Sequential topologies and quotients of Milnor K-groups of higher local fields

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1 Sequential topologies and quotients of Milnor K-groups of higher local fields Ivan esenko Abstract. or a higher local field algebraic properties of K m ( )/ l lk m ( ) of the Milnor K-group are studied by using topological and arithmetical considerations. In particular, standartness of the torsion in the latter group and divisibility of the group l lk m ( ) if the last residue field of is finite are proved. It is shown that l lk m ( ) coincides with the intersection of all neighbourhoods of zero in K m ( ) (and if is an m-dimensional local field, it equals to the kernel of the reciprocity map K m ( ) Gal( ab / ) ). A description of K m ( )/l in the language of topological generators and relations is provided. The equality of several topologies on K m ( ) at the level of subgroups is shown and their difference is discussed. Appendix of this paper written by O. Izhboldin describes a construction of a field containing a primitive p th root and such that p-torsion of K m ( )/ l lk m ( ) is not generated by p-torsion in. He works with the field of rational functions of an infinite product of certain Severi Brauer varieties.. Introduction An n-dimensional local field = k n is a complete discrete valuation field with residue field k n being (n )-dimensional; 0-dimensional local field is a perfect field k 0 of positive characteristic p. Thus to every n-dimensional field corresponds n + fields k n =, k n,..., k 0. Lifting prime elements of k n,..., k to the field one obtains an ordered system of local parameters t n,..., t (t n is a prime element of ). Let s 0 be the minimal integer such that char( ) = char(k s ). If s < n, then is isomorphic to k s ((t s+ ))... ((t n )). Note that the group of principal units of with respect to the discrete valuation of rank n s is divisible if char( ) = 0, and so it is not very interesting from the point of view of class field theory. The field k s if s 0 is called a mixed characteristic field, it is a natural higher dimensional analogue of a p-adic field. rom many points of view Milnor K-groups of are not the most suitable object for a meaningful description of abelian extensions of an n-dimensional local field in higher local class field theory; the structure of Milnor K-groups of is still not completely known. It is more convenient and natural to work with quotients K m ( )/ l lk m ( ) of Milnor K-groups endowed with a special topology. Arithmetical homomorphisms from Milnor K-groups (like a reciprocity map) factorize through such quotients. In this paper we discuss algebraic properties of K m ( )/ l lk m ( ) by using topological and arithmetical considerations which are related to topological and arithmetical properties of higher local fields. In turn, even though applications are not discussed here, the properties of the quotients Algebra i Analiz, 3(200), issue 3, p ; the English translation in St. Petersburg J. Math. 3(2002),

2 2 I. esenko of the K-groups are quite important for the arithmetic of higher local fields due to higher local class field theory. We provide a new short approach which replaces longer approaches in [2 5]. It corrects and clarifies some statements or proofs of [5], [2 5], [2] and links between them. or the reader convenience the text contains almost all definitions. New methods and results of this paper are: () simultaneous work with several topologies, (2) correction of Parshin s theorem on description of the subgroup of topological Milnor K-groups in positive characteristic case, (3) new proofs in the case of finite residue field k 0 of the following results: (a) l lk m ( ) coincides with the intersection Λ m ( ) of all open subgroups in K m ( ) with respect to the topologies on K m ( ) mentioned above. (b) l lk m ( ) is a divisible group. (c) If a primitive rth root of unity is contained in then r-torsion of is generated by r-torsion in. K top m ( ) = K m ( )/ l lk m ( ) Section 2 presents several topologies on the additive and multiplicative group of which are defined by induction on n. They are different from Parshin s topology [5] in characteristic p. Their common feature is that each of them has the same set of convergence sequences. The multiplication is sequentially continuous but not necessarily continuous with respect to some of them. or class field theory sequential continuity seems to be more important than continuity. This is a hidden phenomenon in dimension and 2, where continuity is the same as sequential continuity. In particular, this can affect our understanding of a generalization of harmonic analysis to higher multidimensional fields. Three important pairings of Milnor K-groups are described in section 3. They are in intensive use in section 4 when studying properties of Milnor K-groups of endowed with topologies. We compare various topologies (λ m in 4., ν m, o m, σ m in 4.6) on Milnor K-groups some of which were in use in [2 5] and show that they all coincide at the level of subgroups. In subsections , central in this paper, we prove (a) (c). In fact we prove more general results in the case of a perfect k 0 ; for details see Concerning the structure of Km top ( ): it is completely known in characteristic p (most of results except (a) and (b) were stated by Parshin in [5]) and we supply its (partial) description in characteristic 0. As an application in subsection 4.8 we deduce the standard description of the kernel of the norm map for Km top (which is not currently established for K m of an arbitrary field if m > 2). The results of this paper are important for explicit higher local class field theory as presented in [2 5]; there are further applications, for instance to explicit formulas in class field theory, study of abelian extensions, ukaya s map for Milnor K 2 -groups of complete discrete valuation fields with residue field having one element p-basis. Appendix of this paper written by late O. Izhboldin describes a construction of a field containing a primitive pth root and such that p-torsion of K m ( )/ l lk m ( ) is not generated by p-torsion in. He works with the field of rational functions of an infinite product of certain Severi Brauer varieties. Quite a different construction for irregular prime numbers p follows from works of G. Banaszak [24]. irst, it is easy to see that if p-torsion of K m ( )/ l lk m ( ) is generated by p-torsion in, then l lk m ( ) = p l lk m ( ) and hence the group l lk m ( ) is either zero or infinite.

