Ketu* and the Second Invariant of a Quadratic Spaceypandey/R.Sridharan/93ktheory.pdf · KETU AND THE SECOND INVARIANT 505 order zero at every codimension 1 generization of y belonging

K-Theory 7: 501-515, 1993. 501 © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

Ketu* and the Second Invariant of a Quadratic Space

M. O J A N G U R E N Section de Mathdmatiques, UNIL, CH-IO15 Lausanne, Switzerland

R. P A R I M A L A and R. S R I D H A R A N School of Mathematics, TIFR, Bombay 400005, India

(Received: February 1993)

Abstract. Using the Chern classes defined by Grothendieck, we map a K-theoretic invariant of invertibte symmetric matrices defined by Giffen to the Clifford invariant.

Key words. Chern classes, quadratic forms, Clifford invariant.

1. Introduction

An invariant for quadratic forms over a commutative ring A with values in a quotient of K2(A) has been defined by Giffen [6]. He also verified that if A is a field, his invariant maps to the Clifford invariant of the quadratic form under the well-known symbol map. to the Brauer group.

Using Grothendieck's theory of equivariant Chern classes [7], Shekhtman [13] and Soul6 [14] defined homomorphisms

c~i: Ki(A) ~ H~t~-~(Spec A, #~)

for all possible i, j and m. We show that Giffen's invariant maps under c22 to the Clifford invariant in H2t(Spec A,/~2), defined by the connecting map associated to the sequence

t ~ P2 ~ Spin ~ SO --* 1.

The method of proof involves basically two steps. The first is a general reduction to rank 2 quadratic spaces, via some calculations of &ale cohomology groups of affine quadrics. The second is an explicit calculation of the invariants for rank 2 quadratic spaces over smooth affine curves. In this calculation we use Suslin's results on K-cohomology of smooth affine varieties. Since no explicit description of c22 is known, except on elements generated by symbols, our results can be interpreted

*According to Herrmann GraBman's W6rterbuch zum Ri#-veda, Ketu may not be connected with K2 but 'es bezeichnet das, was sich sichtbar oder kenntlich macht..."

502 M. OJANGUREN ET AL.

as a computation of c22 on that part of K2 arising from invariants of quadratic forms.

In what follows A will denote a commutative ring in which 2 is invertible.

2. The Giffen Invariant

Let A be a commutative ring in which 2 is invertible, WG(A) the Witt-Grothendieck group of A (see, for instance, I-9]) and Wo(A) its quotient by the subgroup generated by the hyperbolic plane H(A). We denote by Wl(A) the kernel of the forgetful map

So: Wo(A) ~ Ko(A)/2Z.

Every class of WI(A) is represented by a symmetric matrix of even rank. We define a homomorphism (called discriminant)

st: WI(A) ~ KI(A)/Tr(KI(A)),

where Tr(a) = a + a t. Let S be a symmetric 2n x 2n matrix representing an element of W~(A). We set st(S) = ( - 1)" (class of S). Let Wz(A) be the kernel of sl. Every class of Wz(A) can be represented by an elementary symmetric matrix whose rank is a multiple of 4. Let ES(A) be the subset of E(A) consisting of symmetric matrices of rank divisible by 4 and trivial discriminant.

We say that a and fl in ES(A) are equivalent if there exists an elementary matrix e and integers m, n, such that

where Jk is the matrix with k diagonal blocks equal to

These equivalence classes of elementary matrices form an Abelian group EWz(A) in which the class of J2,, is the identity and the class of - 0~ is the inverse of the class of e. Clearly, W2 is a quotient of EWz(A). We define a map

s2 : EWz(A ) --* Kz(A)/Tr(Kz(A)).

This time, Tr is the map e ~ c~ - c(, where c~ ~ c~ t is the involution induced on Kz(A) by the unique involution of the Steinberg group St(A) that maps each generator x~s(2 ) to xs~(;.).

Let ~ ~ ES(A) c~ GL4,(A) represent an element of EWz(A) and ~ denote a lift of in St(A). We set

s2(00 = n ( - 1, - 1) + (class of ~-l~t).

Clearly this respects the equivalence in ES(A) and passes down to a homomorphism s2.

Remark. The Steinberg group is written multiplicatively, but Kz additively.

