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Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
3
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2answers
43 views

Continuity of Galois representations from cohomology

One of the most standard way to construct Galois representations is the geometric way: one starts from a variety $X$ defined over $\bf Q$ say; the Galois group acts on ${\overline X}:= X \times {\rm ...
1
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1answer
40 views

Cohomology of Severi-Brauer varieties

What can be said about Galois-module structure of $l$-adic cohomology of a Severi-Brauer variety over a local field? In particular, I'm interested in the proof of the proposition given at the top of ...
5
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1answer
33 views

Why $R^q(\Gamma \circ \eta_{*}) (\Bbb G_{m, \eta}) = H^q(\eta_{ét}, \Bbb G_{m, \eta})$?

Let $X$ be a smooth, projective and connected curve over an algebraically closed field, and let $\eta \rightarrow X$ be its generic point (we also call the inclusion as $\eta$). I want to understand ...
2
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0answers
51 views

Are any two $\ell$-adic Tate twists (non-canonically) isomorphic?

Recall that for a prime $\ell$, the $\ell$-adic Tate twist is defined by $$\mathbb Z_{\ell}(n) := \varprojlim_r \mu_{\ell^r}^{\otimes n}.$$ As abelian groups, we have a (non-canonical) isomorphism ...
2
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1answer
67 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
1
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0answers
77 views

Analogue of locally constant sheaf in algebraic geometry

If I take just the definition of locally constant sheaves for algebraic varieties I get something pretty trivial (basically due to irreducibility); so how can one make a set up similar to what happen ...
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2answers
117 views

Lack of “good covers” in the étale topology

Disclaimer: This question might be terribly naive and almost certainly reflects my own ignorance. If $X$ is a topological space admitting a finite triangulation, then it admits a "good covering," ...
2
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0answers
79 views

Etale cohomology approach on $\tau(n)$

Ramanujan's $\tau$ conjecture states that $$\tau(n)\sim O(n^{\frac{11}2+\epsilon})$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in Status of $\tau(n)$ before Deligne tell ...
0
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0answers
62 views

Pushforward of a constant sheaf in the etale topology

This is a very basic question in etale cohomology. Let $\mathcal{F}$ be the constant sheaf $\mathbb{Z} /(n)$ on $\mathbb{A}^1$ over an algebraically closed field $k$ with $n$ coprime to the ...
3
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1answer
121 views

Cohomology with compact support for sheaves in separated schemes of finite type over a Noetherian scheme: three different definitions

usually there are three notions of cohomology with compact (proper) support. The first one usually done in the étale site. However the second one is used in Verdier duality. The third one is done in ...
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1answer
163 views

In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic ...
2
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0answers
20 views

Semisimplicity of Frobenius action on H^1(X,Q_p)

Let $X$ be a smooth projective curve over a finite field of characteristic $p$, with Hasse-Witt invariant $\lambda>1$. For $\ell\neq p$, the Frobenius action on the etale cohomology group ...
3
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0answers
71 views

Analogue of the Hecke operator for a general curve

Background: The eigenvalues of the Hecke opeator $T_p$ on the space of cusp forms $S_2(\Gamma_0(N))$ (of dimension $g=\text{genus}(X_0(N))$) are analysed using reduction mod $p$ and the use of the ...
8
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1answer
83 views

Three meanings of étale sheaf on X

When I am studying stacky stuffs, I am always confused by the notion of étale abelian sheaves on $X$, because conceivably there might be three different meanings of that: Take the global étale site ...
4
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0answers
84 views

What happens when you drop “étale” from the construction of étale fundamental groups

Haven't thought about this deeply; it just came up at lunch and I wondered if anyone knew the answer. To define the étale fundamental group of a scheme $X$, we fix a point $x_0 \in X$, look at the ...
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1answer
60 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
32 views

isomorphism in étale cohomology

Let $X$ be a proper $k$-variety with $k$ algebraically closed. Why is $H^2_{et}(X,\mathbf{Z}_\ell) \otimes_{\mathbf{Z}_\ell} (\mathbf{Q}_\ell/\mathbf{Z}_\ell) = ...
1
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0answers
26 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?
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0answers
38 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 ...
8
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1answer
260 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
135 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 ...
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0answers
106 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
63 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 $H^q(X,F) = H^q(X_1,F)$?
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25 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 ...
2
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0answers
59 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
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1answer
76 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
39 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
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2answers
232 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
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1answer
209 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 ...
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1answer
883 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.
1
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1answer
45 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: ...
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1answer
87 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
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1answer
177 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 ...
11
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2answers
576 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
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1answer
78 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 ...
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2answers
358 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 ...
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1answer
297 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
63 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
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1answer
85 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
votes
0answers
77 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
72 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 ...
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438 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
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1answer
501 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 ...
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1answer
430 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 ...