3 Sequential topologies and quotients of Milnor K-groups 3 Now using [23, p.289] for each irregular prime number p there is a totally real field K (maximal totally real subfield of Q(µ p )) such that the p-primary part of the group l lk 2 (K) is non-zero. Let = K(µ p ). Then the p-primary part of the finite [22, Th.8.9] group l lk 2 ( ) is non-zero and therefore p-torsion of K 2 ( )/ l lk 2 ( ) is not generated by p-torsion in. I am grateful to G. Banaszak for correspondence on this subject. Throughout the text we denote by µ m the group of mth roots of unity. or an abelian group A we denote m-torsion points of A by Tors m A. 2. Topology on the multiplicative group By O we denote the ring of integers of with respect to the discrete valuation of rank n associated to t n,..., t ; O doesn t depend on the choice of a system of local parameters. Denote by V = + (t n,..., t )O = + t O the group of principal units of as an n-dimensional local field. Denote by O 0 the subring in corresponding to the last residue field k 0 if char( ) = p and the image in of the ring of Witt vectors of k 0 if char( ) = 0. The ring O 0 contains the set of multiplicative representatives R of k As an example, for a 2-dimensional local field with a system of local parameters t 2, t define a base of neighborhoods of as + t i 2 O + t j O 0[[t, t 2 ]] (e.g. [7]). Then every element α can be expanded as a convergent with respect to the just defined topology product α = t a 2 2 ta θ ( + θ i,j t i 2 tj ) with θ R, θ i,j R, a i Z. The set S = {(j, i) : θ i,j 0} satisfies the property: for every i there is j i such that (j, i) S implies j j i. Call such a set admissible We provide a short descrition of topology λ on the additive group of in the case of char(k n ) = p. or a more detailed discussion see [26]. Definition. The topology λ on k 0 is discrete. ix a system of local parameters t i. The residue field k n can be identified with the field k 0 ((t ))... ((t n )). Let 0 be a complete discrete valuation subfield of which contains the elements t n,..., t and a prime element t = t n if is of characteristic p and t = p if is of characteristic 0, and which has the residue field k 0 (t )... (t n ). ix a lifting (and thus a set of representatives S) of k 0 (t )... (t n ) in 0 such that elements of k 0 are mapped to their multiplicative representatives in O 0, and the residues t i k n, i n, are mapped to t i in 0. By linearity this determines the lifting of k 0 (t )... (t n ) to 0. Given the topology λ on the additive group k n, introduce the topology λ on the additive group. irst, an element α 0 is said to be a limit of a sequence of elements α v 0 if and only if given any series α v = i θ v,it i, α = i θ it i with θ S, for every set {U i, < i < + } of neighbourhoods of zero in k n and every i 0 for almost all v the residue of θ v,i θ i belongs to U i for all i < i 0. Second, a subset U in 0 is called open if and only if for every α U and every sequence α v 0 having α as a limit almost all α v belong to U. This determines the topology λ on 0.

4 4 I. esenko Now one proves by induction on n that the completion of 0 with respect to λ is a subfield E of. The field is in fact a finite dimensional vector space over E and thus is endowed with the topology λ. This topology doesn t depend on the choice of a system of local parameters [4]. It is not difficult to deduce the following properties (for some relevant details see [4]): () α is a limit of α v if and only if the sequence α v converges to α with respect to the topology λ; (2) a limit is uniquely determined; (3) each Cauchy sequence with respect to the topology λ converges in ; (4) the limit of the sum of two convergent sequences is the sum of their limits; Remark. Let be of characteristic p. The topology λ on the additive group is different from that introduced by Parshin in [5] for n 2: for example, the set W = \ {t a 2 t c + t a 2 tc : a, c } in = p ((t ))((t 2 )) is open in the just defined topology, ie for each convergent sequence x v x W almost all x v belong to W. If for some open subgroups U i in the additive group of p ((t )) such that U i = p ((t )) for i a the group {x = a i t i 2 : x, a i U i } were contained in W, then for any positive c such that t c U a we would have t a 2 t c + t a 2 tc W, a contradiction. However, a sequence of elements in converges to x with respect to λ if and only if it converges with respect to the topology introduced by Parshin. In fact, the topology λ is the finest topology in which the set of convergent sequences is the same as in the topology introduced by Parshin. So λ can be viewed as the sequential saturation of the Parshin topology Definitions. If char(k n ) = p, then define a topology λ on as the product of the induced from topology on the group of principal units V, the discrete topologies on the cyclic groups generated by t i and the discrete topology on R. If char( ) = char(k s ) = 0, char(k s ) = p, then define a topology λ on as the product of the trivial topology on + (t n,..., t s+ )O (which is a divisible subgroup of ), the discrete topology on the cyclic groups generated by t i with i > s and the topology λ on k s The following properties of the topology λ on can be deduced by induction on dimension n. Properties. () Each Cauchy sequence with respect to the topology λ converges in, the limit of the product of two convergent sequences is the product of their limits. The multiplication in is sequentially continuous. (2) or a 2-dimensional local field its multiplicative group is a topological group and it has a countable base of open subgroups (for example see [7]). In the case of a with n 3 and s 2 both assertions don t hold. or example, let be a two-dimensional field and let L = ((t 3 )). If Z = + W t 3 + t 2 3 [[t 3]] is an open subgroup in + t 3 [[t 3 ]], W, then W is an open subgroup in. It is easy to see that then W W =. So the coefficient of t 2 3 in ZZ can be any element of. Plenty of open subgroups Y of + t 3 [[t 3 ]] don t satisfy the property that the coefficient of t 2 3 in Y can be any element of. Therefore for those open subgroups Y there is no open subgroup Z of L such that ZZ Y. (3) or every open neighbourhood U of in there is r such that V pr.6]). (4) or any sequence a i V the sequence a pi i sequence a i + M i converges to. (5) Subgroups V pr are closed in V. The product of V pr U (see [2, Lemma converges to. If M = (t n,..., t l )O then any and a closed subgroup in V is closed.

