the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.

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6
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153 views

Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids

Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ ...
8
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258 views

What is an example of a non-mixed $\ell$-adic sheaf?

$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
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0answers
58 views

A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and $$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
6
votes
0answers
98 views

Gerbes on the multiplicative group

Let $k$ be an arbitrary field with absolute Galois group $\Gamma$. The group $\text{Hom}(\Gamma,\mathbb{Q}/\mathbb{Z})$ injects into $H^2(\mathbb{A}^1 \setminus \{ 0 \},\mathbb{G}_m)$, as one can see ...
7
votes
2answers
260 views

Group cohomology of fundamental group of a curve

The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this ...
6
votes
0answers
110 views

What is the etale cohomology of a product?

Let $X$ be a (smooth, projective if you wish) variety over a field $k$. I'm mostly interested in $k=\mathbb{F}_q$. What can one say about the etale cohomology ring $H_{et}^*(X \times_k X)$, say for ...
3
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0answers
118 views

What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, \mathbb{...
13
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1answer
583 views

Resolution of singularities in étale cohomology

The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely ...
8
votes
1answer
270 views

Etale cohomology of $\mathrm{Spec}(k\{X,Y\})\backslash\langle0,0\rangle$

Illusie in "Grothendieck et la cohomologie étale" says Artin's Harvard notes on Grothendieck Topologies prove: The étale cohomology with coefficients in $Z/nZ$ of the variety $\mathrm{Spec}(k[X,Y])\...
12
votes
1answer
294 views

Independence of $\ell$ of Betti numbers

When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. ...
23
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2answers
839 views

Steenrod operations in etale cohomology?

For $X$ a topological space, from the short exact sequence $$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$ we get a Bockstein homomorphism $$H^i(X,...
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0answers
79 views

Reference: Relative cohomology of a morphism

Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence $$ \cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots $$ where the ...
2
votes
1answer
205 views

A homotopy argument in etale topology

Suppose everything below is defined over $k=\overline{\mathbb{F}}_q$. Let $H$ be a connected algebraic group acting on a separated variety $Y$. Denote the morphism $H\times Y\rightarrow H\times Y; (h,...
18
votes
1answer
482 views

Flat versus etale cohomology

Although the definition of etale ($\ell$-adic) cohomology is scary, I have at least some intuition for how it should behave: for instance, when it makes sense, I expect that it should be ``similar'' ...
3
votes
1answer
212 views

Étale coverings of cubics and gluings

Consider the plane nodal cubic $X$ given by the equation $y^2=x^2(x+1)$ over a field $k.$ It is not too hard to show that $Y$ has a finite étale covering $X$ of degree $2.$ One does this in the ...
3
votes
1answer
94 views

Bounding the number of top dimensional irreducible components of a variety defined over a finite field

Let $V\subset \mathbb{A}^n_{\mathbb{F}_q}$ be a closed subvariety defined by simultaneously vanishing of $r$ polynomials $f_1,\cdots,f_r\in \mathbb{F}_q[x_1,\cdots,x_n]$, each of degree at most $d$. ...
4
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151 views

Varieties with only top $\ell$-adic cohomology not vanish?

Let $X$ be a $d$-dimensional connected smooth variety over $\overline{\mathbb{F}}_q$. It is well known that if $X$ is isomorphic to an affine space, then all the $\ell$-adic compactly supported ...
2
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1answer
284 views

When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?

I am trying to learn character sheaf theory, and encounter the following question: (*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}_q$, ...
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0answers
122 views

Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
3
votes
1answer
317 views

Galois cohomology of a non-abelian group over a function field

Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, ...
5
votes
1answer
195 views

$\mathbb{A}^1$-invariance of categories of Finite Etale Covers

Let $k$ be algebraically closed with characteristic $0$. For a scheme $X$, let $FEt(X)$ be the category of finite etale covers of $X$. What can be said about $FEt(X \times \mathbb{A}^1)$ and the ...
2
votes
1answer
139 views

Is an $\ell$-adic local system on $\mathbb A^2$ that restricts to a trivial local system on every vertical line a pullback via the first projection?

