Østvær, Paul Arne

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REND. SEM. MAT. UNIV. PADOVA, Vol. 107 (2002)Periodicity in K-groups of Certain Fields.PAUL ARNE ØSTVÆR (*)ABSTRACT - Let k be a field of characteristic different from p . We study the p-tor-sion and the p-cotorsion in the higher algebraic K-groups of k . Under a cer-tain hypothesis we find that these groups are periodic. Some (co)-descentproperties are also pointed out.1. Introduction.Let k be a field of characteristic different from p . In the main part ofthis paper we will assume that the p-cohomological dimension cdp (k) of kis less than three. Additionally, we will assume that the groupHét2 (k ; Qp /Zp (i) ) is trivial for iF2. For such a k we first prove some pe-riodicity results for its algebraic K-groups. Second we discuss some (co)-descent properties for the same groups. These results are easily deducedfrom the Bloch-Lichtenbaum spectral sequence, denoted by BLSS fromnow on, with finite coefficients. We claim no originality whatsoever forthis part. The BLSS for a field such as above resembles the BLSS for acomplex surface. That example was first considered by Suslin [Su2].There are several versions of the BLSS, cf. [BL], [FS], [Le2], [RW]and [We]. Assume k has characteristic zero. The mod p n BLSS for k is athird quadrant cohomological spectral sequence with input the higherChow groups of k with mod p n coefficients, and abutment the mod p n al-gebraic K-groups of k . Suslin [Su3] has proved that the higher Chowgroups of k are isomorphic to the motivic cohomology groups of k . We let(*) Indirizzo dell’A.: Department of Mathematical Sciences, The NorwegianUniversity of Science and Technology, Trondheim, Norway. E-mail: ostvarHma-th.ntnu.noPaul Arne Østvær2subscript M indicate motivic cohomology. From the mentioned results,the mod p n BLSS for k takes the form:E2m , n4HMm2n (k ; Z/p n (2n) ) ¨ K2m2n (k ; Z/p n ) .The outcome of Weibel’s valuation trick from [We] is a mod p n BLSS forfields of positive characteristic. The idea is to replace k by a field F(k) ofcharacteristic zero, and whose motivic cohomology and algebraic K-the-ory groups are naturally isomorphic to the same groups for k . Assume khas positive characteristic l , where lcp . Define R0 (k) to be the Cohen l-ring of k , and define inductively Rn (k) to be Rn21 (k)[t] /(t l2p) where pis a uniformizing parameter for Rn21 (k) and nF1. The quotient field ofthe unioncolim (R0 (k) %R1 (k) %R2 (k) %R)has the desired properties of F(k).Next we explain the relation between the motivic cohomology groupsand the étale cohomology groups of k . The Bloch-Kato conjecture [BK]at the prime p predicts that the Galois symbolKnM (F) /p nKHétn (F ; Z/p n (n) )is an isomorphism for every field F of characteristic different from p . Vo-evodsky proved this conjecture in [Vo] for the prime p42. For p42 theBloch-Kato conjecture was originally formulated by Milnor [Mi]. Suslinand Voevodsky proved in [SV] that if the Bloch-Kato conjecture is trueat the prime p , then there exists natural isomorphismsHMn (k ; Z/p n (i) ) `./´Hétn (k ; Z/p n (i) )0for 0 GnG i ,otherwise .By specialization we get the following result (for two groups A and Bwe let AJB denote an Abelian extension of B by A).THEOREM 1.1. Assume cdp (k) G2. If p is an odd prime, we also as-sume that the Bloch-Kato conjecture holds at p .