The Stickelberger splitting map and Euler systems in the $K$--theory of number fields

THE STICKELBER GER SPLITTING MAP AND EULER SYSTEMS IN THE K –THEOR Y OF NUMBER FIELDS GRZEGORZ BANASZAK* AND CRISTIAN D. POPESCU** Abstract. F or a CM ab elian extension F /K of an arbitrary totally real n um- ber field K , w e co nstruct the Stic ke lb erger splitting maps (in the sense of [1 ]) for b oth the ´ etale and the Quillen K –theory of F and we use these maps to construct Euler systems in the even Quillen K –theory of F . The Stic k elb er ger splitting maps give an imm ediate proof of the annihilation of the groups of divisible elements divK 2 n ( F ) l of the even K –theory of the top field by higher Stic k elb er ger elements, for all odd primes l . This generalizes the results of [1], which only deals with C M ab elian extensions of Q . The techn iques in- v olve d in constructing our E ul er systems at this leve l of generality are quite differen t from those used in [3], where an Euler system in the o dd K –theory with finite co efficien ts of abeli an CM extensions of Q was given. W e wo rk under the assumption that the Iwasa w a µ –in v ariant conjecture holds. This perm its us to make use of the r ecen t results of Greither-Popescu [ 16] on the ´ et ale Coates-Sinnott conjecture for arbitrar y abeli an extensions of totally r eal n umber fields, which are conditional up on this assumption. In up coming work, we will use the Euler systems constructed in this pap er to obtain information on the groups of divisible elemen ts divK 2 n ( F ) l , for all n > 0 and o dd l . It is known that the structure of these groups is intimately related to some of the deepest unsolv ed pr oblems in algebraic num b er theory , e.g. the Kummer - V andiver and Iw asaw a conjectures on class groups of cyclotomic fields. W e mak e these connections explicit in the introduction. 1. Introduction Let F /K be an abelia n CM extension of a totally real n umber field K. Let f be the conductor of F /K and let K f /K b e the ray–class field extensio n with conductor f . Let G f := G ( K f /K ) . F or all n ∈ Z ≥ 0 , Coates [10] defined higher Stic kelberger elements Θ n ( b , f ) ∈ Q [ G ( F /K )], for integral ideals b of K coprime to f . Deligne and Rib et [12] prov ed that Θ n ( b , f ) ∈ Z [ G ( F /K )], if b is also copr ime to w n +1 ( F ) := card H 0 ( F, Q / Z ( n + 1)). A detailed dis cussion o f the Stic kelberger elements and their basic prop er ties is given in § 2 b elow. In 1974, Co ates and Sinnott [11] formulated the following conjecture. Conjecture 1.1 (Coates-Sinnott) . F or al l n ≥ 1 and al l b c oprime t o w n +1 ( F ) , Θ n ( b , f ) annihilates K 2 n ( O F ) . This should b e viewed as a higher analo g ue of the class ical conjecture of Brumer. 2000 Mathematics Subje ct Classific ation. 19D10, 11G30. Key wor ds and phr ases. K -theory of n umber fields; Special V alues of L -functions; Euler Systems. *P artially supp orted by grant NN201607440 of the Polish Minis try of Science and Education. **P artially supp orted by NSF grants DMS-901447 and DMS-0600905. 1 2 G. BANASZAK AND C. D. POP ESCU Conjecture 1.2 (Brumer) . F or al l b c oprime to w 1 ( F ) , Θ 0 ( b , f ) annihilates K 0 ( O F ) tors = C l ( O F ) . Coates and Sinnott [11] prov ed that for the bas e field K = Q the ele ment Θ 1 ( b , f ) annihilates K 2 ( O F ) for F / Q ab elian a nd b coprime to the order of K 2 ( O F ) . More- ov er, in the case K = Q , they proved that Θ n ( b , f ) annihilates the l –a dic ´ etale cohomolog y groups H 2 ( O F [1 /l ] , Z l ( n + 1)) ≃ K et 2 n ( O F [1 /l ]) for any o dd prime l , and any o dd n ≥ 1. One of the ingredients used in the proo f is the fact that Brumer’s conjecture holds true if K = Q . This is the classical theor em of Stick el- ber ger. The pass a ge from annihilation o f ´ etale cohomo logy to that o f K –theory in the ca se n = 1 was p os s ible due to the following theor em (see [28], [8] and [9].) Theorem 1.3 (T ate) . The l –adic Chern map gives a c anonic al isomorphism K 2 ( O L ) ⊗ Z l ∼ = − → K et 2 ( O L [1 /l ]) , for any n umb er field L and any o dd prime l . The following deep conjecture aims a t gener alizing T ate’s theorem. Conjecture 1.4 (Quillen-Lich tenbaum) . F or any numb er field L , any m ≥ 1 and any o dd prime l ther e is a n atur al l –adic Chern map isomorph ism K m ( O L ) ⊗ Z l ∼ = − → K et m ( O L [1 /l ]) (1) V ery recently , Greither and the se cond author used Iwasa wa theo retic techniques to prov e the following results for a general a b elian CM extens io n F / K of a n arbi- trary tota lly rea l field K (see [16].) Theorem 1.5 (Greither -Popescu) . Ass u me that l is o dd and the Iwasawa µ – invariant µ F, l asso ciate d to F and l vanishes. Th en, we have the fol lowing. (1) Q l (1 − ( l , F / K ) − 1 · N l ) · Θ 0 ( b , f ) annihilates C l ( O F ) l , for al l b c oprime t o w 1 ( K ) l , wher e t he pr o duct is taken over primes l of K which divide l and ar e c oprime to f . (2) Θ n ( b , f ) annihilates K et 2 n ( O F [1 /l ]) , for al l n ≥ 1 and al l b c oprime to w n +1 ( F ) l . In fact, stronger results are proved in [16], inv o lving Fitting ideals rather than annihilators and, in the case n = 0, a refinement of Br umer’s conjecture, known a s the B r umer-Stark conjecture (see Theorems 6 .5 a nd 6 .11 in lo c.cit.) Results similar to the Fitting ideal version o f par t (2) o f Theor em 1.5 w ere also obtained with different metho ds by Burns–Gr either in [5 ] a nd by Nguyen Quang Do in [19], under so me extra hypotheses . Note that a well known conjecture of Iw asaw a states that µ F, l = 0 , for all l and F as ab ove. This conjecture is known to hold if F is an ab elian extensio n of Q , due independently to F errero-W ashing ton and Sinnott. Consequently , if the Quillen- Lich tenbaum conjecture is prov ed, then, for all o dd pr imes l , the l –primary pa r t of the Coates-Sinnott conjecture is established unconditiona lly fo r all ab elian exten- sions F / Q and for genera l extensions F / K , under the as sumption that µ F, l = 0. It is hop ed that recent w ork o f Suslin, V oy evodsky , Rost, F r ie dlander, Morel, Levine, W eibel and others will lea d to a pro of of the Quillen-L icht enbaum conjecture. STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 3 In 1992, a differe nt appr o ach tow ards the Coates- Sinnott conjectur e was used in [1], in the case K = Q . Namely , for all n ≥ 1, all b coprime to w n +1 ( F ), a nd l > 2, the firs t author constructed in Ch. IV of lo c.cit. the Stick elb er ger splitting map Λ := Λ n of the b oundary map ∂ F in the Quillen lo ca lization sequence 0 − → K 2 n ( O F ) l − → K 2 n ( F ) l ∂ F − → Λ ← − M v K 2 n − 1 ( k v ) l − → 0 . By definitio n, Λ is a homomor phism such that ∂ F ◦ Λ is the mu ltiplication b y Θ n ( b , f ) . Ab ov e, k v denotes the res idue field of a prime v in O F . The ex istence of such a map Λ implies that Θ n ( b , f ) annihilates the group div ( K 2 n ( F ) l ) of divisible elements in K 2 n ( F ) l (see lo c .cit. a s w ell as Theor e m 4.23 b elow.) This gro up is contained in K 2 n ( O F ) l , which is obvious from the exa ct sequence a b ove and the finiteness of K 2 n − 1 ( k v ) l , fo r all v . The co nstruction of Λ in lo c.cit. was done without app ealing to ´ etale cohomolo g y and the Quillen- Lich tenbaum conjectur e . How ever, it was based on the fact that Brumer’s Conjecture is known to hold for ab elian extensions of Q (Stic kelberger ’s theorem). Since Brumer’s conjecture was not yet proved ov er ar bitrary totally real base fields (a nd it is still not proved unconditiona lly a t that level o f genera lity), the construction o f Λ in lo c.cit. could not b e generalized. A lso, it should b e men tioned that in lo c.cit. v ario us tec hnical difficulties a rose at pr imes l | n and the map Λ was constructed only up to a certa in p ow er l v l ( n ) in tho s e cas es. In 1996, in joint work with Ga jda [3], the fir st autho r discov ered a new, pe r haps deep e r a nd farther reaching a pplication of the existence of Λ for ab elian ex tensions F / Q . Namely , Λ was used in [3] to co nstruct sp ecia l elements which give rise to Euler systems in the K – theory with finite co efficients { K 2 n +1 ( L, Z /l k ) } L , where L runs ov er all a be lia n extensions of Q , such that F ⊆ L and L /F has a s q uare-free conductor c o prime to f l . Now, it is hop ed that these E uler sys tems ca n b e used to study the structure o f the g roup of divisible elements di v K 2 n ( F ) l , fo r all n ≥ 1. This is a g oal truly worth pursuing, as this g r oup structure is linked to s ome of the deep e st unsolved pro blems in algebra ic num b er theor y , as shown at the end o f this int ro duction. The main goal of this pap er is to g eneralize the r esults obtained in [1] a nd [3] to the case of CM ab elian extensions F /K of a rbitrary totally rea l n umber fields K . Moreover, in terms of constructing Euler systems, we go far beyond [3] in that we construct Euler systems in Quillen K –theory rather than K –theory with finite co efficients o nly . Roughly sp eaking, our strategy is a s follows. Step 1. W e fix a n integer m > 0 and ass ume that the m – th Stick elb er ger ele - men ts Θ m ( b , f k ) annihilate K 2 m ( O F k ) l (resp ectively K et 2 m ( O F k ) l ) for each k , where F k := F ( µ l k ) and f k is the conductor of F k /K . Under this a s sumption, we constr uc t the Stick elb erge r splitting maps Λ m (resp ectively Λ et m ) for the K –theory (r esp ec- tively ´ etale K – theory) of F k , fo r all k ≥ 1. (See Lemma 4 .5 and the constructions which lead to it.) Note that, if combined with Theor em 1 .3, Theorem 1 .5 shows that Θ m ( b , f k ) annihilates K 2 m ( O F k ) l , for m = 1 and l o dd, under the a ssumption that µ F, l = 0 (and unconditiona lly if F / Q is ab elian.) Also, in [20], the first a uthor constructs an infinite clas s o f ab elian CM extensions F /K of an arbitra ry totally real num b er field K for which the annihilation of K 2 m ( O F k ) l by Θ m ( b , f k ), for m = 1 a nd l o dd is proved unconditiona lly . 