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K-Theory 6: 29-44, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands, 29

Graded Witt Ring and Unramified Cohomology

R. P A R I M A L A and R. S R I D H A R A N School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabba Road, Bombay 400 005, India

(Received: October 1991)

Abstract. It is proved that for a smooth alfine curve X over a local ring or global field, the graded Witt ring of X is isomorphic to the graded unramified cohomology ring of X. If X is projective and has a rational point, the same result holds if and only if every quadratic space defined on the complement of a rational point extends to X. Such an extension is possible, for instance, if the canonical line bundle on X is a square in Pic X.

Key words. Witt ring, unramified &ale cohomology, signature, real spectrum, cohomological dimension.

O. Introduction

Let k be a field of characteristic different from 2 and X a smooth variety over k. Let k(X) denote the function field of X. Let 0: W(k(X))~ @x~xlW(k(x)) denote the map induced by the second residue homomorphism for a choice of local parameters at codimension one points X 1 of X. The kernel of 0 is independent of the choice of the local parameters and is defined to be the unramified Witt group of X (cf. [CTPa]) and denoted by W.r(X). In view of results of Pardon [P] and Barge-Sansuc-Vogel, W,r(X) consists of classes of forms over k(X) which extend, at every point x of X 1, to a nonsingular quadratic space over the local ring (gx,x. The

groups

I.(X) = I"(k(X)) n W..(X), n >1 O, I"(k(X))

denoting the nth power of the fundamental ideal I(k(X)), give a filtration on W,~(X). The aim of this paper is to compare the associated graded unramified Witt ring @,>~ol,(X)/I,+I(X) with the graded unramified cohomology ring ~.>~oH°(X, ;,ug,) (cf. [BO]) where ,¢g" denotes the Zariski sheaf associated to the presheaf U H~(U, ~).

If X = Spec k, and P,(k) denotes the set of n-fold Pfister forms over k, Arason [A] proves that there is a well defined map e,:P,(k)~ H"(k), such that e ,(((al , a2 .... a.}}) = (al) u (a2) t3 -.. u (a,), (a) ~ HI(k) denoting the square class of a ~ k* (here we abbreviate H"(k) = H~"t(Spec k,/~2))- For a general field k, it is an open question whether e, extends to a homomorphism l"(k) ~ H"(k) and if so, whether it

30 R. PARIMALA AND R. SRIDHARAN

is surjective with kernel I "+ l(k). This question has an affirmative answer for local and global fields as well as for function fields in one variable over a local or global field (cf. [-AEJ 2]). We call a variety X good if for every point x ~ X of codimension ~< 1, the graded Witt ring of k(x) is isomorphic to the graded cohomology ring of k(x). Examples of good varieties are provided by curves over local and global fields and varieties of dimension ~< 3 over R or C ([AEJ 2] and [AEJ t]). If X is good, we have a well defined homomorphism

e,: I . (x) --, n ° ( x , .~")

with kernel 1,+ I(X) (cf. [Pa]). We thus have an induced homomorphism

e = {~,}: (~ In(X)/I,+I(X)-+ ~ H°(X, ~") n>~O n>~O

of the graded unramified Witt ring of X into the graded unramified cohomology ring of X.

Question (Q). If x is good, is the map e an isomorphism?

The above question may be thought of as a global version of the open question on quadratic forms over fields, stated above. In this paper, we answer this question in the affirmative, for smooth affine curves over a local or a global field. The result is true for projective curves X as well if the canonical line bundle f~x is a square in Pic X. The line bundle f~x is a square in Pic X if X is a hyperelliptic curve of genus g >~ 2 with a rational point of ramification over p a or if X is an elliptic curve. We show that f~x is a square in Pic X, if all the 2-torsion points of Pic X~ are defined over k, k-denoting the separable closure of k. The condition f~x is a square in Pic X is imposed to ensure that every quadratic space over Xk{xo}, x0 a rational point of X extends to X (cf. [GHKS]) .

The following question seems to be of independent interest: Does a smooth projective curve X over k have 'extension property' for quadratic spaces, namely: There exists xo~X(k) such that every quadratic space over Xk{xo) extends to X. In fact, we prove that for a local field k and a smooth projective curve X over k, with a rational point, the map ez:Iz(X).-+H°(X,~ 2) is surjective if and only if X has an extension property for quadratic spaces. It is shown in [PaSc] that the extension property for hyperetliptic curves over local fields is equivalent to f~x being a square in Pic X. Thus, in view of the examples in [PaSc], one sees that the map e2 is not in general surjective, even for smooth projective curves over a local field. Thus, Merkurjev's theorem does not admit a globalisation to curves.

We exploit a construction of the 'mod-2 signature' homomorphism on the unramified cohomology defined in [CTPa] to prove the main theorem. We also use results of Kato on two-dimensional class field theory.

GRADED WITT RING AND UNRAMIFIED COHOMOLOGY

1. Some Known Results

31

In this section, we list a few results which we shall use in the paper. We denote by k a field with char k ¢ 2.