KETU AND THE SECOND INVARIANT 503

3. Catch 22 and Transposition

We recall a few facts about equivariant Chern classes, from [7], [13], and [14]. Let X be a G-scheme, i.e. a scheme X with a discrete group G operating on it. A

sheaf f f over X is a G-sheaf if for every g ~ G there is a morphism of sheaves

% : g * ~ - ~ such that (Doh = (ph°h*((Po) for all g, h s G and (Pc = Ida. A global section s of f f is invariant if %(s) = s for every g. This condition makes sense because g * ~ ( X ) = ~ ( g - I ( X ) ) = i f (X) . Associating to every Abelian G-sheaf the group of its global invariant sections, we get a functor whose nth derived functor is, by definition, the nth equivariant cohomology group Hn(X, G, i f ) .

Suppose now that g is a G-vector bundle of rank r over X. The projective bundle P(¢) associated to E is a G-scheme. For every integer m invertible in (9 X let/~m be the sheaf of ruth roots of unity on X or P(g). The sheaf ~ determines a canonical element [0~(1)] ~ HI(P(~) , G, Gin) which maps, under the connecting map 8 associated to the Kummer sequence

1 "-~/,I 2 ---~G m 2--~ Gm-"~ 1,

to an element ~(~) = 810p(¢)(1)] ~ HE(p(¢), G, Pro). By [7],

H*(P(~), G, ~m ~r) = @ H*(X, G,,u~('-s))~ s, O<~j<~r-1

where ~s is the j th cup-product power of ( = ((g). Thus, ~" is a linear combination of its lower powers and there is a relation

~r ._[_ Cl(~o)~r- 1 ..]_ .. . + Cr(~ ) = O,

where c~(g) ~ H2i(X, G, p~) .

If G operates trivially on X, the Kfinneth formula gives, for any j, a natural homomorphism

H2J(X, G,//m ~j) ~ @ Hom(H~(G, Z), H2s-'(X, p~J)) O<~i<<.2j

which, for i = j = 2, maps c2(6 ~) to a homomorphism

(P22(6~): H2(G, 7/) --, H2(X, pm~2).

Taking now X = Spec A, 8 = W, m = 2 and G = GL,(A) with the trivial action on X and the natural action on 8, we get, for any r, homomorphisms

q¢*)" Hz(GL,(A), 7/) ~ H2(Spec A, P2) 22"

which are compatible with the inclusions GL~ = GL,+~. We therefore have a homomorphism

c22: K2(A) = H2(E(A), Z) --* U2(Spec A, P2).

P R O P O S I T I O N 3.1. For any a ~ K2(A), c22(a) = c22(~').

504 M. OJANGUREN ET AL

Proof. We denote by # the inverse of the transpose of matrices and of dements of K1, K2, etc. On K2(A) = H2(E(A), Z) the involution # is induced by # on E(A), hence we only have to show that ~p~2r)2(e '~) = q~)2(c 0 for any e E H2(GLr(A),Z). Let G = GL,(A). The isomorphism # : G ~ G induces # : HZt(x, G, p2) ~ HZi(X, G, Pz) with the property that, for any G-bundle ~ on X,

#ci (~) = ci(~#), (1)

where ¢# is ~ with the action of G twisted through # (see [71, Section 2.3). For 8 = A ~, 8 # is the dual ~ of ~, in the category of G-bundles. For any scheme Y

with a G-action and any G-bundle ~ on Y,

Ci(~ ) = (-- 1) ic i (~) = Ci (~ ) in H2i(Y~ G,/t2).

(This can be proved by the splitting principle as in [7], p. 247, noting that the statement is clear for line bundles with G-action.) From (1) we get #ci(¢) = ci(oz). Since #ci(¢) induces the map ~01~)o # from Hj(GL,(A), Z) t o n2i-J(SpecA, lt2), it follows that

c ~ ( ~ ) = c 2 ~ ( ~ ~) = -c2~(~ ' ) = c22(. ')

and the proposition is proved.

4. An Interlude

THEOREM 4.1. Let X be a Noetherian reduced scheme and U an open subscheme containing all the singular points and all generic points of X, Then the restriction map

H2(X, 6~) ~ H2(U, G~)

is injective. Proof. Let i: U ~ X be the inclusion. Since U contains the generic points of X, the

sheaf Gm on X injects into i,G~. Let the sheaf D be defined by the exact sequence

1 ~ G,~ ~ i , G ~ D --* O.

