Stiefel-Whitney classes, united K-theory and real embeddings of number rings Stephen A. Mitchell* December 2003 1 Introduction Let F be a number field with ring of integers O F, and let A = S-1O F for some * *set S of primes in O F. Each complex embedding fi : F -! C induces a map BGLA-! BGLC ~=* *BU, and hence by pulling back the universal Chern classes we obtain Chern classes c* *n(fi) 2 H2nBGLA. It turns out that for any such A, the mod 2 Chern classes are independ* *ent of the complex embedding fi, and for fixed fi these classes are algebraically inde* *pendent. Now suppose that F admits a real embedding ff : F -! R. Then Stiefel-Whitney* * classes wn(ff) 2 Hn(BGLA; F2) are defined in the analogous way. These classes, however,* * definitely will depend on the choice of real embedding. For example, the first Stiefel-Wh* *itney class can be identified with the induced homomorphism Ax- ! Rx=(Rx )2. Furthermore, a* *lthough the wn(ff) are again algebraically independent for fixed ff, various relations * *hold as ff varies. This raises the question: What are the relations among the wn(ff)? In addition to its intrinsic interest, this question is relevant to the prob* *lem of computing H*(BGLA; F2). For example, if A is a ring of S-integers with 1_22 A, there is a* * fibre sequence of 2-completed spaces ff_ r1 X- ! BGLA+ -! (BO) , where ff_= (ff1, ..., ffr1) corresponds to the r1 distinct real embeddings and * *X is a space whose homotopy-type is accessible_thanks to the validated Lichtenbaum-Quillen conject* *ures [8] [6] and the results of [4]. To make use of the Serre spectral sequence of this * *fibre sequence, one needs to at least know the kernel of H*(ff_; F2), and this is exactly the p* *roblem posed by our question. The relations among the Stiefel-Whitney classes depend on classical number-t* *heoretic invariants of A, such as the narrow ideal class group NP ic A, and unfortunatel* *y we are able to obtain a complete answer only under a certain technical hypothesis on the na* *rrow class group. The proofs, however, involve a result of independent interest: We comput* *e the 2-adic complex, real and self-conjugate K-theory of the algebraic K-theory spectrum KA* *. These ________________________________ *Supported by a grant from the National Science Foundation 1 three groups, together with the various maps between them, constitute the unite* *d K-theory defined by Bousfield [1]. A more detailed outline of our results follows. Unless otherwise indicated, * *all homology and cohomology groups in this paper have Z=2-coefficients, and all class groups* *, K-groups, etc. are localized (usually completed) at the prime 2. All spectra and spaces a* *re completed at the prime 2. 1.1 Relations among Stiefel-Whitney classes Since the mod 2 Chern classes of a complexified real bundle are just the square* *s of the Stiefel-Whitney classes, it is immediate that the w2n(ff) are independent of th* *e choice of real embedding. In Theorem 2.7 we show: Proposition 1.1 If A = F , the relations w2n(ffi) = w2n(ffj) generate all rela* *tions among the Stiefel-Whitney classes of the real embeddings. Note that we can define Stiefel-Whitney classes for any map BGLA-! BO, and * *that the above relations can also be written as w2n(ffi- ffj) = 0. For a general A, * *however, there are more relations. Before going further, it will be convenient to reformulate * *the problem in terms of homology. The existence of the relations stated in the above propo* *sition is equivalent to having a factorization 2r1H*BUH*BO ppp3 | ppp | ppp | pp | ppp |? H*BGLF ______2r1F2H*BO-ff_ where 2 denotes the cotensor product or pullback in the category H of bicommuta* *tive Hopf algebras over F2; note that the cotensor product over F2 is just the tensor pro* *duct. The assertion that these are the only relations is equivalent to the assertion that* * the lifted map is surjective. Note that for general A, the proposition gives an upper bound for the image * *of H*ff_. We obtain a lower bound as follows: Let BSC denote the classifying space for self-* *conjugate K-theory. Then the complexification map factors canonically as `` BO -ffl!BSC -! BU, leading to inclusions 2r1H*BSCH*BO 2r1H*BUH*BO 2r1F2H*BO. There are short exact sequences in H H*(U=O)-! H*BO- ! H*BU and 2 H*(Sp=U)-! H*BO- ! H*BSC. It follows that 2r1H*BUH*BO ~= H*BO (H*(U=O))r1-1 and 2r1H*BSCH*BO ~= H*BO (H*(Sp=U))r1-1. Proposition 1.2 Assume 1_22 A. Then 2r1H*BSCH*BO Im H*ff_ 2r1H*BUH*BO. The problem is to determine where Im H*ff_fits between these two extremes; w* *e will see that already for real quadratic fields and A = O F[1_2], both extremes can occu* *r. Let RF denote the product of r1 copies of R, indexed by the real embeddings.* * Thus RxF=(RxF)2~=(Z=2)r1. Then Proposition 1.1 is valid for any A such that Ax- ! Rx* *F=(RxF)2is surjective. But in general there is an exact sequence 0-! Apos-! Ax- ! RxF=(RxF)2-!NP ic A-! P ic A-! 0, where NP ic A is the narrow ideal class group of divisors modulo totally positi* *ve principal divisors. We write ~A for the kernel of NP ic A-! P ic A, the principal divis* *ors modulo totally positive principal divisors. Note that ~A is a Z=2-vector space, and le* *t b = dimZ=2~A. The precise statement of our main theorem can be found in Theorem 2.11 below* *; here we will state it somewhat imprecisely: Theorem 1.3 Suppose that the short exact sequence 0-! ~P ic R- ! NP ic R- ! P ic R- ! 0 splits. Then Im H*ff_= H*BO (H*(U=O))r1-b-1 (H*(Sp=U))b. Here we have left it ambiguous exactly how the various factors H*(U=O), H*(S* *p=U) are situated in the image. Note that Stiefel-Whitney classes wnfi are defined for any formal Z-linear c* *ombination fi of real embeddings. More precisely, there is a natural homomorphism Z Hom (F, R)-! HomH (H*BO, H*BGLA) ____________ that_factors_through the quotient ZHom (F, R) by the relations 2(ffi - ffj) = 0* *. Thus ZHom (F, R) = Z (Z=2)r1-1, where the generators of the indicated summands can* * be taken as ff1 and fii= ff1 - ffi, 2 i r1. Let V = (Z=2)r1-1denote the torsio* *n subgroup, and let OE denote the composite V ,! Z=2Hom (F, R)-! Hom (Ax, Rx=(Rx )2). Corollary 1.4 The relations among the Stiefel-Whitney classes of the real embed* *dings are generated by the relations (i) w2n(ffi- ffj) = 0 for all n 1; (ii) wn(fi) = 0 for n odd and fi 2 Ker OE. 3 The proof of Theorem 1.3 proceeds roughly as follows: In the diagram followi* *ng Propo- sition 1.1, the factorization comes about because the composites ffi-ffj H*BGLF -! H*BO- ! H*BU are null in H , and since the fibre sequence U=O- ! BO- ! BU induces a short ex* *act sequence in H , we get a lift H*(U=O) 3p pppp | pp | ppp | pp |? H*BGLF ______H*BO-ffi-ffj In fact the maps ffi- ffj can be modified slightly, without changing the ind* *uced map on homology, so that this lifting exists on the level of infinite loop spaces. In * *Theorem 1.3, the idea is to produce enough further liftings of the form H*(Sp=U) 3p pppp | pp | ppp | pp |? H*BGLA ______H*BO-f for suitable maps f constructed from the real embeddings. We will see that the * *map from real to self-conjugate K-theory is injective, so there is no hope of getting su* *ch lifts on the geometric level. Nevertheless, we find enough maps f such that the composite f H*BGLA -! H*BO- ! H*BSC is null in H . This yields a lowering of the upper bound on Im H*ff_. Up to this point we have not used the splitting hypothesis, which now enters* * as follows: The number-theoretic part of our analysis has focused on the image of the natur* *al map æ : Ax=(Ax)2-! R xF=(RxF)2. But Ax=(Ax)2 fits into a short exact sequence 0-! Ax=(Ax)2-! H1'et(A; F2)-! P ic A[2]-! 0, where P ic A[2] denotes the elements annihilated by 2 in the Picard group, and * *æ is just the restriction to Ax=(Ax)2 of the natural map H1'et(A; F2)-! H1'et(RF ; F2). * *This latter map induces a map @ : P ic A[2]-! ~A, and if @ = 0 then our upper and lower bounds* * can be seen to coincide. Since the vanishing of @ is equivalent to the splitting hypothesis* *, this completes the proof of Theorem 1.3. 1.2 United K-theory of KA We compute the complex, real and self-conjugate K-theory of the algebraic K-the* *ory spec- trum KA, at least up to certain extensions. These three cohomology theories can* * be packaged 4 into a single object called united K-theory [1], incorporating the Adams operat* *ions, the coef- ficient rings and the maps between the three theories. Our complete results are* * too technical to state here; we give two sample calculations that play a key role in the Stie* *fel-Whitney class results. The ring KO0KO of degree zero operations in 2-adic real K-theory can be iden* *tified with * * aF the Iwasawa algebra = Z2[[T ]]. Here T corresponds to _5 - 1. LetpTF_= (1 + T* * )2 - 1, where 2aF is the order of the group of 2-power roots of unity in F -1 . Theorem 1.5 KO0KA is generated as a -module by the real embeddings ffi, subj* *ect only to the relations 2TFffi= 0 and 2(ffi- flijffj) = 0 for certain elements flij2 * *. In particular, KO0KA ~= =2TF ( =2)r1-1as -module. Note that KO0KA depends only on F , not on A. For general n, KOnKA will depe* *nd on the basic Iwasawa module M associated to A (see x3.1 for the definition). The ring KSC0KSC of degree zero operations in 2-adic self-conjugate K-theory* * can be identified with 0= [oe], where oe is the group of order two generated by the * *self-conjugate version of _-1. Theorem 1.6 There are short exact sequences of 0-modules 0-! eM+(-1)-! KSC0KA-! eM+ -! 0 and 0-! KO0KA-! KSC0KA-! eM+ (-1)-! 0. Here M is the basic Iwasawa module mentioned above, e is the appropriate ext* *ension of scalars functor, and (-)+, (-)+ refer to oe-invariants and oe-coinvariants, res* *pectively. Organization of the paper: x2 states the results on Stiefel-Whitney classes, an* *d takes the proofs as far as possible without invoking the results on united K-theory. x3 c* *omputes the real K-theory and x4 the self-conjugate K-theory of KA. In x5 the K-theory res* *ults are used to complete the proof of the main Theorem 2.11 on Stiefel-Whitney classes. We assume throughout that F has a real embedding, and that a basepoint embed* *ding ff1 : F R has been fixed. 2 Stiefel-Whitney classes of real embeddings We begin by considering the mod 2 Chern classes associated to a complex embeddi* *ng of F , and deduce the relations w2nffi = w2nffj for Stiefel-Whitney classes. In x2.2 w* *e reformulate the problem in terms of homology. In x2.3 we digress to review the homology of* * BSC, the classifying space for self-conjugate K-theory. In x2.4 we give a general es* *timate on the size of the image of H*ff_, where ff_is the natural map BGLA-! (BO)r1 induced * *by the real embeddings. In x2.5 we digress to review the narrow Picard group NP ic A a* *nd related invariants. Our main theorem is stated in x2.6; we completely determine the ima* *ge of H*ff__ and hence also the relations among the Stiefel-Whitney classes of the real embe* *ddings_under a certain technical hypothesis on NP ic A. We give an outline of the proof, pos* *tponing the details to x5. 5 2.1 Characteristic classes for real and complex embeddings Let ff : F -! Cbe a complex embedding. Then ff induces a map BGL(ff) : BGLF -! * *BGLCffi, where Cffimeans C with the discrete topology. Let A be a subring of F . Pulling* * back the universal mod 2 Chern classes along the composite BGLA-! BGLF -! BGLCffi-!BGLCtop~=BU, where Ctopmeans C with the classical topology, we obtain mod 2 Chern classes cn* *(ff) 2 H2nBGLA. The following result is well-known. Proposition 2.1 The mod 2 Chern classes cn(ff) are independent of the choice o* *f complex embedding ff. Furthermore, for fixed ff the cn(ff) are algebraically independen* *t. Proof: For the first statement, it suffices to take A = F . Suppose fi is anoth* *er complex em- bedding. Then there is a field automorphism OE of C with fi = OEff. Hence it is* * enough to show that OE induces the identity map on H*BGLCffi. By Suslin's theorem [9] BGLCffi-* *!BGLCtop induces an isomorphism on mod 2 homology. Since the map RP 1-! BU classifying * *the complexification of the canonical line bundle is a generating complex for BU, i* *t follows that the natural map i : RP 1-! BGLCffiinduced by the inclusion 1 Cx is a genera* *ting complex for BGLCffi. But OEi = i, so H*BGL(OE) is the identity, as desired. For the second statement, it suffices to take A = Z. Note that i factors ca* *nonically through BGLZ, and induces a canonical map of Hopf algebras S(H~*RP 1)-! H*BGLZ. Then the composite S(H~*RP 1)-! H*BGLZ- ! H*BU is surjective, and dualizing yields the result. Similarly, each real embedding ff : F -! R defines Stiefel-Whitney classes* * wn(ff) 2 HnBGLA. In this case, however, wn(ff) may depend on ff. In fact w1(ff) can be i* *dentified with the homomorphism ffx=2 : F x-! Rx=(Rx )2, and it is clear that this homomo* *rphism depends on ff. For fixed ff the wn(ff) will be algebraically independent, by t* *he argument used for Chern classes, but there will be relations