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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*} \...
The Thin Whistler's user avatar
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 ...
TCiur's user avatar
  • 679
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. ...
gdb's user avatar
  • 2,923
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 ...
user127776's user avatar
  • 5,901
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 ...
curious math guy's user avatar
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 ...
Jesse Silliman's user avatar
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) :...
user avatar
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\...
user avatar
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 ...
user avatar
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 "...
David Loeffler's user avatar
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 $...
Mikhail Bondarko's user avatar
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_{...
David Loeffler's user avatar
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. ...
None's user avatar
  • 33
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 ...
Hiro's user avatar
  • 945
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 ...
Mikhail Bondarko's user avatar
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 ...
Mikhail Bondarko's user avatar
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 ...
Leonid Positselski's user avatar