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0
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1answer
39 views

etale sheafification of a presheaf

Consider the following presheaf on the big etale site of smooth schemes over a field $k$: to every smooth $k$-scheme $U$, associate $$F(U):= \{f: U \to \mathbb A^1_k ~|~ \text{$f$ factors through the ...
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0answers
20 views

isomorphism in étale cohomology

Let $X/k$, $k$ algebraically closed, be a proper variety. Why is $H^2_{et}(X,\mathbf{Z}_\ell) \otimes \mathbf{Q}_\ell/\mathbf{Z}_\ell = (\mathbf{Q}_\ell/\mathbf{Z}_\ell)^{b_2(X) - \rho(X)}$, where ...
1
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0answers
18 views

finiteness in étale cohomology (ell-adic sheaves)

Let $X/k$, $k$ separably closed, be a proper variety. Is then $H_{et}^p(X,\mathbf{Z}_\ell(q))$ finitely generated?
0
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0answers
32 views

$H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$?

Let $X/\mathbf{F}_q$ be a smooth projective geometrically integral variety. Does it follow that $H^1(\mathbf{F}_q, Pic(\bar{X})) = 0$? Isn't this just Lang-Steinberg? I think Lang-Steinberg gives us ...
6
votes
1answer
84 views

What is the intuition behind the Fontaine-Mazur Conjecture?

The Fontaine-Mazur conjecture (over $\textbf{Q}$ for simplicity) says that a (continuous irreducible) Galois representation $$ \rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to ...
6
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1answer
85 views

Affine varieties, the Brauer group and gerbes

For an affine variety $V$, the Brauer group $Br(V)$ and the cohomological Brauer group $Br'(V)$ (i.e. the torsion subgroup of $H^2_{et}(V,\mathbb{G}_m)$) coincide, by results of various authors ...
0
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0answers
96 views

Compare étale morphisms in Top and Schemes

Perhaps because I find the algebraic definition of étale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of étale morphisms of topological spaces. In topological ...
0
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1answer
49 views

proper base change theorem

Let $f: X \to S$ be proper with $S$ the spectrum of a stricly henselian DVR and $X_0$ the special and $X_1$ the generic fibre. Why do we have $H^q(X,F) = H^q(X_0,F)$, but not $H^q(X,F) = H^q(X_1,F)$?
0
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0answers
16 views

vanishing in étale cohomology

Let $S = Spec(R)$ be a strictly henselian ring and $f: X \to S$ a morphism with generic fibre $X_\eta \to \{\eta\}$ and inclusion $i': X_\eta \hookrightarrow X$. Do we have ...
2
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0answers
21 views

Dual of etale fundamental group, field functoriality

Suppose $k$ is an arbitrary field and $X/k$ is a finite-type and integral scheme (hence connected). Let $K$ be the fraction field of $X$. Via the canonical morphism $$\phi: \rm{Spec}(K)\rightarrow ...
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0answers
40 views

Étale cohomology and Picard group of curves

Say we have $X$ a smooth projective curves over $\mathbb{Q}$, then I know there is an isomorphism $H_{ét}^1(X\times_\mathbb{Q}\overline{\mathbb{Q}},\mathbb{G}_m)\cong ...
1
vote
1answer
45 views

Pushforward of pullback of an etale sheaf

I hope this question is not too elementary, but I'm a bit lost : Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the ...
3
votes
1answer
47 views

question about $\ell$-adic local systems

This question came up while reading about the equivalence between smooth $\mathbb{Q}_{\ell}$-sheaves and finite dimensional representations of $\pi_{1}(X,s)$. For a brief description of the situation, ...
2
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0answers
33 views

Etale cohomology and restricted direct product

[migrated to mathoverflow: http://mathoverflow.net/questions/161734/etale-cohomology-and-restricted-direct-product] $\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an abelian ...
4
votes
1answer
121 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
3
votes
1answer
153 views

SGA 4.5 proof of Hilbert 90 and semilinear Galois action

In SGA 4.5's proof of Hilbert 90, proposition 1.5.2(that the inclusion $V'^G \otimes_k k' \rightarrow V'$ is an isomorphism) is deduced from faithfully flat descent as stated in 1.4.5. The way that I ...
30
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1answer
592 views

Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.
0
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1answer
28 views

Finite coverings are closed.

I'm working on solving as many of the exercises in Lenstra's Galois Theory for Schemes as possible, but there is one problem I'm partially stuck on. The statement of the problem is: ...
3
votes
1answer
74 views

Etale cohomology and algebraic closure

$\DeclareMathOperator{\h}{H}$Apologies in advance if this is overly stupid. Let $k$ be a field and $X$ a variety over $k$. Let $n$ be an integer which is invertible in $k$. One often looks at the ...
3
votes
1answer
122 views

Reference for Hasse-Witt invariant

I'm looking for a good reference for learning about Hasse-invariants ($p$-ranks) for curves of arbitrary genus over a field characteristic $p$. All the usual suspects I've searched (Milne's Étale ...
9
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2answers
307 views

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be ...
3
votes
1answer
66 views

Importance of Et(X) not being small?

It was asked earlier (Why Et(X) is not small?) why $Et(X)$, the category of schemes etale over a fixed scheme $X$ is not small. I was wondering how does this issue come up in practice? I'm guessing ...
17
votes
2answers
270 views

What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the ...
7
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1answer
205 views

Comparison between etale and singular cohomology for a singular variety

In his Lectures on Etale Cohomology Milne proves in Theorem 21.1, that for all $r\geq 0$ $$ H^r_{\acute{e}t}(X,\Lambda)\cong H^r(X_{cx},\Lambda) $$ with $X$ a nonsingular $\mathbb{C}$-variety and ...
1
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0answers
56 views

Morphisms from the group variety of $n$-th roots

In Milne - Lectures on étale cohomology, example 6.10 i came across the following. We fix a variety $X$ and work in the category $Var/X$ of varieties over $X$ (so with fixed morphisms to $X$!) and ...
3
votes
1answer
82 views

Union of categories and $\ell$-adic local systems

I have a question about the definition of $\ell$-adic local systems. I understand how to define local systems over any finite extension of $\mathbb{Q}_{\ell}$, but not how to take the "union" of these ...
2
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0answers
71 views

Pullback of commutative group schemes viewed as etale sheaves.

If $f:X\rightarrow Y$ is a morphism of schemes, we get an induced morphism of (small) etale sites $X_{et}\rightarrow Y_{et}$ whose underlying functor is base change along $f$. Any commutative ...
2
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0answers
66 views

Higher direct image of the inclusion of the generic point

$\newcommand{\Spec}{\operatorname{Spec}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Gal}{\operatorname{Gal}}$ I am trying to understand the proof of the following statement: Let ...
21
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0answers
355 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
5
votes
1answer
432 views

étale fundamental group as unification of galois theory and covering theory

what is a good reference if one wants to learn the basic theory of étale cohomology, étale fundamental group and in particular relationships between galois theory and covering theory (unified via the ...
9
votes
1answer
383 views

Étale cohomology of projective space

I have some very basic question about étale cohomology. Namely I would like to compute the étale cohomology of of the projective space over the algebraic closure of $\mathbb F_q$ along with its ...