as ff varies. For example, * *Proposition 2.1 yields the following corollary. Corollary 2.2 Let ff, fi : F -! R be real embeddings. Then w2n(ff) = w2n(fi) fo* *r all n. 2.2 A reformulation in terms of homology Corollary 2.2 can be reformulated in terms of the Hopf algebras H*BO, H*BU and * *their duals. These are connected bicommutative Hopf algebras of finite type over the* * field F2; recall that the category H of all such Hopf algebras is an abelian category. G* *iven objects A1, ..., An, B 2 H and morphisms B- ! Ai, we can form the multiple pushout or * *tensor product BAi= A1 B A2... B An. 6 Dually, we can form the multiple pullback or cotensor product 2B*A*i= A*12B*A*2...2B*A*n. Note that when B = F2, the cotensor product and tensor product coincide. Corollary 2.3 Let ff1, ..., ffr1 denote the distinct real embeddings of F . Th* *en the natural map (ff1, ..., ffr1)* : H*BGLA-! r1H*BO factors: 2r1H*BUH*BO ppp3 | ppp | ppp | pp | ppp |? H*BGLA ______2r1F2H*BO- Dually, (ff1, ..., ffr1)* : r1H*BO- ! H*BGLF factors through r1H*BUH*BO. The structure of the tensor and cotensor products in this last corollary can* * be made more explicit. Let P denote the Hopf kernel of H*BO- ! H*BU. It is a polynomial* * algebra on odd-dimensional primitive generators s1, s3, .... In fact, the fibre sequen* *ce of spectra '' KO - ! KO- ! K induces a fibre sequence of spaces U=O- ! BO- ! BU that in turn induces a short exact sequence in H P = H*(U=O)-! H*BO- ! H*BU. Dually, P* is an exterior algebra on the universal Stiefel-Whitney classes. Now* * it is a trivial fact, valid in any abelian category, that if A-! B is any morphism, with kerne* *l K, then the r-fold multiple pullback A xB x... xB A is naturally isomorphic to A x Kr-1. So* * here we conclude: Proposition 2.4 There are natural isomorphisms 2r1H*BUH*BO ~=H*BO Pr1-1 and r1H*BUH*BO ~=H*BO P*r1-1. 2.3 A digression on self-conjugate K-theory The spectrum KSC of self-conjugate K-theory is the homotopy fibre of the map _-* *1 - 1 : K- ! K. Thus there is a fibre sequence -1fl `` -1K -! KSC -! K. There is also a well-known fibre sequence 7 ''2 ffl 2 ^ KO -! KO -! KSC. The notation for the maps follows [1]. Passing to basepoint-components of zero-th spaces, we get fibre sequences of* * spaces U- ! BSC- ! BU and Sp=U- ! BO- ! BSC. Since iffl = c : KO- ! K, the map H*BSC- ! H*BU is onto and hence the Serre * *spectral sequence of the first fibre sequence collapses. On the other hand, H*(Sp=U) is * *a polynomial algebra on primitive generators in degrees congruent to 2 mod 4; dimension coun* *ting then shows that the Serre spectral sequence of the second fibre sequence also collap* *ses. We conclude that there is a short exact sequence in H H*Sp=U- ! H*BO- ! H*BSC in which the first map is an isomorphism onto F P. Here F denotes the Frobenius* * endomor- phism x 7! x2 on objects of H . Thus: Proposition 2.5 H*BSC = H*BO=F P . Corollary 2.6 Let qk = w2k-1+ w1w2k-2+ ...wk-1wk. Then H*BSC is a polynomial algebra on the even Chern classes c2k and the qk.* * (Note that q2k= c2k-1+ c1c2k-2+ ...ck-1ck.) The corollary follows by a standard calculation; see [7], where this same Ho* *pf algebra arises as H*BGLF3. 2.4 Stiefel-Whitney classes for rings of S-integers, I Our next step is an estimate on the size of Im H*ff_. Theorem 2.7 Suppose 1_22 A. Then 2r1H*BSCH*BO Im H*ff_ 2r1H*BUH*BO. Moreover, equality holds for the second inclusion if and only if Ax- ! RxF=(* *RxF)2is sur- jective. In particular, equality always holds for the second inclusion when A =* * F . 8 Proof: The inclusion H*ff_ 2r1H*BUH*BO was shown above. If equality holds, the* *n H1ff_is surjective and hence Ax- ! RxF=(RxF)2is surjective. Conversely, suppose Ax- ! RxF=(RxF)2is surjective. The canonical map j : RP * *1-! BGLF induces a homomorphism of Hopf algebras S(H~*RP 1) ~=H*BO- ! H*BGLA. It follows* * that the diagonal H*BO 2r1H*BUH*BO is in the image of (ff1, ..., ffr1)*. The proo* *f of Propo- sition 2.4 shows that 2r1H*BUH*BO is generated by H*BO together with the subal* *gebra r1P generated by all primitives. Hence it suffices to show that P rim (H*BO)r1* *is in the image. In order to do this, we need some additional structure. All of the homology * *Hopf algebras considered in this paper are Hopf modules over the Hopf ring H*BO, and induced * *maps are Hopf module maps. For a detailed exposition of this point, see [2]; for present* * purposes the following ad hoc construction will suffice. Tensor product of vector bundles leads to a ring space structure on BO x Z, * *so that BO = BO x {0} becomes a ring space without identity. This yields a second produ* *ct on H*BO, denoted a O b, Similarly, the tensor product of projective modules leads * *to a natural ring space structure on BGLA+ for any commutative ring A. Now the canonical Hop* *f algebra homomorphism S(H~*RP 1)-! H*BGLZ discussed earlier is in fact a homomorphism o* *f Hopf rings. This can be seen by algebraic calculation, or by observing that the H-sp* *ace structure on RP 1 makes QRP+1 a ring space, and QRP+1-! BGLZ+ x Z is a ring map. In any c* *ase, the conclusion is that for any module space X over BGLZ+ , H*X is naturally a H* *opf module over the Hopf ring H*BO. Furthermore, the primitives in H*X always form a submo* *dule for this structure. Returning to the proof, it is easy to show that P rim H*BO is generated by H* *1BO under the Hopf module structure (see for example [2]). Since H*BGLA-! (H*BO)r1is a m* *ap of Hopf modules, we conclude that the image contains P r1, as desired. The Approximation Theorem (see for example [5], II, 3.4) shows that F x-! Rx* *F=(RxF)2is surjective, proving the last assertion of the theorem. It remains to show that 2r1H*BSCH*BO Im H*ff_. This is where the assumptio* *n 1_22 A will be used. As in the argument above, it suffices to show that FP rim H*BOr1_* *the squares of the primitives_lie in Im H*(ff_). Since F P rim H*BO is generated by s21as H* *opf module over H*BO, it suffices to show that the primitives of dimension 2 lie in the im* *age. Since ß2BO maps isomorphically onto the primitives in H2BO, we have reduced to the fo* *llowing standard lemma. Lemma 2.8 If 2 is a unit in A, then K2A=2-! (K2R=2)r1is surjective. Briefly, the Mercurjev-Suslin theorem implies that there is a commutative di* *agram K2A=2 _____-(K2R=2)r1 | | | | | | |? |? Br A[2] ______(Br-R)r1 9 in which the left vertical map is onto and the right vertical map is an isomorp* *hism. Class field theory tells us that the bottom map is surjective. This