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4
votes
0answers
69 views

The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continous map $f: (X,x)\rightarrow (Y,y)$ induces a ...
0
votes
1answer
37 views

Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
2
votes
1answer
41 views

etale vs zariski cohomology for coherent sheaves

Let $X$ be a scheme and $F$ a coherent sheaf on $X$. Does the etale cohomology of $F$ (i.e the cohomology of $F$ on the etale site of $X$) agree with the cohomology of $F$ on the Zariski site?
0
votes
2answers
47 views

question on a concrete example of etale cohomology groups

I don't understand something basic about the following answer to this question on mathoverflow: Angelo Vistoli writes: Let $X = \mathop{\rm Spec}R$, where $R := k[x,y]/(y^2 - x^3 + x^2)$. Consider ...
0
votes
1answer
40 views

Is etale cohomology invariant under purely transcendental extensions?

Let $X$ be a scheme over a field $k$ of characteristic $0$ and $\mathcal F$ is a torsion sheaf on $X$. Let $K/k$ be a purely transcendental extension. Is the natural map $H^i_{et}(X, \mathcal F) \to ...
1
vote
0answers
20 views

Is pullback functor form $Qcoh_{X_{Zar}}$ to sheaves of abelian groups $Sh_{X_{et}}$ exact?

Let $M$ be a quasi-coherent sheaf of $O_X$ modules on $X$, then $X'\to \Gamma(X',M\otimes_{O_X}O_{X'})$ is an abelian sheaf on $X_{et}$, (it is even a fpqc sheaf), denote $i\colon T_{X,Zar}\to ...
1
vote
0answers
53 views

isomorphism of étale cohomology

Let $i: C_0 \hookrightarrow C$ be a nil-immersion (a closed immersion defined by a nilpotent ideal sheaf, so it is a homeomorphism of the underlying topological spaces). Denote by $\mathbb{G}_m$ the ...
1
vote
0answers
38 views

Is “$k[X]^\ast= k^\ast$” preserved under finite surjective etale morphisms $Y \to X$ (for $k$ alg closed)?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a smooth and connected variety over $k$, with $k[X]^\ast = k^\ast$. Suppose that $f: Y \to X$ is a surjective finite etale map ...
2
votes
1answer
101 views

How does Galois group acts on etale cohomology?

I know this may be a trivial question, but I can't find the answer on, for example, Milne's online notes and Danilov's Cohomology of Algebraic Varieties. Suppose $K$ is a number field (say), ...
3
votes
0answers
46 views

Is this intuition for the etale topology essentially correct?

Suppose I have an etale morphism $f : X \to Y$. If $X$ and $Y$ are Riemann surfaces, then this means that $f$ is a local isomorphism, so at any $y \in Y$ I can find a local inverse $g : U \to X$ ...
4
votes
1answer
139 views

Relationship between Galois cohomology and etale cohomology.

Why is étale cohomology a natural generalization of Galois cohomology ? I would like to inform you that I have a few quite sufficient prerequisites Galois cohomology and its application to solve ...
3
votes
0answers
26 views

Compactifying affine algebraic families

Suppose I have a smooth morphism $f:X\to S$ of affine varieties over an algebraically closed field of arbitrary characteristic. I want to regard this as a family of varieties parametrized by $S$ and ...
3
votes
1answer
52 views

Etale fundamental group action on set-theoretic fiber

Let $f: Y \to X$ be a finite etale cover of schemes. Fix a geometric point $x \in X$. I would like $\pi_1(X,x)^{et}$, the etale fundamental group, to act on the set-theoretic fiber of $x$. This set is ...
1
vote
0answers
41 views

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
votes
2answers
57 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
vote
1answer
53 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
votes
1answer
41 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
votes
0answers
55 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
votes
1answer
77 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
vote
0answers
89 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 ...
8
votes
2answers
171 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
votes
0answers
81 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
votes
0answers
99 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
votes
1answer
214 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 ...
11
votes
1answer
326 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
votes
0answers
26 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
votes
0answers
75 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
votes
1answer
97 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
votes
0answers
93 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 ...
0
votes
1answer
76 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 ...
1
vote
0answers
33 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
vote
0answers
30 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?
9
votes
1answer
396 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
votes
1answer
190 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 ...
1
vote
0answers
114 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
votes
1answer
70 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 $H^q(X,F) = H^q(X_1,F)$?
2
votes
0answers
32 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
votes
0answers
69 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
98 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
48 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
votes
0answers
42 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
2answers
272 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 ...
4
votes
1answer
242 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
votes
1answer
1k 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
vote
1answer
47 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: ...
4
votes
1answer
96 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
207 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 ...
12
votes
2answers
759 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
83 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 ...
19
votes
2answers
419 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 ...