4 G. BANASZAK AND C. D. POP ESCU Step 2. W e use the ma p Λ m (resp ectively Λ et m )) of Step 1 to co nstruct sp ecial el- ement s λ v, l k (resp ectively λ et v, l k ) in the K –theory with co efficients K 2 n ( O F, S v ; Z /l k ) (resp ectively ´ etale K –theory with co efficients K et 2 n ( O F, S v ; Z /l k )), for all n > 0, a ll k ≥ 0 and a ll primes v in O F , where S v is a sufficiently lar ge finite set of primes in F . (See Definition 4 .7 .) Step 3. W e us e the s pe c ial element s of Step 3 and a pro jective limit pro cess with res pe c t to k to constr uct the Stickelberger splitting maps Λ n and Λ et n taking v a lues in K 2 n ( F ) l and K et 2 n ( F ), resp ectively , for all n ≥ 1 . (See Definition 4 .16 and Theor e m 4.1 7.) This step ge ne r alizes the co ns tructions in [1 ] to a b e lia n CM extensions of arbitrary totally real fields . It also eliminates the extra-fa c to r l v l ( n ) which app ea red in lo c.cit. in the case l | n , for ab elia n CM extensions of Q . Step 4. W e use the sp ecia l elements of Step 2 as well a s the maps Λ n of Step 3 to construct Euler Sys tems { Λ n ( ξ v ( L ) ) } L in the K -theory without co efficients { K 2 n ( F L ) l } L , for e very n > 0, where L runs thro ugh the square fr ee idea ls o f O F which ar e copr ime to f l , F L is the r ay clas s field of F corres po nding to L and S is a s ufficient ly large finite s e t of primes in O F . (See Definitions 5.4 and 5.5 a s well as Theorem 5 .7.) A similar construction o f Euler s ystems in ´ etale K –theory can b e done witho ut difficulty . This step g eneralizes the constructions of [3] to the ca se of ab elian CM extensions of totally real nu mber fields. It is also worth no ting that while [3] co nt ains a construction of Euler systems o nly in the case of K –theory with co efficients, w e dea l with b oth the K –theor y with and without co efficients in the more gener al setting discussed in this pap er . In the pro cess, as a co nsequence of the construction of Λ n (Step 3), we o btain a dire c t pro of that Θ n ( b , f ) a nnihilates the g roup di v ( K 2 n ( F ) l ), for arbitra ry CM ab elian extensions F /K of totally real base field K and all n > 0, under the assumption tha t l > 2 and µ F, l = 0 (see Theor e m 4.26.) In our up co ming work, we a r e planning on using the Euler s ystems describ ed in Step 4 ab ove to study the structure of the gr oups of div isible elements div K 2 n ( F ) l , for a ll n > 0 and all l > 2 . W e conclude this introduction with a few par agraphs showing that the gr oups of divisible elements in the K –theory of num b er fields lie a t the heart of several impo rtant conjectures in num b er theory , whic h justifies the effor t to understand their s tructure in ter ms of sp ecial v alues of global L –functions. In 19 88, W ar ren Sinnott po int ed out to the first author that Stick elb erger’s Theorem for an ab elian extension F / Q or , mor e gener a lly , B rumer’s co njecture for a CM extension F /K of a totally real n umber field K is equiv a le n t to the existence of a Stic kelberg er splitting map Λ in the following bas ic exac t sequence 0 − → O × F − → F × ∂ F − → Λ ← − M v Z − → C l ( O F ) − → 0 . This means that Λ is a gro up homomor phis m, such that ∂ F ◦ Λ is the multiplication by Θ 0 ( b , f ) . Obviously , the ab ove exact sequence is the low e r pa rt of the Quillen lo calization sequence in K –theor y , since K 1 ( O F ) = O × F , K 1 ( F ) = F × , K 0 ( k v ) = Z , K 0 ( O F ) tors = C l ( O F ) a nd Q uillen’s ∂ F is the direct sum of the v a luation ma ps in this case . F urther, by [2] p. 29 2 we obser ve that for any pr ime l > 2 , the annihilatio n of div ( K 2 n ( F ) l ) b y Θ n ( b , f ) is equiv a lent to the existence of a “splitting” map Λ in STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 5 the following exact sequence 0 − → K 2 n ( O F )[ l k ] − → K 2 n ( F )[ l k ] ∂ F − → Λ ← − M v K 2 n − 1 ( k v )[ l k ] − → div ( K 2 n ( F ) l ) − → 0 such that ∂ F ◦ Λ is the multiplication by Θ n ( b , f ), for any k ≫ 0. Hence, the gr oup o f divisible ele men ts di v ( K 2 n ( F ) l ) is a dir ect a nalogue o f the l –pr ima ry part C l ( O F ) l of the clas s gro up. An y t wo such “s plittings ” Λ differ by a homomorphism in Hom( L v K 2 n − 1 ( k v )[ l k ] , K 2 n ( O F )[ l k ]) . Moreover, the Coates-Sinno tt co njecture is equiv alent to the exis tence of a “splitting” Λ, such that Λ ◦ ∂ F is the multiplication by Θ n ( b , f ) . If the C o ates-Sinnott co njecture ho lds, then such a “splitting” Λ is unique a nd satisfies the prop er t y that ∂ F ◦ Λ is equa l to the multiplication b y Θ n ( b , f ). This is due to the fact that div ( K 2 n ( F ) l ) ⊂ K 2 n ( O F ) l . Clea rly , in the case div ( K 2 n ( F ) l ) = K 2 n ( O F ) l , our map Λ als o has the pro p erty that Λ ◦ ∂ F equals m ultiplication by Θ n ( b , f ) . O bserve tha t if the Quillen-Lich tenbaum conjecture holds, then by Theorem 4 in [2], we hav e div ( K 2 n ( F ) l ) = K 2 n ( O F ) l ⇔      Q v | l w n ( F v ) w n ( F )      − 1 l = 1 . In particular, for F = Q and n o dd, we ha ve w n ( Q ) = w n ( Q l ) = 2. Hence, according to the Q uille n- Lich tenbaum conjecture, for an y l > 2 we should hav e div ( K 2 n ( Q ) l ) = K 2 n ( Z ) l . Now, let A := C l ( Z [ µ l ]) l and let A [ i ] denote the eigenspa c e co rresp onding to the i –th p ower of the T eichm uller character ω : G ( Q ( µ l ) / Q ) → ( Z /l Z ) × . Consider the following cla ssical conjectures in cyclotomic field theory . Conjecture 1.6 (Kummer- V andiver) . A [ l − 1 − n ] = 0 for al l n even and 0 ≤ n ≤ l − 1 Conjecture 1.7 (Iwasaw a) . A [ l − 1 − n ] is cyclic for al l n o dd, such that 1 ≤ n ≤ l − 2 W e ca n s tate the Kummer-V andiver and Iwasaw a co njectures in ter ms of divisible elements in K –theory o f Q (see [3] and [4]): (1) A [ l − 1 − n ] = 0 ⇔ di v ( K 2 n ( Q ) l ) = 0 , for all n e ven, with 1 ≤ n ≤ ( l − 1 ) . (2) A [ l − 1 − n ] is cyclic ⇔ di v ( K 2 n ( Q ) l ) is cyclic, for all n o dd, with n ≤ ( l − 2) . Finally , we would like to p oint out tha t the gro ups of divisible elements discussed in this paper are also related to the Quillen-Lich tenbaum conjecture. Namely , by comparing the exact sequence of [24], Satz 8 with the exact sequence of [2], Theor em 2 we conclude tha t the Q uillen-Lich tenbaum conjecture for the K -gro up K 2 n ( F ) (for any num b er field F a nd any prime l > 2) holds if and only if div ( K 2 n ( F ) l ) = K w 2 n ( O F ) l where K w 2 n ( O F ) l is the wild kernel defined in [2 ]. 6 G. BANASZAK AND C. D. POP ESCU 2. Basic f acts abo ut the Stickelberge r ideals Let F /K b e an ab elian CM extension of a to tally r e al num b er field K . Let f be the conductor of F /K and let K f /K b e the ray c lass field extension corres po nding to f . Let G f := G ( K f /K ) . Every e lement of G f is the F rob enius morphism σ a , fo r some ideal a of O K , coprime to the conductor f . Le t ( a , F ) denote the image of σ a in G ( F /K ) via the natural surjection G f → G ( F /K ) . Cho o se a prime num be r l . With the usual notations, we let I ( f ) /P 1 ( f ) b e the ray class group of fractiona l ideals in K c o prime to f . Let a and a ′ be t wo fractiona l ideals in I ( f ) . The symbol a ≡ a ′ mo d f will mea n that a and a ′ are in the sa me class modulo P 1 ( f ) . F or every a ∈ I ( f ) we consider the par tial zeta function of [10], p. 29 1 , given by (2) ζ f ( a , s ) := X c ≡ a mo d f 1 N c s , Re( s ) > 1 , where the sum is taken ov er the in teg r al idea ls c ∈ I ( f ) and N c denotes the usual norm of the integral ideal c . The par tial zeta ζ f ( a , s ) ca n b e meromorphically contin ued to the co mplex pla ne with a single pole at s = 1 . F or s ∈ C \ { 1 } , consider the Sick elb er ger element of [C], p. 29 7, (3) Θ s ( b , f ) := ( N b s +1 − ( b , F )) X a ζ f ( a , − s )( a , F ) − 1 ∈ C [ G ( F /K )] where b is an integral ideal in I ( f ) and the summation is ov er a finite se t S of ideals a o f O K coprime to f , chosen such that the Ar tin map S − → G ( K f /K ) , a − → σ a is bijective. The element Θ s ( b , f ) can be written in the following way (4) Θ s ( b , f ) := X a ∆ s +1 ( a , b , f )( a , F ) − 1 , where (5) ∆ s +1 ( a , b , f ) := N b s +1 ζ f ( a , − s ) − ζ f ( ab , − s ) . Arithmetically , the Stick elb erg er elements Θ s ( b , f ) a re mo st in teresting for v alues s = n , with n ∈ N ∪ { 0 } . If a , b , f ar e in tegral ideals, s uch that ab is coprime to f , then Deligne and Ribe t [12] pr ov ed that ∆ n +1 ( a , b , f ) are l -a dic integers for a ll primes l 6 | N b and all n ≥ 0. Moreov er , in lo c.cit. it is pr ov ed that (6) ∆ n +1 ( a , b , f ) ≡ N ( ab ) n ∆ 1 ( a , b , f ) mo d w n ( K f ) . As usual, if L is a num b er field, then w n ( L ) is the la rgest num b er m ∈ N such tha t the Ga lois gro up G ( L ( µ m ) /L ) has ex po nent dividing n. Note that w n ( L ) = | H 0 ( G ( L/L ) , Q / Z ( n )) | , where Q / Z ( n ) := ⊕ l Q l / Z l ( n ) . By Theore m 2.4 of [C], the results in [1 2] lead to Θ n ( b , f ) ∈ Z [ G ( F /K )] , whenever b is coprime to w n +1 ( F ) . The ideal o f Z [ G ( F / K )] generated b y the elements Θ n ( b , f ), for a ll integral ideals b co prime to w n +1 ( F ) is called the n -th Stic kelberger ideal for F /K. STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 7 When K ⊂ F ⊂ E is a tow er o f finite ab elian extensions then Res E /F : G ( E /K ) → G ( F /K ) , Res E /F : C [ G ( E /K )] → C [ G ( F /K )] denote the re striction map and its C –linear extensio n at the level of g roup rings, resp ectively . If f | f ′ and f and f ′ are divisible by the same prime idea ls o f O K then, for all b coprime to f , we have the following equality (see [1 0] Lemma 2 .1, p. 292). (7) Res K f ′ /K f Θ s ( b , f ′ ) = Θ s ( b , f ) . Let l is a prime ideal of O K coprime to f . Then, we hav e (8) ζ f ( a , s ) := X c ≡ a mod f l ∤ c 1 N c s + X c ≡ a mod f l | c 1 N c s . Observe that we also hav e (9) X c ≡ a mod f l ∤ c 1 N c s = X a ′ mod lf a ′ ≡ a mod f X c ≡ a ′ mo d lf 1 N c s = X a ′ mod lf a ′ ≡ a mod f ζ lf ( a ′ , s ) Let us fix a finite S of in tegral ideals a in I ( f ) as ab ove. Obser ve that e very class corres p o nding to an integral ideal a modulo P 1 ( f ) can be written uniquely as a class la ′′ mo dulo P 1 ( f ), for some a ′′ from our s et S of chosen integral idea ls. This establishes a o ne–to–one corre s po ndence betw een class es a modulo P 1 ( f ) and a ′′ mo dulo P 1 ( f ) . If l | c , w e put c = l c ′ . Hence , we hav e the following equa lity . (10) X c ≡ a mod f l | c 1 N c s = 1 N l s X c ′ ≡ a ′′ mo d f 1 N c ′ s = 1 N l s ζ f ( a ′′ , s ) F ormulas (8), (9) and (10) lead to the following equality: (11) ζ f ( a , s ) − 1 N l s ζ f ( l − 1 a , s ) = X a ′ mod lf a ′ ≡ a mod f ζ lf ( a ′ , s ) . F or a ll f coprime to l and for a ll b coprime to lf , equality (11) gives: (12) Res K lf /K f Θ s ( b , lf ) = (1 − ( l , F ) − 1 N l s ) Θ s ( b , f ) Indeed we eas ily chec k that: Res K lf /K f ( N b s +1 − ( b , F )) X a ′ mo d lf ζ lf ( a ′ , − s )( a ′ , F ) − 1 = ( N b s +1 − ( b , F )) X a mo d f X a ′ mod lf a ′ ≡ a mod f ζ lf ( a ′ , − s )( a , F ) − 1 = ( N b s +1 − ( b , F )) X a mo d f ( ζ f ( a , − s ) − N l s ζ f ( l − 1 a , − s ))( a , F ) − 1 = ( N b s +1 − ( b , F ))( X a m o d f ζ f ( a , − s )( a , F ) − 1 − ( l , F ) − 1 N l s ζ f ( l − 1 a , − s )( l − 1 a , F ) − 1 ) = 8 G. BANASZAK AND C. D. POP ESCU (1 − ( l , F ) − 1 N l s )( N b s +1 − ( b , F )) X a m o d f ζ f ( a , − s )( a , F ) − 1 Lemma 2 .1. L et f | f ′ b e ide als of O K c oprime t o b . Then, we have the fol lowing. (13) Res K f ′ /K f Θ s ( b , f ′ ) =  Y l ∤ f l | f ′ (1 − ( l , F ) − 1 N l s )  Θ s ( b , f ) Pr o of. The lemma follows from (7) and (12).  Remark 2.2. The prop er ty o f higher Stickelberger elements given b y the abov e Lemma will tra nslate naturally into the E uler System prop er t y of the sp ecia l ele- men ts in Quillen K –theory constructed in § 5 b elow. In what follows, for any given a b e lia n extension F /K of c o nductor f , we consider the field ex tensions F ( µ l k ) / K , fo r all k ≥ 0 and a fixed prime l , where µ l k denotes the group of ro ots of unity o f order dividing l k . W e let f k denote the conductor of the ab elia n extension F ( µ l k ) /K. W e suppress fro m the notatio n the explicit depe ndence of f k on l , s ince the prime l will be chosen and fixed o nce a nd for all in this pap er. 3. Basic f acts abo ut algebraic K -theor y 3.1. The Bo c kstei n sequence and the Bott ele men t. Let us fix a prime num- ber l . F or a ring R we c o nsider the Quille n K -groups K m ( R ) := π m (Ω B Q P ( R )) := [ S m , Ω B QP ( R )] (see [21]) and the K - groups with co efficients K m ( R, Z /l k ) := π m (Ω B Q P ( R ) , Z /l k ) := [ M m l k , Ω B QP ( R )] defined by Browder a nd Karoubi in [6]. Q uillen’s K – groups can also b e computed using Quillen’s plus co nstruction as K n ( R ) := π n ( B GL ( R ) + ) . Any unital homo - morphism o f rings φ : R → R ′ induces natura l homomo rphisms φ R | R ′ : K m ( R, ♦ ) − → K m ( R ′ , ♦ ) where K m ( R, ♦ ) deno tes either K m ( R ) or K m ( R, Z /l k ) . Quillen K -theory a nd K -theo ry with co efficients admit pr o duct structur es: K n ( R, ♦ ) × K m ( R, ♦ ) ∗ − → K m + n ( R, ♦ ) (see [21] and [6].) These induce g r aded ring structures on the gro ups L n ≥ 0 K n ( R, ♦ ) . F or a top olog ical space X , there is a Bo ckstein exa ct sequence − → π m +1 ( X, Z /l k ) b − → π m ( X ) l k − → π m ( X ) − → π m ( X, Z /l k ) − → In pa rticular, if we ta ke X := Ω B QP ( R )), we get the Bo ckstein exa ct sequence in K - theory given by (14) − → K m +1 ( R, Z /l k ) b − → K m ( R ) l k − → K m ( R ) − → K m ( R, Z /l k ) − → STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 9 F or a n y discrete gr oup G , w e have: π n ( B G ) =  G if n = 1 0 if n > 1 . Consequently , for a commutativ e g roup G and X := B G the Bo ckstein map b gives an is omorphism b : π 2 ( B G, Z /l k ) ∼ = − → G [ l k ]. Here, G [ m ] denotes the m – torsion subgroup of the commutativ e g roup G , for all m ∈ N . F or a commutativ e ring with identit y R w e have GL 1 ( R ) = R × . Assume that µ l k ⊂ R × . Then R × [ l k ] = µ l k . Let β deno te the natural comp osition of ma ps: µ l k b − 1 / / π 2 ( B GL 1 ( R ); Z /l k ) / / π 2 ( B GL ( R ); Z /l k )   π 2 ( B GL ( R ) + ; Z /l k ) = / / K 2 ( R, Z /l k ) W e fix a g enerator ξ l k of µ l k . W e define the Bott element (15) β k := β ( ξ l k ) , β k ∈ K 2 ( R ; Z /l k ) as the image of ξ l k via β . F urther, we let β ∗ n k := β k ∗ · · · ∗ β k ∈ K 2 n ( R ; Z /l k ) . The Bott elemen t β k depe nds of course on the ring R . Ho wever, we s uppress this depe ndence from the notation since it will be alwa ys clear wher e a given Bott element lives. F or example, if φ : R → R ′ is a homomorphism of commutative rings containing µ l k , then it is clea r fro m the definitions that the map φ R | R ′ : K 2 ( R ; Z /l k ) − → K 2 ( R ′ , Z / l k ) transp orts the Bott element for R in to the Bott element for R ′ . By a slig ht a buse of notatio n, this will be written as φ R | R ′ ( β k ) = β k . Dwyer and Fiedla nder [13] constructed the ´ eta le K -theory K et ∗ ( R ) and ´ etale K - theory with co efficients K et ∗ ( R, Z /l k ) for a ny commutativ e, No etherian Z [1 /l ]– algebra R . Moreover, they pr ov ed that if l > 2 then there a re natural gra ded ring homomorphisms, ca lled the Dwyer-F riedlander maps: (16) K ∗ ( R ) − → K et ∗ ( R ) (17) K ∗ ( R ; Z /l k ) − → K et ∗ ( R ; Z /l k ) . If R has finite Z /l - cohomolog ical dimension then there ar e Atiy ah-Hirz e bruch type sp ectral sequences (see [13], Prop os itions 5.1, 5 .2 ): (18) E p, − q 2 = H p ( R ; Z l ( q / 2)) ⇒ K et q − p ( R ) . (19) E p, − q 2 = H p ( R ; Z /l k ( q / 2)) ⇒ K et q − p ( R ; Z /l k ) . Throughout, we will deno te by r k ′ /k the r eduction maps at the level of co efficients r k ′ /k : K ∗ ( R ; Z /l k ′ ) → K ∗ ( R ; Z /l k ) , r k ′ /k : K et ∗ ( R ; Z /l k ′ ) → K et ∗ ( R ; Z /l k ) , for a ny R as ab ov e and k ′ ≥ k . 