(1.1) Let X be a smooth irreducible variety over k. We abbreviate H~t(X, #2)= H"(X). Let l = k [ X ] / ( X Z + a) be a quadratic extension (we include the case l - ~ k x k) and ~z: S p e c / ~ S p e c k the projection. Let X t = X XspeckSpec/ and )~,, e HZ(k) the character corresponding to the element a e k*/k .2. The short exact sequence of 6tale sheaves

0 -~ ( m ) , ~ r c . ( (m)3 --, (m)k - ' 0

yields a long exact sequence in 6tale cohomology:

H"(X) ~xa H.+I(X) i_~ H.+Z(Xl) s_L ) H.+I(X).

(1.2) Let I/k be as above. Let s:W(1)~ W(k) be the Scharlau transfer map (cf. ILl , p. 169) and ( ( - a ) ) : W ( k ) ~ W(k) the map obtained by tensoring with ( ( - a)) = (1, - a ) . Whenever Speck and Spec I are good, there exists a long exact sequence of Witt groups compatible with the long exact sequence in cohomology defined in (1.1) through the maps e, (cf. [AEJ 1]).

I"(k) ((-a)), I"+ l(k) i.,I"+1(1) s -~ I"+ ~(k)

le. Ie.+, le.+~ Ie.*,

H"(k) t.)z, , H,,+l(k) i , H,,+I(I) s , H,,+l(k)

(1.3) Let X be a smooth integral curve over k. There is a residue homomorphism

0: H"(k(X)) -> (~ H ~- ~(k(x)), x ~ X

(by x s X, for a curve X, we mean x running over the set of closed points of X) with ker g -~ H°(X, our"), coker a --% H I(X, J/f"). The group H l(X, 3/f") is the subgroup of H"+~(X) defined by elements whose restriction to a non-empty open set is trivial. Further, the sequence

0 --) H~(X, ~ ) -~ H"+Z(X) ~ H°(X, .)f.+a) _> 0

is exact ([B], p. 85).

(1.4) Let X be any smooth variety over k which is good. For a codimension one point x e X ~, let

a ~ : I " ( k ( X ) ) , I " l(k(x))


denote the second residue homomorphism with respect to a choice of a parameter at x. Then, the diagram

I"(k(X)) ~=(~x) ~ ) i . _ l ( k ( x ) ) x e X 1

le, l(e,-1)

H"(k(X)) 0 , ~ H,_l(k(x) ) x ~ X 1

is commutative and induces maps e,: I,(X) -~ H°(X, ~¢~') (cf. [Pa]).

(1.5) We recall the following restflts from [AEJ 2], part of which rely on Kato's results in [Ka]. Let X be a smooth itegral curve over k. If k is a non-Archimedean local field, for n ~ 4,1"(k(X))= O,H"(k(X))= 0 and e3:13(k(X)) ~ H3(k(X)) is an isomorphism. If k is a global field which is totally imaginary, the injectivity of

I3(k(X))-+ G I3(kv(x)),

v running over all places of k, yields that I4(k(X))=0, H4(k(X) ) - -0 and e3: 13(k(X))~ H3(k(X)) is an isomorphism ([AEJ 2]).

(1.6) The map eo:W(X)~H°(X ,~g° ) -~Z /2 is the rank homomorphism and el: I ( X ) ~ H°(X, j r l )= H X(X) is the diseriminant homomorphism. These are al- ways surjective. Thus, to answer (Q) positively for any good variety X, one needs to prove the surjeetivity of e,: 1 , (X)~ H°(X, 3¢f ") for n >f 2.

2. An Extension Lemma

In this section, we prove the following:

P R O P O S I T I O N 2.1. Let X be a smooth projective curve over a field k with char k # 2. Let Xo be a k-rational point of X. Suppose every element in the 2-torsion of Pic X~, k denoting the separable closure of k, is defined over k. Then, every quadratic space over Xk{xo} extends to X.

In view of a residue theorem of [-GHKS], it suffices to prove the following proposition.

P R O P O S I T I O N 2.2. Let X be as in (2.1) and let S(X~) denote the set of all square roots of the canonical bundle f~x in Pic X~. Then S(X~) is defined over k.

Proof. We assume without loss of generality, that genus X ~> 2. Then S(X~) is a principal homogeneous space over 2PicX~-%HI(X~). The function ~o:S(X~)~F2 defined by q~(£~') = dim H°(X~, L~')mod 2 is a quadratic function on S(X~) (in the sense of [At], p. 48) which has a nontrivial zero, since genus X >~ 1 (cf. [Mu]). Let a ~ S(X~) be such that ~0(a)= 0. Then, the map ~0a: 2Pic X~ ~ F 2 defined by q~a(x) = q~(a + x) is a nondegenerate quadratic form on the F2-vector space 2Pic X~

GRADED WITT RING AND UNRAMIFIED COHOMOLOGY 33

whose associated bilinear form'is simply the cup-product on HI(X~,/Zz) (cf. t-Mu]).

We transform the Galois action of G(k/k) on S(XD into an affine action on 2Pic (XD

through the map S(X~) ~ 2Pic X~, x -~ x - a. More explicitly, for a • G(k/k) ,

x ~ 2Pic X~, x ~ = ax + aa - a = x + aa - a, since G(E/k) acts trivially on 2Pic X~,

by assumption. Since dim H°(X~, S~) = dim H°(X~, a~Cf) for any ~gq ~ Pic X~, a • G(k/k), a preserves the quadratic function ~o, on 2Pic X~. By a lemma of Serre quoted in ([At], Lemma 5.1), o" has a fixed point, i.e. for some x • aPicX~,x = x " = x + aa - a; i.e., aa = a and, hence, a(a + x) = a + x, for each x ~ 2PicX~. Thus, a acts trivially on S(X~); i.e. S(X~) a = S(X~). Since X(k) ~ ¢, Pic X = (Pic X~) a~/k)

and every element of S(X~) is defined over k. This completes the proof of (2.2).

