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1
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1answer
44 views

Galois group action on etale cohomology groups

Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism $$H^r_{et}(X\times_{\mathbb{Z}_p}\...
0
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0answers
30 views

Weil conjectures - If two varieties have the same of Fq^d - valued points for all d >> 0, then they have the same Hasse - Weil function

I was working on the following exercise for fun, and I haven't really gotten anywhere with it. Let Z( X; t) be defined as exp ( $\sum_{r= 1}^{\infty} N_r t^r/r$), where $N_r$ is the size of X($\...
2
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1answer
28 views

How to prove that $H^q(X_{et},F) \cong \prod_{i=1}^n H^q(X_{i_{et}}, F|_{X_i})$ for $X = X_1 \sqcup … \sqcup X_n$

Let $X = X_1 \sqcup ... \sqcup X_n$ be a disjoint union of schemes, and let $F \in \text{Ab}(X_{et})$ be an abelian sheaf on an etale site of $X$. I need to show that $H^q(X_{et},F) \cong \prod_{i=1}^...
3
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1answer
83 views

Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
3
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0answers
31 views

Cohomology of affine hypersurface

I'm interested in calculating cohomology of a smooth affine hypersurface (over an arbitrary field). I know I can use (algebraic) de Rham cohomology, which is "easy" in the complex case, but even there ...
1
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0answers
19 views

What is alias etale map?

I think that etal map, is a local homeomorphism(or a local diffeomorphism for manifolds). I want a reference that explain alias etale map.
2
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0answers
25 views

Fixed points of Frobenius

Let $X$ be a scheme over $\mathbb{F}_q$ and $F$ the Frobenius morphism. I want to understand why $X(\mathbb{F}_q)$ are the fixed points of $F$ on $X(\overline{\mathbb{F}}_q)$. Edit. Is the following ...
6
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0answers
96 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
48 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
69 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
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2answers
58 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
46 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
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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 T_{X,et}...
1
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0answers
54 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 ...
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0answers
40 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
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1answer
165 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
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0answers
52 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
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1answer
228 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
1answer
61 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
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0answers
50 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
61 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
56 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 ...
6
votes
1answer
43 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
58 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
80 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 $(\Gamma,proj_{...
1
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0answers
93 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
217 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," i.e....
2
votes
0answers
83 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
120 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
318 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 ...
12
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1answer
438 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 ...
3
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0answers
32 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 $H^1(X,\...
3
votes
0answers
77 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
108 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
99 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
86 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
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0answers
34 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) = (\mathbf{Q}_\ell/\mathbf{Z}_\ell)^{b_2(...
1
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0answers
33 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
458 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 GL_n(\overline{\...
6
votes
1answer
222 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
123 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
75 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
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0answers
34 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 $$\phi: \rm{Spec}(K)\rightarrow X,$$...
2
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0answers
74 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 Pic(X\times_\mathbb{Q}\overline{\...
1
vote
1answer
121 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
45 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
291 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
262 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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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.