invariant. Let A = Co(M, q)e, where e is an idempotent of Co (M, q) generating its centre over R and let P = C l f , where f = l - e . We have an isomorphism ~p: A| defined by ~o(a| z denoting the standard involution on C(M, q) ([KPS]). Let Pf(P) denote the pfaffian defined with respect to ~0. There exists an isomorphism 2: Pf(P)-~R such that (M, q) is isometric to (Alt(P), Pfz) under the map 0: M ~ A l t ( P ) given by O(x)=xf (see [KPS]).

Lemma 6.1 Let ~IER', aeA'. The map ~(,,a)=tl~(a| restricted to Alt(P) is a special similitude with multiplier t l 2 n(a), n denoting the reduced norm of A.

Proof To prove that ~r is a special similitude, with multiplier t/2n(a), we may replace R by a faithfully flat extension. We therefore assume, without loss of generality, that A=EndR(V) with V free, P = V | <o: EndR(V) |174 V) the canonical map, Al t (P)=A 2 V and {el}, 1 <i_<4 a basis of V such that ea ê2 ê3 ê4=J.-I (1)ePf(P)=Aav. Clearly, Pf~p(,,,) = qa n(a) P f where n(a) is the determinant (reduced norm) of a~Endn(V). Thus O(n,a) is a similitude with multiplier q2n(a). We show now that it is special. Let p =qZn(a) and let g: M ~ M be the similitude 0 - ' ~ ( , , , ) 0 with multiplier #, 0 being the isometry of (M, q) with (Alt(P), Pfz) defined earlier. The map M -~ C 1 (M, q) defined by x ~-~ g (x) f + # - 1 g (x) e extends, by the universal property of the Clifford algebra, to an isomorphism g of C(M, q) whose restriction to Co(M, q) is Co(g). We shall show that Co(g) is identity on Z = R e x R f We first verify that g = Int(a + q- ' z(a)- ~). Writing elements of C(M, q) as matrices

Co e Cl f~ C, e Co f ]


we need to check that

( ; t/-~z(a) -1 ) (x O ; f ) ( a ; ~ t/v~a))=(# - ~ ( x ) e g(o)f)"

By definition, we have g(x)f=qaxfz(a) and by the following Lemma 6.2, we get t/- 1 ~ (a) - 1 x e a - 1 = t/- 2 n (a)- 1 g (x) e. Thus ff = Int (a + t/- 1 z (a) - 1), whose restriction Co(g) to the centre Re x Rfis identity. []

Lemma 6.2 Let g: M - ~ M be an R-linear isomorphism and tlER" , a~A'=Co e" be such that for all x~M, g(x) f =tla(xf) z(a). Then, for all xeM,

g(x) e=qn(a) z(a)-' xea- ' ,

where n denotes the reduced norm on A.

Proof It suffices to prove this in a splitting. We therefore assume M=Alt4(R) with the form given by 2p f, 2eR'. The Clifford algebra is M2(M4(R)) and the embedding M ~ C 1 (M, q) is given for xeM by

(where Pl is as in Sect. 3). Let a=(V 00) 0 cA=C~ =(~ tlVoV ) and

e, v e M4(R). Then g(x) f

0 0 g(x)e=(pl(qOxv,) O)=(rldet(v)(vt)-lpl(xv-lvxv t) ~)

so that g(x) e =tln(a) z(a)- 1 xea- 1. This proves the lemma. []

We thus have a homomorphism

~p" R'xA'--*GO+(M,q)

given by ~k(q, a)=O-l~(,,a)0 (we recall that O(x)=xf for x~M). We claim that the kernel of this homomorphism is the subgroup {(q2, r/-l)lq~R.}. To prove this, it suffices to show that the condition q ~o(a | a)= 1e implies that a is central in A. This again can be proved by going over to a faithfully flat extension and assuming that A--C o e = EndR(V) with V free. Let {ei}, 1 < i<4, be a basis of V such that e I ^ e 2 A e 3 A e4 = 2- 1 (1)e Pf (P). If u is the matrix of a e End R (V) with respect to this basis, qq~(aQa): A l t ( V | 1 7 4 given by x~-*rluxu t, xeAlt4(R ). If tluxut=x, Vx~Alt4(R), u is central in EndR(V) is a consequence of the following

Lemma 6.3 Let u~Gl4(R) and tieR" be such that x=rluxu t for all x in Alt4(R ). Then ueR'.


