I need the theorems that prove that Nisnevich cohomology can be computed by the Cech complex similar to what happens in the étale topology. I know this has to be true since (More …

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or …

Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at: http://imrn.oxfordjournals.org/conten …

I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between scheme …

In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_ …

Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are t …

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not. STATEMENT: Let $X$ be a smooth and projective scheme ov …

For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see http://books.google.ru/books?id=TzUmk87bN9c …

I will begin by giving a rough sketch of my understanding of motives. In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motiv …

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"): $E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} …

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension f …

Voevodsky reformulated the Milnor-Bloch-Kato conjecture as a change-of-topology morphism from the Zariski to the étale topology of a field with torsion coefficients being an isomor …

As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is co …

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{ …

I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, ove …