arXiv subjects
Publications and source records attributed to Charles Weibel.
We classify all étale 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.
If I is a nilpotent ideal in a $\mathbb{Q}$-algebra $A$, Goodwillie defined two isomorphisms from $K_*(A,I)$ to negative cyclic homology, $HN_*(A,I)$. One is the relative version of the absolute Chern character, and the other is defined using rational homotopy theory. The question of whether they agree was implicit in Goodwillie's 1986 Annals paper. In this paper, we show that the two isomorphisms agree. Here are three applications. 1.Cathelineau proved that the rational homotopy character is compatible with the $λ$-filtration. It follows that the relative Chern character is also compatible with this filtration for nilpotent ideals. 2.This agreement, together with Cathelineau's result, was used by the authors and Haesemeyer to show that the absolute Chern character, from $K(A)$ to $HN(A)$, is compatible with the $λ$-filtration for every commutative $\mathbb{Q}$-algebra. This is the main result of Infinitesimal cohomology and the Chern character to negative cyclic homology, arXiv:math/0703133v1. 3.This agreement can be used to strengthen Ginot's results in "Formules explicites pour le charactere de Chern en $K$-théorie algébrique", Ann. Inst. Fourier 54 (2004).
Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.
Let V be a smooth variety defined over the real numbers. Every algebraic vector bundle on V induces a complex vector bundle on the underlying topological space V(C), and the involution coming from complex conjugation makes it a Real vector bundle in the sense of Atiyah. This association leads to a natural map from the algebraic K-theory of V to Atiyah's ``Real K-theory'' of V(C). Passing to finite coefficients Z/m, we show that the maps from K_n(V ; Z/m) to KR ^{-n}(V(C);Z/m) are isomorphisms when n is at least the dimension of V, at least when m is a power of two. Our key descent result is a comparison of the K-theory space of V with the homotopy fixed points (for complex conjugation) of the K-theory space of the complex variety V(C). When V is the affine variety of the d-sphere S, it turns out that KR*(V(C))=KO*(S). In this case we show that for all nonnegative n we have K_n(V ; Z/m) = KO^{-n}(S ; Z/m).
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.
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.