Associating to a rational function its order at a codimension 1 regular point induces a map of (&ale) sheaves

i, Gm~ ~) ix, Z, xcX~I)W

where ix: x ~ X is the inclusion of the point x = Spec k(x) into X. This map vanishes on Gm and induces a surjective map

d : D ~ ( ~ ix, Z. xeX(l)\U

We show that d is injective; let V be an 6tale neighborhood of a point x e X. Suppose that f ~ i,G~(V) maps to zero in i~,Z for every x ~ Xtl) \U. By definition f is a unit at every point of V Xx U. Every point y e V outside V Xx U is regular and f has


order zero at every codimension 1 generization of y belonging to V \ V xx U because it maps to zero under d. On the other hand, f has order zero at the generizations of y belonging to V Xx U because f is a unit on V Xx U. Hence, f is a unit at y and d is an isomorphism.

By Lemma 1.9 of [8] we have H I ( X , i , Z ) = O and hence H I ( X , D ) = O . We therefore have an injection

H2(X, Gin) ~ H2(X, i , Gm).

There is a spectral sequence

HP(X, Rqi, Gm) ~ Hn(U, Gin).

Since the 6tale sheaves Rli ,Gm and R2i,Gm are zero we have an isomorphism

H2(X, i , Gm) ~.~ H2(U, Gin).

Remark. It follows easily from Theorem 4.1 that for any commutative Noetherian ring A such that Spec A has finitely many singularities the Giffen invariant and the Clifford invariant have the same image in the Brauer group of A. To see this, it suffices to replace A by its semilocalization at the singular primes and use the fact that, for a semilocal ring, K2(A) is generated by symbols. This was our first approach to the comparison of the two invariants. For the result of the last section Theorem 4.1 is only needed in the well-known special case of a regular scheme, but since it does not seem to appear anywhere in print, we decided to record it here.

5. Quadrics

Let ~ = (aij) e GL.(A) be a symmetric matrix,

u ~ GIn(A), q(X1,.. . , X , ) = ~ avXiX~, i<~,i

A[XI,..., X,] C =

(q(X1,..., X,) - u)'

and B = Cxl. For any A-algebra A', we write BA,, CA, for B ®n A', C ®A A'. Let u: Spec B ~ Spec A b c the structure map.

PROPOSITION 5.1. Let A be reduced and n >f 3. The homomorphisms ~_ ~ n, Gm given by n ~ X~ and Gm ~ n,Gm induce an isomorphism of dtale sheaves

Gm x 7 / - - ~ n,G,n.

Proof. Since Gm commutes with direct limits we may assume that A is noetherian. Suppose first that A is a separably closed field. If n/> 4, X1 is a prime in C and GIn(C) = GIn(A), hence G,~(B) = GIn(A) × Z. If n = 3, by Proposition 5.3, either X1 is a prime or

B -~ (A[X1, X2, X3]/(X1X2 - X23 - 1))x, "~ A[X1, X-~ i, X3],


SO that GIn(B) is again as above. If A is any field and K its separable closure, since

/ ( c~ B = A, we have again GIn(B) = GIn(A) x Z.

Suppose A is any domain and K its field of fractions. Since B is faithfully fiat over A, GIn(B)c~ K = G,,(A), hence

G,~(A) x Z ___ GIn(B) _c GIn(B) c~ Gm(BK)

= Gin(B)n (GIn(K) x Z) = GIn(A) x Z.

Suppose now that A is strictly Henselian with maximal ideal m and minimal prime

ideals Pi . . . . , Ps. Any unit u of B can be written, in BA/p, as viX~', with v, ~ Gm(A/p~). If = wX'~ in Ba/,, with w ~ G,,(A/m), then ni = m for all i and thus u X [ r" maps into

Gm(A/p3 for each i. Since A is strictly Henselian, there is a retraction r/: B ~ A. The element u X [ m - t l (uX[ m) maps to zero in Ba/pl X . . . X BAIp, and, A being reduced,

the map B ~ BA/pl x ... x Ba/p, is injective. Thus, u X [ m ~ GIn(A).