3. Fields with Finite 2.-Cohomological Dimension

Let Z be a smooth integral curve over a field k. Throughout, we assume that char k ~ 2 and Z is good. We define a homomorphism

en: H°(Z, ~n) _, H I(Z, ~ n + 1 )

whose kernel contains en(In(Z)). In other words,

In(z) ~ H °(Z, ~ n ) ~ H l(Z, ~ " ÷ 1) (,n)

is a complex. We also give a set of sufficient conditions under which (,n) is exact. Let ~ e H°(Z, o~fn). Since Z is good, the map en: ln(k(Z)) ~ H ' ( k ( Z ) ) is surjective

so that there exists r ~ P ( k ( Z ) ) such that e,(r) = ~. Since

en- 1 ° ~(r) = ~ o en(r) = ?(o0 = O,

we have, 0(r)~ @x~zIn(k(x)). Let the map gin be defined by the exact sequence

H , + I ( k ( Z ) ) e , ( ~ H~(k(x)) _ ~ H i ( Z , ~¢(n+1) ~ O, x ~ Z

i.e., fin identifies H I(Z, 34f n+l) as the cokernel of ~ (cf. (1.3)). Set Cn(~) = gin o en o ~(r). We verify that Cn is well-defined. In fact, if rl , r2 e In(k(Z)) are such that en(rl) =

en(r2) = ~, then, r12 = rl - r : belongs to In+l(k(Z)) , en+l(r12)~ Hn+I(k(Z)) and

6noen o~(r12) = t~n oc3oen+ l(r12 )

= 0 .

Thus ¢~ is a well-defined homomorphism. For fl s In(Z), ~bn o en(fl) = 6n ° e, o ~(fl) = 0

sO that (*n) is a complex. Further, from the definition of en, it follows that it commutes with kJ~(_ 1; i.e., the diagram

HO(Z, ~ . ) ~ ,,,,,~ n~ (z , ~ + 1)

H°(Z, 31f n+1) ¢"'~'~ H l ( Z , j ~ t n+2)


is commutative, where kJZ_ 1 on the right is the restriction of the map LJZ-I : H"+ 2(Z) ~ H"+ 3(Z).

We define the condition H(n) for a curve Z as follows:

H(n): The map k.) Z_ 1 :H°(Z, oW") ~ H°(Z, J/f"+ 1) is surjective.

Let l = k[X]/(X 2 + 1). Let l' be the extension of k by adjoining ~ 1 . Then either l ' = l o r l ' = k .

We assume in the rest of the section that Z~ is also good.

LEMMA 3.1. Suppose Z satisfies H(n). Then the sequence

HI(Z, 3eF ") i , Hl(Zt,#f") S> Hl(Z,~et°n)

is exact, where i and s are the restrictions of the map i: H"+I(Z)~H"+I(ZI) and s: H"+ l(zt) --* H"+ I(Z) (cf. (1.1)).

Proof By (1.3), we have the following commutative diagram with exact columns:

H"(Z)

l H°(Z, ~")

1 0

0 0 0

1 1 1 Hi(z , ,,~¢f") _--_., HI(Zt, ~ " ) __,, Hi(Z, ~" )

1 1 1 , H n+ l ( z ) , Hn+I(Z,) > nn+l (Z)

1 LAZ-I> HO(z, ~ , + 1 )

l 0

Exactness of the middle row (cf. (1.1)) and the surjectivity of the map H°(Z, ~,~") Uz-l> HO(z, Xf,+ 1) imply the exactness of the top row.

LEMMA 3.2. Let cd2(l') <~n and let ~ e H ° ( Z , ~ ") be such that ak_)Z_le e,+l(I,+l(Z)). Then there exists an element r eI"(k(Z)) such that e,(r)= ~ and 0(r) ® << 1>> = 0.

Proof. If I' = k, any lift r will do, since <<1>> = 0 in Hi(k) and 8(r) ® <<1>> = 0. We therefore assume that l' = I. Since cd2(l) <~ n, cd2(l(Z)) ~< n + 1 so that H"+2(I(Z))=O. Since Z r is good, e,+2:I"+2(I(Z))oH"+2(I(Z)) is onto with kernel I" + 3(I(Z)). Thus, I "+ 2(I(Z)) = N k >~ z I"+k(l(Z)) = 0, by a theorem of Arason- Pfister ([L], p. 290). Let f leI , ,+~(Z)be such that e ,+a ( f l )=~L)X-1 . Then, e,+l(fl,) = (ct UZ-1)t = 0, by (1.2), suffix l denoting images in the corresponding

GRADED WITT RING AND UNRAMIFIED COHOMOLOGY

groups over Z~. Thus, ~zeI".+2(l(Z))= 0. Therefore,

I"(k(Z)) such that rl ® ( (1) ) = ft. We have

by (1.2),

35

there exists r~

(e.(ri) - c0 ~J)~-I = e .+ l ( r l ® ( (1)) ) - c~ L)X_ i

= e .+l( f l ) - c~ ~J)~- 1

= 0 .