10 G. BANASZAK AND C. D. POP ESCU 3.2. K -the o ry of finite fie lds. Let F q be the finite field with q elements. In [Q3], Quillen proved that: K n ( F q ) ≃    Z if n = 0 0 if n = 2 m and m > 0 Z / ( q m − 1 ) Z if n = 2 m − 1 and m > 0 Moreov er, in lo c.cit, pp. 5 83-58 5, it is als o showed that fo r an inclusion i : F q → F q f of finite fields and a ll n ≥ 1 the natur a l map i : K 2 n − 1 ( F q ) → K 2 n − 1 ( F q f ) is injective and the transfer map N : K 2 n − 1 ( F q f ) → K 2 n − 1 ( F q ) is surjective, where we simply write i instead of i F q | F q f and N instead of T r F q f / F q . F urther (see lo c.cit., pp. 583- 585), i induces an isomor phism K 2 n − 1 ( F q ) ∼ = K 2 n − 1 ( F q f ) G ( F q f / F q ) and the q – power F ro b enius automor phism F r q (the ca nonical ge ner ator o f G ( F q f / F q )) acts o n K 2 n − 1 ( F q f ) via multiplication by q n . O bserve tha t i ◦ N = f − 1 X i =0 F r i q . Hence, we have the equalities Ker N = K 2 n − 1 ( F q f ) F r q − I d = K 2 n − 1 ( F q f ) q n − 1 since Ker N is the kernel o f multiplication by P f − 1 i =0 q ni = q nf − 1 q n − 1 in the cyclic gro up K 2 n − 1 ( F q f ) . In par ticular, this shows that the norm map N induces the following group iso morphism K 2 n − 1 ( F q f ) G ( F q f / F q ) ∼ = K 2 n − 1 ( F q ) . By the Bo ckstein exact se quence (14) and Q uillen’s r e s ults ab ove, we observe that K 2 n ( F q , Z / l k ) b − → K 2 n − 1 ( F q )[ l k ] is an isomorphism. Hence, K 2 n ( F q , Z / l k ) is a cyclic group. Let us as sume that µ l k ⊂ F × q (i.e. l k | q − 1.) In this case, B r owder [6] pr ov ed that the element β ∗ n k is a generator of K 2 n ( F q , Z / l k ). Dwyer and F riedlander [1 3] prov ed tha t there is a na tural isomor phism of graded rings: K ∗ ( F q , Z / l k ) ∼ = − → K et ∗ ( F q , Z / l k ) . By abuse of no tation, let β k denote the image of the B ott element defined in (15) via the natura l isomorphism: K 2 ( F q , Z /l k ) ∼ = − → K et 2 ( F q , Z /l k ) . Then by Theor em 5.6 in [1 3] multiplication with β k induces isomorphisms: × β k : K i ( F q , Z / l k ) ∼ = − → K i +2 ( F q , Z / l k ) , × β k : K et i ( F q , Z / l k ) ∼ = − → K et i +2 ( F q , Z / l k ) . In par ticular, if l k | q − 1 and α is a gener ator of K 1 ( F q , Z / l k ) = K 1 ( F q ) /l k , then the ele ment α ∗ β ∗ n − 1 k is a generato r of the cyclic gr oup K 2 n − 1 ( F q , Z /l k ) . STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 11 3.3. K -the o ry of n umber fields and their ring s of in tege rs. Let F b e a nu mber field. As usual, O F denotes the ring o f integers in F and k v is the r esidue field fo r a prime v of O F . F or a finite set of primes S of O F the ring of S -integers of F is denoted O F, S . Quillen [22] prov ed tha t K n ( O F ) is a finitely genera ted gro up for every n ≥ 0 . Borel computed the ranks of the gr oups K n ( O F ) a s follows: K n ( O F ) ⊗ Z Q ≃            Q if n = 0 Q r 1 + r 2 − 1 if n = 1 0 if n = 2 m and n > 0 Q r 1 + r 2 if n ≡ 1 mo d 4 and n 6 = 1 Q r 2 if n ≡ 3 mo d 4 W e ha ve the following localiza tion exact sequences in Quillen K - theory and K - theory with co efficie nts [21]. − → K m ( O F , ♦ ) − → K m ( F, ♦ ) ∂ F − → M v K m − 1 ( k v , ♦ ) − → K m − 1 ( O F , ♦ ) − → Let E /F be a finite ex tension. The na tural maps in K - theory induced by the embedding i : F → E and σ : E → E , for σ ∈ G ( E /F ), will b e denoted for simplicity by i : K m ( F, ♦ ) − → K m ( E , ♦ ) and σ : K m ( E , ♦ ) − → K m ( E , ♦ ) . Observe that i := i F | E and σ := σ E | E , acco rding to the notation in sec tio n 3.1. In additio n to the natural maps i, σ, ∂ F , ∂ E , and pro duct structur e s ∗ for K -theory of F and E int ro duced ab ov e, we have (see [21]) the tr ansfer map T r E /F : K m ( E , ♦ ) − → K m ( F, ♦ ) and the reduction map r v : K m ( O F, S , ♦ ) − → K m ( k v , ♦ ) for a ny prime v / ∈ S. The maps discussed ab ove enjoy many compatibility prop erties. F or ex a mple, σ is naturally compatible with i , ∂ F , ∂ E , the pro duct structure ∗ , T r E /F and r w and r v . See e.g. [1] for e x planations of some of these compatibility prop er ties. Let us men tion b e low two nontrivial such compatibility prop erties which will b e used in what follows. By a result of Gillet [17], we hav e the following commutativ e diag rams in Quillen K -theory and K -theory with co efficients: (20) K m ( F, ♦ ) × K n ( O F , ♦ ) ∂ F × id   ∗ / / K m + n ( F, ♦ ) ∂ F   L v K m − 1 ( k v , ♦ ) × K n ( O F , ♦ ) ∗ / / L v K m + n − 1 ( k v , ♦ ) Let E /F b e a finite extension unramified ov er a prime v of O F . Let w b e a prime of O E ov er v . F rom now on, we will write N w/v := T r k w /k v . The fo llowing dia g ram shows the compatibility of transfer with the b oundary map in lo caliza tion sequences for Q uillen K -theory and K -theory with co efficients. 12 G. BANASZAK AND C. D. POP ESCU (21) K m ( E , ♦ ) T r E /F   ∂ E / / L v L w | v K m − 1 ( k w , ♦ ) L v L w | v N w/v   K m ( F, ♦ ) ∂ F / / L v K m − 1 ( k v , ♦ ) where the direct sums a re taken with re s pe ct primes v in F and w in E , r esp ectively . 4. Construction of Λ and Λ et and first applica tions In this section, w e cons truct sp ecia l ele men ts in K -theory and ´ etale K -theory with co efficients, under the ass umption that for some fixed m > 0 the Stick elb erger elements Θ m ( b , f k ) annihilate K 2 m ( O F l k ) for all k ≥ 0 . This will pro duce sp ecia l elements in K -theory a nd ´ etale K - theory without co efficients which are o f primary impo rtance in our construction of Eule r systems given in § 5. These constructions will also give us the Stick elb er ger s plitting maps Λ := Λ n and Λ et := Λ et n announced in the intro duction. As a byproduct, w e obtain a direc t pro of of the annihilation of the groups divisible element s div ( K 2 n ( F ) l ), for all n > 0, genera lizing the r esults of [1]. All the r esults in this section ar e stated for both K -theor y a nd ´ etale K -theory . How ever, detailed pro ofs will b e given only in the case of K - theory sinc e the pro ofs in the case o f ´ etale K -theory are very similar. The key idea in transferring the K – theoretic constructions to ´ eta le K –theor y is the following. Replace Gillet’s result [17] for K -theory (commutativ e diagra m (20) of § 3 ) with the compatibilit y o f the Dwyer-F riedlander sp ectr al seq uence with the pro duct structure ([13], Pr op osition 5.4) combin ed with Soul ´ e ’s observ atio n (see [25], p. 27 5) that the localiza tion sequence in ´ etale co homology (see [25], p. 268 ) is compatible with the pr o duct by the ´ etale cohomolo gy of O F, S . 4.1. Constructing sp ecial el ement s in K –theory with co efficients. Let L be a num ber field, such that µ l k ⊂ L . Let S be a finite set of prime ideals of O L containing all primes over l . Let i ∈ N a nd let m ∈ Z , such that i + 2 m > 0. Then, for R = L o r R = O L,S there is a na tural gr oup iso mo rphism (see [13] Theo rem 5.6): (22) K et i ( R ; Z /l k ) ∼ = − → K et i +2 m ( R ; Z /l k ) which s ends η to η ∗ β ∗ m k for an y η ∈ K et i ( R ; Z /l k ) . If m ≥ 0 this isomorphism is just the multiplication by β ∗ m k . If m < 0 a nd i + 2 m > 0 , then the isomorphis m (22) is the in verse o f the multiplication by β ∗ − m k isomorphism: (23) ∗ β ∗ − m k : K et i +2 m ( R ; Z /l k ) ∼ = − → K et i ( R ; Z /l k ) . Now, let us consider Q uillen K -theory . If m ≥ 0 , there is a natura l ho momorphism (24) ∗ β ∗ m : K i ( R ; Z /l k ) → K i +2 m ( R ; Z /l k ) which is just multiplication b y β ∗ m k . The homo mo rphism (24) is compatible with the isomorphism (22) via the Dwy er-F riedlander map. If m < 0 and i + 2 m > 0, STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 13 then take the homomor phism (25) t ( m ) : K i ( R ; Z /l k ) → K i +2 m ( R ; Z /l k ) to b e the unique homo morphism which ma kes the following diag ram co mm utative. K i ( R ; Z /l k )   t ( m ) / / K i +2 m ( R ; Z /l k ) K et i ( R ; Z /l k ) ( ∗ β ∗ − m k ) − 1 / / K et i +2 m ( R ; Z /l k ) O O The left vertical a rrow is the Dwy er- F riedlander ma p, while the r ight vertical arrow is the Dwyer-F riedlander splitting map (see [13], Prop os itio n 8.4.) The latter map is o btained as the multiplication of the inv er se of the isomo rphism K i ′ ( R ; Z /l k ) ∼ = − → K et i ′ ( R ; Z /l k ) , for i ′ = 1 or i ′ = 2 , by a nonnegative p ower of the Bott element β ∗ m ′ k , w ith m ′ ≥ 0 (see the pro of of Pro p osition 8.4 in [13].) Remark 4. 1. It is c lear that the Dwy er-F riedla nder splitting from [13], Pro po sition 8.4 is compa tible with the maps Z /l j → Z / l j − 1 at the lev el o f co e fficient s, for all 1 ≤ j ≤ k . Consequently , the map t ( m ) is na turally compatible with these maps. In a ddition, t ( m ) is naturally compatible with the ring embedding R → R ′ , w her e R ′ = L ′ or R ′ = O L ′ ,S for a num b er field e x tension L ′ /L. L et t et ( m ) := ( ∗ β ∗ − m k ) − 1 . It is clear from the ab ov e diagr am that t ( m ) a nd t et ( m ) are natura lly co mpatible with the Dwyer-F riedlander maps. Lemma 4 . 2. L et L = F ( µ l k ) a nd let i > 0 and m < 0 , such t hat i + 2 m > 0 . Then, for R = L or R = O L,S , the n atur al gr oup homomorph isms t et ( m ) and t ( m ) have the fol lowing pr op erties: (26) t et ( m )( α ) σ a = t et ( m )( α N a m σ a ) (27) t ( m )( α ) σ a = t ( m )( α N a m σ a ) for any ide al a of O F c oprime to f k and for α ∈ K et i ( R ; Z /l k ) and α ∈ K i ( R ; Z /l k ) , r esp e ctively. Lemma 4.3. If i ∈ { 1 , 2 } , α ∈ K i ( R ; Z /l k ) and n + m > 0 then (28) t et ( m )( α ∗ β ∗ n k ) = α ∗ β ∗ n + m k . (29) t ( m )( α ∗ β ∗ n k ) = α ∗ β ∗ n + m k . Pr o of of L emmas 4.2 and 4.3. The pro p e r ties in Lemmas 4.2 a nd 4.3 follow directly from the definition of the maps t et ( m ) a nd t ( m ) .  