The map Gm x 7 / ~ re.Gin is an isomorphism on stalks because the strict Henseliz-

ation of a reduced local ring is reduced. Hence, this map is an isomorphism.

C O R O L L A R Y 5.2. With the same notation as above, 1~2 ~ zc,lt2 is an isomorphism.

Proof. Restrict the isomorphism of the above proposition to the 2-torsion subsheaves.

P R O P O S I T I O N 5.3. Let A be a local domain and C = A [ X , Y, Z ] / ( X Y - Z 2 - u), u

a unit o f A. Le t l = a X + b Y + cZ with Aa + Ab + Ac = A. I f l is not a prime in C, Cl ~ A [ X i , X-~ l, Z] .

Proof. If a is invertible in A, we may assume that a = 1 and take 1 = X + b Y +

cZ = X1 as a variable, so that

A [ X i , Y, Z i ] C =

( X 1 Y - - bi Y2 - Z 2 - u)

with Z1 = Z + ½cYand b~ = b - ½ e .

Since A is a domain and X i is not a prime,

bi = 0 and B ~ _ A [ X I , X r l , Z] .

Suppose a and b are both nonunits. Then c is a unit and we may assume e = 1. Let

Z~ = Z + a X + bY. Then

C/(Z~) = A [ X , Y, Z 1 ] / ( X Y - (aX + bY ) 2 - u, Z1)

= A [ X , Y ] / ( X Y - (aX + bY ) 2 - u)

is a domain since 4ab # 1, contradicting the assumption that l is not a prime.

P R O P O S I T I O N 5.4 Let q = (aij) be a symmetric stably elementary matrix o f size

n >1 3. Let Q = Spee B where

B = Cx~ and C = A [ X 1 .. . . . X .] / (~ ,a i~XiXj +_- 1).


Let re: Q ~ Spec A be the structure map. The induced homomorphism

H2(A,/~2) ~ H2( B,/~2)

is injective.

We postpone the proof of this proposition to the end of the section.

LEMMA 5.5. Suppose that A is local and reduced and that q is split. For n >~ 2, Pic B has no 2-torsion.

Proof. Suppose that A is a normal domain. If n >1 4, Xt generates a prime ideal in C and Pic B injects into CI(B)/CI(A) which in turn injects into CI(BK)= CI(CK)= Pic CK = 0, K being the field of fractions of A.

If n = 3 and X1 generates a prime ideal, arguing as above we get that Pic B injects into P icC r = Z. If XI is not a prime, by Proposition 5.3, B ~-A[X1, Xll, X2], hence P icB = PicA = 0. If n = 2, B ~- A [ X I , X ? I , I - 1 ] , where l = aX~ + b, with a, b e A and Aa + Ab = A. In particular, B is faithfully flat over A. An invertible B-module is equivalent to the image of a divisorial ideal J of A. Since B is faithfully flat over A, J must be projective over A, hence free. Thus Pic B = 0.

To deal with the general case, we may assume that A is essentially of finite type over Z and proceed by induction on its dimension. The zero-dimensional case is already proved. Since A is essentially of finite type over Z, its integral closure A is a finite A-module. Let c be the conductor of A in A. The cartesian square

B , B

l l B/cB ~ 12/cB

gives an exact sequence ([2], IX, 5.4)

Gm(B/cB) x Gin(12) ~ G,~(B/cB) ~ Pic B ~ Pic(B/cB) x Pic 12,

where t2 = B ®A A. Since dim A/c < dim A, by induction, Pic(B/cB) has no 2-torsion. Since A is a product of normal domains, the argument above shows that Pic 12 has

no 2-torsion. It remains to show that Coker e has no 2-torsion. Let A0 = (A/C)red,

Bo = B ®a A0, A0 = (ff/C)red, 120 = B ® -4o- Let u be a unit in G,,(B/cB) which represents a 2-torsion element in Coker ~. Its

image v in Gm(12o) can be lifted to an element of G,.(Bo) x G,,(B). In fact, by 5.1, on each connected component of Spec 12o, v is of the form voX'~, Vo e G,,(Ao). The map GIn(A/c) ~ G,,(Ao) is surjective and, Ale being local, the map G,,(A/c) × G,,(A)-~ Gin{A/c) is also surjective. On each component, X~ lifts to a unit of B/cB, thus, modifying u by an element coming from G,,(B/cB) × G,,(12), we may assume that u maps to 1 in G,,(Bo), so that u e t + nit(B/cB). Since this group has a finite filtration whose quotients, being isomorphic to (nil(12/c12))~/(nil(12/c12)) ~+I, are uniquely 2-divisible, it follows that the image of u in Coker ~ is zero. Thus the lemma is proved.