proves the lemma,* * and completes the proof of the proposition. Theorem 2.7 has the following corollary for Stiefel-Whitney classes. Corollary 2.9 The ideal of all relations among the Stiefel-Whitney classes wnff* *i contains the relations w2nffi = w2nffj, and if Ax- ! RxF=(RxF)2is surjective then these * *are the only relations. In particular this holds for A = F . In the general case there are at most the further relations qk(ffi) = qk(ffj* *), or equivalently wn(ffi) = wn(ffj) for n odd. 2.5 The narrow Picard group and related invariants Let F posdenote the group of totally positive elements of F ; that is, the elem* *ents that map to a positive number under every real embedding of F . Note that there is a sh* *ort exact sequence 0-! F pos-!F x-! RxF=(RxF)2-!0. Similarly, let Aposdenote Ax \ F pos, the group of totally positive units of* * A. Again there is a short exact sequence 0-! Apos-! Ax- ! RxF=(RxF)2, p __ but now the last map need not be onto. For example, when F = Q m is real qua* *dratic and A = O F, the map O xF-!R xF=(RxF)2= (Z=2)2 will have rank one precisely whe* *n the fundamental unit ffl is totally positive. In the range 2 m 10, for instance* *, one can easily check by hand that this happens precisely when m = 3, 6, 7. Now let Div A, P rin A denote the divisor group and principal divisor group * *respectively, so that P ic A = Div A=P rin A. A principal divisor is said to be totally posit* *ive if it admits a totally positive generator. The group of all such totally positive principal* * divisors will be denoted P rinposA. Let ~A = P rin A=P rinposA. Then ~A has exponent 2, and i* *n fact ~A ~=coker (Ax- ! RxF=(RxF)2). Thus there is an exact sequence 0-! Apos-! Ax- ! RxF=(RxF)2-!~A-! 0, where the last map is the obvious one arising from the identification RxF=(RxF)* *2= F x=F pos. Finally, the narrow Picard group NP ic A is defined by NP ic A = Div A=P rinposA. Hence there is a short exact sequence 0-! ~A-! NP ic A-! P ic A-! 0. Under the isomorphisms of class field theory, NP ic A corresponds to the max* *imal abelian extension ~HFof F that is unramified at all finite primes and in which the prim* *es in S split completely. 10 We can splice the last two exact sequences together to get 0-! Apos-! Ax- ! RxF=(RxF)2-!NP ic A-! P ic A-! 0. Here the map RxF=(RxF)2-!NP ic A has the following Galois-theoretic interpretat* *ion: There is a canonical identification RxF=(RxF)2= H1(RF ; Z2) = r1H1(R; Z2). Each real* * embedding ffi determines a well-defined öc mplex conjugationö ei in G(Fab=F ), and ffi m* *aps to oei in G(H~F=F ). For our purposes, there is a further complication that must be taken into ac* *count. Assume 1=2 2 A, and recall the short exact sequence 0-! Ax=(Ax)2-! H1'et(A; Z=2)-! P ic A[2]-! 0. Then the map Ax=(Ax)2-! R xF=(RxF)2considered above is the restriction of th* *e natural map H1'et(A; Z=2)-! H1'et(RF ; Z=2). In particular, there is an induced map P* * ic A[2]-! ~A. The rank of this map is b - a, where a = dimZ=2coker (H1'et(A; Z=2)-! RxF=(RxF)2) and b = dimZ=2coker (Ax- ! RxF=(RxF)2) = dimZ=2~A. Proposition 2.10 a = b if and only if the short exact sequence 0-! ~A-! NP ic A-! P ic A-! 0 splits. Proof: The nonzero elements of H1'et(A; Z=2) correspond to quadratic extensions* * E of F that are unramified away from S and thepinfinite_places. If [I] 2 P ic A with I2 = (* *x), we obtain such an extension by taking E = F x. Of course x is only defined up to units i* *n A_that is, as an element of P rin A_but in any case we see that the map j : P ic A[2]-* *! ~A has the following description: Given [I] 2 P ic A such that I2 is principal, choose a g* *enerator x for I2 as above. Regarding x as a principal divisor, we then get a well-defined elemen* *t [x] 2 ~A. In other words, j coincides with the boundary map @ in the six-term exact sequence* * obtained from the short exact sequence by reducing mod 2. Since the short exact sequence* * splits if and only if @ = 0, this completes the proof. We conclude this section with a few special cases and examples. Example 1: Take A = O F[1_2], and suppose that (i) P ic A = 0 = P ic A0, and (i* *i) there is a unique prime P0 over 2 in A0. (These conditions hold, for example, when F is th* *e maximal real subfield of Q(~2n).) Then NP ic A = 0. To see this, suppose P ic A = 0 but NP ic A 6= 0. Then there is a quadratic * *extension L of F that is ramified only at 1 and with the unique prime P over 2 in O F split* *ting in L. In particular, L \ F0 = 0. Hence L0 = LF0 is a quadratic extension of F0, unram* *ified at all finite primes and with P 0split. Hence P ic A0 6= 0. 11 p __ p __ Example 2: Let F = Q 7. Then the fundamental unit ffl = 8 + 3 7 is totally po* *sitive.pSince_ P ic OF = 0, we have ~O F = NP ic OF = Z=2. The ideal over 2 is generated by 3* * + 7, which again is totally positive. Hence for A = O F[1_2], we again find ~A = NP * *ic A = Z=2. Example 3: (The authorplearned_this example from Ralph Greenberg.) Consider a* * real quadratic field Q m , and suppose (i) the fundamental unit is totally positive* *; and (ii) the prime divisors of m are all congruent to 1 mod 4. Then in the case A = O F, it * *can be shown that the short exact sequence of Proposition 2.10 does not split. If in additio* *n m = 5 mod 8, the splitting still fails for A = O F[1_2]. For an explicit example, one can ta* *ke m = 205. 2.6 Stiefel-Whitney classes for rings of S-integers, II Objects in H with a compatible H*BO-module structure form a category H H*BO; if* * there is also a compatible action of the Steenrod algebra A we get a category H H*BOA* * (see [2]). In the category H H*BO, Hom (P , H*BO) = Z=2 = Hom (F P , H*BO), where the isomorphisms are given by restricting to P1, F(P 1), respectively. He* *nce subobjects of (H*BO)r1 of the form P r (F P )s will be uniquely determined by specifying * *a pair of independent subspaces of (H1BO)r1, of dimensions r, s respectively. The H*BO su* *mmand occurring below always comes for the diagonal embedding, and contains a disting* *uished copy of P corresponding to the diagonal subspace of (Z=2)r1. We remark also that in * *the category H H*BOA , End H*BO = Z2, and hence Aut (H*BO)r1= GLnZ2. Theorem 2.11 Suppose that the short exact sequence 0-! ~P ic A-! NP ic A-! P ic A-! 