5 Sequential topologies and quotients of Milnor K-groups 5 Remark. If char( ) = p and k 0 is finite then the topology λ on the multiplicative group is different from that introduced by Parshin in [5]. or example, for n 3 each open subgroup A in with respect to the topology introduced in [5] possesses the property: + t 2 no ( + t 3 no )A (indeed, it is easy to see that the product of two open subgroups of k n is equal to k n ; this implies the indicated property). However, the subgroup in + t n O topologically generated by + θt in n... t i with (i n,..., i ) (2,,..., 0) (all zeros except the first two components), i n (ie the sequential closure of the subgroup generated by these elements), is open in λ and doesn t satisfy the above-mentioned property. So the topology λ and the topology of [5] are distinct even at the level of subgroups. Another, third topology on is discussed in subsection Definition. Call a subset X of elements of (Z) n greater than 0 admissible if for every < m n and every (i m,..., i n ) there is j(i m,..., i n ) such that (i,..., i n ) X implies i m j(i m,..., i n ), and there is j such that all (i,..., i n ) X implies i n j. If char(k n ) = 0 then every element α is a product of an infinitely divisible element, a power of t n and an element of k n. If char(k n ) = p then every element α is a convergent product α = t an n... t a θ ( + θ in,...,i t in n... t i ), θ R, θ in,...,i R with {(i,..., i n ) : θ in,...,i 0} being an admissible set (see [4], [4]) Include a principal unit ε V into an arbitrary topological basis {ε α } of V and consider the subgroup topologically generated by t an n,..., t a, R, principal units ε α ε and ε ps. It is an open subgroup of finite index of. Denote the shift-invariant topology ν on which has these open subgroups as the base of neighbourhoods of. A sequence of elements in converges to with respect to λ if and only if it converges with respect to ν. The group is a topological group with respect to ν. Since for every subgroup H of and an element α such that there is an open subset U of which includes H and doesn t contain α there is an open subgroup V which includes H and doesn t contain α, every subgroup of which is the intersection of some open subsets is the intersection of some open subgroups. In particular, the intersection of all open subgroups of finite index containing a closed subgroup H coincides with H. Hence for any subgroup H of a sequence of elements of /H converges with respect to the quotient topology of ν if and only if it converges with respect to the quotient topology of λ. Thus, if two sequences converge in /H, then their product converges to the product of their limits. or other definitions and details see [7] (n = 2), [5], [2 5], [4]. 3. Pairings of K-groups of higher local fields Let G ur,ab,p be the Galois group of the maximal abelian unramified p-extension of over (which corresponds to the maximal p-extension of the last residue field k 0 ). 3.. Let be an n-dimensional local field of characteristic p. or α,..., α n, a Witt vector (β 0,..., β r ) W r ( ) and ϕ G ur,ab,p put (α,..., α n, (β 0,..., β r )] r (ϕ) = (ϕ )(γ 0,..., γ r )

6 6 I. esenko where (rob )(γ 0,..., γ r ) = (λ 0,..., λ r ); and the ith ghost component λ (i) of (λ 0,..., λ r ) is res k0 (β (i) α dα αn dα n ). In fact, to make the previous definition precise one needs, as usual with Witt vectors, to pass to Witt vectors over a ring of characteristic zero, like Z p ((t ))... ((t n )) and then return back. This is a sequentially continuous and symbolic (ie satisfies the Steinberg property) in the first n coordinates map. It defines the Artin Schreier Witt Parshin pairing [5] (finite k 0 case), [4]: K n ( )/p r W r ( )/(rob )W r ( ) Hom ( G ur,ab,p, W r ( p ) ) where rob is the robenius map Let be an n-dimensional local field of characteristic 0 and let a primitive p r th root of unity ζ be contained in. Suppose that p is odd (if p = 2 then formulas become much more complicated). Suppose first that char(k n ) = p. Let X,..., X n be independent indeterminates over the quotient field of O 0 (the latter is defined at the beginning of section 2). or an element of, with θ R, θ in,...,i R put α = t an n... t a θ ( + θ in,...,i t in n... t i ) α(x) = X an n... X a θ ( + θ in,...,i X in n... X i ). The formal power series α(x) O 0 ((X ))... ((X n )) depends on the choice of local parameters and the choice of a power series expression of α. Denote z(x) = ζ(x), s(x) = z(x) pr. Define the action of the operator on θ s and on X i as raising to the pth power. or α put l(α) = p log α(x) p. Now for elements α,..., α n+ define Φ(α,..., α n+ ) as ( ) n+ ( ) n+ i d α l(α i ) d α i p d α i+ α α i αi+ p d α n+ αn+. as i= Define the Vostokov map [9] (case of finite k 0 ), [4]: V r (α,..., α n+ )(ϕ) = ζ (ϕ )δ, V r : ( ) n+ Hom(G ur,ab,p, µ p r) where (rob )δ = res Φ(α,..., α n+ )/s(x) for ϕ G ur,ab,p. This map V r doesn t depend on the attaching formal power series to elements of and the choice of a system of local parameters. The map V r is sequentially continuous and symbolic, so it induces a homomorphism V r : K n+ ( )/p r Hom(G ur,ab,p, µ p r). The latter is in fact an isomorphism, as follows from the results of section 4.7. or χ Hom(G ur,ab,p, µ p r) denote by E(χ) any principal unit such that V r ({t,..., t n, E(χ)} = χ. The elements E(χ) in their explicit form introduced by Vostokov as primary elements in [9, sect. ] in the case of finite k 0 play an important role in deducing the explicit formula. If char( ) = char(k s ) char(k s ) then define the Vostokov pairing K m ( )/p r K n+ m ( )/p r Hom(G ur,ab,p, µ p r) using the canonical surjections K ( )/p r K (k s )/p r.

7 Sequential topologies and quotients of Milnor K-groups The composition of border homomorphisms in K-theory (eg, [9, Ch.IX, sect. 2]) supplies a homomorphism m : K m ( ) K m (k 0 ) K 0 (k 0 ). The kernel of n is the subgroup V K n ( ) defined in 4.. We also get pairings K m ( ) K n+ m ( ) K n+ ( ) n+ K m (k 0 ) K 0 (k 0 ). In particular, the tame symbol c is the composition of n+ with the projection to K (k 0 ) and the lifting K (k 0 ) R can be explicitly described as follows. or an element α and its expression as in 3.2 put v (j) (α) = a j for j n. or elements α,..., α n+ of the value c(α, α 2,..., α n+ ) is the element of R whose residue is equal to the residue of α b... αb n+ n+ ( )b in the last residue field k 0, where b = s,i<j v(s) (α i )v (s) (α j )b s i,j and b j is the determinant of the matrix obtained by omitting the jth column with the sign ( ) j from the matrix A = (v (i) (α j )), and b s i,j is the determinant of the matrix obtained by omitting the ith and jth columns and sth row from A. 4. K top -groups 4.. Definition. Let λ m be the finest topology on K m ( ) for which the map φ: m K m ( ), φ(α,..., α m ) = {α,..., α m } is sequentially continuous with respect to the product of the topology λ on (section 2) and for which the subtraction in K m ( ) is sequentially continuous. Define K top m ( ) = K m ( )/Λ m ( ) with the quotient topology where Λ m ( ) is the intersection of all neighbourhoods of 0 with respect to λ m (and so is a subgroup). Remark. Every sequentially open (closed) subset with respect to λ m is open (resp. closed). The topology λ m coincides with the finest topology on K m ( ) for which the map φ is sequentially continuous with respect to the product of the topology ν on (defined in 2.6) or Parshin s topology [5] and for which the subtraction in K m ( ) is sequentially continuous. rom the definition it follows that λ m is a shift invariant topology. We have λ = λ and K top ( ) coincides with the quotient of by the maximal divisible subgroup of V. If the first residue field k n is of characteristic p and the last residue field k 0 is finite, then K top ( ) = K ( ). Definition. Denote by V K m ( ) the subgroup of K m ( ) generated by V ; similarly introduce V K top m ( ). Remark 2. The induced topology on V K m ( ) by λ m coincides with the the finest topology on V K m ( ) for which the restriction of φ on V m V K m ( ) is sequentially continuous with respect to the product of the topology λ on and for which the subtraction in V K m ( ) is sequentially continuous. Indeed, the map φ and subtraction in K m ( ) is sequentially continuous with respect to the shift invariant topology on K m ( ) extended from the finest topology on V K m ( ) as above.