Thus, there exists q eH"(l(Z)) such that s ( t / )= e . (h ) -c~ . Let e.(t/i ) = q and

r = rl - s(th). Then, s(t/i ) ® ( ( t ) ) = 0 by (1.2) and

O(r) ® ( (1) ) = c~(r~ ® ((1>>) - ~)(s(~h) ® ((1)))

= ~(~)

= 0 .

e.(r) = e . ( r i ) - e.(s(ql))

= e . ( r l ) - soe.(th)

= en(rl) - - S(tl)

P R O P O S I T I O N 3.3. Suppose cd2(l') <~ n. (A) I f e . + l : I . + l ( Z ) ~ H ° ( Z , ~ "+i) is surjective, then, image ¢. c image(s:

H~(Z. ~,+1) __, Hi(z, ~.+1)). (B) I f Z satisfies H(n + 1) and (*n + 1) is exact, then (*n) is exact. Proof (A) Let e eH° (Z , gf"). Since e .+ l is surjective, ~k_)Z_ 1 e e.+l(l.+i(Z)).

By (3.2), there exists r e I"(k(Z)) with e.(r) = e and 0(r) ® {{1)) = 0. Since

e ._ i o O(r) = O o e.(r) = ~(~) = O, O(r) e (~ I"(k(x)) x e Z

and O(r)® ( (1)) = 0 implies that O(r) = s(~) for some ~/e @x~z~I"(l(x)). We have,

(o.(cO = 6. o e. o ~(r)

= 5. oe.os(n)

= sof .oe.(q)

with 6. o e . (q )e Hl(Zz, j g . + i ) . The fact that s commutes with e. and 6. follows f rom

([A], 4.11 and 4.14).

(B) Let c~ ~ H°(Z, o~") with ¢,,(~) = 0. We have

¢.+1(~ w z - , ) = 4~.(=) ~ z - ~ = 0

and (*n + 1) being exact, a UX- i~e .+i ( I .+ l (Z) ) . As in the p roof of (A), one obtains r~I"(k(Z)) with e . ( r ) = a , ~ ( r ) ® ( ( l > > = O and an q~Gx~z,I"(l(x))

36

such rOWS,

that s(r / )= 0r. We have the following

R. PARIMALA AND R. SRIDHARAN

commutat ive diagram with exact

Hn+~(k(Z)) ~-~ @ Hn(k(x)) ~ , H~(Z, AP n+~) , 0 x~Z

1' l' l' Hn+l(l(Z)) ~ @ nn(l(x)) ~Sn, Hl(Zt,~f~n+l) - -~0

xeZi

i s i' I s Hn+~(k(Z)) O, @ Hn(k(x)) f i " ,H~(Z,o~ n+~) ,0 .

x~Z

We have

so 3no en(tl) = g)n o en(~(r))

= 4).(~)

= 0 .

Since Z satisfies H(n + 1), by (3.1), there exists 0 e O~zHn(k (x ) ) such that 6n°en(rl) = 6noi(O). Exactness of the middle row gives an t/1 e Hn+l(l(Z)) with O(th)=en(rl) - i (O). Let r l2eln+l( l (Z)) be such that en+ l ( t / 2 )= th . We have,

- - = - - = = - - I n en- l(O(r S(tl2))) ~en(r) Oen(s(tl2)) 0(c0 0. Thus ~(r S(tl2)) e ~x~Z (k(x)). Further ,

en(O(r -- s(T/2)) ) : en o 0(r) -- c~o en+ 1 ° s(r/2)

= enoS(r) -- Oos(~h)

= en o 8(r) - s(en(tl) - i(O))

= en° ~(r) - so en(tl)

= 0 .

Thus, ~(r - s(tI2)) e @x~Z

~(r - s(~2)) ® ((1))

I "+ l(k(x)). Further , by (1.2),

= - so a(t/2 ) ® (/~1>)

= 0 .

Hence

3 ( r - s ( t l 2 ) ) e i m a g e { s : ( ~ l n + l ( l ( x ) ) ~ l ~ l " + l ( k ( x ) ) . } xeZl xeZ

By our assumption, cd21' <<. n, so that In+l(l(x)) = 0 for x ~ Zl (see proof of (3.2)). Thus, 8(r - s(t/2)) = 0 and r - s(t/2) e In(Z), with en(r - s(t/2)) = e.

C O R O L L A R Y 3.4. Suppose ed2(l') <% n, Z satisfies H(n + 1) and H"+2(Zt) = 0. Then surjectivity of e.+l implies surjectivity of e,.


Proof Surjectivity of e.+l implies that (*n + 1) is exact. By (3.3) (B), we have, (,n)

exact. Since HI(ZI, ~ , + 1 ) ¢._~H,+2(ZI) = 0, by (A), ~b, = 0 so that e, is surjective.

4. Local Fields

LEMMA 4.1. Let k be afield with cd2k <~ n. Let Z be a smooth integral curve over k, which is good. The map em:I,~(Z)-~ H°(Z, o~m) is an isomorphism for m >~ n + 1.

