Witt rings and almost free pro-2-groups

JOURNAL

OF ALGEBRA

132, 377-383 (1990)

Witt Rings and Almost

Free Pro-2-Groups

ROGER WARE * Departmen

of Mathematics, The Pennsylvania Stare University, University Park, Pennsylvania 16802 Communicuted by A. FrBhlich Received July 16, 1988

Let F be a field of characteristic not 2, WF the Witt ring of quadratic forms over F, IF the maximal ideal consisting of even dimensional forms, and I*F its square. Let G,(2) =Gal(F(2)/F), where F(2) is the maximal 2-extension (=quadratic closure) of F. In this note we prove a decomposition theorem for WF in the casethat Z’F is torsion free and use it to obtain a new proof of a theorem of Ershov on the structure of almost free pro-a-groups [E, Theorem 41. In what follows W,F denotes the torsion subgroup of WF and WredF= WF/W, F is the reduced Witt ring. The set of orderings on F will be denoted by X, and will be given the Harrison topology, with subbasis consisting of the sets H,(a) = { < E X,1 0
377 0021-8693/90 $3.00 CopyrIght C 1990 by Academic Press. Inc. All rights of reproductjon in any form reserved.

378

ROGER WARE

algebraic extensions of R(t) (cf. [S, Section 51). It should be noted that for formally real fields, pseudo real closed * generalized Hilbert * H, * quasiPythagorean. THEOREM 1. Let F be a formally real quasi-Pythagoreanfield. Then there exists a nonreal 2-extension L/F such that

(1) 12L = 0 and the map W,F -+ IL is an isomorphism. (2) The restriction G,(2) -+ Gal(F,,/F) is an isomorphism, where FpY is the Pythagorean closure of F. (3) The natural map WF --) WredF x WL is an isomorphism. Proof: Let L/F be a maximal 2-extension such that TF/p2 + i/i’ is injective, where TF denotes the set of nonzero sums of squares in F. The maximality of L shows that TF/p’ + t/i’ is an isomorphism. This, is turn, shows that L is nonreal and W,F+ IL is surjective. Since Z2F is torsion free, 12L = 0. Plister’s isomorphism gives t/i’z IL and TF/F2 z W,F. Hence the bottom arrow in the following commutative diagram is an isomorphism: TF/h’&

=I

W,F -

i/i’

I=

IL

This implies that G,(2) -+ Gal(F,,/F) is an isomorphism [W2, Theorem 2.101 and that WF -+ WredFx WL is injective. To prove surjectivity, let r, s denote (respectively) the maps WF-, WredF, WF+ WL and let (a,, a2)E W,,,Fx WL. Case 1. (a,, a,) l Zre,,x IL. Then choose b E IF, c, dE W,F such that then r(a)=r(b)=a, and r(b) = a,, s(c) = a,, s(d)=s(b). If a= b+c-d s(a) = s(c) = a,. Case 2. a, $ Ired, a, $ IL. Then (a, - 1, a2 - 1) E Iredx IL so there exists aE WFsuch that aH(a,--,a,-1). Then a+lb(a,,a,). Remarks. (1) Marshall proved that a finitely generated abstract Witt ring R with R # R, and Z’, torsion free decomposes as a product of a reduced factor together with a finite number of factors of the form 2/4;2 [M, Proposition 6.261. The Witt ring WL of a field with Z2L = 0 is completely determined by i/i’ via Pfister’s isomorphism (see [L, p. 36; M, p. 491 for details) and in the special case that It/t21 = 2’~ cc (i.e., WL is finitely generated) then WL is isomorphic to r copies of 2142 if - 14 L2 and r copies of the group

WITT

RINGS

AND

PRO-2-GROUPS

379

ring 2/2Z[C], ICI = 2, if - 1 EL’. This distinction does not affect the structure of WFr W,,,F x WL since in the proof of Theorem 1 L can be chosen with - 1 E L2 or with - 1 $ L2 (compare [M, Lemma 5.111). (2) A field L satisfies Z2L = 0 if and only if G,(2) is a free pro-2group [Wl, Theorem 3.11. Hence we obtain another proof that Gal(Fp;,,/F) is a free pro-Zgroup when Z2F is torsion free [W2, Theorem 3.11 (compare [E, Theorem 4]), (3) The isomorphism G,(2) z Gal(F,,/F) implies that L . FpY= F(2) and LnF,,=F. (4) If Z2F is not torsion free one cannot (in general) expect to obtain the type of decomposition described in the theorem. For example, if F is a formally real algebraic number field then there does not exist a 2-extension LJF such that WF + WL induces an isomorphism WFz WredF x WL. Indeed, such an isomorphism would imply that the neutral map Z,F+ IL is an isomorphism, whence G,(2)= Gal(F,,/F), contrary to [W2, Corollary 4.41. Ershov in [E] calls a pro-2-group almost free if it contains a free pro-2group as an open subgroup and he proves a structure theorem for such groups. By [ Wl, Proposition 3.21, G,(2) is almost free if and only if F satisfies H, (compare [S, Section 51). The next theorem describes part of this structure and extends it to a slightly larger class of pro-2-Galois groups. For this theorem we assume that F is either a formally real field satisfying H, or a quasi-Pythagorean field such that the order space has only finitely many Marshall components [M, p. 1611 and that each of these components has the form X,[M, p. 1533 for some character y: k/p2 + ( + 11. This latter assumption is satisfied, for example, when F has at most a finite number of orderings or when WredF is not basic [M, p. 114, p. 1461. THEOREM 2. Suppose F satisfies one of the above assumptions. Then there is a Pythagorean 2-extension K/F such that