8 8 I. esenko A sequentially continuous symbol homomorphism from the tensor product of m copies of to a topological Hausdorff group G induces a continuous homomorphism from Km top ( ) to G. Therefore the Artin Schreier Parshin, Vostokov pairings and tame symbol defined in section 3 are factorized through topological K-groups. ε. rom property (5) of 2.4 one deduces that {θ, ε} = 0 in K top 2 ( ) for θ R and a principal unit Suppose that k 0 is algebraic over p. Then () {θ, θ } = 0 in K 2 ( ) for θ, θ R. More generally, if the absolute Galois group of k 0 is procyclic then {θ, θ } l lk 2 ( ) for θ, θ R (this is due to the fact that the norm map for a finite extension of k 0 is surjective). (2) Using the tame symbol defined in 3.3 one easily shows that K m ( ) splits into the direct sum of V K m ( ), several copies of Z and of the group R. (3) Km top ( ) splits into the direct sum of V Km top ( ), several copies of Z and of the group R. In particular, Kn top ( ) is isomorphic to the direct sum of the cyclic group generated by {t,..., t n }, n copies of R, and the subgroup V Kn top ( ). Remark 3. In the general case of a perfect residue field k 0 the group Km top ( )/p splits into the direct sum of V Km top ( )/p and several copies of Z/p or two principal units ε, η in K 2 ( ) the following formula is straightforward: {ε, η} = { ε, + (ε )η} { + (ε )(η )ε, η}. The principal unit +(ε )(η )ε is of higher order than that of ε, η. Now for β (t n,..., t )O and θ R, we get { θt in n... t i, + β} = {θt in n... t i, θt in n... t i ( + β)} { θt in n... t i β( θt in n... t i ), + β}. The first symbol of the right hand side can be written as the sum of symbols {t i, λ i } with principal units λ i which sequentially continuously depend on θt in n... t i and + β, and the first element of the second symbol is closer to than θt in n... t i is. or an arbitrary principal unit ε factorize it into the convergent product of θt in n... t i and then apply the previous argument. One can verify following Parshin s method [5, sect.2] (put η = + β) that the symbol {ε, η} can be written in K top 2 ( ) as i {ρ i, t i } + {ρ, η} with principal units ρ i, ρ which sequentially continuously depend on ε and η and ρ closer to than ε is. This implies (repeating the argument) that for arbitrary principal units ε, η the symbol {ε, η} can be written in K top 2 ( ) as {ρ i, t i } with principal units ρ i which sequentially continuously depend on ε and η. Therefore, every element x of V K m ( ) can be written as a sum of an element of Λ m ( ) plus a fixed number of elements of the form {α i } {some local parameters} with α i. Remark. Since lx for x V K m ( ) can be written as the sum of symbols {α l } {local parameters} (α V ) and an element of Λ m ( ), from property (4) of 2.4 we get r p r V K top m ( ) = {0}.

9 Sequential topologies and quotients of Milnor K-groups rom the previous subsection we immediately deduce that the group V Km top ( ) is topologically generated (with admissible sets playing the same role as in the case of ) by symbols { + θt in n... t i, t j,..., t jm } with θ R. Topological relations among these generators (modulo p r for each r in the case of char( ) = p, modulo p r in the case char( ) = 0 and a primitive p r th root of unity belongs to, modulo p if char( ) = 0 and a primitive pth root of unity doesn t belong to ) are revealed using the Artin Schreier Witt Parshin and Vostokov pairings, for details see [5], [2 5]. Simultaneously one verifies that appropriately modified pairings are nondegenerate. or example, if µ p r, then the Vostokov pairing V r is very useful in the study of the structure of Km top ( )/p r in terms of topological generators and relations between them, see [2, sect. 3 and 4] (the quotient filtration on K m ( )/p induced by the standard filtration on as a discrete valuation field can be also described in terms of differential forms over k n by Bloch Kato s result []). Using explicit calculations with the Vostokov pairing one shows that p r Km top ( ) coincides with the intersection of open subgroups of Km top ( ) containing p r Km top ( ). One can also prove the following lemma (the method of the proof is entirely similar to that of [2]). Lemma. Let be an n-dimensional local field containing a primitive pth root. Let L = (µ p r) and p s = L :. Let σ be a generator of Gal(L/ ). Then the annihilator of i /L K top n+ m ( ) with respect to the Vostokov pairing V r is equal to (σ )Km top (L) + p r s i /L Km top ( ) + p r Km top (L) Definition. or topological spaces A i, i d, introduce the following -product topology on A i : first, an element (x,..., x d ) A i is a limit of a sequence (x v,,..., x v,d ) A i if x v,i x i when v + for all i d. Now a subset Y A i is called an open subset if for every y Y and every sequence y v A i having y as a limit almost all y v belong to Y. The -product topology is in general strictly finer than the topology of the product of topological spaces; it is the sequential saturation of the product topology. or example, if n = 3, char( ) = p, then the set \ {( + t 3 t a 2 t c, + t 3t a 2 tc ) : a, c } is open in with respect to the -product topology, but is not open in the product topology. Compare this definition with the definitions in In subsections we study relations between Λ m ( ) V K m ( ) and l lv K m ( ) in the general case of a perfect k 0. Remark. If k 0 is algebraic over p then Λ m ( ) V K m ( ) and l lk m ( ) = l lv K m ( ) as follows from 4.. In general l lk m ( ) V K m ( ) (because the divisible part of R can be nontrivial; see also () in 4.). Theorem. If char( ) = p then Λ m ( ) V K m ( ) coincides with l lv K m ( ); the group Λ m ( ) V K m ( ) is p-divisible. Proof. According to Bloch Kato Gabber theorem [] the differential symbol d : K m ( )/p Ω m, d {α,..., α m } = d α α d α m α m is injective. Ω m is a finite-dimensional vector space over / p, so the intersection of all neighborhoods of zero in Ω m with respect to the induced by λ topology is trivial. The differential symbol d is continuous. Therefore its injectivity implies Λ m ( ) pk m ( ).