0 splits. Then the image of H*ff_: H*BGLA-! (H*BO)r1has the form H*BO Pr1-b-1 (F P )b, where the tensor products are to be interpreted as internal direct sums in the * *category H H*BOA . In particular, Im H*ff_is completely determined by H1ff_. The proof of this theorem will yield partial results even when the splitting* * hypothesis on NP ic R does not hold. To restate Theorem 2.11 in terms of Stiefel-Whitney classes, consider the na* *tural homo- morphism Z =2[Hom (F, R)]-! Hom (Ax, Rx=(Rx )2) that maps ff : F -! R to the induced map Ax- ! Rx=(Rx )2, and note that the ker* *nel has dimension b. As explained in the introduction, there is a composite map OE: _____________ V ,! Z[Hom (F, R)]-!Z=2[Hom (F, R)]-! Hom (Ax, Rx=(Rx )2), 12 _____________ where Z[Hom (F, R)]is Z[Hom (F, R)] mod 2(ffi- ffj), and V ~= (Z=2)r1-1is gener* *ated by the ffi- ffj. Corollary 2.12 The relations among the Stiefel-Whitney classes are generated b* *y the rela- tions (i) w2n(ffi- ffj) = 0 and (ii) wn(fi) = 0 for all odd n and fi 2 Ker OE. Alternatively, (ii) may be replaced by (iii) qn(fi) = 0 for all n, where fi ranges over a basis for Ker OE. The rest of this section is devoted to an outline of the proof of Theorem 2.* *11. The details are postponed to x5. We assume the splitting hypothesis of Theorem 2.11* * only where indicated. Lemma 2.13 H*BO Pr1-b-1 (F P )b Im H*ff_. Proof: By Theorem 2.7 we have H*BO (F P )r1-1 Im H*ff_. On the other hand, s* *ince r1 - b = dimZ=2Im (Rx- ! RxF=(RxF)2), the argument of Theorem 2.7 shows that P * *r1-b Im H*ff_. Recall that one of the copies of P lies in the diagonal H*BO. Now by Corollary 2.3 we know that Im H*ff_ H*BO Pr1. The problem is to sharpen this upper bound until it agrees with the lower bo* *und of Lemma 2.13. Let us temporarily write f1 = ff1 and for i 2 fi= ff1-ffi, regarded as map* *s KR- ! KO. Then for i 2 the composite fi c H*BGLR -! H*BO -! H*BU is null in H , and hence there is a lift H*(U=O) 3p pppp | pp | ppp | pp |? H*BGLR ______H*BO-f i In other words, if f = (f1, ..., fr1), then we have a lift of the form H*BO (H*(U=O))r1-1 *p ppppp | pppp | ppp | ppp |? H*BGLR _________-(H*BO)r1f 13 Remark: Let i 2. The fi's themselves do not satisfy cfi = 0, but they can be * *modified slightly to arrange this; see Proposition 3.7. We will show that up to a further automorphism of (KO)r1, we may write ff_: * *KR- ! (KO)r1 as g = (g1, ..., gr1) so that there is a further lifting H*BO (H*(U=O))r1-a-1 (H*(Sp=U))a pp* | pppp | pppp | pppp | ppp | pppp | pppp | ppp |? H*BGLR ______H*BO-g (H*BO)r1-a-1 (H*BO)a Now if the splitting hypothesis of Theorem 2.11 holds, then a = b by Proposi* *tion 2.10 and hence the lower and upper bounds agree, proving Theorem 2.11. To obtain the refined lifting displayed above, the main point is to produce * *enough maps gi2 KO0KR such that the composite gi ffl H*BGLR -! H*BO -! H*BSC is null. Here it will definitely not be possible to arrange that fflgi= 0 in KS* *C0KR, and in fact we will see that ffl : KO0KR- ! KSC0KR is injective. Instead we will proceed as* * follows: The ring of operations KSC0KSC is a commutative local ring, abstractly isomorph* *ic to K0K ~= 0. We will choose the gi's so that for r1 - a + 1 i r1 there is a li* *ft KSC pp`| pp | pp |` pp | pp | p |? KR ______KSC-fflgi where ` lies in the maximal ideal of KSC0KSC. On the other hand, it is easy to * *see that any such ` has the property Im H* 10` F H*BSC. But for all i 2, the gi's w* *ill be chosen so that Im H* 10` P H*BSC. Since F H*BSC \ PH*BSC = F2, this will show that H* 10fflgiis null, as desired. 3 Real K-theory of KA In this section we compute KO*KA as a module over KO*KO, up to certain extensio* *ns. In fact, we compute (K*KA, KO*KA) as an object in Bousfield's category ACR [1];* * this entails keeping track of the realification map r and the complexification map c. We assume throughout that A = S-1O F with S finite and 1_22 A. The results * *can be extended to the case S infinite by passing to inverse limits. As usual, all* * spectra are completed at the prime 2. 14 3.1 Notation The ring of operations K0K is isomorphic to the pro-group ring 0= Z2[[ 0]], wh* *ere Z2x ~= 0. Here k 2 Z2x corresponds to _k 2 K0K. To avoid superfluous notation, we w* *ill not distinguish between a group of order two and its unique nontrivial element, usu* *ally denoted oe. Thus 0 = x oe, where oe corresponds to the Adams operation _-1 and ~=* * Z2has topological generator fl corresponding to _5. In fact fl is topologically gener* *ated by _-1 and _k where k is any integer congruent to 3 mod 8; topologists usually take k = 3* *, but for Galois-theoretic reasons we take _5 as our standard generator unless otherwise * *mentioned. Then 0= [oe] with ~=Z2[[T ]], T = _5 - 1. The full ring of operations K*K is a twisted tensor product 0~K*S0, where K* **S0 = ß-*K = Z2[fi, fi-1], fi 2 ß2K. The twisting is given by _kfi = kfi_k; in the no* *tation of Tate twisting this is written ß2nK = Z2(n). Similarly, KO*KO ~= 0~KO*S0, where corresponds to the real Adams operatio* *ns and KO*S0 = ß-*KO = Z2[j, ,, fiR , fi-1R]=(2j, j3, ,2 - 2fiR , j,). Here j 2 ß1KO, , 2 ß4KO, fiR 2 ß8KO. Remark: In the computations below, we will usually not bother to record the ,-m* *ultiplications, because of the formula , = rfi2c in KO-4KO. The group 0 is also canonically isomorphic to Aut ~1 (C), where ~1 (-) deno* *tes the group of all 2-power roots of unity. Hence if we fix a real embedding of the nu* *mber field F, and let F1 = F (~1 C) denote the 2-adic cyclotomic extension, we have a monomor* *phism of finite indexp_0F G(F1 =F )-! 0. Moreover, 0F= F x oe, where F = G(F1 =F0)* *. Here F0 = F -1 is the first stage in the usual filtration F F0 F1 ... F1 . Let aF = 2 | ~1 (F0) |. Then F has index 2aF-2 in , and 0F 0is generat* *ed by the elements oe, flF corresponding to _-1, _q; here q is any integer such that q = * * 1 mod 2aF but aF-2 not mod 2aF+1. We have 0 0 Z 2= Z 2= =!F, where !F = (1 + T )2 - 1. F F Let ES denote the maximal abelian 2-extension of F1 that is unramified away* * from S. Then for A = S-1O F, the basic Iwasawa module M associated to A is M = G(ES=F1 * *). If L is a 0F-module, we will write eL for 0 0 L (e is for "extension of s* *calars"). We F are thinking of e as a functor from compact 0F-modules to compact 0-modules. * *It is easy to see that e commutes with Tate twisting, so that the expression eL(n) is unam* *biguous. Now write L+, L+ respectively for the kernel and cokernel of 1 - oe. Similar* *ly, write L-, L- respectively for the kernel and cokernel of 1 + oe. Then e commutes with all* * four of these functors, so that expressions such as eL+ are unambiguous. Furthermore, e is e* *vidently an exact functor, and so commutes with both ordinary and Tate homology/cohomolo* *gy: Hp(oe; eL) = eHp(oe; L), and so on. Finally, if n is even then (L+)(n) = (L(n))+, while if n is odd (L+)(n) = (L* *(n))-. The notation L+(n) will always mean (L+)(n). Similar remarks apply to L-, L . 