COROLLARY 5.6. Let n >f 3. The sheaf Rix,/t2 is constant, isomorphic to 7//2, generated by the image of X1.

Proof. Let s be a global section of R~rc,p2, represented on an &ale open set SpecA' by an element ~H~(BA,,pz). In view of 5.5 and 5.1 we may assume, by shrinking Spec A', that ~ = X~.

COROLLARY 5.7. Let n >t 3. I f A is connected, the group H°(A, Rite,p2) is generated by Xi.

Proof of Proposition 5.4. Since ~2(R)--/12(Rred) for every ring R in which 2 is invertible, we may assume that A is reduced. The Leray spectral sequence for n: Q ~ Spec A gives an exact sequence

Hi(B, g2) ~-~ H°(A, R1;rc,/t2) --> H2(A, zc,p2) /~ HZ(B, ]~2)"

By Corollary 5.7, ~ is surjective and, hence, /3 is injective. Since A is reduced, it follows from Corollary 5.2 that P2 ~ zc,P2 is an isomorphism and this completes the proof of the proposition.

6. The Generalized Rees Ring of an Invertible Module

, I o = i . Let I be an invertible A-module and L = , ~ z I , where A, is the n-fold tensor product l ® A ' " ® a I if n is positive and ( i - , ) - i if n is negative. The canonical isomorphism Im®aI" -* I ~+" defines on L the structure of a graded A-algebra, usually called the generalized Rees ring of I.

PROPOSITION 6.1. Let A be reduced. The A-algebra L is faithfully flat and the kernel of the map Pie A ~ Pie L is the cyclic group generated by the class of I.

Proof. Since locally I is free and L isomorphic to A[-t, t - i ] , L is faithfully fiat over A. Clearly, L @a I ~ L as L-modules. Let P be an invertible A-module and ~p: L --* L ®A P an isomorphism of L-modules. The fact that ~p(1) belongs to some homogene- ous component I" @a P and that (o induces an isomorphism A ~, 1" ®a P may be verified locally on A, using the fact that A is reduced (compare with [4], III.3). Thus P ~- I -n for some n.

PROPOSITION 6.2. Let A be reduced. The kernel of H2(A, P z ) ~ H2(L, P2) is generated by 0[I], where [ I ] is the class of I in Hi(A, Gin) and 0 is the connecting homomorphism of the Kummer exact sequence.

Proof. Let re: Spec L ~ Spec A be the structure map. The natural map P2 ~ z~,pz is an isomorphism since, for a local ring A, the ring L is just A[t, t- l] . Hence, the Leray spectral sequence gives an exact sequence

Hi( A, P2) HI(~) ~ Hi(L, P2) 6 ~ H0(A, Rl~,p2) ~ HZ(A, Pz) ~ H2(L, Pz).

For any open set Spec A I over which I is free there is a split exact sequence

0 --~ Hi(A/-, P2) ~ Hi (L f , P2) 6y, H0(As, Rln,pz) ~ 0


with L i ~ - A y [ t , t - 1 ] . We claim that H°(Ay, Rlrc, l~2)= Z/2. Let s be a global section of R 17r,p2, x e Spec A I and Ax the strict henselisation of A at x. On Spec Ax,

- L * UL* ~2 s is represented by an element ~ H l ( L a x , p 2 ) "~ ax/~ AxJ X 2PiCLAx. Since Lax --- A~[t, t - 1] and A~ is reduced, L]~ ~- A* x 7/. Further, by 5.5, 2Pic LA~ = O, So that ~ = t ~, e = 0 or 1. Thus, H°(AI, Rlrc,pz) = 7//2, generated by t and hence H°(A, R lzc, p2) is either Z/2 or trivial.