(a) The mod p n algebraic K-groups of k are given up to extensions byKn (k ; Z/p n ) `./´Hét1 (k ; Z/p n (i) )Hét2 (k ; Z/p n (i11) ) JHét0 (k ; Z/p n (i) )for n42 i21,for n42 iD0.Periodicity in K-groups of certain fields 3(b) The extension above is split by the anti-Chern classes of Kahnif p is odd, or p42 and k contains a primitive fourth root ofunity.REMARK 1.2. Part (b) of Theorem 1.1 is due to Kahn, see Theorem3.1 in [Ka2]. The results from [FS] and [Le2] make it plain that Theo-rem 1.1, and hence some of the results in this paper may be generalizedto certain schemes with mod p étale cohomological dimension less thanthree.In Section 2 we prove results which appear to be new. For this we willonly consider fields with the properties stated in the beginning of the in-troduction. The assumptions on k can often be checked in practice. Ourresults reveal a periodicity phenomena for the p-torsion and the p-cotor-sion in the algebraic K-groups of such a field. The proofs are very ele-mentary and straightforward. However, the results might be useful inspecific examples. The same remarks apply to the results in Section 3.Let k 8 /k be a Galois extension of fields as above. In Proposition 3.3 wepoint out the connection between the Galois (co)-invariants of the alge-braic K-groups of k 8 and the algebraic K-groups of k .2. Periodicity in K-groups.Assume cdp (k) G2. Then the long exact sequence in étale cohomolo-gy induced by the coefficient extension 0 KZ/p(n) KQp /Zp (n) KKQp /Zp (n) K0 shows that the group Hét2 (k ; Qp /Zp (n) ) is divisible. Weimpose the additional assumption that the latter group is trivial for nFF2. For an Abelian group A we let A]p(40n pn A be its maximal p-torsionsubgroup. Let k be an algebraic closure of k .First we translate the additional assumption into a statement aboutthe K-groups of k . Consider the diagramK2n (k ; Qp /Zp ) K K2n (k ; Qp /Zp )bI IbK2n21 (k)]p( K K2n21 (k)]p(where the vertical maps are the Bockstein maps. From Theorem 1.1; theupper horizontal map is injective, since it can be identified with the natu-ral injective map Hét0 (k ; Qp /Zp (n) ) KHét0 (k; Qp /Zp (n) ). We know thePaul Arne Østvær4Bockstein map for k is an isomorphism from [Su]. Hence the Bocksteinmap for k is an isomorphism, and it follows that K2n (k)7Qp /Zp is thetrivial group for all nF1. Note also that K2n21 (k)]p( injects intoK2n21 (k)]p(.The previous remarks combined with Theorem 1.1 give an isomor-phism:Hét0 (k ; Qp /Zp (n) ) K`K2n21 (k)]p( .(2.1)Let en denote the exponent of the multiplicative group (Z/p n )3 , and letm n (k) denote the group of nth roots of unity in k .LEMMA 2.2. Let m , nF1. Then p n K2n21 (k) is isomorphic top n K2(n1men )21 (k) and there is an exact sequence0 KHét0 (k ; Z/p n (n) ) KK2n21 (k) Kp nK2n21 (k) .