Proof Since CdEk <~ n, we have cdz(k(Z)) ~< n + 1 and cd2(k(x)) <~ n for dosed points x e Z. Since Z is good, the maps e,,: Im(k(Z)) -~ H"(k(Z)) , era- 1 : I " - l(k(x)) H " - ~(k(x)), x ~ Z are both isomorphisms for m/> n + 1. (All these groups are zero for m > n + t.) The commutative diagram

0 , Ira(Z) ' lm(k(Z)) , ~ Im-l(k(x)) xeliZ

I e" l ~ l ~

0 , H°(Z, 3¢") , Hm(k(Z)) ~, ( ~ H " - ~(k(x)) x ~ Z

yields that em:Im(Z)-'~ H°(Z, Win) is an isomorphism for m ~> n + 1.

LEMMA 4.2. Let k and Z be as in (4.1). Suppose that Z is affine. Then the map e.: I , (Z) ~ H°(Z, ~ " ) is surjective.

Proof Since cd2k <<. n, cd2k(Z) ~< n + 1, so that H"+2(k(Z)) = 0 and the Zariski sheaf W,+z = 0 on Z. Hence, Z satisfies H(n + 1), By (4.1), e,+l being surjective,

(*n + 1) is exact. Therefore, by (3.3) (B), (,n) is exact. To show that e, is surjective, it suffices to show that H I(Z, W,+I ) = 0. The spectral sequence ([Mil], Remark 2.2.1 (b), p. 106)

HP(G, Hq(zt)) ~ HP+q(Z),

G = G(T:/k), ~: denoting the separable closure of k, yields, in view of Hq(Z~) = 0 for q > 2,HP(G, - ) = 0 for p > n, cd2k being less than n + 1,

H"+ Z(Z) -% H"(G, H2(Z~)) ~ H"(G, Pic Z~/2).

Since Z~ is not projective, it misses at least one point of the completed curve so that PicZk is divisible. Since H~(Z, Jt ~"+1) c ~H,+2(Z) = 0, the lemma follows.

LEMMA 4.3. Let k be a field with cdzk <~ n. Let X be a smooth projective integral curve over k, which is good. Let xo ~ X(k) be a k-rational point. The map e,: I , (X) H ° ( X , ~ ") is surjective if and only if the restriction map I , ( X ) ~ I , ( X \ { x o } ) is surjective.

Proof The sequence

H'(k(X)) o_% G I-I~- l(k(x)) ~r~s, 14'- ~(k) x E X


is a complex ([A], Satz 4.16), so that if Xo e X(k) and a ~ Hi(k(X)) is an element such that ~(~) has one possible nonzero component at Xo, then ~ e ker ~. Thus, in view of [BO] the restriction map H ° ( X , ~ i) ~ H ° ( X \ { x o } , ~ i) is an isomorphism for

every i. In the following commutative diagram

I ,+I(X) - ' H ° ( X , ~ " + I )

1 1' I,+ a(X\{xo}) ~ H°(X\{xo} , 2/g "+ a),

horizontal arrows are isomorphisms, by (4.1), and the right vertical arrow is an isomorphism. Hence, the left vertical arrow is also an isomorphism. In the following commutative diagram

0 0 0

1 1 1 0 , I,+ l(X) , I , (X) , H°(X, ~ " )

0 ---, Z.+l(x\{xo}) , z.(X\{xo}) H°(X\{xo}, xe.) ,0

the rows are exact (4.2) and the extreme vertical arrows are isomorphisms. The lemma follows immediately.

T H E O R E M 4.4. Let k be a local field or a totally imaginary global field. For any smooth curve Z over k, the map e,: I , (Z) --* H°(Z, ~ " ) is surjective for n ~ 2. The map e 2 is surjective if Z is affine. I f Z is projective, and Z(k) :~ 4), e2 is surjective if and only if Z has extension property for quadratic spaces: There exists Xo ~ Z(k) such that every quadratic space over Z\{xo} extends to Z.

Proof. If k is a local or a global field, any smooth integral curve Z over k is good ([AEJ 2], p. 649, corollary). If k = R, the map H°(Z, ~ i ) __. HO(z, ~ i + a ) is surjective for i >~ 1 ([W], Remark 2.4.4). The surjectivity of I ( Z ) ~ H°(Z, ~ 1 ) implies the surjectivity of I , (Z) ---, H°(Z, ~ " ) for n/> 1. If k = C, a non-Archimedean local field or a totally imaginary global field, we have, cdzk ~< 2 so that e, is surjective for n ¢ 2 by (4.1). The map e2 is surjective if Z is not projective by (4.2). Suppose Z is projective. By (4.3), e2 is surjective if and only if the map I2(X)--* I2(X\{xo}) is surjective. If X satisfies extension property for quadratic spaces with respect to Xo E X(k), then clearly I2(X) -+ I z (X\{xo}) is surjective. Suppose conversely that the map I2(X)--. I 2 ( X \ { x o } ) i s surjective. Let q be a quadratic space on X\{xo}. To show that q extends to X, we need to check that the second residue ~?~o(q) = 0. If rank q is odd, we may replace q by q_l_<l> and assume that rank q is even. We have disc q E Ha(X\{xo} , #2) = Ha(X,/ t2) so that disc q is integral on X. Replacing q by


q l ( 1 , disc q), we may assume that disc q is trivial. Thus, [q] e I2(X\{xo}) = lz(X). Therefore O~o(q) = 0 and q extends to X.