10 0 I. esenko Let J consist of j,..., j m and run all (m )-elements subsets of {,..., n}, m n+. Let E J be the subgroup of V generated by + θt in n t i, θ O 0, with restrictions that p doesn t divide gcd(i,..., i n ) and the smallest index l for which i l is prime to p doesn t belong to J. Consider the induced by λ topology on E J. Applications of the Artin Schreier Witt Parshin pairing provide a proof of a corrected theorem of Parshin (the proof goes in the same way as in [5]) which claims that there exists an isomorphism and homeomorphism ψ from the group J E J (with the -product of the induced topology by Λ on E J ) onto V Km top ( ). This provides an explicit and satisfactory description of the topology on Km top ( ) in the positive characteristic case. Therefore the group V K m ( )/Λ m ( ) V K m ( ) = ψ( E J ) doesn t have nontrivial p-torsion. The group K m ( )/V K m ( ) has no nontrivial p-torsion due to 3.3 (since Tors p K i (k 0 ) = {0}). Now from Λ m ( ) pk m ( ) we deduce that Λ m ( ) V K m ( ) = p(λ m ( ) V K m ( )). Since V is l-divisible for every l prime to p, we get or the inverse inclusion see Remark in 4.2. Λ m ( ) V K m ( ) l lv K m ( ) or m n +, d = ( n m ) define the homomorphism g: V d V K m ( ), β J {β J, t j,..., t jm }, where J = {j,..., j m } runs over all m elements subsets of {,..., n}. Theorem. (i) im(g) + Λ m ( ) V K m ( ) = V K m ( ). The homomorphism g 0 : V d /g (Λ m ( )) V K top m ( ) is a homeomorphism between V d /g (Λ m ( )) with the quotient topology of the -product topology of the induced by λ topology on V and V Km top ( ) with the topology λ m. (ii) Λ m ( ) V K m ( ) is the intersection of all open subgroups of finite index in V K m ( ). Proof. rom the definitions and 4.2 it follows that for α V, α 2,..., α m there exist elements β J V, J = {j,..., j m }, which sequentially continuously depend on α,..., α m such that the symbol {α,..., α m } can be written as {βj, t j,..., t jm } mod Λ m( ). So there is a sequentially continuous map f: V m V d such that its composition with g coincides with the restriction of the map φ on V m modulo Λ m ( ). Let U be an open subset in V K m ( ). Then g (U) is open in the -product of the topology λ on V d. Indeed, otherwise for some J there were a sequence α(i) J g (U) which converges to α J g (U). Then the properties of the map φ imply that the sequence φ(α (i) J ) U converges to φ(α J ) U which contradicts the openness of U. Thus, g (Λ m ( )) is the intersection of open subsets of V d. The product of two convergent sequences x i, y i in V d /g (Λ m ( )) converges to the product of their limits by 2.6. Hence using Remark 2 of 4. we deduce that the quotient topology of λ m on V K m ( )/Λ m ( ) V K m ( ) is the quotient topology of the -product of λ on V d via g. Thus, g: V d /g (Λ m ( )) V K m ( )/Λ m ( ) V K m ( )

11 Sequential topologies and quotients of Milnor K-groups is a homeomorphism of V d /g (Λ m ( )) with the quotient of the -product topology of the λ topologies on V d and of V K m ( )/Λ m ( ) V K m ( ) with the quotient topology of λ m. Note that Λ m ( ) is closed: if x i Λ m ( ) tends to x, then x = x i + y i where x i, y i converges to 0, so x converges to 0 and hence belongs to Λ m ( ). To deduce (ii) use the fact that every closed subgroup in V d is the intersection of certain open subgroups of finite index (see 2.6), hence every closed subgroup in V K m ( ) is the intersection of certain open subgroups of finite index. Remark { } {. Using property (4) of 2.4 and the previous theorem we deduce that for sets of subgroups Zi = p i V Km top ( ) } or { } { Z i = { + M i }K top m ( ) } where M = (t n,..., t l )O then the natural map V K top m ( ) lim V K top m ( )/Z i is surjective. Remark 2. Introduce on V Km top ( ) the topology ν m induced by the topology ν on V via the map g. Due to the properties of ν in 2.6 the group V Km top ( ) is a topological group with respect to ν m and by property (ii) of the preceding theorem the intersection of all neighbourhoods of zero in V Km top ( ) with respect to λ m is zero. Since ν and λ have the same sets of convergent sequences, ν m and λ m have also the same sets of convergent sequences. Remark 3. rom the proof of the theorem we deduce that λ m coincides with the finest topology o m on K m ( ) such that the map ( m ) d K m ( ), (β j,i ) {β,i,..., β m,i } is sequentially continuous. i d Remark 4. rom the proof of the theorem we deduce that λ m coincides with the finest topology σ m on K m ( ) such that for all α i, i m, the map from to K m ( ), α {α, α,..., α m }, is sequentially continuous and the subtraction in K m ( ) is sequentially continuous Theorem. (i) Λ m ( ) V K m ( ) coincides with l lv K m ( ). (ii) Let l be any prime number if k 0 is algebraic over p and l = p otherwise. If contains a primitive lth root ζ l of unity and lx = 0 for x Km top ( ), then x = {ζ l } y for some y K top m ( ). (iii) The group l lv K m ( ) is divisible. The sequence splits. 0 Λ m ( ) V K m ( ) V K m ( ) V K top m ( ) 0 (iv) If k 0 is algebraic over p then (i) and (iii) hold if Λ m ( ) V K m ( ) is replaced by Λ m ( ) and l lv K m ( ) by l lk m ( ). Proof. To deduce (i) we use Remark of 4.2 which implies that l lv K m ( ) Λ m ( ). In view of Theorem 4.5 it remains to treat the case of fields of characteristic 0; there it is sufficient to consider the case of a field of characteristic 0 with residue field k n of characteristic p. In the course of the proof we use Bloch Kato s theorem which says that the norm residue homomorphism for a henselian discrete valuation field is an isomorphism [].

