All Questions
Tagged with motivic-cohomology etale-cohomology
17 questions
16
votes
1
answer
1k
views
Motivic cohomology with finite coefficients for singular varieties
Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in ...
- 15.6k
11
votes
1
answer
917
views
Motivic cohomology and pushforward maps
I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel "...
- 37k
8
votes
1
answer
426
views
When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?
For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For $...
- 16.9k
8
votes
0
answers
574
views
Reference request: Motivic Cohomology and Cycle class maps
For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
- 592
7
votes
1
answer
689
views
Two motivic complexes, compared
Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
user120812
5
votes
0
answers
167
views
Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra
By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...
- 5,901
5
votes
0
answers
512
views
Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
user92332
4
votes
1
answer
249
views
Suspension Theorem in $\mathbb{A}^1$-homotopy
In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...
- 4,418
4
votes
0
answers
255
views
On (the cohomology of) Hensel pairs
I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$.
I would like the following facts to be true (and to make sense!:)).
a.) The motivic ...
- 16.9k
3
votes
0
answers
205
views
Beilinson-Lichtenbaum conjecture for algebraic extensions of $\mathbb{Z}/m$
Let $X$ be smooth over some field $k$ and $m\in\mathbb{Z}$ so that $m$ maps to a unit in $k^{\times}$. By Beilinson-Lichtenbaum one has an isomorphism of cohomology groups
\begin{equation*}
\...
- 1,805
3
votes
0
answers
168
views
Symmetrical monoidal $2$-category of cohomological correspondences
My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature.
Let me be more precise about what I mean.
...
- 2,923
3
votes
0
answers
178
views
Regulator maps for ordinary varieties
Let $K$ be a finite extension of $\mathbf{Q}_p$ and $\mathcal{X}$ a smooth proper scheme over the ring of integers $\mathcal{O}_K$. For $i, j$ integers with $i \ne 2j$, there's a regulator map
$$ H^i_{...
- 37k
2
votes
1
answer
615
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}} H_{ Y_{i_{0} \cdots ...
- 945
2
votes
0
answers
309
views
representability of étale and motivic cohomology by schemes
One has $\mathrm{H}^1_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^2(X,\mathbf{Z}(1)) = \mathrm{Pic}(X)$ (étale and motivic cohomology).
Is étale or motivic cohomology in other dimensions also ...
user19475
1
vote
0
answers
179
views
Interpretation of Tate conjecture using motivic homotopy
For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...
- 679
1
vote
0
answers
566
views
Homotopy theory of schemes
I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...
- 33
1
vote
0
answers
232
views
What can one say about a smooth variety whose lower cohomology is trivial?
Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...
- 16.9k