(2.3)In particular, the group K2men21 (k) contains an element of order pn .PROOF. From (2.1) we find an isomorphism Hét0 (k;Z/p n(n))K`K`p n K2n21 (k). Now employ the Gal (k s /k)-module isomorphismZ/p n (n) `Z/p n (n1en ) where k s is a separable closure of k . The lastclaim follows from p n K2men21 (k) `p n K2en21 (k) `Hét0 (k ; Z/p n (0) ) and thefact that the absolute Galois group of k acts trivially on Z/p n (0) by defi-nition of the Tate twist. rREMARK 2.4. If k contains a primitive p nth root of unity, then:m p n (k) `p n K3 (k) `p n K5 (k) `RThis follows since Z/p n (i) is independent of the twist i under the givenassumption.We claim the Bockstein exact sequence in K-theory and Theorem 1.1combine to make a commutative diagram:0KK2n (k) /p nKK2n (k ; Z/p n )Kp n K2n21 (k)K0I V I`0 KH 2ét (k ; Z/p n (n11) ) KK2n (k ; Z/p n ) KH 0ét (k ; Z/p n (n) ) K0For k there is a unique choice of isomorphism on the right hand side thatPeriodicity in K-groups of certain fields 5makes the diagram commutative. For k we choose the isomorphism thatis compatible with the inclusion into k. This gives a natural isomor-phism:K2n (k) /p nK`Hét2 (k ; Z/p n (n11) )(2.5)LEMMA 2.6. Let m , nF1. Then K2n (k) /p n is isomorphic toK2(n1men ) (k) /pn and there is an exact sequenceK2n22 (k) Kp nK2n22 (k) KHét2 (k ; Z/p n (n) ) K0 .(2.7)PROOF. Given (2.5), the proof is a verbatim copy of the argument forLemma 2.2. The periodicity can be decreased according to Remark2.4. rThe mod p n Bockstein exact sequence in K-theory and Theorem 1.1give the short exact sequence0 KK2n21 (k) /p nKHét1 (k ; Z/p n (n) ) Kp n K2n22 (k) K0 .(2.8)The sequence (2.8) splits if n is a multiple of en and k is a number fieldwhich satisfies the assumptions in Theorem 1.1. These assumptions aresatisfied unless k is real and p42, cf. Theorem 4.5 [RW]. Indeed, Lem-ma 2.2 shows that the mod p n reduction of K2men21 (k) is a full subgroupof Hét1 (k ; Z/p n (men ) ), hence a direct summand. These remarks motivatethe following observation.LEMMA 2.9. If (2.8) splits for n and n1men , then:K2n21 (k) /p n5p n K2n22 (k) `K2(n1men )21 (k) /pn5p n K2(n1men )22 (k) .In particular, if K2n21 (k) /p n is finite and isomorphic toK2(n1men )21 (k) /pn , then p n K2n22 (k) ` p n K2(n1men )22 (k). Likewise, ifp n K2n22 (k) is finite and isomorphic to p n K2(n1men )22 (k), thenK2n21 (k) /p n`K2(n1men )21 (k) /pn .PROOF. The first claim is clear from periodicity of Hét1 (k ; Z/p n (n) ).The remaining claims follow from the cancellation property of finitegroups, see [Hi]. rThe exact sequences (2.3), (2.7) and (2.8) imply the next result.Paul Arne Østvær6THEOREM 2.10. Let nF2. Then we have the exact sequence(2.11) 0KHét0 (k; Z/p n (n))KK2n21 (k)Kp nK2n21 (k)KHét1 (k; Z/p n (n))KKK2n22 (k) Kp nK2n22 (k) KHét2 (k ; Z/p n (n) ) K0 .REMARK 2.11. Sequence (2.11) inserted n42 and with K2 (k) re-placed with its indecomposable part is known from [Le1] and[MS].3. (Co)-descent.Let k 8/k be a Galois extension of fields with group G . We keep the as-sumptions that cdp (k) G2 and H 2ét (k ; Qp /Zp (n) ) 40 for all nF2, andlikewise for k 8 . Consider the Hochschild-Serre spectral sequenceE2s , t4H s (G , H tét (k 8 ; Qp /Zp (n) ) ) ¨ H s1 tét (k ; Qp /Zp (n) )(3.1)and the Tate spectral sequence:E22s , t4Hs (G , H tét (k 8 ; Qp /Zp (n) ) ¨ H 2s1 tét (k ; Qp /Zp (n) ) .