Questions tagged [etale-cohomology]

for questions about etale cohomology of schemes, including foundational material and applications.

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22 votes
2 answers
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Etale cohomology with coefficients in the integers

Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\...
Benjamin Antieau's user avatar
41 votes
2 answers
9k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
Qiaochu Yuan's user avatar
16 votes
0 answers
848 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
Tom163's user avatar
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12 votes
1 answer
2k views

Semisimplicity of Frobenius operation on etale cohomology?

Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$. Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? ...
Jan Weidner's user avatar
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45 votes
3 answers
5k views

"Cute" applications of the étale fundamental group

When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
Libli's user avatar
  • 7,210
32 votes
2 answers
2k views

Etale cohomology can not be computed by Cech

It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
SashaP's user avatar
  • 7,027
29 votes
4 answers
8k views

Etale cohomology and l-adic Tate modules

$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$ Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
Jonah Sinick's user avatar
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27 votes
2 answers
2k views

Etale site is useful - examples of using the small fppf site?

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here: Points in sites (etale, fppf, ... ) There, ...
LMN's user avatar
  • 3,525
25 votes
1 answer
2k views

Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
Jim Humphreys's user avatar
22 votes
2 answers
2k views

Comparing cohomology over ${\mathbb C}$ and over ${\mathbb F}_q$

I have the following (probably well-known) question: let $X$ be a regular scheme over $\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$. Let also $X_{\mathbb C}$...
Alexander Braverman's user avatar
19 votes
5 answers
4k views

Equivalent statements of the Riemann hypothesis in the Weil conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
Brandon Levin's user avatar
16 votes
2 answers
1k views

Cohomological dimension-doubling

I'm sure this is a question which has been asked many times, if not necessarily on this site: Why does a (smooth, projective) scheme over a field, with dimension d, behave as though it were a ...
Ryan Reich's user avatar
  • 7,173
15 votes
2 answers
1k views

Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...
Tom Lovering's user avatar
14 votes
1 answer
7k views

A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
Mikhail Bondarko's user avatar
12 votes
2 answers
2k views

Étale cohomology of morphism whose fibers are vector spaces

Let $X\rightarrow Y$ be a morphism (may not be smooth) of varieties such that the fibres are vector spaces. Are the $l$-adic cohomologies of $X$ and $Y$ equal? If not, under what condition (other ...
user avatar
11 votes
1 answer
1k views

Etale cohomology of localizations of henselian rings

Let $R$ be a ring (say noetherian of finite Krull dimension, possibly with additional hypotheses) henselian along the ideal $(p)$, and let $\hat{R}$ be the $p$-adic completion. Is it true that the ...
Akhil Mathew's user avatar
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11 votes
1 answer
568 views

“Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$. Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
user avatar
10 votes
2 answers
2k views

Is there a "universal" cohomology theory for varieties over p-adic fields?

Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, \mathbb{...
David Loeffler's user avatar
10 votes
2 answers
2k views

locally constant constructible sheaves and finite etale coverings

Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way. We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
Lei's user avatar
  • 304
9 votes
1 answer
411 views

Is there a Poincare residue in characteristic $p$?

The Poincare residue I mean is there one here: https://en.wikipedia.org/wiki/Poincar%C3%A9_residue Basically, I would like a nice way to use a meromorphic $n$-form on $\mathbf{P}^n_{\mathbf{F}_p}$ ...
Kim's user avatar
  • 4,034
9 votes
1 answer
2k views

Under what conditions is the induced map of etale fundamental groups surjective?

Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
Yellow Pig's user avatar
  • 2,490
8 votes
1 answer
940 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 ...
user's user avatar
  • 81
8 votes
2 answers
1k views

Group cohomology of fundamental group of a curve

The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this ...
User0829's user avatar
  • 1,378
7 votes
1 answer
785 views

Etale and Algebraic K-theory with rational coefficients

We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
user97282's user avatar
6 votes
1 answer
346 views

Adèlic points and algebraic closure

Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$. Let $F/K$ vary over all finite Galois number ...
user avatar
6 votes
0 answers
868 views

Étale cohomology with support and functoriality

Suppose we have a scheme $X$ and a closed subscheme $Z$, with complement $U$. Then, for any étale sheaf $F$ on $X$, we get a long exact sequence in cohomology $\cdots H^i(X,F) \to H^i(U,F) \to H^{...
Martin Bright's user avatar
6 votes
1 answer
970 views

Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$?

Let $F$ be a $p$-adic field. Let $(G_{F}, W_{F}, I_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$. Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne p$...
jmc's user avatar
  • 5,434
5 votes
1 answer
400 views

Does the étale topos determine the Hodge numbers?

Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the ...
user avatar
5 votes
1 answer
1k views

Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology? Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$. What if $G = \...
user12832's user avatar
  • 407
5 votes
1 answer
2k views

The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...
Mikhail Bondarko's user avatar
5 votes
1 answer
519 views

Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves

I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...
Mikhail Bondarko's user avatar
5 votes
1 answer
389 views

On universally closed morphisms of reduced schemes

In this question I'd like to examine some properties of universally closed morphisms. The question is self-contained. It can also be seen as a follow-up to this question. Let $R$ be a discrete ...
user avatar
5 votes
1 answer
650 views

Fiberwise acyclic, locally acyclic morphisms

The quick definition of a map $f \colon X \to B$ of schemes being acyclic is that the natural unit of adjunction $\def\id{\operatorname{id}}\id \to f_* f^*$ is an isomorphism, where we take $f_*$ to ...
Ryan Reich's user avatar
  • 7,173
4 votes
0 answers
264 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. ...
user avatar
4 votes
1 answer
238 views

$l$-dependence of the group of homologically zero cycles

Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
SashaP's user avatar
  • 7,027
4 votes
2 answers
493 views

An $\ell$-adic local system which is trivial on every fiber of a morphism

Let $f: X \to Y$ be a morphism with connected fibers, where $X, Y$ are smooth algebraic varieties (I am specifically interested in the case when $f$ is a Zariski locally trivial fibration with ...
Yellow Pig's user avatar
  • 2,490
4 votes
1 answer
273 views

Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
user avatar
4 votes
0 answers
380 views

Comparison of complex and p-adic Hodge structure

Let $X$ be a smooth projective variety over a p-adic local field $K$, and let $\bar{K}$ be the algebraic closure of $K$. Fix an isomorphism $\sigma:\bar{K}\to\mathbb{C}$. Do $\sigma$ induces an (iso)...
Bonbon's user avatar
  • 806
4 votes
2 answers
842 views

Twisted forms and $\check{H}^1$

I am reading Milne's Étale cohomology, III.4. A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in ...
Dima Sustretov's user avatar
3 votes
0 answers
165 views

Cycle maps as edge maps

Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space. The exponential sequence on $X$: $$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...
user avatar
3 votes
1 answer
410 views

Characterization of étale locally constant sheaves over a normal scheme

I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156: Assume $X$ be a normal & connected scheme with generic point $g: \eta \to X$. Then ...
user267839's user avatar
  • 5,948
3 votes
1 answer
465 views

Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...
lime's user avatar
  • 85
2 votes
0 answers
158 views

Interpretation of some maps involving cohomology groups

I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
oleout's user avatar
  • 865
2 votes
1 answer
557 views

commutative diagram with Yoneda pairing, Weil pairing and edge morphism

Why does the following diagram commute?$\require{AMScd}$ \begin{CD} H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\ H^1(...
user avatar
1 vote
0 answers
160 views

Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$ is an inductive system ...
user267839's user avatar
  • 5,948
0 votes
0 answers
125 views

A stalk criterion for unit map to be an isomorphism on étale site

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
user267839's user avatar
  • 5,948
0 votes
0 answers
265 views

Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the Weil Conjecture by Freitag/Kiehl)

I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15. It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. ...
user267839's user avatar
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