(1) O+ W,F+ (2)

WF+

WK-+O is exact and

G,(2) is the free pro-2-product of G,(2) and Gal(Fp’,,/F).

ProoJ: First assume F satisfies H,. Then F is a SAP field and by [Cl, Theorem 7 and Remark (b)] there is a Pythagorean 2-extension K/F such that the restriction p: X, + X, is a homeomorphism. Hence the sequence 0 + W,F-+ WF -I, WK is exact. We assert that r is surjective. Indeed, if b E k then p( H,(b)) = U is a closen subset of X,. Since F is a SAP field, there exists a in P such that U = H,(a). Then, as K is Pythagorean, a = b (mod k”). Hence r is surjective, proving (1). (2) follows from (1)

380

ROGER WARE

Theorem 1, and Jacob’s theorem on free pro-2-products [JW, Theorem 3.41. Now assume that X, has only finitely many Marshall components, each of the form Xi = X,,, i = 1,2, .... k. Then, as in the proof of Theorem 6.23 in [M] we have an injective homomorphism WredFL

R, x ... x R,,

where R, is the Pfister quotient associated to the component X, [M, pp. 147-1511. Moreover, for each i, either R,gZ or Ri is not basic. We assert that 4 is surjective. To prove this it suffices to show that given one dimensional ‘forms’ ai E Ri, i = 1, .... k, there exists a in W,,,F with #(a) = (a,, .... ak). Since X,= uf= 1Xi and each ai is an element of R, E C(X,, Z) there is a continuous function a: X, + { k 1) such that a = ai on X,. It remains to show that a E W,,,F. By the Becker-Brocker Representation Theorem it is enough to prove that II,, y u(g)= 1 for all fans VEX, with IV1 =4 [M, Theorem 7.171. If V is such a fan then by [M, Corollary 8.151, I’c X, for some i and, because USER,, it follows that n,, ,, a(a) = n,, y u,(a) = 1. Hence WredF g R, x x R, so Realization Theorem 4.8 and Remarks 4.9 (i) in [AEJ] complete the proof of (1) in this case.As before, (2) now follows from Theorem 1 and [JW, Theorem 3.43. Remark. Let K/F be a 2-extension with K Pythagorean and such that p: X,+X, is bijective. If F is not a SAP field then the map WF+ WK may fail to be surjective. For example, there exists a Pythagorean 2-extension K of F = Q( (x))( (y)) such that K is a SAP field and X, -+ X, is bijective (see [C2] for details). Since F is not a SAP field, WredF-+ WK is not an isomorphism. COROLLARY 1. If F satisfies any of the hypotheses of Theorem 2 then G,(2) is the free pro-2-product of a free pro-2-group and a pro-2-group (topologicully) generated by involutions.

ProoJ: A field K is Pythagorean iff G,(2) is topologically generated by involutions [B, Korollar 21. Remark. A (partial) converse to Corollary 1 holds. Namely, if G,(2) = G, * G, is the free pro-2-product, where G, is (top) generated by involutions and G, is a free pro-Zgroup, then F is a quasi-Pythagorean field. In fact, if F, is the fixed field of G, then the isomorphisms H’(G, Z/22) + H’(G, , Z/22 @ H’(G,, Z/22), i = 1, 2, induce an isomorphism WF g WF, x WF* (see [JW, Remark 3.53; compare [S, Proposition 1.4.1(i)]). Since G, is generated by involutions, WF, is torsion free and since G, is free (pro-2) Z2Fz = 0. Hence Z2F is torsion free; i.e., F is quasi-Pythagorean.

WITT RINGS AND PRO-2-GROUPS COROLLARY

381

2. (cf. [E, Corollary]). Let F, be a quasi-Pythagorean field

either satisfying H4 or having (at most) a finite number of orderings. For any $eld F2, the following statements are equivalent:

(1)

WF,z

(2)

G,,(2) z G,(2).

