All Questions
27 questions
17
votes
2
answers
1k
views
Hodge standard conjecture for étale cohomology
It is known that Hodge standard conjecture is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\...
- 797
16
votes
1
answer
3k
views
Tate twists and cohomology of $\mathbf{P}^1$
I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to ...
user113452
15
votes
2
answers
2k
views
Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...
- 5,504
12
votes
0
answers
811
views
Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
user113452
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
1k
views
motivic t-structure and realisations
Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $
\mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...
- 81
8
votes
1
answer
1k
views
Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?
$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$
Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...
- 16.9k
8
votes
1
answer
1k
views
If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?
Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...
- 16.9k
8
votes
0
answers
333
views
Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
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
8
votes
0
answers
603
views
A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures
Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite.
A weakening of this conjecture states that the $\ell$-...
- 15.4k
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
7
votes
2
answers
775
views
actions of the absolute Galois group and the motivic Galois group on étale cohomology
Let $K$ be a field of characteristic $0$; let $\ell$ be any prime; and let $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ be a Tannakian category of motives over $K$ with coefficients in $\mathbb{Q}_{\ell}$. So,...
- 1,298
7
votes
1
answer
512
views
Is there any theory of "étale cohomology" with algebraic coefficients?
For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...
- 371
6
votes
1
answer
302
views
Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus
Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...
- 7,952
6
votes
1
answer
525
views
Functoriality for $\ell$-adic cohomology - a question
This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...
user134132
5
votes
1
answer
348
views
Spectral sequence in Betti cohomology
Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X_{\rm an}\to X_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $...
user119470
5
votes
0
answers
513
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
0
answers
426
views
In which "sense" unramified Milnor-Witt K-groups are unramified
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
- 6,028
4
votes
0
answers
342
views
Voevodsky's motives and Deligne's systems of realizations
$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...
- 15.4k
4
votes
0
answers
265
views
Explicit linear object underlying $l$-adic cohomology for almost all $l$
If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...
user158636
4
votes
0
answers
306
views
Etale cohomology of projective spaces in the rigid analytic setting
Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
3
votes
0
answers
307
views
Semisimplicity conjecture
In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...
user92332
2
votes
0
answers
245
views
What unramified Galois representations come from geometry?
I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
user158636
1
vote
2
answers
392
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 16.9k
1
vote
0
answers
290
views
Coniveau in étale motivic cohomology
Let $X$ be a smooth variety over a field.
Is there a spectral sequence:
$$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...
user92332