# Two-primary algebraic K-theory of two-regular number fields

Abstract: We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order.

Topics: Quadratic field (69%), Quadratic integer (66%), Tensor product of fields (66%), Field (mathematics) (65%), Algebraic integer (64%)

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01 Jan 2013-

Abstract: Projective modules and vector bundles The Grothendieck group $K_0$ $K_1$ and $K_2$ of a ring Definitions of higher $K$-theory The fundamental theorems of higher $K$-theory The higher $K$-theory of fields Nomenclature Bibliography Index

225 citations

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Abstract: We relate the algebraic K-theory of a totally real number field F to its étale cohomology. We also relate it to the zeta-function of F when F is Abelian. This establishes the two-primary part of the “Lichtenbaum conjectures.” To do this we compute the two-primary K-groups of F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory. Introduction In the early 1970’s, Lichtenbaum [L1, L2] made several distinct conjectures about the relation between the algebraic K-theory, étale cohomology and zeta function of a totally real number field F . This paper confirms Lichtenbaum’s conjectural connection between the two-primary K-theory and étale cohomology of F , and (when Gal(F/Q) is Abelian) to the zeta function. Up to a factor of 21 , we obtain the relationship conjectured by Lichtenbaum in [L2, 2.4 and 2.6]. In the special case F = Q, this result was obtained in [W3]. Our methods depend upon the recent spectacular results of Voevodsky [V2], Suslin and Voevodsky [SV], and Bloch and Lichtenbaum [BL]. Together with Appendix B to this paper, they yield a spectral sequence, starting with the étale cohomology of any field of characteristic zero and converging to its 2-primary Ktheory. For number fields, this is essentially the spectral sequence whose existence was conjectured by Quillen in [Q4]. The main technical difficulties with this spectral sequence, overcome in this paper, are that it does not degenerate at E2 when F has a real embedding, and that it has no known multiplicative structure. To describe our result we introduce some notation. If A is an Abelian group, we let A{2} denote its 2-primary torsion subgroup, and let #A denote its order when A is finite. We write Kn(R) for the nth algebraic K-group of a ring R, and H ét(R;M) for the nth étale cohomology group of Spec(R) with coefficients in M . Theorem 0.1. Let F be a totally real number field, with r1 real embeddings. Let R = OF [ 1 2 ] denote the ring of 2-integers in F . Then for all even i > 0 21 · #K2i−2(R){2} #K2i−1(R){2} = #H ét(R; Z2(i)) #H ét(R; Z2(i)) . (It is well known that all groups appearing in this formula are finite.)

120 citations

### Cites methods from "Two-primary algebraic K-theory of t..."

...Simple examples of totally real elds with r1 =2s uch that K 8k+4(R) has no 2torsion (so = j = 0) are given by Rognes and stvr in [ RO ]....

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01 Jan 2005-

Abstract: The problem of computing the higher K-theory of a number field F , and of its rings of integers OF , has a rich history. Since 1972, we have known that the groups Kn(OF ) are finitely generated [48], and known their ranks [7], but have only had conjectural knowledge about their torsion subgroups [33, 34, 5] until 1997 (starting with [76]). The resolutions of many of these conjectures by Suslin, Voevodsky, Rost and others have finally made it possible to describe the groups K∗(OF ). One of the goals of this survey is to give such a description; here is the odd half of the answer (the integers wi(F ) are even, and are defined in section 3):

58 citations

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Abstract: Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic Kgroups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-groups, analogous to orthogonal and symplectic topological K-theory. The proofs use in an essential way higher KSC -theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitian K-groups in terms of higher algebraic K-groups. We also relate periodicity to etale hermitian K-groups by proving a hermitian version of Thomason’s etale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

9 citations

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01 Jan 2003-

Abstract: We identify the 2-adic homotopy type of the algebraic K-theory space for rings of integers in two-regular exceptional number fields. The answer is given in terms of well-known spaces considered in topological K-theory.

7 citations

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380 citations

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Abstract: Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:

336 citations

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Abstract: We relate the algebraic K-theory of a totally real number field F to its étale cohomology. We also relate it to the zeta-function of F when F is Abelian. This establishes the two-primary part of the “Lichtenbaum conjectures.” To do this we compute the two-primary K-groups of F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory. Introduction In the early 1970’s, Lichtenbaum [L1, L2] made several distinct conjectures about the relation between the algebraic K-theory, étale cohomology and zeta function of a totally real number field F . This paper confirms Lichtenbaum’s conjectural connection between the two-primary K-theory and étale cohomology of F , and (when Gal(F/Q) is Abelian) to the zeta function. Up to a factor of 21 , we obtain the relationship conjectured by Lichtenbaum in [L2, 2.4 and 2.6]. In the special case F = Q, this result was obtained in [W3]. Our methods depend upon the recent spectacular results of Voevodsky [V2], Suslin and Voevodsky [SV], and Bloch and Lichtenbaum [BL]. Together with Appendix B to this paper, they yield a spectral sequence, starting with the étale cohomology of any field of characteristic zero and converging to its 2-primary Ktheory. For number fields, this is essentially the spectral sequence whose existence was conjectured by Quillen in [Q4]. The main technical difficulties with this spectral sequence, overcome in this paper, are that it does not degenerate at E2 when F has a real embedding, and that it has no known multiplicative structure. To describe our result we introduce some notation. If A is an Abelian group, we let A{2} denote its 2-primary torsion subgroup, and let #A denote its order when A is finite. We write Kn(R) for the nth algebraic K-group of a ring R, and H ét(R;M) for the nth étale cohomology group of Spec(R) with coefficients in M . Theorem 0.1. Let F be a totally real number field, with r1 real embeddings. Let R = OF [ 1 2 ] denote the ring of 2-integers in F . Then for all even i > 0 21 · #K2i−2(R){2} #K2i−1(R){2} = #H ét(R; Z2(i)) #H ét(R; Z2(i)) . (It is well known that all groups appearing in this formula are finite.)

120 citations

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66 citations

### "Two-primary algebraic K-theory of t..." refers background in this paper

...A number field F is called2-regular [ GJ ] if the 2-Sylow subgroup R2(F) of the kernel in K2(F) of the regular symbols attached to the non-complex places of F , is trivial....

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