(3.2)Here (3.1) is a first quadrant cohomological spectral sequence. More-over, (3.2) is discussed in Chapter I Appendix 1 [Se] and in Proposition3.1.1 [Ka1]. This is a second quadrant cohomological spectral sequence.The following result is now trivial to prove.PROPOSITION 3.3. Let M q denote H qét (k 8 ; Qp /Zp (n) ), and let nF2.We have the exact sequences0KH 1 (G, M 0)KK2n21 (k; Qp /Zp)KK2n21 (k 8;Qp /Zp)GKH 2 (G,M 0)K0and:0KH2 (G, M 1)KK2n22 (k 8; Qp /Zp)GKK2n22 (k; Qp /Zp)KH1 (G, M 1)K0 .In addition we have the naturally induced isomorphismsK2n22 (k ; Qp /Zp ) K`K2n22 (k 8 ; Qp /Zp )Gand:K2n21 (k 8 ; Qp /Zp )GK`K2n21 (k ; Qp /Zp ) .Periodicity in K-groups of certain fields 7The d 2-differentials in (3.1) and (3.2) give isomorphismsH q (G , K2n21 (k 8 ; Qp /Zp ) ) K`H q12 (G , K2n22 (k 8 ; Qp /Zp ) )andHq12 (G , K2n21 (k 8 ; Qp /Zp ) ) K`Hq (G , K2n22 (k 8 ; Qp /Zp ) )for all qF1.REMARK 3.4. It follows that K2n21 (k)]p(K`K2n21 (k 8 )]p(G , andthe transfer map induces a surjection K2n22 (k 8 )]p(G ˆ K2n22 (k)]p(.That surjection is an isomorphism if K2n22 (k 8 )]p( is reduced. Thefirst claim follows from the diagram displayed in the beginning of Sec-tion 2, and the second claim follows from an obvious Bockstein se-quence argument.R E F E R E N C E S[BK] S. BLOCH - K. KATO, p-adic étale cohomology, Inst. Hautes Études Sci.Publ. Math., 63 (1986), pp. 107-152.[BL] S. BLOCH - S. LICHTENBAUM, A spectral sequence for motivic cohomology,UIUC K-Theory preprint server (1995).[FS] E. M. FRIEDLANDER - A. A. SUSLIN, The spectral sequence relating alge-braic K-theory to motivic cohomology, UIUC K-Theory preprint server(1999).[Hi] R. HIRSHON, On cancellations in groups, Amer. Math. Monthly, 76 (1969),pp. 1037-1039.[Ka1] B. KAHN, Deux théorèmes de comparaison en cohomologie étale; applica-tions, Duke Math. J., 69 (1993), pp. 137-165.[Ka2] B. KAHN, On the Lichtenbaum-Quillen conjectures, Algebraic K-theoryand algebraic topology (P. G. Goerss and J. F. Jardine, eds.), NATO ASISeries C, vol. 407, Kluwer, 1993, pp. 147-166.[Le1] M. LEVINE, The indecomposable K3 of fields, Ann. Scient. Éc. Norm.Sup., T.22 (1989), pp. 255-344.[Le2] M. LEVINE, K-theory and motivic cohomology of schemes, UIUC K-The-ory preprint server (1999).[MS] A. S. MERKURJEV - A. A. SUSLIN, The group K3 for a field, Math. USSRIzv., 36 (1991), pp. 541-565.[Mi] J. MILNOR, Algebraic K-theory and quadratic forms, Invent. Math., 9(1970), pp. 318-344.[RW] J. ROGNES - C. A. WEIBEL, Two-primary algebraic K-theory of rings ofintegers in number fields, J. Amer. Math. Soc., 13 (2000), pp. 1-54.[Se] J.-P. SERRE, Galois cohomology, Springer Verlag, 1997.Paul Arne Østvær8[Su1] A. A. SUSLIN, On the K-theory of algebraically closed fields, Invent.Math., 73 (1983), pp. 241-245.[Su2] A. A. SUSLIN, Algebraic K-theory and motivic cohomology, Proc. Intern.Congress Math., Zürich 1994, Birkhäuser, 1995, pp. 342-351.[Su3] A. A. SUSLIN, Higher Chow groups and étale cohomology, Cycles, trans-fers and motivic homology theories, Annals of Math. Studies, 143, Prince-ton University Press, 1999.[SV] A. A. SUSLIN - V. VOEVODSKY, The Bloch-Kato conjecture and motivic co-homology with finite coefficients, The arithmetic and geometry of alge-braic cycles, NATO ASI Series C, vol. 548, Kluwer, 2000, pp. 117-189.[Vo] V. VOEVODSKY, The Milnor conjecture, Preprint (1996).[We] C. A. WEIBEL, Lecture IX, Bloch’s higher Chow groups, Algebraic K-the-ory and its applications, World Scientific, 1999, pp. 95-104.

Periodicity in $K$ -groups of certain fields

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Periodicity in KKK-groups of certain fields

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Periodicity in $K$ -groups of certain fields