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Questions tagged [etale-cohomology]

For questions on the étale cohomology groups of an algebraic variety or scheme, algebraic analogues of the usual cohomology groups with finite coefficients of a topological space.

2
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1answer
52 views

Constant sheaves on the étale site of a scheme

I am learning étale cohomology with Tamme's book. When talking about example of abelian sheaves on the étale site, he mentions the following equality for an abelian group $A$ : let $A_X$ be the ...
2
votes
1answer
75 views

The splitting of Galois representations

Suppose $X$ is a smooth projective variety defined over a number field $K$, then the etale cohomology $H^i_{et}(X,\mathbb{Q}_\ell)$ defines a continuous representation of the absolute Galois group $\...
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3answers
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Kummer sequence etale topology

Consider the category $C=Sch/S$ of schemes over $S$ and let $n \in \Gamma(S,\mathcal{O}_S)^{*}$. It is possible to show that $$0 \rightarrow \mu_{n,S} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m ...
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0answers
59 views

Is there a fiber bundle/espace-etale interpretation of sheaves on a Grothendieck site

Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to ...
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0answers
16 views

A decomposition property of constructible sheaves

I am trying to understant this proof on a decomposition property of constructible sheaves on a Noetherian scheme. But I start to get lost from line 6:"Since F is constructible, there is a finite ...
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0answers
35 views

There are no $\operatorname{Sp}$-invariant linear forms

Let $V, W$ be finite dimensional $\mathbb{Q}_\ell$ vector spaces, with $\dim W = 1 $, $G$ be a profinite group which acts on $V, W$, $\psi : V \times V \to W $ a bilinear slew-symmetric perfect $G$-...
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0answers
58 views

$H^0_c (X_{et}, \mathscr{F}) = 0$ for any affine smooth curve $X$

Let $k$ be an algebraically closed field, $X$ an affine connected smooth $k$ curve, and $\mathscr{F}$ a locally constant sheaf on $X_{et}$. Then $H^0_c (X, \mathscr{F}) = 0$? (This is 2.10.(a) of ...
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0answers
38 views

of finite presentation morphism

A morphism $f:X\rightarrow Y$ of schemes is called locally of finite presentation if for any $x\in X$, there exists an affine open neighborhood $V=\mathrm{Spec} B$ of $f(x)$ in Y, and an affine open ...
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vote
1answer
63 views

Proper morphism induces a map between compact support etale cohomology groups

Let $f : Y \to X$ be a proper morphism between noetherian separated schemes of finte type (over a noetherian scheme $S$), and $\mathscr{F}$ a sheaf on the small etale site on $X$. Then does $f$ induce ...
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vote
0answers
45 views

Calculating etale cohomology of Picard stack

I would like to try to calculate say $H_{\acute{e}tale}^*(B\mathbb{G}_{m, k}, \mathbb{Z}/p)$. Intuitively, since $B\mathbb{G}_m$ classifies line bundles, its topological analogue should be $\mathbb{CP}...
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0answers
80 views

Proposition 1.6.6, Etale Cohomology theory, Lei Fu

I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu. $\mathrm{Proposition} 1.6.6$ Let $g:S'\rightarrow S$ be a quasi-compact ...
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0answers
38 views

Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?

Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $\mathbb{G}_m$ and its subgroup $\mu_p$, the $p$-th roots of unity. It is well known that the quotient ...
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votes
0answers
57 views

Cech cohomology does not compute étale cohomology - Explanation of an example

In the first answer to this MO post the author says that the $H^2$ of $X$ can be compute using the Cech-to-derived functor spectral sequence, i.e. in that case the Mayer-Vietoris sequence. I'm having ...
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0answers
60 views

The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
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0answers
32 views

Computation of etale cohomology group

Let $X$ be a closed subvariety of $\mathbb{A_{1}}\times\mathbb{A_{1}}\times\mathbb{A_{1}}$ over $\mathbb{F_{p}}$ defind as solution set of $xy-z^{p}-z=0$. I want to compute etale cohomology group $H^{...
16
votes
1answer
251 views

Why é​t​a​l​e​?

Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions ...
4
votes
1answer
63 views

Does a sheaf of abelian groups on a scheme $X$ induce a sheaf of abelian groups on the étale site $X_{ét}$?

Fixed a scheme $X$, étale cohomology $H^\bullet(X,F)$ is defined for all sheaves of abelian groups $F$ over the étale site $X_{ét}$. Now, just to understand, I tried to see if this covers sheaves of ...
0
votes
1answer
81 views

Explaination of “Smooth of relative dimension d”.

