The celebrated Lichtenbaum-Quillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓ-adic algebraic K-groups of X are closely related to the ℓ-adic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral sequence of Atiyah-Hirzebruch type H p q ét (X; Zℓ ()) ⇒ πq−p(KX 2 ℓ) but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the ℓ-adic integers, and are to be interpreted as zero if q is odd. Throughout this paper, étale cohomology is continuous étale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield ℓ-adic completion of the spectrum KX, not the naive ℓ-adic completion of the K-groups. In a remarkable paper [42], Thomason proved the Lichtenbaum-Quillen conjectures for a certain localized form of K-theory- so-called “Bott-periodic” K-theory. The first step was the development of an elaborate theory of hypercohomology spectra H ·(X; E) associated to étale presheaves of spectra E- or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally Supported by a grant from the National Science Foundation 1 equipped with a suitable descent spectral sequence, and there is a natural map E(X)−→H · (X; E). In particular, the K-theory of a scheme X maps to its associated hypercohomology spectrum H · ét(X; K). Fix a prime ℓ, which we will assume is odd in order to simplify the discussion.Let L(−) denote Bousfield localization with respect to complex K-theory, and let ˆ L(−) denote its ℓ-adic completion. The main theorem of [42] can be stated as follows. Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull dimension, with sufficiently nice residue fields of characteristic different from ℓ. Then the natural map KX−→H · ét(X; K) induces a weak equivalence

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

Abstract not found

this paper is to extend the results of the third author to this wider context. Theorem 4.1 asserts that after localization at ` the maps

Abstract. We survey briefly some of the K-theoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic K-theory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain

the prime 2

Abstract. In this paper we verify the strong Quillen–Lichtenbaum conjecture for integers in real number fields at the prime two. That is, we prove that the Dwyer– Friedlander map from mod two algebraic K–theory to mod two étale topological K–theory is a weak equivalence on zero–connected covers for two–integers in real number fields. The proof is given by comparing two explicit calculations. Contents

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. 1. Introduction. Let E be a number field, OE its ring of algebraic integers, and RE = OE [ 1] the

Abstract. This paper gives a short and purely homotopical proof of the triviality of the product map K3(Z) ⊗ K1(Z) → K4(Z). The argument is based on the study of the external product for the homotopy groups of the K-theory spectrum and the fact that the 4-dimensional Hurewicz homomorphism in the algebraic K-theory of Z is an isomorphism. The purpose of this short note is to provide a simple proof of the following assertion concerning the 4-th algebraic K-group of the ring of integers Z. Theorem. The product map K3(Z) ⊗ K1(Z) − → K4(Z) is trivial. Recall that J. Rognes detected recently a gap in his proof of the vanishing of K4(Z). Consequently, it may be interesting to consider the product of elements of K3(Z) ∼ = Z/48 with elements of K1(Z) ∼ = Z/2 in K4(Z). Notice that the statement of the above theorem could be deduced from Bökstedt’s result on the factorization of the algebraic K-theory space BGL(Z) + after completion at the prime 2: (BGL(Z)+) ̂ 2 ≃ JK(Z) ̂ 2 × ? (see [B]). However, C. Soulé asked me during the Poznań conference on algebraic K-theory for a direct proof of the theorem using only topological arguments. It will be an immediate consequence of Propositions 3 and 5 below. Throughout the paper, we shall consider all homology groups with coefficients in Z and use the following notation: for any abelian group G, let us write K(G, n) for the Eilenberg-MacLane space whose n-th homotopy group is G and HG for the Eilenberg-MacLane spectrum whose 0-th homotopy group is G; for a CWcomplex or a spectrum X and an integer n, αn: X → X[n] denotes the n-th Postnikov section of X, i.e., αn induces an isomorphism on the i-th homotopy group for i ≤ n and πiX[n] = 0 for i> n. Lemma 1. Let S denote the sphere spectrum, X any spectrum, n an integer and m a positive integer. Then the image of the external product ∧ : πnX ⊗ πmS → πn+mX

. This paper gives a short and purely homotopical proof of the triviality of the product map K3 (Z)\Omega K1 (Z) ! K4 (Z) . The argument is based on the study of the external product for the homotopy groups of the K-theory spectrum and the fact that the 4-dimensional Hurewicz homomorphism in the algebraic K-theory of Z is an isomorphism. The purpose of this short note is to provide a simple proof of the following assertion concerning the 4-th algebraic K-group of the ring of integers Z . Theorem. The product map K 3 (Z)\Omega K 1 (Z) \Gamma! K 4 (Z) is trivial. Recall that J. Rognes detected recently a gap in his proof of the vanishing of K 4 (Z) . Consequently, it may be interesting to consider the product of elements of K 3 (Z) ¸ = Z=48 with elements of K 1 (Z) ¸ = Z=2 in K 4 (Z) . Notice that the statement of the above theorem could be deduced from Bokstedt's result on the factorization of the algebraic K-theory space BGL(Z) + after completion at the prime 2 : (BGL(Z) ...