Homotopy Ends and Thomason Model Categories

In the last year of his life, Bob Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. In this paper we explain and prove Thomason's results, based on his private notebooks. The first half presents Thomason's ideas about homotopy ends and its generalizations. This material may be of independent interest. Then we define Thomason model categories and give some examples. The usual proof shows that the homotopy category exists. In the last two sections we prove the main theorem: functor categories inherit a Thomason model structure, at least when the original category is enriched over simplicial sets and fibrations are preserved by limits.Comment: 39 pages, AMS-TeX file using picte

Some surfaces of general type for which Bloch's conjecture holds

We give many examples of surfaces of general type with $p_g=0$ for which Bloch's conjecture holds, for all values of $K^2$ except 9. Our surfaces are equipped with an involution

Cotensor products of modules

Let C be a coalgebra over a field k and A its dual algebra. The category of C-comodules is equivalent to a category of A-modules. We use this to interpret the cotensor product M \square N of two comodules in terms of the appropriate Hochschild cohomology of the A-bimodule M \otimes N, when A is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.Comment: 16 pages, LaTe

Relative Cartier divisors and K-theory

We study the relative Picard group $Pic(f)$ of a map $f:X\to S$ of schemes. If $f$ is faithful affine, it is the relative Cartier divisor group $I(f)$. The relative group $K_0(f)$ has a $\gamma$-filtration, and $Pic(f)$ is the top quotient for the $\gamma$-filtration. When $f$ is induced by a ring homomorphism $A\to B$, we show that the relative "nil" groups $NPic(f)$ and $NK_n(f)$ are continuous $W(A)$-modules

Slices of Co-Operations for KGLKGL

We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic $K$-theory.Comment: 20 page

Twisted K-theory, Real A\mathcal{A}-bundles and Grothendieck-Witt groups

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted $K$-theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties

Unstable operations in \'etale and motivic cohomology

We classify all \'etale cohomology operations on H_\et^n(-,\muell{i}), showing that they were all constructed by Epstein. We also construct operations $P^a$ on the mod-$\ell$ motivic cohomology groups $H^{p,q}$, differing from Voevodsky's operations; we use them to classify all motivic cohomology operations on $H^{p,1}$ and $H^{1,q}$ and suggest a general classification.Comment: 26 page

Principal ideals in mod-ℓ\ell Milnor KK-theory

Fix a symbol $\underline{a}$ in the mod-$\ell$ Milnor $K$-theory of a field $k$, and a norm variety $X$ for $\underline{a}$. We show that the ideal generated by $\underline{a}$ is the kernel of the $K$-theory map induced by $k\subset k(X)$ and give generators for the annihilator of the ideal. When $\ell=2$, this was done by Orlov, Vishik and Voevodsky.Comment: 14 page

K-theory of line bundles and smooth varieties

We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.Comment: 11 page

Norm Varieties and the Chain Lemma (after Markus Rost)

The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,\mu_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rost's 1998 preprints, his web site and Rost's lectures at the Institute for Advanced Study in 1999-2000 and 2005