Suppose I is not a square in Pic A. Then, under the connecting map ~ of the Kummer sequence, a [ I ] is a nontrivial element in ker(HZ(A, pz)-~ H2(L, kl2)) by Proposition 6.1 and, hence, it generates the whole kernel.

Suppose I = j z for some J 6 PicA. If {SpecAf, f ~ S} is a covering which trivializes J, we can choose local generators t f for I such that tf = u}ot o, usa ~ A~o denoting the cocycle associated to J. The set {t s, f ~ S} represents a nonzero section of R tT"c,tl2 since it is nonzero even locally on A. This section is the image under 6 of the discriminant module J L ® L J L ~ - J Z L = I L ~ - L over L. In this case H2( A, Pz) ~ Hz( L,/g2) is injective.

7. Rank 2 Forms over an Afline Curve

Assume that A is a smooth affine algebra of dimension 1 over a field of characteristic not equal to 2 and let H(I) be the hyperbolic plane over an invertible A-module I. We can represent H(I) by a symmetric matrix

with b z - ac = 1. We assume that (8 s °) is stably elementary, so that it represents a class in EWz(A ). In this case, since dim A = t, this matrix is in fact elementary.

PROPOSITION 7.1. c22 ° s2 (~ sO) = 0 if and only if the class of I is a square in Pic A. Proof. By 4.2.1 of [15] the map c22:K2(A)~HZ(A, p2) factors through

H°(X, )ffz), where X'2 is the Zariski sheaf associated to the presheaf U ~ K2(U). Since the transposition on K2 coincides locally with the inverse, this induces a factoring of c22 as a composite

K2(A)/Tr(Kz(A)) ~ H°(X, .~(2)/2 -* H2(A, P2).

Since A is smooth, by 4.2.1 and Theorem 4.3 of [15] the second map is injective. Thus, ¢22 °s2(q)= 0 if and only if the image of s2(q) in H°(X, ~Y~2)/2 is zero, q denoting the class of (~ sO) in EW2(A). Let q / = {L], 1 ~< i ~< N} be an affine open covering of X = Spec A which trivializes I and let

{ufj ~ Gm((gx(U ~ c~ Uj)), 1 ~ i, j ~< N}

be a cocycle representing L Let 5 be a lift (~ sO) in St(A). Then

s2(q) = class of ( - 1 , - 1) + ~- lg t in Kz(A)/Tr(K2(A)).


By shrinking og, we may assume that

on U~, with flie E4(Cx(U~)). Let ~ be a lift of fl~ in St((gx(U~)). Let

] = w21(- 1)w~3(- 1)hla(- 1),

where

wd;O = xd~.)xj~ ( _ ,~- 1 )x&~), hd;O = wv(.~.)w d - 1)

(cf I'10] page 71). Then .7 is a lift of J2 and j - 1 j , = ( _ 1, - 1). On ui, ~ = ci][iJ~ with cl ~ K2((Ox(Ui)). By further shrinking q/, we may assume that each c~ is a sum of symbols, so that -c~ = c~. Then

(-- 1 , - 1 ) + ~ - * ~ t = _ 2c~

on Ui so that 2ci = 2cj on Ui ~ Uj. The image of s2(q) in H°(X, ~"2) is represented by the cocycle { -2c~ , l~< i~<N} . Since 2 ( c ~ - c j ) = 0 on U~nUj, {c~-c j , 1 ~< i, j ~< N} defines a 1-cocycle on Spec A with values in 2J(=.

L E M M A 7.2. The class of the 1-cocycle {ci - c i, t <<. i, j <~ N} in Hi(X, 2a~f'2) is equal to the class of the 1-cocycle { ( - 1, uij), 1 <<. i, j <<. N}.

We grant this lemma. Suppose the image of s2(q) in H°(X, af2)/2 is zero. After possibly shrinking q/, we may assume that there exists a cocycle {c~, 1 ~<i~< N} in H°(X, ~2 ) such that 2ci = 2c~ on Ui. Then 2(ci - c~) = 0 and ci - cj = (ci - c~) - ( c j - c~.), which shows that the 1-cocycle { c ; - c j} with values in 2s¢2 is a co- boundary. By Lemma 7.2, the cocycle { ( -1 , uij)} is trivial in Hi (x , 2~2).

