All Questions
14 questions
1
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138
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Syntomic f-cohomology for open varieties
Syntomic cohomology $H^{i+j}_{\mathrm{syn}}(X,n)$ of a proper variety $X$ with good reduction over a $p$-adic field $K$ is computed via a spectral sequence in terms of $H^i_{\mathrm{f}}(G_K;H^j_{\...
- 15.4k
8
votes
0
answers
333
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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
6
votes
1
answer
487
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Reference request: good reduction equivalent to crystalline étale cohomology
Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
- 785
1
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0
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125
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Is the Frobenius semisimple on the de-Rham cohomology?
Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
- 785
1
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0
answers
215
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$p$-adic étale cohomology group of open smooth varieties
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$.
Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...
- 349
5
votes
0
answers
387
views
Calculating étale fundamental groups from the usual fundamental group
$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...
- 541
8
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0
answers
433
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Intuition for de Rham comparison theorem in $p$-adic Hodge theory
The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...
- 1,949
3
votes
0
answers
232
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$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
user158636
1
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0
answers
272
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$p$-adic Galois representation and Étale homology
Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...
- 4,428
14
votes
2
answers
2k
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Is there a version of algebraic de Rham cohomology that can be used to calculate torsion classes?
Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.
Say $X$ is an algebraic ...
- 1,347
2
votes
0
answers
232
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Berthelot’s comparison theorem and functoriality
Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
- 181
5
votes
0
answers
659
views
Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory
Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
- 806
3
votes
0
answers
518
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Comparison theorem between étale and de Rham cohomology for local system
This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles"
Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a rational representation of $G$ (I guess it means ...
- 31
5
votes
0
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278
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Tate's conjecture and symmetry of Hodge-Tate weights
I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...
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