15 3.2 Complex K-theory In this notation we have [4]: Theorem 3.1 K-2nKA ~=eZ2(n) K-2n-1KA ~=eM(n) Of course these isomorphisms are determined by the two cases with n = 0. We need to take a closer look at the isomorphism K0KA ~=eZ2. First of all, t* *he proof of the theorem above shows: Proposition 3.2 Let fi : F -! C be any field embedding. Then the induced map K* *^fi gen- erates K0KA as 0-module (after pulling back to KA). Second, the relation between generators coming from different embeddings fi * *can be explained as follows: Let Hom (F, C) denote the set of all such field embedding* *s. Note that a fixed basepoint embedding fi1 determines a bijection Hom (F, C) ~=GQ =GF. Proposition 3.3 There is a natural isomorphism of 0-modules Z 2[GQ 1\Hom (F, C)] ~=K0KA, or equivalently, fixing fi1 as above, Z2[GQ 1\GQ =GF] ~=K0KA. Proof: Up to homotopy, the natural action of Aut Con KC factors through the epi* *morphism Aut C-! G(Q 1=Q ) = F. To see this, recall Suslin's equivalence KC ~= bu (reme* *mber that all spectra are completed at 2), which implies that the homotopy action is dete* *rmined by its effect on ß*. In fact, since the action is by automorphisms of ring spectr* *a, it is even determined by its action on ß2KC ~=Hom (Z=21 , ~21). This proves the claim. It follows that the natural map Z2[GQ =GF]-! K0KA determined by fi1 factors through a map from the double-coset module OE : Z2[GQ 1\GQ =GF]-! K0KA. Now GQ 1\GQ =GF = 0=H, where H is the image of GF in 0; i.e., H = 0F. He* *nce the source and target of OE are finitely-generated Z2-modules of the same rank;* * since OE is surjective by Proposition 3.2, this completes the proof. At the risk of belaboring the point, we can partially rephrase the last prop* *osition as follows. 16 Corollary 3.4 Define an equivalence relation on Hom (F, C) by fi ~ fi0if Kfi = * *Kfi0. Then 0 acts transitively on the set of equivalence classes, with isotropy 0F. Wr Now recall that KrelA is the fibre of the natural map KA-! 1bo. Proposition 3.5 KO2nKrelA = 0 for all n, and ( eN+ (n) if n even KO-2n-1KrelA ~= - eM (n) if n odd Proof: By [4] Theorem 4.9, which applies here thanks to [6], we have ( eN(n) if m = -2n - 1 Km KrelA ~= 0 if m even Furthermore, K*KrelA is oe-acyclic. Hence ~= m rel + c : KOm KrelA -! (K K A) . This proves the proposition for m even or m = -2n - 1 with n even. If m = -2* *n - 1 with n odd, we get KO-2n-1KrelA ~=eN- (n). Then the short exact sequence 0-! M- ! N- ! r1F-!0 shows that M- = N- , completing the proof. W r rel Let ffi : 1bo-! K A denote the connecting map in the cofibre sequence * *defining KrelA. Since K*KrelA has projective dimension one as 0-module ([4], Theorem 4.* *9) and is concentrated in odd degrees, while K*bo is concentrated in even degrees, ffi* * is uniquely determined by K0ffi. Furthermore K0ffi factors as `r1 eN- ! eN=eM- ! K0( bo) ~= r1, where eN=eM ~= r although we do not have a canonical basis. On the other hand* *, the computation of K0KR above shows: Wr Proposition 3.6 There is a -basis for eN=eM such that eN=eM- ! K0 1bo has d* *iagonal matrix (TF, 1, ..., 1). 3.3 Computation of KOm KA for m even By Corollary 3.4, for 2 i r1there are elements fli2 , well defined mod F,* * such that cK(ffi) = c(fliK(ff1)) 2 K0KA. Define fi2 KO0KA by f1 = K(ff1), fi= K(ffi) - fl* *iK(ff1). Proposition 3.7 KO0KA ~= =2TF ( =2)r1-1. In fact KO0KA has a presentation* * as -module with generators fi and relations 2TFf1 = 0, 2fi= 0 (2 i r1). 17 Proof: Note that by construction, c(TFf1) = 0 = c(fi), i 2. Hence the formul* *a rc = 2 shows that the fi's satisfy the indicated relations. Furthermore, since K0Krel* *A = 0, the resulting homomorphism =2TF ( =2)r1-1-! KO0KA is surjective. Hence it is en* *ough to show that KO0KA is abstractly isomorphic to =2TF ( =2)r1-1. To see this con* *sider the coboundary map KO0ffi : eN+ -! r1, and recall that there is an exact sequence 0-! M+ -! N+ -! r1-! ( =2)r1-! 0. It follows from Proposition 3.6 that eN+ =eM+ ~=( )r1has a -basis such that t* *he map eN+ =eM+ -! r induced by KO0ffi is a diagonal matrix (2TF, 2, ..., 2). This co* *mpletes the proof. Note that c : KO0KA-! K0KA is the obvious epimorphism =2TF ( =2)r1-1-! * *=TF, while r : K0KA-! KO0KA is the obvious monomorphism =TF =2TF =2TF ( =2)r1-1. Proposition 3.8 KO-2KA ~=eF2(1), generated by j2f1, and j2fi= 0 for 2 i r1. Proof: Consider the commutative square r1` K-2 bo _______-K-2KA | | | r| |r | | |? | r1` |? KO-2 bo ______KO-2KA- in which the horizontal maps and the left vertical map c are onto; hence the ri* *ght vertical map is also onto. This shows that KO-2KA is generated by rficf1 = j2f1, and tha* *t rficfi= 0 = j2fi, 0 i 2. On the other hand Proposition 3.6 implies that the image o* *f the lower map is isomorphic to eF2(1). Note that c : KO-2KA-! K-2KA is the zero map, while r : K-2KA-! KO-2KA is the obvious epimorphism eZ2(1)-! eF2(1). Proposition 3.9 KO-4KA ~=eZ2(2), generated by rfi2c(f1) = ,f1. Proof: Consider the diagram analogous to the one used in the previous propositi* *on. Alge- braically it has the form ( )r1 _______eZ2(2)- | | r| |r | | |? |? ( )r1 ______KO-4KA- 18 where now the left vertical map is an isomorphism and both horizontal maps are * *surjective. Hence the right vertical map is surjective and KO-4KA is generated by ,f1. As i* *n the pre- vious proposition, Proposition 3.6 shows that KO-4KA is abstractly isomorphic t* *o eZ2(2), and the result follows. Note that c : KO-4KA-! K-4KA is multiplication by two, while r : K-4KA-! K* *O-4KA is an isomorphism. The final case is trivial: Proposition 3.10 KO-6KA = 0. Note that the results of this section imply: Corollary 3.11 For m = 2, 3, 4, 5, 6 mod 8, j : KO-m KA-! KO-m-1 KA is zero. 3.4 Computation of KOm KA for m odd Proposition 3.12 KO-3KA ~=((1 - oe)eM)(1) and KO-5KA ~=((1 + oe)eM)(2) = eM+ (2). In each case the maps c, r are given respectively by the evident inclusion and * *norm maps. Proof: Consider the commutative squares r K-m KA _________-KO-m KA | | | | 1+f|f |c+ | | |? |? (1 + oe)K-m KA ______(K-m-KA)+i where c+ is the natural factorization of c and i is the inclusion. For m = 3, 5* *, Corollary 3.11 implies that r is onto and c+ is injective. Hence c+ is an isomorphism onto (1 * *+ oe)K-m KA. The remaining claims of the proposition follow easily from this, where in the c* *ase m = 5 we use the fact that ^H0(oe; M) = 0 (see [4], 4.8). Proposition 3.13 There are short exact sequences 0-! ( =2)r1-!KO-1KA-! eM+ -! 