COROLLARY 4.5. Let Z be a smooth projective integral curve over k, which is a local field or a totally imaginary number field. Suppose f~z is a square in Pic Z and Z(k) ~ (a. Then ~, is an isomorphism for all n.

5. Surjectivity of e4 for Curves over Number Fields: Mod-2 Signatures

Let k be a number field and Z a smooth curve over k. Let H~(k(Z)) denote the ' - 1 torsion' subgroup of H~(k(Z)): H~(k(Z)) = {4 ~ H"(k(Z), ~ u (X_I) ~ = 0 for some positive integer r}. Let h,: H"(k(Z)) -~ C~(Pk(z), 2/2) be the map defined in [-AEJ 1], (Pk(z) denoting the space of orderings of k(Z)). Then ker h, = H~(k(Z)), in view of the results of ([AEJ 1], Lemma 2.2). We recall the rood-2 signature homomorphism

h,: H°(Z, ~g") - ) <g(Specr Z, 2/2)

defined in [CTPa]. If (x, v) e Spec, Z, x s Spec Z, v: k(x) --+ L an injection into a real closed field L, and e E H°(Z, ~ " ) , then h.(e)(x, v) = H"(v)(g~), ~ denoting the restriction of e~ to the fibre at x, ~ ~ H"~((gz,~) being a lift of~; we identify H"(L) -~ 2 /2 for a real closed field L. The diagram

H°(Z, 2/£") h., cg(Spec~ Z, 2/2) t ~res

H"(k(Z)) h. , rC(pk(z) ' 2/2)

is commutative, where Pk(z) is the set of generic orderings of Z, treated as a subspace of Specr Z. Further, the total signature homomorphism sgn: W ( Z ) ~ cg(Specr Z, Z) maps I . (Z) into 2"Z and we have a commutative diagram

I , (Z) sgn , cg(Spec, Z, 2"2)

l l h, H°(Z, ~ " ) ---* ~(Spec, Z, 2"2/2 "+ 12) = cg(Spec~ Z, 2_/2).

LEMMA 5.1. Ht4(k(Z)) = O. Proof. Suppose first that k is totally imaginary. Since cdzk <, 2, cd2k(Z) ~< 3, so

that H4(k(Z))= 0. Suppose k is not totally imaginary. Let l = k(x/ -~f ). Then H"(I(Z)) = 0 for n >i 4 and the exact sequence

0 = H4(I(Z)) -* H ' (k (Z) ) uz-1 HS(k(Z)) --+ Hs(I(Z)) = 0

yields that k.) X_ 1 is an isomorphism. Thus, H~(k(Z)) = O.


L E M M A 5.2. Let Y be a smooth affine curve over a number field k and let Spec, Y

denote the real spectrum of Y. I f Spec, Y = U1 u U2 is decomposition into disjoint open sets, then there exists a quadratic space q in I4(Y) with sgn q = 24 on U~ and s g n q = 0 o n U2.

Proof. By a theorem of Mah6 [M] , there exists an integer N /> 1, and a quadratic space q on Y such that sgnq = 2 N on U ~ , s g n q - - 0 on U 2 , s g n : W ( Y ) ~ U(Spec, Y, Z) being the total signature homomorphism. Replacing q by ((1)) ®4 ® q,

we may assume that the class of q in W(Y) belongs to I4(Y). Suppose N >7 5. In view of the commutative diagram

I4(Y) sgn , U(Spec, Y, 247/)

I e' 1 HO(z, ~ 4 ) _ ~ U(Spec, Y, 24;7//25~) = U(Spec, Y, Z/2),

since N >~ 5, sgn q - 0 mod 257/so that h4e4(q) = 0. By [AEJ 1], e4(q) e H4t(k(Y)) = 0 by (5.1). Thus, q ~ I5(Y). We have a commutative diagram

0 ' I4(Y) ' I4(k(Y)) ------~ (~ I3(k(x)) x e Y

1®<<,>> 1®<<1>> 1®<<,>> 0 , Is(Y) , I5(k(Y)) -----~ (~ I4(k(x))

x ~ Y

The two right vertical maps are isomorphisms since by (1.5), In@Y))= O, ln-I(I(x)) = 0 for x s Y, for n >~ 4. Hence, ® ( ( 1 ) ) : I 4 ( Y ) ~ Is(Y) is an isomor-

phism. Thus, there exists q~ ~ I4(Y) with q~ ® ((1)) = q, sgn ql = 2N-1 on U~ and

sgn q~ = 0 on U2. Repeating this process, one obtains q e I4(Y) with sgn q = 2 4 on

U~ and sgnq = 0 on U2.

L E M M A 5.3. Let Z /k be a smooth affine curve over a number field k. Then the map e4: I4(Z) --* H°(Z, ~ 4 ) is surjective.