Let f : $X \rightarrow Y$ be a morphism of smooth schemes of finite type over $k$. Then I want to understand the meaning of a morphism is "smooth of relative dimension $d$". In particular if Z is a ...
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0answers
18 views

The dimension of stalks for flat morphisms

I want to prove the conclusion below: If $f:X\to Y$ is a quasi-finite flat morphism, then $\dim \mathcal{O}_{X,x}=\dim \mathcal{O}_{Y,f(x)}$. For the part $\dim \mathcal{O}_{X,x}\leq\dim \mathcal{O}...
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1answer
85 views

Strict Epimorphism of Schemes

I am reading Milne's Etale Cohomology and ran across this problem which has so-far eluded me. According to Milne, in any category with fiber products, we say that a morphism $f:Y \to X$ is a strict ...
4
votes
2answers
220 views

Comparison of the mod p etale cohomology of special fiber and generic fiber

I am new to etale cohomology theory and recently learnt some base change theorems like proper base change and smooth base change. However, I am somehow confused about the etale cohomology group of $\...
1
vote
1answer
77 views

About the definition of l-adic Tate-twist

In the J. Tate's paper "Relations Between $K_2$ and Galois Cohomology" Let F any field $F^{\text{sep}}$ the separable closure of F $G_F=\text{Gal}(F^{\text{sep}}/F$) he defines the ($\mathbb{Z}_l,G_F$)...
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vote
1answer
49 views

Descent datum for a module

Below is a definition of a descent datum on stacks project: It then says that if $N = B \otimes_AM$ for some $A$-module $M$, then it has a canonical descent datum given by the map $$N \otimes_AB \to ...
7
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1answer
167 views

Computing étale cohomology group $H^1( \text{Spec}(k), \mu_n)$ and $H^1( \text{Spec}(k), \underline{\Bbb{Z}/\mathord{n \Bbb{Z}}})$

I am starting to learn about étale cohomology and would like to compute a simple example. Let $k$ be a field with a fixed separable extension $k^s.$ I want to compute $H^1( \operatorname{Spec}(k), \...
1
vote
1answer
62 views

The étale topos is coherent: does the scheme need to be quasicompact?

It is known that the little étale topos of a scheme $X$ is coherent, i.e. the Grothendieck topology of the étale site is generated by a basis of finite coverings. For example, Butz and Moerdijk say ...
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0answers
36 views

Homotopy classes of maps inside a Grothendieck topos

Moerdijk, Classifying spaces and classifying topoi, defines the Verdier cohomology of a Grothendieck topos $\mathcal E$ by taking a limit on $HC$, the homotopy of hypercoverings in $\mathcal E$. I ...
1
vote
1answer
65 views

Three definitions of hypercoverings

The notion of hypercovering (or hypercover) was introduced by J.L. Verdier in SGA 4. In his definition, a hypercovering $K_\bullet$ is a semirepresentable semisimplicial presheaf on a site $C$ such ...
3
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0answers
74 views

Computing $\pi_{et}(X,x)^{\operatorname{ab}}$

Let $X$ a smooth projective algebraic curve of genus $g$ over $k$. ($k$ is an algebraically closed field of characteristic $0$). I want to compute $\pi_{et}(X,x)^{\operatorname{ab}}$. I'm trying to ...
4
votes
0answers
156 views

Galois cover ramified at some points

I have seen statements like this " Fermat curve is a wild Galois cover of $\mathbb{P}_{k}^{1}$ ramified at $\{0,1, \infty\}$." Here, $k$ is a field of characteristic $p>0$. I would like to ...
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votes
0answers
34 views

Comparison of top degree and $0$-th etale cohomology with compact support

Let $k$ be an algebraically closed field, $X$ be a finite type separated reduced scheme over $k$ whose irreducible components are all of dimension $d$. Do we have $\operatorname{dim}_kH^{2d}_{c}(X,\...
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vote
0answers
41 views

Multiplication by $n$ map on $Pic(X)$?

Let $S$ be a scheme over the base field $k$. Then there exists an exact sequence in the fppf topology $$0 \to \mu_{n, S} \to \mathbf{G}_{m, S} \xrightarrow{(\cdot)^n} \mathbf{G}_{m, S} \to 0$$ The ...
3
votes
1answer
134 views

Soulé vanishing for $H^2$ of $\mathbb{Q}_p(n)$

In his paper "The motivic fundamental group of the projective line minus three points and the theorem of Siegel", M. Kim uses the following result: let $p$ be an odd prime, $T$ be a finite set of ...
2
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0answers
104 views

How to compute $H^i_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p)$?