If v is a prime of O L,S , m < 0 and i + 2 m > 0, then we construc t the morphism (30) t v ( m ) : K i ( k v ; Z /l k ) → K i +2 m ( k v ; Z /l k ) 14 G. BANASZAK AND C. D. POP ESCU in the same way as we have done for O L,S or L. Namely , t v ( m ) is the ho momorphism which makes the following diagram co mmu te. K i ( k v ; Z /l k ) ∼ =   t v ( m ) / / K i +2 m ( k v ; Z /l k ) K et i ( k v ; Z /l k ) ( ∗ β ∗ − m k ) − 1 / / K et i +2 m ( k v ; Z /l k ) ∼ = O O The rig h t vertical arr ow is the in verse of the Dwyer-F riedlander map which, in the case of a finite field, is clear ly seen to b e equal to the Dwyer-F riedlander splitting map descr ibe d ab ov e . Similarly to t et ( m ) we can co nstruct t et v ( m ) := ( ∗ β ∗ − m k ) − 1 . W e observe that the maps t ( m ) and t v ( m ) are compatible with the reduction maps a nd the bo undary maps. In other words, w e have the following commutativ e diagrams . K i ( O L,S ; Z / l k ) t ( m )   r v / / K i ( k v ; Z /l k ) t v ( m )   K i +2 m ( O L,S ; Z /l k ) r v / / K i +2 m ( k v ; Z / l k ) K i ( O L,S , Z / l k ) t ( m )   ∂ / / L v ∈ S K i − 1 ( k v ; Z / l k ) t v ( m )   K i +2 m ( O L,S ; Z / l k ) ∂ / / L v ∈ S K i − 1+2 m ( k v ; Z / l k ) Let us point out that we have similar comm utative diagra ms for ´ etale K -theory and the maps t et ( m ) a nd t et v ( m ) . As observed a b ove, the map t ( m ) for m < 0 has the same prop erties as the m ultiplication by β ∗ m for m ≥ 0 . So, we ma ke the following. Definition 4.4. F or m < 0, we define the symbols α ∗ β ∗ m := t ( m )( α ) , α v ∗ β ∗ m := t v ( m )( α v ) , for all α ∈ K i ( O L ; Z /l k ) and α v ∈ K i ( k v ; Z /l k )), resp ectively . F or m ≥ 0, the symbols α ∗ β ∗ m and α v ∗ β ∗ m denote the usual pro ducts. Let m > 0 b e a natural num b er. Througho ut the res t of this section we a ssume that Θ m ( b , f k ) annihilates K 2 m ( O F l k ) for all k ≥ 0 . F or a prime v of O F , let k v be its r e sidue field and q v the car dinality of k v . Similar ly , for any pr ime w of O F l k , we let k w be its res idue field. W e put E := F l k . If v 6 | l , we obs erve that k w = k v ( ξ l k ), since the corr e sp onding lo cal field extensio n E w /F v is unr amified. F or a n y finite set S of pr imes in O F and any k ≥ 0, there is an exac t sequence (see [22]): 0 − → K 2 m ( O F l k ) − → K 2 m ( O F l k , S ) ∂ − → M v ∈ S M w | v K 2 m − 1 ( k w ) − → 0 Let ξ w, k ∈ K 2 m − 1 ( k w ) l be a genera tor of the l -to rsion par t of K 2 m − 1 ( k w ). Pick an element x w, k ∈ K 2 m ( O F l k ,S ) l such tha t ∂ ( x w, k ) = ξ w, k . Obviously , x Θ m ( b , f k ) w, k do es not dep end on the c hoice of x w, k since Θ m ( b , f k ) annihilates K 2 m ( O F l k ) . If STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 15 ord( ξ w, k ) = l a , then x l a w, k ∈ K 2 m ( O F l k ) . Hence, ( x Θ m ( b , f k ) w, k ) l a = ( x l a w, k ) Θ m ( b , f k ) = 0 . Consequently , there is a well defined map: Λ m : M v ∈ S M w | v K 2 m − 1 ( k w ) l − → K 2 m ( O F l k , S ) l , (31) Λ m ( ξ w, k ) := x Θ m ( b , f k ) w, k . If R is either a num b er field L or its ring of ( S, l )– int egers O L,S [1 /l ], for some finite s e t S ⊆ Sp ec( O L ), T a te proved in [28] that there is a natur al iso morphism: K 2 ( R ) l ∼ = − → K et 2 ( R ) . Dwyer a nd F riedlander [13] prov ed that the na tur al maps: K j ( R ; Z /l k ) − → K et j ( R ; Z /l k ) , are sur jections for j ≥ 1 and iso morphisms for j = 1 , 2 . As explained in [2], for a ny nu mber field L , any finite s et S ⊂ Sp ec( O L ) and any j ≥ 1, we hav e the fo llowing commutativ e dia grams with exact rows and (surjective) Dwyer-F riedlander maps as vertical ar rows. 0 / / K 2 j ( O L ) l     / / K 2 j ( O L,S ) l     ∂ / / L v ∈ S K 2 j − 1 ( k v ) l ∼ =   / / 0 0 / / K et 2 j ( O L [1 /l ]) / / K et 2 j ( O L,S [1 /l ]) ∂ et / / L v ∈ S K et 2 j − 1 ( k v ) / / 0 F or j = 1 , the left a nd the middle vertical ar rows in the ab ov e diagr am ar e also isomorphisms, according to T a te’s theor e m. If the Q uillen-Lich tenbaum conjecture holds, then these are is omorphisms for a ll j > 0 . Our assumption that Θ m ( b , f k ) annihilates K 2 m ( O F l k ) for all k ≥ 0 implies that Θ m ( b , f k ) annihilates K et 2 m ( O F l k [1 /l ]), for all k ≥ 0 . In the diagram ab ov e, let y w, k and ζ w, k denote the images of x w, k and ξ w, k via the middle vertical and right vertical a rrows, res pe c tively . Then, we define Λ et m ( ζ w, k ) := y Θ m ( b , f k ) w, k . Clearly , the following diagr am is co mm utative K 2 m ( O F l k ,S ) l   L v ∈ S L w | v K 2 m − 1 ( k w ) l Λ m o o ∼ =   K et 2 m ( O F l k ,S [1 /l ]) L v ∈ S L w | v K et 2 m − 1 ( k w ) l Λ et m o o where the vertical maps are the Dwyer-F riedlander maps . Lemma 4.5. The m aps Λ m and Λ et m satisfy t he fol lowing pr op erties ∂ Λ m ( ξ w, k ) := ξ Θ m ( b , f k ) w, k , ∂ et Λ et m ( ζ w, k ) := ζ Θ m ( b , f k ) w, k . Pr o of. The lemma follows immediately b y co mpatibility o f ∂ and ∂ et with the G ( E / F ) action.  16 G. BANASZAK AND C. D. POP ESCU Let us fix a n n ∈ N . Let v be a prime in O F sitting abov e p 6 = l in Z . Let S := S v be the finite set primes of O F consisting of all the primes ov e r p and all the primes ov e r l . Let k ( v ) b e the natura l num b er for which l k ( v ) || q n v − 1 . Observe that if l | q v − 1 then k ( v ) = v l ( q v − 1 ) + v l ( n ) (see e.g . [1, p. 336].) Definition 4.6. As in lo c.cit. p. 335, let us define: γ l := Y l 6 | f l | l (1 − ( l , F ) − 1 N l n ) − 1 = Y l 6 | f l | l (1 + ( l , F ) − 1 N l n + ( l , F ) − 2 N l 2 n + · · · ) . If l | f for every l | l then na turally we let γ l := 1 . Obs erve that γ l is a well defined op erator on a ny Z l [ G ( F /K )]-mo dule which is a tor sion ab elian g roup with a finite exp onent. Definition 4.7. F or a ll k ≥ 0 and E := F ( µ l k ), let us define elemen ts: λ v, l k := T r E /F (Λ m ( ξ w, k ) ∗ β ∗ n − m k ) N b n − m γ l ∈ K 2 n ( O F, S ; Z / l k )) . Similarly , define elements: λ et v, l k := T r E /F (Λ et m ( ζ w, k ) ∗ β ∗ n − m k ) N b n − m γ l ∈ K et 2 n ( O F, S ; Z / l k )) . Obviously , λ et v, l k is the imag e of λ v, l k via the Dwyer-F riedlander map. Let us fix a prime sitting ab ov e v in ea ch of the fields F ( µ l k ), s uch that if k ≤ k ′ and w and w ′ are the fixed primes in E = F ( µ l k ) and E ′ := F ( µ l k ′ ), res pe c tively , then w ′ sits a bove w . By the surjectivity of the transfer maps for K -theory o f finite fields (se e the end of § 3), w e can asso cia te to ea ch k and the chosen prime w in E = F ( µ l k ) a genera tor ξ w, k of K 2 m − 1 ( k w ) l and a gener ator ζ w, k of K et 2 m − 1 ( k w ) l , such that N w ′ /w ( ξ w ′ ,k ′ ) = ξ w, k , N w ′ /w ( ζ w ′ ,k ′ ) = ζ w, k , for all k ≤ k ′ , wher e w a nd w ′ are the fix ed primes in E = F ( µ l k ) and E ′ = F ( µ l k ′ ), resp ectively . Lemma 4.8. With notations as ab ove, for every k ≤ k ′ we have r k ′ /k ( N w ′ /v ( ξ w ′ ,k ′ ∗ β ∗ n − m k ′ )) = N w/v ( ξ w, k ∗ β ∗ n − m k ) , r k ′ /k ( N w ′ /v ( ζ w ′ ,k ′ ∗ β ∗ n − m k ′ )) = N w/v ( ζ w, k ∗ β ∗ n − m k ) . Pr o of. First, let us conside r the case n − m ≥ 0 . The formula follows by the co mpa t- ibilit y o f the elements ( ξ w, k ) w with res p ect to the nor m maps, by the co mpatibilit y of Bo tt elements with resp ect to the co efficient r eduction map r k ′ /k ( β k ′ ) = β k , and by the pro jection fo rmula. Mor e precis e ly , we hav e the following equalities : r k ′ /k ( N w ′ /v ( ξ w ′ ,k ′ ∗ β ∗ n − m k ′ )) = N w ′ /v ( r k ′ /k ( ξ w ′ ,k ′ ∗ β ∗ n − m k ′ )) = N w ′ /v ( ξ w ′ ,k ′ ∗ β ∗ n − m k )) = N w/v ( N w ′ /w ( ξ w ′ ,k ′ ) ∗ β ∗ n − m k )) = = N w/v ( ξ w, k ∗ β ∗ n − m k ) . Next, let us consider the case n − m < 0 . Observe tha t the Dwyer-F riedlander maps comm ute with N w/v and N w ′ /v . Hence w e can argue in the s ame w ay as in the case n − m ≥ 0 by using the pro jectio n for m ula for the negative t w is t in ´ etale STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 17 cohomolog y , since for any finite field F q with l 6 | q , we hav e natural isomorphisms coming fr o m the Dwyer-F riedlander sp ectral sequence (cf. the end of § 3.1): (32) K et 2 j − 1 ( F q ) ∼ = H 1 ( F q ; Z l ( j )) (33) K et 2 j − 1 ( F q ; Z /l k ) ∼ = H 1 ( F q ; Z / l k ( j )) .  Lemma 4.9. F or al l 0 ≤ k ≤ k ′ , we have r k ′ /k ( λ v, l k ′ ) = λ v, l k r k ′ /k ( λ et v, l k ′ ) = λ v, l k . Pr o of. Consider the following c ommut ative diagram: K 2 m ( O E ′ ,S ) T r E ′ /E   L w ′ ∈ S ∂ w ′ / / L w ′ ∈ S K 2 m − 1 ( k w ′ ) L w ∈ S L w ′ | w N w ′ /w   K 2 m ( O E ,S ) L w ∈ S ∂ w / / L w ∈ S K 2 m − 1 ( k w ) It fo llows that w e have T r E ′ /E ( x w ′ ,k ′ ) Θ m ( b , f k ) = x Θ m ( b , f k ) w, k . Hence the case n − m ≥ 0 follows by the pro jection formula: r k ′ /k ( T r E ′ /F ( x Θ m ( b , f k ′ ) w ′ ,k ′ ∗ β ∗ n − m k ′ ) N b n − m γ l ) = = T r E /F ( T r E ′ /E ( x Θ m ( b , f k ′ ) w ′ ,k ′ ∗ β ∗ n − m k )) N b n − m γ l = = T r E /F ( x Θ m ( b , f k ) w, k ∗ β ∗ n − m k ) N b n − m γ l . Now, co nsider the ca se n − m < 0 . W e o bs erve that T r E ′ /E commutes with the Dwyer-F riedlander map. Hence T r E ′ /E also commut es with the splitting of the Dwyer-F riedlander map since the s plitting is a monomorphism. By the Dwyer- F r iedlander sp ectra l sequence for any num b er field L and any finite se t S of prime ideals o f O L containing all primes ov er l , we hav e the following isomorphism (34) K et 2 j ( O L,S ) ∼ = H 2 ( O L,S ; Z l ( j + 1 )) and the following exact sequence (35) 0 → H 2 ( O L,S ; Z / l k ( j + 1 )) → K et 2 j ( O L,S ; Z / l k ) → H 0 ( O L,S ; Z / l k ( j )) → 0 . Since x Θ m ( b , f k ) w, k ∈ K 2 m ( O F k ,S ) , its image in K et 2 m ( O F k ,S ; Z /l k ) lies in fact in in the ´ etelae cohomolo gy gr oup H 2 ( O F k ,S ; Z /l k ( m + 1)) . Hence, one can settle the cas e n − m < 0 as well by using the pro jection formula for the ´ etale cohomolog y with negative twists.  