12 2 I. esenko To complete the proof of (i) we show by induction on r that p r V K m ( ) is equal to the intersection of open subgroups of V K m ( ) which contain p r V K m ( ); then certainly p r V K m ( ) Λ m ( ) V K m ( ) (note that lv K m ( ) = V K m ( ) for l prime to p). irst of all, we can assume that µ p is contained in applying the standard argument by using (p, (µ p ) : ) = and l-divisibility of V K m ( ) for l prime to p. If r =, then we can use the description of V Km top ( )/p provided by the Vostokov symbol of 3.2 and 4.3 and compare it with the description of the quotients U i K m ( ) + pk m ( )/U i+ K m ( ) + pk m ( ) (where U i K m ( ) is generated by + t i no ) provided by Bloch Kato s theorem []; one deduces that V K m ( )/p = V Km top ( )/p. rom 4.3 we deduce then that the intersection of open subgroups in V K m ( ) containing pv K m ( ) is equal to pv K m ( ). It follows from the explicit description of the Vostokov symbol that K n+ ( )/p = K top n+ ( )/p Hom(G ur,ab,p, µ p ). If µ p r, then one deduces by induction on r using Vostokov s primary elements E(χ) that Induction Step. or a field consider the pairing K n+ ( )/p r = K top n+ ( )/pr Hom(G ur,ab,p, µ p r). (, ) r : K m ( )/p r H n+ m (, µ n m p ) r Hn+ (, µ n p ) r given by the cup product and the map H (, µ p r). If µ p r, then from Bloch Kato s theorem one can deduce that (, ) r is up to identifications the same as the Vostokov pairing V r (, ). or χ H n+ m (, µ n m p ) put r A χ = {α K m ( ) : (α, χ) r = 0}. Let α belong to the intersection of all open subgroups of V K m ( ) which contain p r V K m ( ). Due to Lemma below α A χ for every χ H n+ m (, µ n m p r ). Set L = (µ p r) and p s = L :. rom the induction hypothesis we deduce that α p s V K m ( ) and hence α = N L/ β for some β V K m (L). Then 0 = (α, χ) r, = (N L/ β, χ) r, = (β, i /L χ) r,l where i /L is the natural map. Keeping in mind the identification between the Vostokov pairing V r and (, ) r for the field L we see that the element β is annihilated by i /L K n+ m ( ) with respect to the Vostokov pairing. rom Lemma of 4.3 using its notation we deduce that β (σ )K m (L) + p r s i /L K m ( ) + p r K m (L), and therefore α p r K m ( ). Since K m ( )/V K m ( ) has no nontrivial p-torsion (see 3.3), we conclude α p r V K m ( ) as required. Thus, the intersection of open subgroups containing p r V K m ( ) is equal to p r V K m ( ) and Λ m ( ) V K m ( ) p r V K m ( ). Lemma. A χ is an open subgroup of K m ( ). Proof. Due to Remark 4 of 4.6 it suffices to show that for every b 2,..., b m the group B = {a : ({a, b 2,..., b m }, χ) r = }

13 Sequential topologies and quotients of Milnor K-groups 3 is open in, i.e. if a i then a i B for all sufficiently large i. Denote by β the image of {b 2,..., b m }, χ with respect to the pairing K m ( )/p r H n+ m (, µ p n m ) r Hn (, µ n p ); r then B is the annihilator of β with respect to the pairing /p r H n (, µ n p ) r Hn+ (, µ n p ). r We argue by induction on r. If r =, then using the link between (, ) and V (, ) and the explicit formula for V we deduce that B is open. Induction step. Let r >. Since a p i, the induction hypothesis implies that ap i B for all sufficiently large i. So (a i, β) Tors p H n+ (, µ n pr ) for all sufficiently large i. The exact sequence µ m p µ m p r µ m induces the exact sequence p r H n (, µ n p ) r Hn (, µ n ) H n+ (, µ n p r p ) H n+ (, µ n p ). r Due to Bloch Kato s theorem the first map is surjective, hence the third one is injective. Therefore, Tors p H n+ (, µ n p ) is isomorphic to a quotient of r Hn+ (, µ n p ) which due to Bloch Kato s theorem is isomorphic to a quotient S of K top n+ ( )/p. Similar to the previous argument Hn (, µ n p )/p r is isomorphic to a subquotient S 2 of K n ( )/p. It remains to show that the annihilator of the image of β in S 2 with respect to the pairing /p S 2 S is open; identifying the latter with the induced one by the Vostokov pairing one completes the induction step. To prove (ii) assume that the field contains a primitive lth root of unity ζ l. If l p, then from Remark of 4.2 and property (3) in 4. we deduce the result. Suppose that l = p. The exact sequence µ m p s µ m µ m p s+ p induces the commutative diagram µ p K m ( )/p K m ( )/p s p K m ( )/p s+ H m (, µ m p ) H m (, µ m p ) s Hm (, µ m and the bottom horizontal sequence is exact, the left and the right vertical homomorphisms are isomorphisms due to Bloch Kato s theorem. Hence if px p r K m ( ), then x = {ζ p } a r + p r c r. Denote by D r the preimage of the closed subgroup p r Km top ( ) with respect to the continuous homomorphism K top m ( ) h top V K m ( ), z {ζ p } z. The kernel of h is equal to D = D r. Let α K top m ( ) \ D. There is a positive integer r such that h(α) pr V Km top ( ). Include α in a topological basis of K top m ( ) modulo p (see 4.3 and Remark 3 of 4.). Let U α + pk top m ( ) be the subgroup of K top m ( ) topologically generated by elements of the basis distinct from multiples of α. Then a r D r U α. Since { } U α is a basis of open neighbourhoods of 0 with respect to the quotient topology of ν m extended naturally to K top m ( )/D (see Remark 2 of 4.5 and Remark 3 of 4.) we deduce that {a r mod D} is a Cauchy sequence in K top m ( )/D. Since D pktop m ( ) there is a closed subgroup D of K top m ( ) such that D/p D /p = K top m ( )/p. Let y be a limit of the image of {a r } in D /p (which exists since D is complete) then z = {ζ p } y. p s+ )