This question is a reformulation of a special case of the question An $\ell$-adic local system which is trivial on every fiber of a morphism (this special case did not receive an answer on ...
3
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2answers
195 views

An $\ell$-adic local system which is trivial on every fiber of a morphism

Let $f: X \to Y$ be a morphism with connected fibers, where $X, Y$ are smooth algebraic varieties (I am specifically interested in the case when $f$ is a Zariski locally trivial fibration with ...
5
votes
1answer
371 views

The Weil numbers and modulus of an elliptic curve

I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then $X(\...
4
votes
0answers
148 views

How to compute the first etale cohomology of a constructible torsion-free sheaf?

I am interested in the following example! Let $k$ be a field, let $X_0$ be the scheme $\mathrm{Spec}R$ with $R_0=k[x,y]/(xy)$, let $R$ be the strict Hensilian localalisation of $R_0$ at the origin ...
4
votes
1answer
175 views

$l$-dependence of the group of homologically zero cycles

Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
6
votes
1answer
301 views

Under what conditions is the induced map of etale fundamental groups surjective?

Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
5
votes
0answers
112 views

Gysin exact sequence for a singular subvariety

Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety. Let $Y \subset X$ be a (possibly) ...
2
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128 views

Specialization theorem for cohomology groups

If $f:X\longrightarrow Y$ is a proper lisse morphism and $\mathcal{F}$ is a torsion sheaf on $X$ then one has: $$ (R^{i}f_{*}\mathcal{F})_{\overline{y}}\cong H_{c}^{i}(X_{\overline{y}},\mathcal{F}_{|...
4
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0answers
89 views

Weil structures and rational structures on $\overline{\mathbb{Q}}_{\ell}$-sheaves

In the literature concerning characteristic functions of varieties over a finite field, there is a notion called Weil structure defined in the following way: Definition Let $X$ be a finite type ...
6
votes
2answers
345 views

Algebraic proof without using comparison theorem for étale cohomology

Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity). The étale cohomology groups of X are therefore equipped with an action of $\mu_n$...
8
votes
1answer
287 views

Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field. According to Mazza--Voevodsky--Weibel "...
4
votes
1answer
203 views

How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic

Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$. If $k= \mathbb C$, there is a natural injective morphism of vector spaces $$H^0(X,\...
6
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0answers
251 views

A problem on universally locally acyclic

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ (...
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0answers
102 views

Cycle with integral coefficients from cycle with $\mathbb Z_l$-coefficients

Let $X$ be a $n$-dimensional ($n>2$) smooth projective variety over $k=\bar k$ of positive characteristic. Take a divisor $D\in Pic(X).$ Suppose we know that $\frac{[D]^2}{2}\in H^4(X,\mathbb Z_l(...
2
votes
2answers
208 views

Birational morphism and first cohomology group

Let $f:X\rightarrow Y$ be a proper birational morphism of smooth projective varieties over $k=\bar k$ ($char(k)>0$). Is it true that $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ($l\...
2
votes
1answer
194 views

Higher direct image of locally constant torsion sheaf (étale cohomology)

Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...
2
votes
0answers
117 views

Cycle map and flat cycle

Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...
1
vote
1answer
252 views

Etale cohomology and topological invariance

Let $X$ be a projective scheme and $X_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on $X_{\...
4
votes
1answer
285 views

Picard groups of Fano varieties in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime. Is the natural morphism $\mathrm{Pic}(X) \otimes \...
3
votes
0answers
196 views

Is there a Hochschild-Serre spectral sequence for unramified cohomology?

Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...
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vote
0answers
61 views

Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
8
votes
1answer
403 views

Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$, ${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles. Let $G$ be an affine ...
2
votes
1answer
105 views

formally etale deformations of algebras

Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong B\otimes_Ak$...
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votes
0answers
51 views

constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
3
votes
2answers
262 views

étale cohomology via Cech cocycles for a quasi-projective scheme

I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.
1
vote
1answer
102 views

Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...
5
votes
0answers
259 views

Is it possible to assume that an étale neighborhood is connected?

I am new to étale topology (though I've seen Grothendieck's sites before). Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...
8
votes
1answer
304 views

When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$). For $...
7
votes
1answer
482 views

Étale cohomology versus classical cohomology

Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology $H^p(X(\mathbb{C}),\mathbb{Z}/n)$...