Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents

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 not found

Abstract. We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove a conditional Gersten resolution, which implies that Hi (Z(n)) = 0 for i> n and that there is a Gersten resolution for Hi (Z/pr(n)), if the residue characteristic is p. We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture, an identification Z/m(n)ét ∼ = µ ⊗n m, for m invertible on the scheme, and a Gersten resolution with (arbitrary) finite coefficients. Over a discrete valuation ring of mixed characteristic (0, p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasi-isomorphism provided the Bloch-Kato conjecture holds. 1.

Introduction One of the central problems of algebraic K-theory is to compute the K-groups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = Spec R, then the homology groups of the zero-th space 1 KX are of interest since they are the homology groups of the innite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex K-theory is another important invariant. Let K denote the periodic complex K-theory spectrum, and let ^ K denote its Bouseld `-adic completion

In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic K-theory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the p-adic K-theory and topological cyclic homology agree in non-negative degrees, [20]. This has been

the prime 2

this paper; in effect we work entirely in the category H

In this paper we combine ideas of Soulé [23] and Deninger [5, 6] to prove a p-adic analogue of Beilinson’s conjectures for motives associated to Hecke characters of imaginary quadratic fields. Let E be an elliptic curve defined over an imaginary quadratic field K with

Abstract not found