14 4 I. esenko To prove (iii) note that for l prime to p Theorem 4.6 (i) shows that V Km top ( ) has no nontrivial l -torsion; hence by part (i) l lv K m ( ) is l -divisible. Let z l lv K m ( ). Then z = px for some x V K m ( ). Suppose that a primitive pth root of unity belongs to. Then from the previous part we get x = {ζ p } y + w with w Λ m ( ) V K m ( ). Hence z = pw and Λ m ( ) V K m ( ) is p-divisible. If doesn t contain a primitive pth root, then pass to (ζ p ) and use the norm map argument. (iv) follows from Remark of 4.5. Remark to the proof. Let O be the discrete valuation ring with respect to t n. Using Kurihara s exponential map [25] Ω m O K m ( )/p r K m ( ), whose definition uses the syntomic complex, and the description by Bloch and Kato of U i K m ( ) + p r K m ( )/U i+ K m ( ) + p r K m ( ) for small i one can prove that p r V K m ( ) is equal to the intersection of open subgroups of V K m ( ) which contain p r V K m ( ) in the same way as in the case r =. Corollary. If k 0 is algebraic over p then for every integer l In general K m ( )/l K top m ( )/l. V K m ( )/l V K top m ( )/l. Thus, one can use topological generators of Km top ( )/l to describe the structure of K m ( )/l. In particular, if contains a primitive lth root of unity or l is prime or l is a power of char( ), then all relations between topological generators of V Km top ( )/l are known (as an application of the Artin Schreir Witt Parshin or Vostokov pairing). Remark. Let be of characteristic zero. The l-adic symbol defined in [8] K n ( ) l H n (, Z l (n)) induces the monomorphism V K top n ( ) l H n (, Z l (n)). Remark 2. Let ρ m be the finest topology on K m ( ) for which the map from m to K m ( ) is sequentially continuous and the intersection of all neighbourhoods of zero in K m ( ) contains l lk m ( ) (the topology ρ m was used in [2 3]). Then ρ m is λ m and the intersection of all neighbourhoods of zero in K m ( ) with respect to ρ m coincides with l lk m ( ). On the level of subgroups λ m and ρ m coincide. Remark 3. or another approach to describe the set of open subgroups of K m ( ) see [9]. Remark 4. or char( ) = 0 I. B. Zhukov found (applying higher class field theory) a complete algebraic description of Kn top ( ) in several cases (see [2]). In particular, if T p Kn top ( ) is the topological closure of the p-torsion in Kn top ( ) and has a local parameter t n algebraic over Q p, then V Kn top ( )/T p Kn top ( ) possesses a topological basis of the form {ε, t n,..., t } with ε

15 Sequential topologies and quotients of Milnor K-groups 5 running free Z p -generators of the group of principal units of the algebraic closure of Q p in modulo its p-primary torsion. Remark 5. Let k 0 be finite (resp. infinite and not p-algebraically closed). rom the point of view of higher local class field theory Theorems 4.6 and 4.7 show that the kernel of the reciprocity map (resp. K n ( ) Gal( ab / ) V K n ( ) Hom(G ur,ab,p, Gal(E/ )) where E is a maximal abelian totally ramified p-extension of ), which due to existence theorem [9], [2 3] coincides with the intersection of all open subgroups of finite index in K n ( ) (resp. normic subgroups of V K n ( ) [4, sect.5]), is equal to Λ n ( ) = l lk n ( ) (resp. Λ n ( ) V K n ( ) = l lv K n ( )). Thus, the induced map from Kn top ( ) (resp. from V Kn top ( )) is a monomorphism (with dense image). Remark 6. It is an open problem to describe the torsion in Λ m ( ). or n = see [] (finite k 0 ), [2] (perfect k 0 ) The norm map on topological K-groups. It follows easily from 4. that for a cyclic extension L/ of a prime degree Km top (L) is generated by L and the image of K top m ( ). or an arbitrary multidimensional local field define the norm on K top ( ) as induced from the norm on Milnor K-groups. Using Theorems , Remark 4 of 4.6 and a description of N L/ : L for a cyclic extension of prime degree [2 3] one directly shows that for a finite extension M/ () the image of an open subgroup in K n (M) with respect to the norm map is an open subgroup, (2) the preimage of an open subgroup in K m ( ) with respect to the norm map is an open subgroup. Remark. In characteristic p if k 0 is finite there is a very simple way to define the norm map on topological K-groups [5 6]: (a) for a cyclic extension L/ of a prime degree introduce N L/ : Km top (L) Km top ( ) as induced by the norm on K ; (b) for an arbitrary abelian extension L/ define the norm presenting L/ as a tower of cyclic extensions of prime degree. Correctness of this definition follows from an application of the Artin Schreier Witt Parshin and tame pairings [5]. Compatibility of the just defined norm with induced from the Milnor K-groups follows then from 4.7. Theorem. If L/ is a cyclic extensions of a prime degree l with a generator σ then the sequence K top n n i /L ( σ) ( )/l K top (L)/l K top (L)/l is exact, where i /L is induced by the field embedding. n N L/ K top ( )/l n This theorem is verified by explicit calculations in Kn top /l-groups whose structure is completely known due to an application of the Artin Schreier Witt Parshin, tame and Vostokov pairings (adjoin if necessary a primitive lth root of unity ζ). Similar calculations show that the index of the norm group N L/ Kn top (L) in Kn top ( ) is finite of order L : if the latter is prime and k 0 is finite [2 3].

