John Rognes

TWO-PRIMARY ALGEBRAIC K-THEORY OF RINGS OF INTEGERS IN NUMBER FIELDS

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 2-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 K-theory. 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 [ 12 ] 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)) #H1 ét(R; Z2(i)) .

Galois extensions of structured ring spectra ; Stably dualizable groups

Galois Extensions of Structured Ring Spectra: Abstract Introduction Galois extensions in algebra Closed categories of structured module spectra Galois extensions in topology Examples of Galois extensions Dualizability and alternate characterizations Galois theory I Pro-Galois extensions and the Amitsur complex Separable and etale extensions Mapping spaces of commutative

Algebraic K-theory of topological K-theory

S

Topology, Geometry and Quantum Field Theory: Two-vector bundles and forms of elliptic cohomology

-algebras Galois theory II Hopf-Galois extensions in topology References Stably Dualizable Groups: Abstract Introduction The dualizing spectrum Duality theory Computations Norm and transfer maps References Index.

The smooth Whitehead spectrum of a point at odd regular primes

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are

Two-primary algebraic K-theory of pointed spaces

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

K 4 (Z) is the trivial group

Let p be an odd regular prime, and assume that the Lichtenbaum{Quillen conjecture holds for K(Z[1=p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh () is described. A suspended copy of the cokernel-of-J spectrum splits o, and the torsion homotopy of the remainder equals the torsion homotopy of the ber of the restricted S 1 -transfer map t : C P 1 ! S. The homotopy groups of Wh () are determined in a range of degrees, and the cohomology of Wh () is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or dieomorphisms of highly connected, high dimensional compact smooth manifolds.

Algebraic K-theory of the two-adic integers

We compute the mod 2 cohomology of Waldhausen’s algebraic K-theory spectrum A(∗) of the category of .nite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh Di1 (∗). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions ∗ 6 18, and up to extensions in dimensions 19 6 ∗ 6 21. As applications we show that the linearization map L:A(∗) → K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher– Waldhausen map hw:G=O → � Wh Di1 (∗) does not admit a four-fold delooping, and there is a 2-complete spectrum map M : Wh Di1 (∗) → �g=o ⊕ which is precisely 9-connected. Here g=o⊕ is a spectrum whose underlying space has the 2-complete homotopy type of G=O. ? 2002 Elsevier Science Ltd. All rights reserved.

Hopf algebra structure on topological Hochschild homology

Abstract We prove that the fourth algebraic K -group of the integers is the trivial group, i.e., that K 4 ( Z )=0 . The argument uses rank-, poset- and component filtrations of the algebraic K -theory spectrum K( Z ) from Rognes (Topology 31 (1992) 813–845; K -Theory 7 (1993) 175–200), and a group homology computation of H 1 (SL 4 ( Z ); St 4 ) from Soule, to compute the odd primary spectrum homology of K( Z ) in degrees ⩽4. This shows that the odd torsion in K 4 ( Z ) is trivial. The 2-torsion in K 4 ( Z ) was shown to be trivial in Rognes and Weibel (J. Amer. Math. Soc., to appear).

Topological cyclic homology of the integers at two

We compute the two-completed algebraic K-groups K∗(Z2)2∧ of the two-adic integers, and determine the homotopy type of the two-completed algebraic K-theory spectrum K(Z2)2∧. The natural map K(Z)2∧ → K(Z2)2∧ is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.