Proof. Let u = ( ~ b~), ~, r, ?, 6~M2(R). Choosing X=(o ~ ) w i t h ~ = ( ? 1 10)'

the condition x=rluxu' yields n a e a ' = e and ~=0. Similarly, setting x = ( 0 ~) 0 '

we get t / f e 6 ' = e and f l=0. Taking x = ( ~ 0)' we get a=fi . Finally, writing x

=rlUXU'forx=(gf f Oh),flarbitraryinM2(R),weseethat~isascalar.

Theorem 6.4 Let (M, q) be a quadratic space of rank 6 with trivial Arf invariant, let e be an idempotent which generates the centre of Co and let A = Co e. There exists an exact sequence

t ~(R" x A')/{(t/2, t/-'), t/eR'} v

GO+(M,q) , Pic(R).

Proof. The map tff is induced by qJ: R ' x Co e ' ~ GO+ (M, q) as defined above and is injective by 6.3. Let gEGO+(M, q) with # as the multiplier. The map M ~ C ( M , q) given by x ~--~g(x)f+p-lg(x)e extends to an automorphism g of C(M, q). Let I~= {x~C(M, q)l~,(x) y= yx, V yeC(M, q)}. Since C(M, q) is an Azumaya algebra, I~ is an invertible R-module and ~ is inner if and only if the class [I~] in Pic(R) is trivial. We put v(g)=[I~]. Assume that this class is trivial, so that there exists ueC(M, q) such that if= Int u. Since the restriction of g to Co(M,q) is identity on its centre R e x R f ue=eu and u f = f u . This implies that ueCo(M,q)'. Let u=a+b, aECoe, beCof We have uxu -1 =g(x) f+ p- l g(x) e for xeM. Since z(g(x)f + # - l g ( x ) e ) = g(x)f+ #- 1 g(x) e, z (uxu-~)=uxu -1 for xeM; i.e. z(u)u commutes with M and hence is central in C(M, q). Let z(u)u=FleR'. Then (z(b)+z(a)) (a+b)=r(b)a+z(a)b=rl with z (b) a e Co e, z (a) b e Co f. Thus, z (b) a = r/e, z (a) b = t/f. Further, (a + r/z (a)- i f ) x =(g(x)f +p-~g(x)e)(a+tlz(a)-l f); i.e. g(x ) f=r l - laxz (a) f +rl-laxfz(a). Since g(x)f=(OgO-1)(xf)=Ot,_,,,~(xf) for all xeM, it follows that g--@(r/-1, a). []

We next compute the group GO + (M, q) if (M, q) is a rank 6 quadratic space with arbitrary Arf invariant. Let Z be the centre of C o and let z denote the restriction of the standard involution z of C(M, q) to Co. Then z is an involution of second kind on the rank 16 Azumaya algebra C O over Z. In particular z induces the (unique) nontrivial R-linear involution of the quadratic algebra Z. We have an isomorphism of Z-algebras Z | 2 1 5 given by x| xz(y)). It follows that Z Q ( M , q) is a quadratic space of rank 6 over Z with trivial Arf invariant. Thus Z @ (M, q) is isometric to a pfaffian arising from a 2-torsion datum. We now describe this datum. We view Ca as a Z-module through the left action of Z. Then ~o: Co| z Co---" Endz(C~) given by ~o(a @ b) (x)=axz(b) is an isomorphism and we claim that (Co, Ca, ~o) is the required datum. Let Pf: Alt(C~)~ Pf(C~) be the pfaffian of the datum (Co, C~, ~o). The multiplication in C induces an isomorphism Z | AIt(C~) and it is shown in [KPS] that there exists an isomorphism 2: Pf(C1)-~ R such that

Z | (M, q) -~ (Alt (C~), Pf~).


Thus there exists a (Galois) descent map 6: (Alt(C0, Pfz)-~(Alt(C0, Pf~) such that the descended quadratic space {xeCl l f (x )=x} is (M, q)(see [KPS] for details). In view of the computation of GO§ for the trivial Arf invariant spaces, we have a homomorphism

O: Z" x Co ~ GO+ (AIt(C,), Pfz).

For any Azumaya Z-algebra B with an involution z of the second kind, let GU(B) = {beB'l z(b) beZ'}.