Proof. Let h:H°(Z, og¢ '4) --* U(Spec, Z ,Z /2 ) be the mod-2 signature homomorphism. The kernel of h:H4(k(Z) )~ U(Pk(z), Z/2) is precisely H4(k(Z)) which is zero by (5.1). Therefore the map h:H°(Z, ~,~4) __+ U(Spec,(Z), Z/2) is injective. Let

~ HO(Z, ~ 4 ) . Let Ut tj U2 = Spec~ Z be a decomposition into disjoint open sets

with h(e) = 1 on U1 and h@) = 0 on U2. By (5.2) there exists q e I4(Y) such that sgn q = 24 on U1 and sgn q = 0 on U2. This implies that h o e4(q) = h(a). Since h is

injective, we have e4(q) = e.

P R O P O S I T I O N 5.4. Let Z be any smooth curve over a number field k. Then the map e4: I4(Z) ~ H°(Z, J4 ~4) is surjective.

GRADED WlTT RING AND UNRAMIFIED COHOMOLOGY 41

Proof Let Y = SpecA c Z be an affine open subscheme such that Yk,(k~)= Xk~(ki) for every real completion ki of k. We have the following commutative diagram

I4(X) - ~ H°(X, ~ '*)

1 t LdY) ~-, U°(Y, ~f4)

with the vertical maps injective and the lower horizontal map surjective, by (5.3). The surjectivity of e4 would follow provided we show that I4(X) = I4(Y). Let fl ~ I4(Y). For x e Y, ~x(fl) = 0. For x e X \ Y , k(x) is totally imaginary so that I3(k(x)) = 0 and

hence ~x(fl)= 0. Thus ~(f l)= 0 and fl ~ I4(X). Thus, it suffices to exhibit such a Y c X. Let X ~ P~ be an embedding of X in a projective space. Let Pk" c__~p,N be the Veronese embedding for the hypersurface Eo~.i.~ x~ in P~, xi, 0 ~< i ~< n, denoting the coordinate planes in P~,. Let H c p~V the hyperplane with H c~ P~, = {Zx~ = 0}. Then Y = X \ H is affine and has the required properties.

6. Main Theorem

We come to the main result of this paper.

T H E O R E M 6.1. Let Z be a smooth quasi-projective integral curve over a global field k of characteristic not 2. Then, the graded Witt ring of Z is isomorphic to the graded (rood-2) cohomology ring of Z in the following cases.

(1) Z is affine. (2) Z is projective, Z has a k-rational point and the canonical line bundle of Z is a

square in Pic Z.

In fact, we prove the following more precise

T H E O R E M 6.2. Let Z be a smooth quasi-projective curve over a global field k of characteristic not 2. Then the maps e , , : I ~ ( Z ) ~ H ° ( Z , ~ ") are surjective for n # 2. The map e2 is surjective ff Z satisfies one of the two conditions (1) and (2) of (6.1).

In view of (4.4) and (4.5) it is enough to prove the theorem in the case where k is a number field with at least one real completion. We set I = k(x ~ 1). We denote by Z a smooth quasi-projective curve over k.

LEMMA 6.3. Let Z be as in (6.2). Then the map e,: I ,(Z) ~ H°(Z, ~ ) is surjective for n >~ 4.

Proof. Since cd2(l(Z)) <% 3,1n(l(Z)) = O, I~-l(l(x)) = 0 for closed points x e Z, for n/> 4. In the following commutative diagram, for n ~> 4, the rows are exact and the

42

tWO right vertical maps are isomorphisms

R. PARIMALA AND R. SRIDHARAN

0 , I , (Z) , I"(k(Z)) , @ I"- i (k(x)) x e Z

i®<<,>> 0 , I,+ , (Z) , I "+ ~(k(Z)) , @ I"(k(x))

x e Z

Hence, the left vertical map is an isomorphism. Since H"(I(Z)) = 0, H"-~(I(x)) = 0 for n >~ 4, x e Z, a similar diagram as above with I replaced by W yields that the map L ) ; ( _ i : H ° ( Z , ~ f " ) - ~ H ° ( Z , og "+i) is an isomorphism for n >/4. The commutative diagram

I , (Z) e. , HO(z ' ~ f , )

l® ((1)) lWZ-,

I ,+i (Z) e.+,, Ho(Z ' 3¢f,+i )

for n/> 4 shows that the surjectivity of e 4 implies the surjectivity of e, for n ~> 4. The surjectivity of e4 is proved in (5.4).

LEMMA 6.4. For Z as in (6.2), the map e 3 : I 3 ( Z ) ~ H°(Z, 3¢ ~3) is surjective. Proof. We have ed21 ~< 2. Since the map L3Z_I:H°(Z, ;~") ~ H°(Z, W , + l ) is an

isomorphism for n ~> 4, Z satisfies H(4). Further, e4 being surjective (5.4), (,4) is exact. Thus, to show that e3 is surjective, it is enough by (3.3) to show that Hl(Zl , W , ) = 0. We have,

Hl(Zl , Y f ~) ~--~ Hs(Zz) = O,

since cd2t ~ 2 and cdzZ I ~< 2, and in view of the convergence of the spectral sequence

HP(C, Hq(Zr)) ~ H"(Z,).

LEMMA 6.5. For Z as in (6.2), the curve Z satisfies H(3); i.e.k.)X_ i: H°(Z, W 3) H°(Z, W 4) is surjective.

