37 votes
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"Cute" applications of the étale fundamental group

Using the étale fundamental group one can construct an injective group homomorphism $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$ which is ...
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26 votes
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Steenrod operations in etale cohomology?

You maybe want to have a look at P. Brosnan and R. Joshua. Comparison of motivic and simplicial operations in mod-$\ell$ motivic and étale cohomology. In: Feynman amplitudes, periods and motives, ...
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22 votes

What is known about the cohomological dimension of algebraic number fields?

By definition, an algebraic number field is a finite extension of the field of rational numbers $\Bbb Q$. An algebraic number field $K$ is called totally imaginary if it has no embeddings into $\...
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19 votes

Clarifying the connection between 'etale locally' and 'formally locally'

For local noetherian rings, the henselization has much more direct algebro-geometric meaning than the completion since it is built from local-etale algebras. Hence, your statement that the completion ...
18 votes
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A short proof for simple connectedness of the projective line

You can deduce this from the classification of vector bundles on $\mathbf{P}^1$. Say $f:C \to \mathbf{P}^1$ is a connected finite etale Galois cover of degree $n$. We must show $n=1$. The sheaf $E := ...
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16 votes

Steenrod operations in etale cohomology?

Your first map fits in an action of the Steenrod algebra. In fact $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$, an element of the derived category ...
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  • 15.6k
16 votes
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Commutative algebra counterexample

Let $R=\mathbb{Z}$ and let $M=\mathbb{Z}$ with $x$ acting by $2$. Then $M\otimes_{R[x]}R[x,x^{-1}]\cong \mathbb{Z}[1/2]$ is not finitely generated over $R$.
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  • 29.9k
16 votes
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Étale cohomology of morphism whose fibers are vector spaces

The answer below is slightly reorganized to incorporate the edits. I thank @PiotrAchinger and @S.D. for their comments correcting and clarifying this answer. Definition 1. An affine space morphism ...
15 votes

Is there a "universal" cohomology theory for varieties over p-adic fields?

I'm seeing this old question just now, and simply wanted to remark that the situation may be slightly better. Namely, enlarging the category of $\mathbb Q$-vector spaces into the larger semisimple $\...
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14 votes
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Etale cohomology of $\mathrm{Spec}(k\{X,Y\})\backslash\langle0,0\rangle$

I'm not sure what "easy" means in the context of etale cohomology but there is a way of passing from Artin's result to the stated one. Let $j$ from $\mathbb A^2 \backslash \langle 0,0\rangle$ to to $\...
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  • 115k
14 votes
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Hodge standard conjecture for étale cohomology

I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic. The proof in ...
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  • 17k
14 votes

"Cute" applications of the étale fundamental group

I think a fantastic application of the étale fundamental group is in extensions of the Chabauty-Coleman(-Kim) method. The original idea was that we could bound the rational points on a curve $C$ by ...
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13 votes

Idea of using etale site

I can tell you how they are related. Before Riemann people would say, for example, the complex square root function (for $z\neq 0$) is two valued, but for any small region of (non-zero) complex ...
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13 votes

The Weil numbers and modulus of an elliptic curve

If $E$ does not have CM, then $\tau$ will be transcendental over $\mathbb Q$, so it's hard to imagine a relation with the eigenvalues of Frobenius, which are integers in an imaginary quadratic field. ...
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13 votes
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Etale cohomology can not be computed by Cech

Let $k$ be an algebraically closed field. Glue two copies of $\text{Spec}(k[[x]])$ along $\text{Spec}(k((x)))$. This gives a scheme $X = U \cup V$ such that any etale covering of $X$ can be refined by ...
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  • 146
13 votes
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Etale cohomology with coefficients in $\mathbb{Q}$

The following is surely expressing whatever is in the core non-formal aspect of Joe Berner's answer (which is above my pay grade); it is offered as an alternative version of the same ideas. Let $X$ ...
13 votes
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Tate twists and cohomology of $\mathbf{P}^1$

The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the ...
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13 votes
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Etale cohomology of localizations of henselian rings

TL;DR: Your expectation is right. In fact, there is a third object to compare with $R[1/p]$ and $\hat R[1/p]$, the affinoid rigid space ${\rm Spf}(\hat R)^{\rm rig}$. The cohomology comparison is ...
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13 votes
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Cohomology of resolution of singularity

In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects ...
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13 votes
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On the definition of the etale site of an adic space

Great question! The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat ...
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13 votes
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Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?

Good question! Let me try to guess what Gabber had in mind there. (Note that he only says "known" (to him), not "well-known"...) The claim is that the extension splits. Note that ...
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12 votes
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Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a ...
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12 votes

Flat versus etale cohomology

Here is a good example of some pathological behaviour that will shatter all your hopes and dreams. Claim. Let $k$ be a perfect field, and let $X$ be a smooth, proper, integral $k$-scheme. Then the ...
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12 votes
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Computing the etale cohomology of spheres

This is true over any algebraically closed field $k$ of characteristic different from $2$. More generally, if $\ell$ is invertible in $k$, then $$H^i(X,\mathbb Z_\ell) = \left\{\begin{array}{ll}\...
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12 votes
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Complete intersections in toric varieties

Any smooth projective toric variety is rational, in particular simply connected. Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete ...
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11 votes
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Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular ...
11 votes
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Cohomology of a constant etale sheaf

The etale cohomology groups $H^i(X_{et},V)$ vanish for $i > 0$ if $X$ is a quasi-compact, quasi-separated, normal scheme, and $V$ is a $\mathbb{Q}$-vector space. (In fact, one may replace "normal" ...
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11 votes
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Under what conditions is the induced map of etale fundamental groups surjective?

There is a very general criterion for a map on $\pi_1$ to be surjective. Recall that for $X$ connected, the category of finite étale covers of $X$ is equivalent to the category $\pi_1(X)\text{ -}\...
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11 votes
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Why define étale cohomological dimension as it is defined?

I think it's a combination of two things: First, we do etale cohomology with torsion sheaves only, because of pathologies with non-torsion sheaves, and so we care about the cohomological dimension ...
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  • 115k
11 votes
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Derived completion of complexes

As I mentioned in a comment, $K \to L$ must also be an isomorphism after rationalizing. For example, if $0 \to R \to F \to \Bbb Q \to 0$ is a free resolution of $\Bbb Q$, then let $$ K = \dots \to 0 \...
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  • 47.4k

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