L E M M A 7.3. Let X be a smooth curve over afield k of characteristic different from 2. There is an isomorphism PicX/2-+ Hi(X, 2)F2), which, in terms of Cech cocycles, is given by mapping the class of {J~j} to that of { ( - 1 , J~j)}.

By Lemma 7.3, [ I ] is trivial in Pic A/2. Conversely, if the class of I in Pic A/2 is trivial, reversing the above arguments, we see that the image of s:(q) in H°(X, X2)/2 is zero. This proves Proposition 7.1, provided we prove the two lemmas.

Proof of Lemma 7.2. We can choose the trivializations fl~ such that

This has a lift h~2(U~g) in St(U/n U~). Putting ulj = u we get, using the identities

wdu)~ = wdu),

h,j(u) t = hi j ( - u)h,j(- 1)- 1


and Corollary 9.4 of [10],

cj - ci = J - ihl2(u)Jhl2(u)'

= hi3( - 1)-lw43 ( - 1)-lw2i ( - 1)-lh12(u)w21 (- 1)w43(- 1)

h la ( - 1)hi2(- u)hia(- 1) - i

= h i3(- 1)-lh21(-u)h21( - 1)-lhla( - 1)hi2(- u)hl2(- 1) - i

= h2i(u)h12(- u)hl2(- 1) - i

= h12(u)-lhl2(- u)hl2(- 1)-1

= hi2(u)hl2(- 1)-lhl2(u) -1 = ( - 1,u).

Proof of Lemma 7.3. We denote by ~f" the Zariski sheaf on X, associated to the presheaf U ~ H"(U, P2). By Theorem 6.1 of [3], Pic X/2 ~- HI(X, ~ x ) is the cokernel of the map

Hi (k(X),/.2)

=k(X)/(k(X),)2 0_~ ~ HO(k(x),p2 ) = ~) Z/2, xeX (1) x~X~l)

where the map is given, at x, by t3(f) = ( - 1) v-re. By [11], 8.7.8.(b), H~(X, / i f z) is the cokernel of the tame symbol map

2Kz(k(X))-* ~ 2K~(k(x))= (~ Z/2. xeX(1) x~XO)

By Theorem 1.8 of [15], mapping f e k(X) to the symbol ( - 1, f ) yields a surjection

k(X)/(k(X)*) 2 ~ 2Kz(k(X)),

such that the diagram

k(X)/(k(X)*) 2 ~ (~Z/2

i l = 2K2(k(X)) , @Z/2

is commutative. This yields an isomorphism Pie X/2 ~ H~(X, 23f~2) on the cokernds. Since all the maps in the resolutions are explicit, it is easy to verify that, in terms of cocydes, the isomorphism is as claimed.

THEOREM 7.4. The class of I in PicA~2 c, H2(A,/'2) is precisely c22os2(~ o). In particular,

Proof Let L = @,~zI". Since IL is trivial in PicL, by Proposition 7.1, C22 o S2( q ~) L) = 0. In view of (5.3), c22 o sz(q) is either zero or the class of I in Pic A/2 c H2/A,/.2). If I is not a square in Pic A, by Proposition 7.1, c22 o sz(q) ¢ 0 so that C 2 2 o s 2 ( q ) = [ 1 ] . If I is a square in PicA, by Proposition 7.1,


c22 o sz(q) = [I] = 0. On the other hand, by [12], Theorem 20, ez(q) = [I] and this completes the proof of the theorem.

8. T h e C o m p a r i s o n

LEMMA 8.1 (Casanova). Let ~ G L 4 . ( A ) represent an element of EW2(A) and 0~ GL,~,(A) be a matrix such that OtO~ E(A). Then

s2(Ot~O) = s2(~) + s2(OtO) + n(-1, -1) .

Proof Let ~ be a lift of ~ and OtO a lift of OtO in St(A). For any ~0~ GL(A), let x ~ x 'p be the unique isomorphism of St(A) lifting the conjugation by <p in E(A). Choosing 0t0 ~0 as a lift of 0t~0, we get

s2(Ot~O) = n ( - 1 , - 1 ) + (Ot'~ • ~°)t(Ot"~ • ~0)-1 = s2(O~O) + (~0),(0 ~- 1)<0t)-1

= s2 (0 ~ 0) + ( ~ - 1 )<0,) -1

since, for any x s St(A), (x°) t = (xt) (°')-1. To show that the second term coincides with ~t~-x and thus complete the proof, it suffices to observe that the action of any q~ s GL(A) on K2(A) is trivial: indeed, by [1], Kz(A) = Hz(E(A), Z) is a direct factor of H2(GL(A), 7/) and an inner homomorphism of any group G induces the identity map on H,(G, 7/).