Theorem 4.10. F or every k ≥ 0 , we have ∂ F ( λ v, l k ) = N ( ξ w, k ∗ β ∗ n − m k ) Θ m ( b , f ) , ∂ et F ( λ et v, l k ) = N ( ζ w, k ∗ β ∗ n − m k ) Θ m ( b , f ) . 18 G. BANASZAK AND C. D. POP ESCU Pr o of. The pro of is similar to the pro ofs of Theorem 1, pp. 3 36-34 0 of [1] and Prop ositio n 2, pp. 22 1-222 o f [3]. The diagr am a t the end o f § 3 gives the following commutativ e diag ram o f K –groups with co efficients K 2 n ( O E ,S ; Z /l k ) T r E /F   ∂ E / / L v ∈ S L w | v K 2 n − 1 ( k w ; Z / l k ) N   K 2 n ( O F, S ; Z / l k ) ∂ F / / L v ∈ S K 2 n − 1 ( k v , ; Z /l k ) , where N := L v L w | v N w/v . Hence we have ∂ F ◦ T r E /F = N ◦ ∂ E . The compati- bilities of some of the natura l maps men tioned in § 3 which will be used next can be express ed via the following commutativ e diagra ms, explaining the a ction o f the groups G ( E /K ) a nd G ( F /K ) o n the K –groups with co e fficient s in the diagram ab ov e. F or j > 0 we use the fo llowing commutativ e diag ram. K 2 j ( O E ,S ; Z / l k ) σ − 1 a   r w / / K 2 j ( k w ; Z / l k ) σ − 1 a   K 2 j ( O E ,S ; Z / l k ) r w σ − 1 a / / K 2 j ( k w σ − 1 a ; Z / l k ) The ab ove diagra m gives the following equality: (36) r w σ − 1 a ( β ∗ n − m k ) = r w σ − 1 a (( β ∗ n − m k ) N a n − m σ − 1 a ) = ( r w ( β ∗ n − m k )) N a n − m σ − 1 a . F o r a ny j ∈ Z , we hav e the following commutativ e diag ram: H 0 ( O E ,S ; Z / l k ( j )) σ − 1 a   r w / / H 0 ( k w ; Z / l k ( j )) σ − 1 a   H 0 ( O E ,S ; Z / l k ( j )) r w σ − 1 a / / H 0 ( k w σ − 1 a ; Z /l k ( j )) If ξ l k := exp ( 2 π i l k ) is the generator of µ l k then the ab ove diagr a m gives (37) r w σ − 1 a ( ξ ⊗ n − m l k ) = r w σ − 1 a ( ξ ⊗ n − m l k ) N a n − m σ − 1 a ) = ( r w ( ξ ⊗ n − m l k )) N a n − m σ − 1 a . W e can write the m –th Stick e lb er ger element as follows (38) Θ m ( b , f k ) = X a mo d f k ′   X c mo d f k , w σ c − 1 = w ∆ m +1 ( ac , b , f ) σ c − 1   · σ a − 1 , where P ′ a mo d f k denotes the sum over a ma x imal set S of idea l classes a mo d f k , such that the primes w σ − 1 a , for a ∈ S , are distinct. By formula (6), for every m ≥ 1 and n ≥ 1 we have ∆ n +1 ( a , b , f ) ≡ N a n − m N b n − m ∆ m +1 ( ac , b , f ) mo d w min { m,n } ( K f ) . It is clear that for all m, n ≥ 1 we get the following cong r uence mo d w min { m,n } ( K f k ) . Θ n ( b , f k ) ≡ X a mo d f k ′ ( X c mo d f k , w σ c − 1 = w N a n − m N c n − m N b n − m ∆ m +1 ( ac , b , f k ) σ c − 1 ) σ a − 1 STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 19 Equalities (3 6), (37), (3 8), Lemma 4.2, Gillet’s result [17] (diagra m (20)), the com- patibilit y of t ( n − m ) and t v ( n − m ) with ∂ and the a bove congruence s satisfied by Stic kelberger ele men ts lead in b oth cases n − m ≥ 0 and n − m < 0 to the fo llowing equalities. ∂ E ( x Θ m ( b , f k ) w, k ∗ β ∗ n − m k ) N b n − m = = X a mo d f k ′ ξ P c mod f k , w σ c − 1 = w ∆ m +1 ( ac , b , f k ) σ ( ac ) − 1 w, k ∗ ( β ∗ n − m k ) ( N a c ) n − m N b n − m σ ( ac ) − 1 = = ( ξ w, k ∗ β ∗ n − m k ) P ′ a mod f k P c mod f k , w σ c − 1 = w ∆ m +1 ( ac , b , f k )( N a c ) n − m N b n − m σ ( ac ) − 1 = = ( ξ w, k ∗ β ∗ n − m k ) Θ n ( b , f k ) . By the firs t comm utative diag r am of this pr o of, the equalities ab ove and Lemma 2.1, we obtain: ∂ F ( λ v, l k ) = N ( ∂ E ( x Θ m ( b , f k ) w, k ∗ β ∗ n − m k ) N b n − m ) γ l = N (( ξ w, k ∗ β ∗ n − m k ) Θ n ( b , f k ) ) γ l = = ( N ( ξ w, k ∗ β ∗ n − m k )) γ l − 1 Θ n ( b , f ) γ l = ( N ( ξ w, k ∗ β ∗ n − m k )) Θ n ( b , f ) .  Theorem 4.11. F or ev ery v such that l | q n v − 1 and for al l k ≥ k ( v ) , ther e ar e homomorph isms Λ v, l k : K 2 n − 1 ( k v ; Z / l k ) → K 2 n ( O F, S ; Z / l k ) , Λ et v, l k : K et 2 n − 1 ( k v ; Z / l k ) → K et 2 n ( O F, S ; Z /l k ) : which satisfy the fol lowing e qu alities: Λ v, l k ( N ( ξ w, k ∗ β ∗ n − m k )) = λ v, l k , Λ et v, l k ( N ( ζ w, k ∗ β ∗ n − m k )) = λ et v, l k . Pr o of. The definition of Λ m (see (31)), com bined with the natural isomor phis m K 2 m − 1 ( k w ) /l k ∼ = K 2 m − 1 ( k w ; Z /l k Z ) and the natural monomor phis m K 2 m ( O E ,S ) /l k → K 2 m ( O E ,S ; Z /l k Z ) , coming fro m the corr esp onding Bo ckstein exa ct sequences, leads to the fo llowing homomorphism: e Λ m : K 2 m − 1 ( k w ; Z /l k Z ) → K 2 m ( O E ,S ; Z /l k Z ) . Multiplying o n the targ et and on the so urce of this homomorphis m with the n − m power of the Bott element if n − m ≥ 0 (resp. applying the map t w ( n − m ) to the source and t ( n − m ) to the tar get if n − m < 0) under the obse rv ation that t he following map is an iso morphism: K 2 m − 1 ( k w ; Z /l k Z ) ∼ ∗ β ∗ n − m k / / K 2 n − 1 ( k w ; Z /l k Z ) (cf. the notatio n of t ( j ) and t w ( j ) ) show that there exists a unique ho momorphism e Λ m ∗ β ∗ n − m k : K 2 n − 1 ( k w ; Z / l k ) → K 2 n ( O E ,S ; Z / l k ) , 20 G. BANASZAK AND C. D. POP ESCU sending ξ w, k ∗ β ∗ n − m k → x Θ m ( b , f k ) w, k ∗ β ∗ n − m k . Next, we comp ose the ho momorphisms e Λ m ∗ β ∗ n − m k defined a b ov e and T r E /F : K 2 n ( O E ,S ; Z / l k ) → K 2 n ( O F, S ; Z / l k ) to obtain the following homomorphism: T r E /F ◦ ( e Λ m ∗ β ∗ n − m k ) : K 2 n − 1 ( k w ; Z / l k ) → K 2 n ( O F, S ; Z / l k ) . W e observe that this homomorphism facto rs throug h the quotient o f G ( k w /k v )– coinv ariants K 2 n − 1 ( k w ; Z / l k ) G ( k w /k v ) := K 2 n − 1 ( k w ; Z / l k ) /K 2 n − 1 ( k w ; Z /l k ) F r v − I d , where F r v ∈ G ( k w /k v ) ⊆ G ( E /F ) is the F r o b enius elemen t of the prime w ov er v. Since F r v acts via q n v –p ow ers on K 2 n − 1 ( k w ), the canonical isomorphism K 2 n − 1 ( k w ; Z /l k ) ∼ = K 2 n − 1 ( k w ) /l k (see § 3) and ass umption k ≥ k ( v ) give K 2 n − 1 ( k w ; Z / l k ) G ( k w /k v ) ∼ = K 2 n − 1 ( k w ; Z / l k ) /l k ( v ) ∼ = K 2 n − 1 ( k w ) /l k ( v ) . The obvious co mm utative diagr am with surjective vertical morphisms (see § 3) K 2 n − 1 ( k w ) /l k N w/v   ∼ = / / K 2 n − 1 ( k w ; Z / l k ) N w/v   K 2 n − 1 ( k v ) /l k ∼ = / / K 2 n − 1 ( k v ; Z / l k ) combined with the last isomorphism ab ov e, gives an isomor phism K 2 n − 1 ( k w ; Z / l k ) G ( k w /k v ) N w/v ∼ / / K 2 n − 1 ( k v ; Z / l k ) Now, the requir ed homomorphis m is: (39) Λ v, l k : K 2 n − 1 ( k v ; Z /l k ) / / K 2 n ( O F, S ; Z /l k ) defined by Λ v, l k ( x ) := [ T r E /F ◦ ( e Λ m ∗ β ∗ n − m k ) ◦ N w/v − 1 ( x )] N b n − m γ l , for all x ∈ K 2 n − 1 ( k v ; Z / l k ). By definition, this map sends N ( ξ w, k ∗ β ∗ n − m k ) on to the ele ment λ v, l k := T r E /F ( x Θ m ( b , f k ) w, k ∗ β ∗ n − m k ) N b n − m γ l .  4.2. Constructing Λ and Λ et for K –theory w i thout co effi cien ts . Let us fix n > 0. In this section, w e use the sp ecia l elemen ts and λ v, l k and λ et v, l k defined ab ov e to constr uct the maps Λ n and Λ et n for the K –theor y (resp ectively ´ etale K – theory) without co efficients. Since n is fixed throughout, we will deno te Λ := Λ n and Λ et := Λ et n . Observe that for every j > 0 and every prime l , the Bo ckstein exact s e quence (14) and res ults of Quillen [22], [2 3] g ive natur a l isomorphis ms (40) K j ( O F, S ) l ∼ = lim ← − k K j ( O F, S ; Z / l k ) , (41) K j ( k v ) l ∼ = lim ← − k K j ( k v ; Z / l k ) . STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 21 Similar is o morphisms hold for the ´ e tale K -theory . Definition 4.12. W e de fine λ v ∈ K 2 n ( O F, S ) l and λ et v ∈ K et 2 n ( O F, S ) to be the elements co rresp onding to ( λ v, l k ) k ∈ lim ← − k K 2 n ( O F, S ; Z / l k ) , ( λ et v, l k ) k ∈ lim ← − k K et 2 n ( O F, S ; Z / l k ) via the isomo rphism (40) and its ´ etale analog ue, resp ectively . Definition 4.13. W e define ξ v ∈ K 2 n − 1 ( k v ) l and ζ v ∈ K et 2 n − 1 ( k v ) to be the ele- men ts co r resp onding to ( N ( ξ w, k ∗ β ∗ n − m k )) k ∈ lim ← − k K 2 n − 1 ( k v ; Z / l k ) , ( N ( ζ w, k ∗ β ∗ n − m k )) k ∈ lim ← − k K et 2 n − 1 ( k v ; Z / l k ) , via the isomo rphism (41) and its ´ etale analog ue, resp ectively . Definition 4.14 . Assume that l | q n v − 1. Since the homo morphisms Λ v, l k , and Λ et v, l k , fro m Theorem 4.11, a re compatible with the co efficient reductio n maps r k ′ /k , for a ll k ′ ≥ k ≥ k ( v ) , we can define ho mo morphisms Λ v := lim ← − k Λ v, l k : K 2 n − 1 ( k v ) l → K 2 n ( O F, S ) l ֒ → K 2 n ( F ) l , Λ et v := lim ← − k Λ et v, l k : K et 2 n − 1 ( k v ) → K et 2 n ( O F, S ) ֒ → K et 2 n ( F ) l , for all v . Here , the rig h tmost arrows a r e the inclusio ns K 2 n ( O F, S ) ⊂ K 2 n ( F ) and K et 2 n ( O F, S ) ⊂ K et 2 n ( F ) l , r e s pe ctively . If l ∤ q n v − 1, then the mo rphisms Λ v and Λ et v are trivia l, by default. Remark 4.15. It is clear from Theorem 4.11 that, for all v , we have Λ v ( ξ v ) = λ v , Λ et v ( ζ v ) = λ et v . Definition 4.16. W e define the maps Λ n and Λ et n as follows: Λ : M v K 2 n − 1 ( k v ) l → K 2 n ( F ) l , Λ := Y v Λ v . Λ et : M v K et 2 n − 1 ( k v ) → K et 2 n ( F ) l , Λ et := Y v Λ et v . Theorem 4.17. The maps Λ and Λ et satisfy the fol lowing pr op erties: ∂ F ◦ Λ ( ξ v ) = ξ Θ n ( b , f ) v , ∂ et F ◦ Λ et ( ζ v ) = ζ Θ n ( b , f ) v . Pr o of. Consider the following c ommut ative diagram. K 2 n ( O F, S ) /l k   L v ∈ S ∂ v / / L v ∈ S K 2 n − 1 ( k v ) /l k   K 2 n ( O F, S ; Z / l k ) L v ∈ S ∂ v / / L v ∈ S K 2 n − 1 ( k v ; Z / l k ) 22 G. BANASZAK AND C. D. POP ESCU The vertical a rrows in the diagram c o me from the Bo ckstein exact seq ue nc e . It is clear from the diagra m that the inv er se limit ov er k of the b ottom horizo ntal arrow gives the b oundary map ∂ F = L v ∈ S ∂ v : ∂ F : K 2 n ( O F, S ) l → M v K 2 n − 1 ( k v ) l . Now, the theore m follows by T he o rems 4.1 0 and 4.11.  