I'm trying to compute $H^i_{\operatorname{fppf}}(\mathbb{P}^1(k),\alpha_p)$, where $\mathbb{P}^1(k)$ is a projective line over the field $k$ of characteristic $p>0$. I start with with following ...
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votes
0answers
99 views

Cycle class map and generalisation of Kummer theory

Suppose $X$ is a variety defined over an algebraically closed field $k$ of characteristic zero, and $Z$ is a closed subvariety of codimension $c$, then there is a long exact sequence of etale ...
2
votes
1answer
110 views

Long exact sequence in etale cohomology with support

Let $\Lambda := \mathbb{Z}/m\mathbb{Z}$, $X$ a scheme, $i : Y\hookrightarrow X$ a closed immersion, $j : U := X-Y\hookrightarrow X$ the open immersion of the complement, and $\mathcal{F}$ an etale ...
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vote
0answers
22 views

Explicit formula to compute the conductor of Etale cohomology?

I think I am stuck at a calculation problem. Suppose there is an elliptic surface $X$ defined over $Q$, take $\overline{X}:=X\times_Q \overline{Q}$, and denote by $\varphi: G_{Q}\to Aut(H^2_{et}(\...
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0answers
48 views

Twists of etale cohomology

In etale cohomology, for a variety $X$ defined over $\mathbb{Q}$ there is an isomorphism \begin{equation} H^n_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell(n)) \simeq H^n_{et}(X_{\overline{\mathbb{Q}}...
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1answer
59 views

Surjectivity of map of étale sheaves

Let $F\to G$ be a map of étale sheaves on a scheme $X$. Suppose that for any closed point $x\in X$, the map between stalks $$F_x\to G_x$$ is surjective. Note that I am not forming stalks at the ...
1
vote
1answer
72 views

Open normal subgroups of Galois

Let $K$ be a number field, $v$ a finite place of $K$. Let $K'$ be the maximal extension of $K$ unramified at $v$. For instance, for $K = \mathbf{Q}$ and $v = p$ a prime, $K' = \mathbf{Q}(\{\zeta_{n}, ...
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votes
1answer
84 views

Clarification about Milne's Etale Cohomology text

I am reading Miln'es Etale Cohomology (1980 Princeton University Press). In page 46 to give some exampeles of $E$-morphisms he writes $E=(et)$ of all etale morphisms of finite-type. Next page he ...
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votes
1answer
83 views

Structure of étale maps

Is it true that any étale map of affine schemes is a composition of finitely many finite étale maps and Zariski localizations?
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2answers
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Idea behind geometric points in étale theory.

Throughout this post, we fix a field $k$ (of characteristic zero if makes our lives easier :)) and $X$ an arbitrary variety (scheme) over $k$. I understand what's the pathology/discrepancy, behind the ...
2
votes
1answer
153 views

Étale sheaves as colimits of representable sheaves

If $X$ is a scheme, then by a representable étale sheaf one means the following: for a scheme $Y\to X$ over $X$, we may consider the presheaf of sets $$U \mapsto \operatorname{Hom}_X (U,Y),$$ and it ...
3
votes
1answer
135 views

Map on étale cohomology in terms of torsors

Let $X$ be the variety $\mathbb{C}\backslash\{0\}$, and consider the map $f:X\to X$, $z\mapsto z^2$. I am trying to compute the induced map $f^*$ on the first étale cohomology of $X$ with coefficients ...
0
votes
1answer
52 views

How to understand the end of the proof of Proposition 2.2 in Exposé V of SGA I?

Proposition 2.2 in Exposé V of SGA I states the following (schemes assumed locally noetherian). Let $X$ be a scheme with an admissible action of a finite group $G$, let $Y$ be the quotient scheme; ...
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0answers
147 views

Computation of an étale cohomology group on the projective line

I have the following problem: Let $U_{0}$ be a a smooth geometrically irreducible affine curve over $\mathbb{F}_{q}$, and let $\mathcal{E}_{0}$ be a constructible étale sheaf of $\mathbb{Q}_{\ell}$-...
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0answers
47 views

How to prove this ‘lemme connu’?

At the end of Exposé I in SGA 1 it is asserted that a well known lemma states the following: Let $k$ be an infinite field and $E$ be a finite product of finite field extensions of $k$. Suppose not ...
4
votes
0answers
83 views

Derived functors of Lower shriek in etale cohomology

Suppose $f:X \longrightarrow Y$ is a morphism (quasi-finite or compactifiable), $\mathcal{F} \in X_{et}$. What do the stalks of $R^{1}f_{!}\mathcal{F}$ look like?
7
votes
1answer
102 views

Is $\pi_1^{et}(\mathrm{Spec}\, k) \simeq \mathrm{Gal}(k^{sep}/k)$? (confusion about profinite completions)

The question is in the title, namely: Suppose that $k$ is a field and $k^{sep}$ its absolute algebraic separable closure. Denote $\bar s :\mathrm{Spec}\, k^{sep} \rightarrow \mathrm{Spec}\, k$ the ...
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votes
0answers
32 views

etale sheafification of exact forms

We know that the set of differential forms together with restrictions is a sheaf. But what happens if we take the subset of exact forms and make its etale sheafification? Do we obtain a distinct sheaf?...