0 and 0-! eM+- ! KO-1KA-! eF2-! 0 19 In fact the first short exact sequence corresponds to 0-! jKO0KA-! KO-1KA-! c(KO-1KA)-! 0 and the second to '' -2 0-! r(K-1KA)-! KO-1KA -! KO KA-! 0. Proof: We have jKO0KA = (KO0KA)=(Im c) ~= ( =2)r1, using Proposition 3.7. Now consider the commutative square in the proof of Proposition 3.12, with m = -1. * *Although r is no longer surjective and c+ is no longer injective, the bottom arrow is an* * equality and c+ is surjective. Hence c(KO-1KA) = eM+ , yielding the first exact sequence. By Corollary 3.11 the sequence fi-1c-1 r -1 '' -2 0-! KO-3KA -! K KA -! KO KA -! KO KA-! 0 is exact. Since r : K-3KA-! KO-3KA is onto, we can compute the image of fi-1c* * by computing the image of the composite fi-1cr = fi-1(1 + oe) = (1 - oe)fi-1. Henc* *e Im (fi-1c) = Im (1 - oe), yielding the second exact sequence. Proposition 3.14 KO1KA ~= eM- (-1). Furthermore, j : KO1KA-! KO0KA maps onto the 2-torsion submodule ( =2)r1, generated by TFf1, f2, ..., fr1, and in f* *act the exact sequence '' 1 K1KA -r! KO1KA -! jKO KA-! 0 can be identified with eM(-1) 1+ff-!e(M(-1))+- ! ^H0(oe; eM(-1)) = eH^1(oe; M)(-1) ~=( =2)r1-! 0. Proof: There is a short exact sequence rfi-1 3 0-! KO1KA -c! K1KA -! KO KA-! 0. Since KO3KA injects into K3KA, the kernel of rfi-1 is the same as the kernel of* * crfi-1 = (1 + oe)fi-1 = fi-1(1 - oe). Hence KO1KA maps isomorphically onto the kernel o* *f 1 - oe, proving the first statement. The remaining assertions follow easily using Proposition 3.7. Note we have shown that c and r are given by inclusion of the fixed points a* *nd the norm 1 + oe, respectively. 4 Self-conjugate and united K-theory of KA Note: For details on united K-theory, see [1]. 20 4.1 (K*KA, KO*KA) as CR-object United K-theory consists of real, complex and self-conjugate K-theory, together* * with the various operations and maps relating the three theories. It has the homological* * advantage that its Adams spectral sequence vanishes above filtration 2. Furthermore, if * *the real K- theory of a spectrum satisfies a certain condition called CR-acyclicity, the se* *lf-conjugate theory is redundant for homological purposes and one can use the "CR" theory co* *nsisting of real and complex K-theory only. Although we do not make use of the united K-* *theory Adams spectral sequence in this paper, it is interesting to note that KA is CR-* *acyclic. Theorem 4.1 KA is CR-acyclic in the sense of Bousfield. Proof: We must show that the chain complex '' -m-1 ...-! jKO-m KA -! jKO KA-! ... is exact. In view of Corollary 3.11, this reduces to showing that '' 0 '' -1 0-! jKO1KA -! jKO KA -! jKO KA-! 0 is short exact. The second j is surjective by Proposition 3.8. By Proposition 3* *.7, jKO1KA is a free =2-module on TFf1, f2, ..., fr1. By Proposition 3.13, jKO0KA is a * *free =2- module on jfi, 1 i r1. Hence the first j above is injective with cokernel i* *somorphic to =(2, TF) = eF2, proving the proposition. 4.2 Self-conjugate K-theory 4.2.1 Operations in self-conjugate K-theory The Adams operations in self-conjugate K-theory (see [1]) lead to a continuous * *group homo- morphism from 0to the group of units in [KSC, KSC]. This in turn yields a cont* *inuous ring homomorphism 0-! KSC0KSC. Proposition 4.2 0-! KSC0KSC is an isomorphism of topological rings. Hence the functor KSC*(-) takes values in graded compact 0-modules, as it d* *oes for K*(-) and KO*(-). Now consider the natural transformations induced by the stan* *dard maps `` fl fi KO -ffl!KSC -! K -! KSC -! KO. Proposition 4.3 For any spectrum X, there are short exact sequences of 0-modu* *les 0-! (Kn-1X)+(-1)-! KSCnX- ! (KnX)+- ! 0 and 0-! KOnX=j2-! KSCnX- ! KOn+3X(1)-! 0. The resulting maps (Kn-1X)+(-1)-! KOn+3X(1) and KOnX=j2-! (KnX)+ are in- duced by rfi-2 and c, respectively. 21 4.3 Self-conjugate K-theory of KA In this section we compute KSC*KA up to extensions, using Proposition 4.3. One * *could also make use of KrelKA, as in x3.2, but this does not seem to help much in res* *olving the extension problems. For the application to Stiefel-Whitney classes, we only nee* *d KSC0KA. All exact sequences and isomorphisms below are as 0-modules. The proofs amoun* *t to substituting the results of our real and complex K-theory calculations into Pro* *position 4.3. Proposition 4.4 There are short exact sequences 0-! eM+(-1)-! KSC0KA-! eM+ -! 0 and 0-! KO0KR- ! KSC0KA-! eM+ (-1)-! 0. Proposition 4.5 There is a short exact sequence 0-! eF2-! KSC-1KA-! eM+ -! 0 and an isomorphism ~= -1 KO-1KA=j2 -! KSC KA. Proposition 4.6 There are isomorphisms ~= -2 eM- -! KSC KA and ~= - KSC-2KA -! eM . Proposition 4.7 There are short exact sequences 0-! eZ2(-1)-! KSC-3KA-! eM- (-1)-! 0 and 0-! (e(1 - oe)M)(1)-! KSC-3KA-! KO0X(1)-! 0. To compute the j multiplications in KSC*KA, we use the formula j = -1flfii.* * In other words, j : KSCnX- ! KSCn-1X is the composite KSCnX- ! KnX- ! Kn-2X(-1)-! KSCn-1X. 22 Proposition 4.8 For n = -1, -2, j : KSCnKA-! KSCn-1KA is zero. For n = 0, j is the evident composite KSC0KA-! eZ2-! eZ=2-! KSC-1KA, where the first two maps are surjective and the last is injective. For n = 1, j is given by the composite KSC1KA-! eM- (-1)-! (eM- =(1 - oe))(-1) = ( =2)r1-! KSC0KA, where again the first two maps are surjective and the last is injective. The proof is by direct inspection. Multiplication by the generator ! 2 ß3KSC can also be computed, at least up * *to exten- sions, by using the formula ! = flfi2i. For example, ! : KSC0KA-! KSC-3KA fac* *tors as KSC0KA-! eZ2-! eZ2(-1)-! KSC-3KA, where a twist has been introduced because ß3KSC ~=Z 2(1) as 0-module. Details * *will be left to the reader. 