In the nex t pro po sition we will co nstruct a Stickelberger splitting map Γ which is complementary to the map Λ cons tructed ab ove. (42) 0 − → K 2 n ( O F ) l i − → Γ ← − K 2 n ( F ) l ∂ F − → Λ ← − M v K 2 n − 1 ( k v ) l − → 0 . The exis tence o f Γ was suggested in 19 88 by Christophe Soul´ e in a letter to the first author and it is a direct conseque nc e of the following mo dule theore tic lemma. Lemma 4.18. L et R b e a c ommu tative ring with 1 and let r ∈ R b e fixe d. L et 0 / / A ι / / B π / / C / / 0 b e an exact se quenc e of R –mo dules. Then, the fol lowing ar e e quivalent: (1) Ther e exists an R –mo dule morphism Λ : C → B , such that π ◦ Λ = r · id C . (2) Ther e exists an R –mo dule morphism Γ : B → A , s uch that Γ ◦ ι = r · id A . Mor e over, if Λ and Γ exist, they c an b e chosen so that Γ ◦ Λ = 0 . Pr o of of L emma. Assume that (1) holds. By the defining pr op erty of Λ, we hav e (43) (Λ ◦ π )( − b ) + r b ∈ Im ( ι ) , ∀ b ∈ B . W e define Γ( b ) := ι − 1 ((Λ ◦ π )( − b ) + r b ), for all b ∈ B , where ι − 1 ( x ) is the preima g e of x via ι , for all x ∈ Im( ι ). One can chec k without difficult y tha t Γ is a n R –mo dule morphism which satisfies Γ ◦ ι = r · id A , Γ ◦ Λ = 0 . Now, assume that (2) holds. Let c ∈ C . T ake b ∈ B , such that π ( b ) = c . Then, by the defining pro pe r ty of Γ , one can chec k that the ele men t ( ι ◦ Γ)( − b ) + r b ∈ B is indep endent on the c hosen b . F o r all c ∈ C , we define Λ( c ) := ( ι ◦ Γ)( − b ) + r b , where b ∈ B , such that π ( b ) = c . It is easily seen that the map Λ defined this wa y is an R –mo dule morphism and it sa tisfies π ◦ Λ = r · id C , Γ ◦ Λ = 0 .  Prop ositi o n 4. 19. The existenc e of a map Λ satisfying the pr op ert y ( ∂ F ◦ Λ)( ξ v ) = ξ Θ n ( b , f ) v is e quivalent to the existenc e of a map Γ : K 2 n ( F ) l → K 2 n ( O F ) l with the pr op erty (Γ ◦ i )( η ) = η Θ n ( b , f ) . Mo r e over, if they exist, t he maps Λ and Γ c an b e chosen so t hat Γ ◦ Λ = 0 . Pr o of. The pro of o f the Pro p o sition follows dir ectly fr om the ab ov e Lemma applied to R := Z [ G ( F /K )], r := Θ n ( b , f ) a nd Quillen’s lo caliza tion ex act s equence (42).  STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 23 Remark 4. 20. F rom the pro of of Lemma 4.18 it is clear that if one of the maps Λ and Γ is given, then the other o ne can b e chosen s uch that r · id B = Λ ◦ π + i ◦ Γ . Remark 4. 21. Observe that the map Λ is defined in the sa me wa y for b oth cases l ∤ n and l | n. If r estricted to the par ticular ca se K = Q , our constr uc tio n improves upo n that of [1]. In lo c.cit., in the cas e l | n the map Λ was cons tructed only up to a facto r of l v l ( n ) . Analogously , there is a Stick elbe rger splitting ma p Γ et which is co mplement ary to the map Λ et such that the ´ etale a nalogue o f the Pro po sition 4.19 holds. 4.3. Annihilating div K 2 n ( F ) l . Now, let us give a s et of immediate applicatio ns of our construction of the Stick elb erger splitting maps Λ n . In what follows, if A ia an ab elian gr oup, div A denotes its subgro up of divisible elements. The a pplica tions which follow concer n a nnihilation of the groups div K 2 n ( F ) l by higher Stick elb e r ger elements of the type proved in [1] in the cas e where the base field is Q . The difference is that while [1] dea ls w ith ab elian extensions F / Q , under certain r estrictions if l | n , we deal with ab elian extensions F /K o f an ar bitrary totally real num b er field K under no restrictive conditions. The desired a nnihilation r esult fo llows from the following. Lemma 4.22. With notations as in L emma 4.18, assume that a map Λ exists. F urther, assume t hat C and A viewe d as ab elian gr oups satisf y div C = 0 and A has finite ex p onent. Then, we have t he fol lowing. (1) div B ⊆ Im( ι ) . (2) r annihilates div B . Pr o of. Since any mor phism maps divisible elements to divisible e le men ts, we hav e π ( div B ) = 0, by our assumption o n C . This concludes the pro of o f (1). Let x ∈ div B . Let m be the exp onent of A a nd let b ∈ B , suc h that m · b = x . Multiply (43) by m to conclude that (Λ ◦ π )( − x ) + r · x = 0 . Now, part (1) implies that π ( x ) = 0 . Consequently , the la st equa lity implies that r · x = 0, which concludes the pro of.  Theorem 4.23 . L et m > 0 b e a n atu r al n umb er. Assume tha t the Stickelb er ger elements Θ m ( b , f k ) annihilate t he gr oups K 2 m ( O F k ) l for al l k ≥ 1 . Then the S tick- elb er ger’s element Θ n ( b , f ) annihilates the gr oup div K 2 n ( F ) l for every n ≥ 1 . Pr o of. The pro of is very similar to that of [Ba1, Cor. 1, p. 3 40]. Let us fix n ≥ 1. Under our annihilation hypo thesis, we have constructed a map Λ := Λ n satisfying the pro pe r ties in Pr op osition 4.19 relative to the Q uillen lo ca lization seq uence (4 2). Note that A := K 2 n ( O F ) l is finite and there fore it has a finite exp onent. Also, note that C := L v K 2 n − 1 ( k v ) l is a dir e ct sum of finite a belia n groups a nd therefo r e div C = 0. Consequently , the e x act sequence (42) together with the map Λ and element r := Θ n ( b , f ) in the ring R := Z [ G ( F /K )] satisfy the hypo theses of Le mma 4.22. Therefore, we hav e Θ n ( b , f ) · div K 2 n ( F ) l = 0 .  24 G. BANASZAK AND C. D. POP ESCU Remark 4. 24. Observe that we can res tr ict the map Λ to the l k –torsion part, for any k ≥ 1 . F o r any k ≫ 0, ther e is an exact sequence 0 − → K 2 n ( O F )[ l k ] − → K 2 n ( F )[ l k ] ∂ F − → Λ ← − M v K 2 n − 1 ( k v )[ l k ] − → div ( K 2 n ( F ) l ) − → 0 By Theorem 4.17, we know tha t ∂ F ◦ Λ is the multiplication by Θ n ( b , f ) . As p ointed out in the Introduction, this implies the annihilation of div ( K 2 n ( F ) l ) and cons e- quently g ives a seco nd pro of for Theo rem 4.2 3 Let us define F 0 := F and: Θ n ( b , f 0 ) = (  Q l ∤ f l | l (1 − ( l , F ) − 1 N l n )  Θ n ( b , f ) if l ∤ f Θ n ( b , f ) if l | f Hence by the formula (1 3) we ge t (44) Res F k +1 /F k Θ n ( b , f k +1 ) = Θ n ( b , f k ) Hence by formula (44) we c a n define the element (45) Θ n ( b , f ∞ ) := lim ← − k Θ n ( b , f k ) ∈ lim ← − k Z l [ G ( F k /F )] . Corollary 4.25 . L et m > 0 b e a natu r al numb er. Assume that the Stickelb er ger elements Θ m ( b , f k ) annihilate t he gr oups K 2 m ( O F k ) l for al l k ≥ 1 . Then the S tick- elb er ger element Θ n ( b , f k ) annihilates the gr oup div K 2 n ( F k ) l for every k ≥ 0 and every n ≥ 1 . In p articular Θ n ( b , f ∞ ) a nnihilates the gr oup lim − → k div K 2 n ( F k ) l for every n ≥ 1 . Pr o of. F ollows immediately from Theo rem 4.2 3.  Theorem 4.2 6. L et F / K b e an ab elian CM extension of an arbitr ary t otal ly r e al numb er field K and let l b e an o dd prime. If t he Iwasawa µ –invariant µ F, l asso ciate d to F and l vanishes, then Θ n ( b , f ) annihilates t he gr oup div ( K 2 n ( F ) l ) , for al l n ≥ 1 and al l b c oprime to w n +1 ( F ) f l . Pr o of. In [16] (see Theore m 6.11 ), it is sho wn that if µ F, l = 0, then Θ n ( b , f ) annihilates K et 2 n ( O F [1 /l ]), for all n ≥ 1 and all b as ab ove. F r om the definition of Iwasa wa’s µ –in v ariant o ne co ncludes right aw ay tha t if µ F, l = 0, then µ F k ,l = 0, for all k . Conseq ue ntly , Θ 1 ( b , f k ) annihilates K et 2 ( O F k [1 /l ]), for all k . Now, one applies T a te’s Theo rem 1 .3 to conclude that Θ 1 ( b , f ) annihilates K 2 ( O F k ) l , fo r all k . Theorem 4.2 3 implies the desired r e sult.  Remark 4.27. It is a class ical conjecture of Iwasaw a that µ F, l = 0 , for all n umber fields F and all primes l . Corollary 4.28 . Le t F / Q b e an ab elian extensions of c onductor f . Then Θ n ( b, f ) annihilates the gr oup div K 2 n ( F ) l for al l n ≥ 1 and al l b c oprime t o w n +1 ( F ) f l . Pr o of. By a w e ll known theorem of F er rero- W a shington a nd Sinnott, µ F, l = 0 for all fie lds F which are ab elian extensions of Q and all l . Now, the Coro lla ry is an immediate conse quence of the previous Theorem.  