0
votes
0answers
59 views
Reference for Cech cohomology for Nisnevich topology
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 …
8
votes
1answer
287 views
Motivic cohomology and cohomology of Milnor K-theory sheaf
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 …
4
votes
1answer
338 views
Voevodsky’s proof in any characteristic (for motivic and Chow)
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 …
3
votes
0answers
97 views
Correspondences of schemes induced by a finite Galois extension
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 …
1
vote
1answer
123 views
Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$
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}_ …
10
votes
1answer
317 views
Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?
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 …
0
votes
0answers
151 views
Vanishing of motivic cohomology with finite coefficients in negative degrees
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 …
2
votes
0answers
247 views
Voevodsky’s ‘split standard triple’ argument: an explanation; does it work with $Z/nZ$-coefficients?
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 …
25
votes
1answer
817 views
What is the relationship between motivic cohomology and the theory of motives?
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 …
1
vote
1answer
165 views
Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?
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}} …
5
votes
2answers
255 views
When do the $\gamma$-filtration and codimension filtration of K-theory agree?
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 …
3
votes
2answers
679 views
Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture
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 …
10
votes
1answer
497 views
Motivic cohomology vs. K-theory for singular varieties
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 …
8
votes
2answers
842 views
State of the art for Gersten’s conjecture for K-theory?
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^{ …
3
votes
0answers
399 views
Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?
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 …