16 6 I. esenko It is well known that a corollary of this theorem and the description of the torsion of Kn top ( ) of 4.7 is the exactness of the sequence Kn top (L) σ Kn top (L) N L/ K top n ( ) for every cyclic extension L/ of prime degree if k 0 is finite and of degree p in the case of perfect k 0. Acknowledgement. The author thanks I. Zhukov and O. Izhboldin for many questions and interest to this work. The work on this paper was partially supported by a Royal Society grant for collaboration between the UK and Russia. References [] S. Bloch and K. Kato, p-adic etale cohomology, Inst. Hautes Études Sci. Publ. Math. 63(986), [2] I. esenko, Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic, Algebra i Analiz 3, issue 3(99), 65 96; English transl. in St. Petersburg Math. J. 3(992), [3] I. esenko, Multidimensional local class field theory. II, Algebra i Analiz 3, issue 5(99), 68 90; English transl. in St. Petersburg Math. J. 3(992), [4] I. esenko, Abelian local p-class field theory Math. Ann. 30(995), [5] I. esenko, Abelian extensions of complete discrete valuation fields, Number Theory Paris 993/94, Cambridge Univ. Press, 996, [6] I. B. esenko and S. V. Vostokov, Local ields and Their Extensions: A Constructive Approach, AMS, 993. [7] K. Kato, A generalization of local class field theory by using K-groups. I, J. ac. Sci. Univ. Tokyo Sect. IA Math. 26(979), [8] K. Kato, A generalization of local class field theory by using K-groups. II J. ac. Sci. Univ. Tokyo Sect. IA Math. 27(980), [9] K. Kato, The existence theorem for higher local class field theory, Preprint IHES, 980. [0] K. Kato, Galois cohomology of complete discrete valuation fields, in Algebraic K-theory, Part II (Oberwolfach, 980), Lect. Notes in Math. vol 967, 982, , Springer. [] B. Kahn, L anneau de Milnor d un corps local a corps residuel parfait, Ann. Inst. ourier 34(984), [2] A. Merkur ev, On the torsion in K 2 of local fields, Ann. Math. 8(983), [3] A. S. Merkur ev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46(982), 0 046; Engl. transl. in Math. USSR Izvestiya 2(983), [4] A. I. Madunts and I. B. Zhukov, Multidimensional complete fields: topology and other basic constructions, Trudy St. Petersburg Mat. Obshchestva 3, 994; English transl. in Proceed. St. Petersburg Math. Society, AMS Translation Series 2, vol 66, 995, 34. [5] A. N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov., vol. 65, 985, 43 70; English transl. in Proc. Steklov Inst. Math. 985, issue 3, [6] A. N. Parshin, Galois cohomology and Brauer group of local fields, Trudy Mat. Inst. Steklov., vol. 83, 990, 59 69; English transl. in Proc. Steklov Inst. Math. 99, issue 4, [7] A. A. Suslin, Torsion in K 2 of fields, K-theory (987), [8] J. Tate, Relations between K 2 and Galois cohomology, Invent. math. 36(976), [9] S. V. Vostokov, Explicit construction in class field theory of a multidimensional local field, Izv. AN SSSR. Ser. Mat. 49(985), ; English transl. in Math. USSR Izv. 26(986).

17 Sequential topologies and quotients of Milnor K-groups 7 [20] I. B. Zhukov, Structure theorems for complete fields, Trudy St. Petersburg Mat. Obshchestva, vol. 3, 994; English transl. in Proceed. St. Petersburg Math. Society, AMS Translation Series 2, 66(995), [2] I. B. Zhukov, Milnor and topological K-groups of multidimensional complete fields, Algebra i analiz, 9(997) issue, 98 46; English transl. in St. Petersburg Math. J., 9(998), [22] D. Arlettaz, Algebraic K-theory of rings from topological viewpoint, K-theory Preprint Archives, 420, [23] G. Banaszak, Generalization of the Moore exact sequence and the wild kernel for higher K-groups, Compos. Math., 86(993), [24] G. Banaszak, private notes. [25] M. Kurihara, The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law, J. reine angew. Math., 498(998), [26] I. Zhukov, Higher dimensional local fields, in Invitation to higher local fields, Part I, sect., pp. 5 8, Warwick School of Mathematics University of Nottingham Nottingham NG7 2RD England ibf@maths.nott.ac.uk

18 Appendix: On the group K 2 ( )/ l lk 2 ( ) Oleg Izhboldin Abstract. We construct a field containing a primitive pth root and such that p-torsion of the group K m ( )/ l lk m ( ) is not generated by p-torsion in. The method of proof is to work with the field of rational functions of an infinite product of certain Severi Brauer varieties using Merkur ev Suslin s theorem, Suslin s theorem on the torsion in K 2 ( ) and Kahn s theorem. Mathematics Subject Classification (2000). 9C30, 9C99 Key words. Merkur ev Suslin theorem, Severi Brauer variety, generic splitting field.. Introduction Throughout the text we denote by µ m the group of mth roots of unity. or an abelian group A we denote m-torsion points of A by Tors m A. ix a prime integer p. In this appendix is a field of characteristic different from p. In the case p = 2 we will suppose that. Let ζ p i be a primitive p i th root and i = (ζ p i), = i. or a field put s( ) = sup{n : n = }... Definition. Set DK n ( ) = l lk n ( ), D p K n ( ) = i p i K n ( ), K t n( ) = K n ( )/DK n ( ). Clearly, K ( t ) = n=0 Kt n( ) is a graded ring. There is a natural question, for which fields the structure of the torsion subgroup in Kn( t ) is "standard". More precisely, for which fields the condition ζ p n implies that Tors p n Kn( t ) = {ζ p n} Kn t ( )? Section 4 of the previous paper contains a description of Km( t ) = Km top ( ) for a higher local field ; in particular it is shown that the answer to the question in the previous paragraph is positive for higher local fields with finite residue field k 0. The main purpose of this note is to construct fields for which the answer is negative. We prove the following.2. Theorem. Let p be a prime integer. Then there exists a field such that ζ p and Tors p K t 2 ( ) {ζ p, }. The proof is given in 3.4; now we deduce some corollaries. Corollary. Let p be a prime integer and m 2. Then there exists a field such that ζ p and Tors p Km( t ) {ζ p }Km t ( ).