Lemma 6.5 Let rleZ', beC'o. Then ~9(,,b) belongs to GO+(M, q) if and only if beGU(Co) and #tl-l =n(b)(z(b)b) -2, n denoting the reduced norm on Co, bar denoting the effect of the non-trivial automorphism ~o of Z over R.

Proof The map Ot~,b) belongs to GO+ (M, q) if and only if it commutes with the descent 6 on (Alt(C1), Pfz) (noting that the descent of a special similitude is automatically a special similitude). Clearly, it suffices to prove the lemma after a base change SIR with S faithfully flat over R. Thus, we assume, without loss of generality, that Co = Endz(V), V free over Z, C1 = V| V Let {ei} be an ordered basis of V such that el AezAe3Ae4=2-1(1)E/k4V =Pf(Al t (V | V)). We identify Endz(V) with M4(Z) and Alt(V| z V) with Alt4(Z) through the choice of this basis. We denote by x~--,~ the extension entrywise o f a o to M4(Z ). Let r on Endz(V ) be given by x~--~u-a~tu, ueGL4(Z). The descent map 6 on Alt4(Z) is, up to a scalar, the map x~--~t - 'p-~O -~. The map O(,.b)is given by ~b~,,b)(x)=qbxb t, xeAlt4(Z). The condition O~,,b)6 ---6 ~(,, b) is equivalent to the following two conditions: 1) The element 7= u- lNub is central; i.e. r ( b ) b e Z ~ (that 7 is central is a consequence of 6.3 since, for all xeAlt4(Z)), 7xr for some peZ. 2) (z(b) b)- 2n(b)=#tl - ~.

This proves the lemma. [ ]

Let H be the subgroup of Z" x GU(Co) defined by

H = {(t/, b)eZ" x GU(Co)I#q- 1 = nn(b)(z(b) b)- 2}.

Theorem 6.6 Let (M, q) be a quadratic space of rank 6 with Arf invariant Z. The association (rl, b)~--~k(,,b ) yields a homomorphism ~k: H ~ GO+(M, q) such that the induced sequence

1--*H/{Olz, rl-x),rlEZ "} ~' , GO+(M,q)--*Pic(Z)

is exact.

Proof. This theorem is immediate from 6.5 and 6.4. []

We now indicate a method of computation of the group of similitudes of rank 4 quadratic spaces. Let (M, q) be a quadratic space of rank 4 and Arf invariant Z over R. The map Z | M--* CI(M, q), (~, x)~-~(x is an isomorphism and (M, q) is in fact a descent of the reduced norm on C1 as a Co-module, the descent map being given by z, the restriction of the standard involution of C(M, q) to Cl. There is a map

R" x Co "-* GO + (M, q)


given by (/7, b)~--~0~,,b), where O~,,b)(x)=rlbxz(b). The multiplier of the map O~,,b) is rl2n(b) n(b), where n is the reduced no rm of Co as a Z-algebra. This map is surjective 'if Pic(Z) is trivial and has kernel {(#/i, # -1 ) , /~eZ '} . We thus have,

Theorem 6.7 Let (M, q) be a rank 4 quadratic space over R with Ar f invariant Z. Then the sequence

1 --* (R" x Co)~{# ~, I~- 1), Ft ~ Z'} ~ GO + (M, q) --* Pic (Z)

is exact.

Theorem 6.8 Let (M, q) have trivial Ar f invariant and e an idenpotent in the centre of Co(M, q) generating the centre over R and A = C o ( M , q)e. Then the sequence

1 --* (A" x A')/{(t/, r/- 1), r/~ R'} --* GO+ (M, q) --* Pic(R)

is exact.

We do not give the details of proofs of 6.7 and 6.8.

Remark. In Theorem 6.4 and Theorem 6.6, the image in Pic(R), resp. Pic(Z), of the last map of the exact sequence, consists, in fact, of elements of torsion 4 and, in 6.7 and 6.8, of elements of order 2. This follows from Theorem 3.1, p. 111, of [KO] . We also observe that the above sequences are parts of c o h o m o - logy sequences induced by exact sequences of sheaves.

Acknowledgements. The first author thanks the Tata Institute for the hospitality accorded during the preparation of this paper.

References

[ART]

[A]

[D]

[J]

[K]

[Kn] [KO]

[KOS]

[KPS]

[KS] [P]

[S]

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Transcript of Math. Z. 206, 589-604 (1991) Mathematische Zeitschriftypandey/R.Sridharan/91.pdf · Z. 206, 589-604...