Proof. Let a e H°(Z,~,~ 4) and let ~i denote its image in H°(Zk~, ~ * ) , where {kl, 1 ~ i ~ r} denote the set of real completions of k. Since k~ is real closed, l~= ki(x/-Z1) is algebraically closed so that Hi(l i(Z))=O, HJ-l( l i (x))= 0 for x e Z~,,j >/2. Thus the commutative diagram

H3(k~(Z)) uz- , , H,(k~(Z))

~l l ~ G H2(k,(x)) u z - , • / t3(k,(x))

X~Zk i x~Zkf


yields an isomorphism

t.)Z - ~ : HO(Zk,, ~¢f3) __% Ho(Zk,, ~ 4 ) .

Let f l i ~ H ° ( Z k i , ~ 3) be such that fl~Ux-1 = cq. In view of [Ka]§4, there exists fl~ H°(Z, 2/f 3) with image fli ~ H°(Zk,, j/f3). Replacing ~ by ~ - fl U)~-I, we may assume that c~ maps to zero in H°(Xkl, ~ 4 ) , 1 ~ i ~< r. This would imply that ct = 0 in view of the following

LEMMA 6.6. For Z as in (6.2), the map

nO(z, ~4) ~ G n°(z~,, ~4) l<~i<~r

is injective. Proof. Let SpecA = Y c Z be chosen as in the proof of (5.4). Let ~ ~ H°(Z, ~ 4 )

map to zero in H°(ZkI,~4), l<<.i<<.r. Let Specr Y = U1 u U2 be such that h(~)i U1 = 1, h(~)l U2 = 0. As in (5.4), we may choose q ~ I4(Y) such that sgn q = 2 ~ on U1, sgn q = 0 on U2 and e4(q) = ~. We have, q @ ki ~ I4(Yg,) mapping to zero in HO(Yki, ~ 4 ) and, hence, q ® ki ~ I5(Yki) and the signature of q ® ki is divisible by 2 5. The projection =i: Yk~ ~ Y induces a map Spec~ Yki-~ Specr Y and a map U(Specr Y, Z) ~ U(Spec~ Yki, Z) with ~h(sgn q) = sgn(q ® k~). Since sgn q is not divisible by 2 5, it follows that sgn(q ®k~)= 0. Since I2(ki(Z)) is torsion free, q ® ki = 0, 1 ~< i ~< r. Since the map I4(k(Z)) ~ @l<,i~I4(ki(Z)) is injective ([AEJ 2]),

[q] = 0 in W(k(Z)) and in particular, e4(q) = e = 0.

LEMMA 6.7. For Z as in (6.2), the map e2 is surjective if either Z is affine or if Z is projective and the canonical line bundle of Z is a square in Pic Z.

Proof. We have cd21 ~ 2 and by (6.5), Z satisfies H(3). Since e3 is surjective (6.4), to prove the surjectivity of e2 it is enough to show by (3.3) that ~o2 is zero. Suppose Z is not projective. Since cd2I ~< 2 and H2(Zt) = Pic Z~/2 = 0, Z1 being not projective, it follows that H4(Zt) = 0 and HI(Zt, ~¢g 3) which is a subgroup of H'~(Zt) is also zero. Thus ~o2 = 0 in this case. Suppose Z is projective and the canonical line bundle of Z is a square in P icZ. Let e s H°(Z, 0~2). The map ~02 commutes with base change and for each completion k~ of k, ~02 is surjective for Zk~, under the given hypothesis (4.4). Thus @2(0~)e Hi(z, ~,¢3)~..~H4(Z) maps to zero in H4(Zk~) for each v. By a

result of Kato [Ka] , the map H4(Z) ~ H~,H4(Zk~.) is injective, so that q)2(e) = 0. This completes the proof of the lemma and the proof of Theorem 6.1.

Remark. The condition (2) on Z in (6.1) can be replaced by the condition that Z(k~) ~ c~ and the canonical line bundle on Zk~ is a square in Pic Zk~ for each completion k~ of k.

COROLLARY 6.8. Let X be a smooth projective curve over a global field k of characteristic different from 2, with a k-rational point. Suppose 2Pic X~ = 2Pic X. Then the graded Witt ring of X is isomorphic to the graded cohomotogy ring of X.

Proof. Immediate from (6.1) and (2.1).


C O R O L L A R Y 6.9. Let k be a global field of characteristic different from 2 and let n: X ~ p1 be a double covering with X smooth (X hyperelliptic). Then if X(k) ~ ~ and genus X <<. 1, or if genus X >1 2 and n has a rational ramification point, the graded Witt ring of X is isomorphic to the graded cohomology ring of X.

Proof. If genus X = 0, and X(k) # ~b, X --% P 1, f ix = (_9(- 2). If genus X = 1, ~ x (9x. If genus X i> 2 and x ~ P1 is a k-rat ional point of ramificat ion for n and if

n(P) = x, then ~ x = (9(2(9 - 1)P). Thus, in all these cases, £~x is a square in Pic X.

Acknowledgement

We thank J. L. Colliot-Th616ne for critical reading of the manuscript . The first named au thor is grateful to the Forschungsinst i tu t f/Jr M a t h e m a t i k for its hospital i ty

when this work was nearing complet ion.

References

[A]

[AEJ 1]

[AEJ 2]

[At]

[B]

[BO]

[CTP]

EGHKS]

[Ka]

ILl [M]

[Mu]

[Mill IV]

[Pa] [PaSc]

[w]

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