Remark 4.29 . Observe that the Co rollar y ab ov e strengthens Corolla ry 1, p. 340 of [1] in the ca se l | n. STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 25 5. Constructing E uler systems out of Λ –elements As mentioned in § 4, in this section we constr uct E uler sys tems for the even K - theory of CM abe lian ex tensions of to tally r eal n umber fields. W e construct an Euler system in the K –theory with co efficients. Then, by passing to a pro jective limit, we obtain an Euler sys tem in Quillen K –theory . Our constr uctions are quite different fro m those in [3], wher e an Euler systems in the o dd K –theory with finite co efficients o f CM ab elian extensions of Q was describ ed. As ab ove, we fix a finite a be lia n CM extensio n F /K of a totally real num be r field o f conductor f a nd fix a prime num b er l . W e let L = l 1 . . . l t be a product of mutually distinct pr ime ideals of O K coprime to l f . W e le t F L := F K L , wher e K L is the ray class field of K for the ideal L . Since F /K has conductor f the CM- extension F L /K has co nductor dividing Lf . As usual, we let F L l k := F L ( µ l k ), for every k ≥ 0 . Let us fix a prime v in O F sitting ab ov e a r a tional prime p 6 = l . Let S := S v be the set consis ting o f all the primes of O F sitting ab ov e p or ab ov e l . F or all L a s ab ov e and k ≥ 0, we fix primes w k ( L ) o f O F L l k sitting ab ov e v , such that w k ′ ( L ′ ) sits ab ove w k ( L ) whenever l k L | l k ′ L ′ . F or simplicity , w e let v ( L ) := w 0 ( L ), for all L a s ab ov e. Also , if k is fixed, we let w ( L ) := w k ( L ), for all L as ab ove. Let us fix integers m > 0. F or all L a s a bove and all k ≥ 0, let Θ m ( b L , Lf k ) denote the m -th Stickelberger element for the integral ideal b L of O F , coprime to Lf l , and the extension F L l k /K . As usual, we assume throughout that Θ m ( b L , Lf k ) annihilates K 2 m ( O F L l k ), for all L a s ab ove and all k ≥ 0 . By the sur jectivity of the tr a nsfer maps for the K -theor y of finite fields we can fix gener ators ξ w k ( L ) ,k of K 2 m − 1 ( k w k ( L ) ) l , fo r all k ≥ 0 a nd L as a b ov e, such that N w k ′ ( L ) /w k ( L ) ( ξ w k ′ ( L ) ,k ′ ) = ξ w k ( L ) ,k , whenever we hav e k ≤ k ′ . Remark 5. 1. Note that the cyclicity of the groups K 2 m − 1 ( k w k ( L ) ) l and the sur- jectivit y of the appropr iate transfer maps implies tha t the element s ( ξ w k ( L ) ,k ) k , ( N w k ( L ′ ) /w k ( L ) ( ξ w k ( L ′ ) ,k )) k , viewed inside of Z l –mo dule lim ← − k K 2 m − 1 ( k w k ( L ) ) l , differ by a factor in Z × l , for a ll L and L ′ as a bove, such that L | L ′ . Ab ove, the pro jectiv e limit is taken with res pe ct to the transfer maps. Let us fix k ≥ 0. F or any L a s a b ove, we hav e the lo calization exa ct sequence: 0 − → K 2 m ( O F L l k ) − → K 2 m ( O F L l k , S ) ∂ − → M v 0 ∈ S M w | v 0 K 2 m − 1 ( k w ) − → 0 , where the direct sum is taken with respec t to all the primes w in O F L l k which sit above primes v 0 in S . Pick an element x w ( L ) ,k ∈ K 2 m ( O F L l k ,S ) l , such that ∂ ( x w ( L ) ,k ) = ξ w ( L ) ,k . T he following element: (46) Λ m ( ξ w ( L ) ,k ) := x Θ m ( b L , Lf k ) w ( L ) ,k do es not dep end on the choice of x w ( L ) ,k since Θ m ( b L , Lf k ) annihilates K 2 m ( O F L l k ) . Observe that by constr uction we hav e the following equa lities: 26 G. BANASZAK AND C. D. POP ESCU (47) ∂ F L l k ( T r F L ′ l k /F L l k  x w ( L ′ ) ,k  ) = N w ( L ′ ) /w ( L ) ( ∂ F L ′ l k  x w ( L ′ ) ,k  = = N w ( L ′ ) /w ( L ) ( ξ w ( L ′ ) ,k ) , (48) N w ( L ) /v ( L ) ( N w ( L ′ ) /w ( L ) ( ξ w ( L ′ ) ,k )) = N v ( L ′ ) /v ( L ) ( N w ( L ′ ) /v ( L ′ ) ( ξ w ( L ′ ) ,k )) = = N v ( L ′ ) /v ( L ) ( ξ v ( L ′ ) , 0 )) W e choo se the ideals b L in s uch a wa y s o that they are coprime to l Lf and N b L ′ ≡ N b L mo d l k . Then, the elements { Λ m ( ξ w ( L ) ,k ) } L form an Euler system in K -theor y without co efficients { K 2 m ( O L l k ,S ) l } L . Namely , we hav e: Prop ositi o n 5. 2. If L ′ = l ′ L , t hen the fol lowing e quality hold s: (49) T r F L ′ l k /F L l k (Λ m ( ξ w ( L ′ ) ,k )) = Λ m ( N w ( L ′ ) /w ( L ) ( ξ w ( L ′ ) ,k )) 1 − N ( l ′ ) m ( l ′ , F L l k ) − 1 . Pr o of. The Prop ositio n fo llows by (47) and Lemma 2.1.  Let us fix an arbitra r y integer n > 0 . Next, we use the Eule r system ab ove to construct Euler systems { λ v ( L ) } L in the K –gr o ups { K 2 n ( O L ,S ) l } L . The gener al idea is as follows. First, o ne constr ucts Euler Sys tems { λ v ( L ) , k } L in the K –theory with co efficients { K 2 n ( O L ,S , Z /l k ) } L , for all k > 0. Then one passes to a pro jectiv e limit with resp ect to k . The cons tructions, ideas and r esults developed in § 4 play a key ro le in what fo llows. F o r every L as ab ov e and ev ery k ≥ 0, we follow the ideas in § 4 and define the elements λ v ( L ) , k ∈ K 2 n ( O F L ,S ; Z / l k ) by: λ v ( L ) , k : = T r F L l k /F L ( x Θ m ( b L , Lf k ) w k ( L ) ,k ∗ β ∗ n − m k ) N b n − m L γ l = (50) T r F L l k /F L (Λ m ( ξ w k ( L ) ,k ) ∗ β ∗ n − m k ) N b n − m L γ l , where the o p e r ator γ l ∈ Z l [ G ( F /K )] is given in Definition 4.6. The following theorem lies at the heart of our constructio n of the Euler system for higher K - groups o f CM ab elian e x tensions o f a rbitrary totally r e al num b er fields. Theorem 5.3. F or every k ′ ≥ k and every L and L ′ = Ll ′ we have: r k ′ /k ( λ v ( L ) , k ′ ) = λ v ( L ) , k ∂ F L ( λ v ( L ) , k ) = ( N L ( ξ w ( L ) ,k ∗ β ∗ n − m k )) Θ n ( b L , fL ) T r F L ′ /F L ( λ v ( L ′ ) ,k ) = ( λ ′ v ( L ) , k ) 1 − N ( l ′ ) n ( l ′ , F L ) − 1 , wher e N L := T r k w ( L ) /k v ( L ) and λ ′ v ( L ) , k is define d by λ ′ v ( L ) , k := T r F L l k /F L (Λ m ( N w k ( L ′ ) /w k ( L ) ( ξ w k ( L ′ ) ,k )) ∗ β ∗ n − m k ) N b n − m L γ l . STICKELBERG ER SPLITTING IN THE K –THEOR Y OF NUMBER FIELDS 27 Pr o of. The first formula follo ws b y Lemma 4.9. The s econd formula follows b y Theorem 4.10. Let us prove the Euler System pro per ty (the third formula in the statement o f the Theorem.) W e apply Lemma 2.1 and definition (50): T r F L ′ /F L ( λ v ( L ′ ) ,k ) = T r F L ′ /F L T r F L ′ l k /F L ′ ( x Θ m ( b L ′ , f k L ′ ) w ( L ′ ) ,k ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L T r F L ′ l k /F L l k ( x Θ m ( b L ′ , f k L ′ ) w ( L ′ ) ,k ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L ( T r F L ′ l k /F L l k x Θ m ( b L ′ , f k L ′ ) w ( L ′ ) ,k ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L ( T r F L ′ l k /F L l k  x w ( L ′ ) ,k  Θ m ( b L ′ , f k L ′ ) ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L ( T r F L ′ l k /F L l k  x w ( L ′ ) ,k  Res K f k L ′ /K f k L Θ m ( b L ′ , f k L ′ ) ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L ( T r F L ′ l k /F L l k  x w ( L ′ ) ,k   1 − ( l ′ , F L l k ) − 1 N ( l ′ ) m  Θ m ( b L , f k L ) ∗ β ∗ n − m k ) N b n − m γ l = T r F L l k /F L ( T r F L ′ l k /F L l k  x w ( L ′ ) ,k  Θ m ( b L , f k L ) ∗ β ∗ n − m k ) N b L n − m γ l  1 − ( l ′ , F L ) − 1 N ( l ′ ) n )  = ( λ ′ v ( L ) , k ) 1 − N ( l ′ ) n ( l ′ , F L ) − 1 . The last equality is a dire c t cons equence of equalities (47) and (48).  Now, let b be a fixed ideal in O K , coprime to f l . Co nsider all L as ab ov e which are co prime to l bf . Naturally , we can choose b L := b , for a ll such L . These choices and the results o f § 4 permit us to define the elemen ts λ v ( L ) ∈ K 2 n ( O F L ,S ) l and ξ v ( L ) ∈ K 2 n − 1 ( k v ( L ) ) l as follows. Definition 5.4. Let λ v ( L ) ∈ K 2 n ( O F L ,S ) l be the element corres p o nding to ( λ v ( L ) , l k ) k ∈ lim ← − k K 2 n ( O F L ,S ; Z / l k ) via the isomo rphism (40) for the ring O F L ,S . Definition 5.5. Let ξ v ( L ) ∈ K 2 n − 1 ( k v ( L ) ) l be the element cor resp onding to ( N L ( ξ w ( L ) ,k ∗ β ∗ n − m k )) k ∈ lim ← − k K 2 n − 1 ( k v ( L ) ; Z /l k ) via the isomo rphism (41) for the finite field k v ( L ) . Remark 5.6. Note that Λ ( ξ v ( L ) ) = λ v ( L ) (see Remark 4.15.) The next result shows that the elements { λ v ( L ) } L provide an Euler System for the K -theor y without co efficient s { K 2 n ( O F L ,S ) l } L . Theorem 5.7. F or every L and L ′ as ab ove, such that L ′ = Ll ′ , we have the fol lowing e qualities: (51) ∂ F L (Λ( ξ v ( L ) )) = ξ Θ n ( b , Lf ) v ( L ) (52) T r F L ′ /F L (Λ( ξ v ( L ′ ) )) = Λ( N v ( L ′ ) /v ( L ) ( ξ v ( L ′ ) )) 1 − N ( l ′ ) n ( l ′ , F L ) Pr o of. This follows directly fro m Theor em 5.3.  28 G. BANASZAK AND C. D. POP ESCU Remark 5.8. It is easy to see that one ca n co nstruct Euler systems for ´ etale K - theory in a similar manner. Remark 5.9. In our up coming work, we will use the Euler s y stems constr ucted ab ov e t o in vestigate the structure of the group of divisible ele ments divK 2 n ( F ) l inside K 2 n ( F ) l . T he str ucture of div K 2 n ( F ) l is of principal interest vis a vis some classical co njectures in alg ebraic num b er theory , as ex plained in the introduction. 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W eib el, Intr o duction to algeb r aic K -the ory , b o ok in progress at Charles W eibel home page, h ttp://www.math.rutgers.edu/ ∼ w eib el/ 30. G.W. Whitehead, Elements of Homotopy The ory , Graduate T exts in Math. 61 (1978) Springer-V erl ag Dep ar tment of Ma thema tics a nd Computer Science, Adam Mickiewicz University, Pozna ´ n 61 614, Poland E-mail addr ess : banaszak@amu.ed u.pl Dep ar tment of Ma themat ics, University of California, San Diego, La Jolla, CA 92093, USA E-mail addr ess : cpopescu@math.u csd.edu