LEMMA 8.2. Let O~ SL2(A). Then OtO is stably elementary and e2 coincides with C2 2 o $2 On

(10 otO0) •

They both are ( - 1) u ( - 1) in HZ(A,/~2). Proof Since

Ot = ( ? l ~ )0_1 (01 ; ) -1 ,

OtO is a commutator and hence stably elementary. To prove the second assertion, it suffices to consider the generic case of the matrix 0 =(~ f) over the ring A = 7/[½, x , y , z , t ] / ( x t - y z - 1 ) . Since x is a prime in A and A is regular, Pic A - Pic Ax = 0, so that H2(A, P2) ~ 2Br(A). Since Br(A) injects (§4!) into Br(K), K the field of fractions of A, it suffices to compare the two invariants in Br(K), which has been done in [6]. Clearly

(1 0) e2 0 030 = e 2 ( 1 4 ) = ( - 1 ) u ( - 1 ) .

THEOREM 8.3. The Giffen invariant maps, under c22, to the Clifford invariant.


Proof. Let a~ GL4,,(A) represent an element of EW2(A) and let q be the corresponding quadratic form.

Replacing A by the A-algebra B of Proposition 5.4, we can make the matrix a elementarily equivalent to (-+01 ~). The comparison of the invariants in HZ(A,/,2) reduces, by Proposition 5.4, to their comparison in HZ(B,/*2) for the form (+-01 ~). Repeating this process 4n - 3 times we are reduced to compare the invariants of the matrix

- 1 0

0 a

0 b

Since a c - b 2 = - 1 , the quadratic space (A 2, (~ ~)) is isometric to H(1) for some invertible module I. By Theorem 20 of [12], e2(H(I)) is the class of ! in Pie A/2 c H2(A,/*2). If we extend the scalars to L = ~3~zI" , I becomes principal and

for some 7 e SLz(L). By Lemma 8.1,

s2 s211 1 + sz 2 7~7) + (-- 1, - 1)

and by Lemma 8.2,

c22 o s2 --- ( - 1) u ( - 1). 7t7

For a matrix of the form

(1 b) --a

- b

a

s2(~) = (a, b) modulo Tr(K2(A)), and c22 o s2(a) = (a) w (b) (see [6]). Since

1 -I j)

514

is elementarily equivalent to

M. OJANGUREN ET AL.

1 - 1 --1 I ) '

it follows that over L, C22oS2(fl)= O. Therefore c22 os2(fi), by Proposition 6.1, is either the dass of I or zero. Suppose

is elementary. Then this holds for the matrix

t l , , . 1 0

B= 0 - 1

x y

y z,

over the ring

Z[½, x, y, z, X~j, Y~j, Z~, T~j] [(det X)- a , . . . , (det T)- 1 ] A =

( x z - y 2 + 1, X Y X - 1 Y - 1 Z T Z - 1 T -1 = r)

where X = (X~i) . . . . , T = (T~j) are generic matrices and 1 ~ i, j ~< 4n + 4. We note that if

- 1 = ~

a

b

is such that ac - b 2 = - I and

is elementary over a ring B, there is a specialization from A to B, specializing fl to y, since over any commutative ring, any elementary matrix is a product of two commutators ([5], Remark 21). If c22 o s2(~) = 0 this invariant would be zero for any matrix of the form fl over any ring. However the computations in Section 7 show the


existence of matrices fl over Dedekind domains C for which it is not zero: we may take C = R[x, y]/(x 2 + y2 _ 1) and

~=

0

--1 0

0 X2 _ y2 2 x y "

0 2xy y2 _ _ X2/

The corresponding ideal I is the generator of Pie C = Z/2. This shows that the two invariants coincide.

Acknowledgements

Financial support from IFCPAR/CEFIPRA and FNRS is gratefully acknowledged. The first author also thanks Sandro di Pippozzo for moral support during the monsoon.

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