5 Proof of Theorem 2.11 We follow the outline given in x2.6. Since a = b under our splitting hypothesis* * (see Propo- sition 2.10), what remains to be shown is: Lemma 5.1 Im H*ff_ H*BO Pr1-a-1 (F P )a. We recall that a = dimZ=2Ker (H1(RF ; Z=2)-! H1(A; Z=2)). Now ffl : KO0KA-! KSC0KA is a homomorphism of 0-modules, and therefore ind* *uces a map ffl=M 0: KO0KA=M -! KSC0KR=M 0of Z=2-vector spaces. Lemma 5.2 dimZ=2Ker (ffl=M 0) a. Assuming this, we may choose a minimal generating set g1, ..., gr1for KO0KR * *such that (i) g1 = f1; (ii) g2, ..., gr1generate Ker c; and (iii) __gr1-a-1, ..., __gr1ar* *e in the kernel of ffl=M 0. Here __gimeans the reduction of gimod M 0. Lemma 5.3 a) 0acts trivially on H*BSC. b) If ` 2 M 0, then Im H* 10` FH*BSC. 23 Proof: a) It is clear that 0fixes the canonical map j : RP 1-! BO, and hence a* *lso fixes the composite fflj : RP 1-! BSC. But fflj is a generating complex for H*BSC, so (a)* * follows. b) By part (a), T and 1 - oe induce the null map on H*BSC. So we need only c* *heck that the image of the H-space squaring map_denoted [2]_lies in the squares F H*BSC. * * But the analogous statement for H*BO is obviously true, and the assertion for H*BSC* * follows immediately. It follows that there is a lift H*BO (H*U=O)r1-a-1 (H*Sp=U)a pp | ppp* | pppp | pppp | pppp | ppp | pppp | ppp |? H*BGLA ______H*BO-g (H*BO)r1-a-1 (H*BO)a completing the proof of Lemma 5.1 and hence also the proof of Theorem 2.11. It remains to prove Lemma 5.2. Let A 0denote Ker (KO0KR=M -! KSC0KR=M 0SC), and let a0 denote its dimensi* *on over Z=2. In the proof that follows we will successively replace A0 by certain* * Z=2-vector spaces A 1, A 2, A 3with dimensions a1, a2, a3, with a0 a1 - 1 and a1 = a2 = * *a3 = a + 1. Step 1: There is a commutative diagram of 0-modules c - 0 ______________-Ker c ___________KO0KA- __________K0KA- _____________ 0 | | | | | | | | | |? |? |? 0 _________-(K-1KA)+(-1) ______KSC0KA- __________K0KA- _____________-0 in which Ker c ~=( =2)r1and the vertical maps are injective. Reducing modulo M* * 0, we get a diagram of the form (Z=2)r1___________-(Z=2)r1_____________Z=2-_______________-0 | | | h1| h0| = | | | | |? |? |? K-1KA=M 0 ______KSC0KA=M- 0 _________-Z=2 _______________-0 Let A 1= Ker h1, a1 = dimZ=2A 1. Then A 1-! A0 is onto, and hence a0 a1- 1* *. (Note that this last inequality holds whether or not the kernel Z=2 of the upper left* * horizontal map in the diagram lies in A 1.) 24 Step 2. We next equate a1 with a purely number-theoretic invariant. Let A 2deno* *te the kernel of the natural map [M- =(1 - oe)M]=M -! M=M 0 and let a2 = dimZ=2A 2. We claim that a1 = a2. To see this, first note that multiplication by j : KO1KA-! KO0KR induces an* * iso- morphism KO1KR=rK1KA ~=Ker c. By Proposition 3.14, the source of this isomorphi* *sm can be identified with e[M- =(1 - oe)M](-1), where the twist is irrelevant sinc* *e the module being twisted is isomorphic to ( =2)r1. On the other hand, fflj = jffl = -1fl* *fic and hence fflj : KO1KA-! KSC0KA can be identified with the composite eM- (-1)-! eM(-1)-! eM+=(-1) = (K-1KA)+=(-1) KSC0KA . In other words, (Ker c)=M -! K-1KA=M 0can be identified with e([M- =(1 - oe)M]=M -! M=M 0), proving that a1 = a2 as desired. Step 3. Recall that we have fixed an embedding ff1 : F R, so that the 2-adic * *cyclotomic extension F1 is a subfield of C. Let Fn+denote the real subfield of Fn for 0 * *n 1, so in particular G(F1+=F ) = F. Similarly, let A+ndenote the corresponding ring of S* *-integers in Fn+. Let Y R n= RF+n= R. Hom(F+n,R) Note that H1(R1 ; Z2) ~=( F =2)r1, and there is a natural augmentation H1(R1* * ; Z2)-! Z=2 given by applying H1(-; Z2) to the composite r1a Spec R1- ! Spec RF = Spec R-! Spec R, where the last map is the folding map. Let ~H1(R1 ; Z2) denote the kernel of th* *is augmentation. The Serre spectral sequence for A+1=A yields a short exact sequence 0-! H1(A+1; Z2)=T -! H1(A; Z2)-! Z2-! 0 which, when reduced mod 2, becomes 0-! H1(A+1; Z2)=M -! H1(A; Z=2)-! Z=2-! 0. Thus the natural map H1(R1 ; Z2)-! H1(A+1; Z2) induces a map h3 : ~H1(R1 ; Z2)=M -! H1(A+1; Z2)=M . Let A 3= Ker h3, a3 = dimZ=2A 3. Claim: a2 = a3. 25 Note first that the Serre spectral sequence of A1 =A+1yields a short exact s* *equence 0-! M+- ! H1(A+1; Z2)-! Z=2-! 0. Hence we get an injective map M- =(1 - oe)M- ! H1(A+1; Z2). We next show that the image of this map coincides with the image of ~H1(R1 ;* * Z2)-! H1(A+1; Z2). Recall that the homology Serre spectral sequence of A1 =A+1is a spectral sequen* *ce of mod- ules over H*(oe; Z2) via the cap product. If x 2 H2(oe; Z2) is a generator, we * *have in particular a commutative diagram 0 __________-H2(oe; M)______H3(A+1;-Z2)________Z-=2 _____________0- | | | | | | | | = | | | | |? |? |? 0 __________-H0(oe; M)______H1(A+1;-Z2)________Z-=2 _____________0- where the vertical maps are given by cap product with x. This shows that the im* *age of M+ in H1(A+1; Z2) coincides with the image of Ker (H3(A+1; Z2)-! Z=2) under the c* *ap product. On the other hand, the commutative diagram ~= H3(R1 ; Z2)______H3(A+1;-Z2) | | | | ~ | | = | | | | |? |? H1(R1 ; Z2)______H1(A+1;-Z2) where again the vertical maps are given by cap product with x, and the top hori* *zontal arrow is an isomorphism by Tate's theorem, shows that this last image can be identifi* *ed with the image of ~H1(R1 ; Z2). It follows that a2 = a3, as claimed. Step 4. In this final step, we show that a3 = a + 1. This will complete the pro* *of of the key lemma, and hence also the proof of Theorem 2.11. Consider the commutative diagram of exact sequences Z=2 __________-H~1(R1 ; Z2)=M______H1(R1-; Z2)=M __________Z-=2 __________* *_______0- | | | | | | | = | | h3| h4| | | | | | |? |? |? |? 0 ___________-H1(A+1; Z2)=M _______H1(A;-Z=2) ____________Z-=2 __________* *_______0- Here we note that H1(R1 ; Z2) = H1(R1 ; Z=2), and that H1(R1 ; Z2)=M = H1(R* *1 ; Z2)=TF. Now let a4 = dimZ=2Ker h4. Then since the arrow Z=2-! ~H1(R1 ; Z2)=M in the u* *pper left is injective, diagram chase shows a3 = a4+ 1. On the other hand, h4 can be identif* *ied with the natural map H1(RF ; Z=2)-! H1(A; Z=2). Hence a4 = a, completing the proof of L* *emma 5.2. 26 References [1]Bousfield, A.K., A classification of K-local spectra, J. Pure Appl. Algebra* * 66 (1990), 120-163. [2]Mitchell, S.A., The algebraic K-theory spectrum of a 2-adic local field, K-* *theory 25 (2002), 1-37. [3]Mitchell, S.A., Topological K-theory of algebraic K-theory spectra, K-theor* *y 21 (2000), 229-247. [4]Mitchell, S.A., K-theory hypercohomology spectra of number rings at the pri* *me 2, Proceedings of the Arolla Conference on Algebraic Topology, Contemporary Ma* *th. 265 (2000), 129-158. [5]Neukirch, J., Algebraic Number Theory, Springer, Berlin, 1999. [6]Østvær, Paul Arne, 'Etale descent for real number fields, Topology 42 (2003* *), 197-225. [7]Quillen, D., On the cohomology and K-theory of the general linear group ove* *r a finite field, Annals of Math. 96 (1972), 552-586. [8]Rognes,J., and Weibel,C., Two-primary algebraic K-theory of rings of intege* *rs in number fields, J. Amer. Math. Soc. 13 (2000), 1-54. [9]Suslin, A., On the K-theory of local fields, J. Pure Appl. Alg. 34 (1984), * *301- 318. 27