WF,

(2)* (1) is [Wl, Theorem 2.21. (1) + (2). If (1) holds then F2 also satisfies the hypothesis so by Theorems 1 and 2 there exist fields Li and Pythagorean fields Ki such that G,(2)z G,(2) * G,(2) (pro-2-product) for i= 1, 2. Since G,(2) is a free pro-2-group of rank GL,(2) = dim,, ii/t: = dim,, TF,/pf (i= 1,2) and TF, /pT z TF,/>z, we have G,,(2) z G,,(2). Moreover, WK, z WredF1 E WredF2 2 WK, so if F, has only a finite number of orderings then G,,(2) g G,(2), by [JW, Remark 4.141. Now assume F, satisfies H4. Then K,, K, are SAP fields. For each i E X,, , let Ei be a euclidean closure of i and let Ci = G, (2) = { 1, oi}. Every 2-extension of a Pythagorean SAP field satisfies SAP so the argument given in [ELI, p. 11871 shows that if H is the closed subgroup of G,,(2) generated by {oil, .... ci,} then the fixed field of H is a Pythagorean SAP field with exactly r orderings. From a theorem of Jacob [J] it follows that H is the free pro-2-product C;, * ... * C,,. Since G,,(2) is (top) generated by the set { uj} it Xh,, the bijection X,, % X,, (induced by WK, r WK,) yields a topological isomorphism G,,(2) -+ G,,(2) (via Cl,,* . * C, 3 Cx(i,) * . . . * C%(,,)). Proof:

Remarks. (1) The proof of Corollary 2 shows that if K is a Pythagorean SAP field then G,(2) is the union (direct limit) of the family { Ps}, where SE X, is finite and P, is the free pro-2-product of the groups C, = { 1, a,}, iE S. Using the universal property of free pro-finite products [BNW] one can set up a projective system (Ps, es, T), S, TG X,, S, T finite, where for SG T, I/I~,~: P, -+ P, is induced by 0, ++oi if iE S and oj H 1 if j E T - S. Then it is easily verified that G,-(2) g @m. P, (compare [E, Th. 33). (2) If K is a Pythagorean SAP field with only finitely many orderings and o,, .... O, are the corresponding involutions (as in the proof of Corollary 2) then it can be shown that G,(2) is the semidirect product x, = gig, Fx{l,a}, where c=gr, 9 is free (pro-2) with basis {xl};::, and o acts on 9 via x’ = x;’ (conversely, any such semidirect product is the free (pro-2) product of r cyclic groups of order 2). (3) If @-lk’ is an infinite group then G,(2) cannot be the free pro-2product ( = coproduct ) of a family of cyclic groups of order 2 (in the sense of [BNW]).

382

ROGER WARE

ProoJ: Suppose there is a topological isomorphism G,(2) g b,,- D/N, where D is the discrete free product of a family { Ci} I E, of cyclic groups of order 2 and ,V is the family of all normal subgroups N of D such that D/N is a finite 2-group and C, G N for all but a finite number of i in I. Let H= Gal(K(2)/K(fl)). Then H is an open subgroup of G,(2) so if H’ is the image of H under the above isomorphism then H’ is an open subgroup of Iim,- D/N. Then, by the definition of the topology on b,, D/N, there exists NE Jt‘ with N c H’. Since F(G) is nonreal, H and hence H’ contains no involution. Thus the set I is finite so that lim + D/N and hence G,(2) is a finitely generated pro-2-group. But then kzi is a finite group. THEOREM

3. For a formally

real ,field F the following

statements are

equivalent:

(1) G,(2) is almost free (i.e., F satisfies H4). (2) G,(2)= G, * G2 (jlree pro-2-product), where G, is a free pro-2group and G2 is the union of a directed system of closed subgroups each of which is a free pro-2-product of a,finite number qf copies of Z/22. Proof (1) = (2) follows from Theorem 2 and Remark 1 following Corollary 2. (2) j (1). Let L, K be (respectively) the fixed fields of G,, G, (acting on F(2)). Then Z2(L) =0 and K is Pythagorean (as G, is topologically generated by involutions). We assert that K is a SAP field. If not, then by [ELl, Theorem 5.31 and [W3, Theorem 2, Corollary I], G, = G,(2) contains a closed torsion free abelian subgroup H of rank 2. Since H is finitely generated, condition (2) implies that G, has a closed subgroup N z Zj2Z * ... * L/2L with Hz N. Then the fixed field E of N satisfies WEr Z x Z x x Z (Witt ring product; seethe remark following Corollary 1). Hence by [W3, Theorem 2, Corollary 11, rank H < 1, a contradiction. Finally, as in the remark following Corollary 1, F is quasi-Pythagorean and we have an isomorphism WFg WL x WK. Since K is a SAP field and L is nonreal F is a SAP field. Hence F satisfies H,, proving (1).

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Cnl [BNW]

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WITT RINGS AND PRO-Z-GROUPS

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[KN]

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383

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