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I'll complement the list of well known books on the subject by some freely available documents, which I find user-friendly.
Here are great lecture notes , from a course that de Jong (of Stacks Project fame) gave in 2009.
Edgar José Martins Dias Costa's short dissertation on the subject .
Evan Jenkins's notes of a seminar on étale cohomology (click ...

answered Nov 10 '11 at 23:10

Georges Elencwajg

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Actually, the semisimplicity should hold with no hypotheses on X, so no example should exist. In fact it is generally expected that, with char. 0 coefficients and over a finite field (both hypotheses being necessary), every mixed motive is a direct sum of pure motives -- so the question for arbitrary varieties reduces to that for smooth projective ones.
...

25

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, Contemporary Math. 648, 2015, 29-55.
There are two sequences of cohomology operations in motivic and étale cohomology which can rightfully be called Steenrod ...

answered Mar 17 '16 at 9:07

Matthias Wendt

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23

The standard example is a copy of $\mathbb A^1_k$, where $k$ is an algebraically closed field, with two points glued. In algebraic terms, $X = \mathop{\rm Spec}R$, where $R := k[x,y]/(y^2 - x^3 + x^2)$. Consider the finite morphism $\pi\colon \mathbb A^1 \to X$, which yields an exact sequence
$$
0 \to \mathbb Z_X \to \pi_*\mathbb Z_{\mathbb A^1} \to i_*\...

19

A Morse function is a map of a manifold to the real line locally equivalent to:
$$f(x_1,\ldots, x_n)=-x_1^2\ldots -x_k^2+ x_{k+1}^2+\ldots+x_n^2$$
for some $k$. In other words,
for which the singularities are as simple as possible.
While a Lefschetz pencil is a map of a smooth projective variety to the projective line local analytically given by $f=x_1^2+\...

19

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 provides more intuition than the henselization is just a matter of having less experience with henselizations. See Example 2 below for why there is no need in ...

18

Let $X$ be a smooth projective variety over a finite field
$\mathbb{F}_{q}$ of caracteristic $p$ and let $l$ be a prime number different from p.
We consider the following statement :
(A) The action of the Frobenius on the etale cohomology $H^{i}_{et}( X_{\overline{\mathbb{F}}_{q}}, \mathbb{Q}_{l})$ is semisimple.
How to suppress the projective hypothesis ...

17

If a (say constant) group $G$ acts on a scheme $X$, you may want to consider the notion of a $G$-sheaf : a sheaf $\mathcal F$ endowed with isomorphisms $\lambda_g: g^* \mathcal F\simeq \mathcal F$, for $g\in G$ satisfying the usual cocycle conditions. Then by functoriality of cohomology for $g:X\to X$ you get an isomorphism $H^i(X, \mathcal F) \to H^i(X, g^*\...

17

Briefly, to understand $p$-phenomena in characteristic $p$ you need to replace the etale site by the fppf site. For example, to understand the $p$-torsion in the Brauer group you need the $p$-Kummer sequence and the cohomology of $\mu_{p^n}$, and the study of the $p$-torsion in the Tate-Shafarevich group entails the study of the cohomology of finite group ...

17

See SGA 4, Exposé VIII, 7.8, which defines an abstract point of a site as a functor from the topos of that site (i.e. the category of sheaves of sets on that site) to the category of sets that commutes with arbitrary inductive (direct) limits and finite projective (inverse) limits.*
One should think of a functor as being the functor that takes a sheaf to ...

16

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 $E := f_* \mathcal{O}_C$ is a rank $n$ vector bundle on $\mathbf{P}^1$, so we can write it as $E \simeq \oplus_{i=1}^n \mathcal{O}(a_i)$ for some integers $a_i$. As $...

16

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 $R$.

15

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 of $\mathbb{F}_2$ which has the property of being an $E_\infty$-algebra.[1]
This means that not only it is a commutative algebra for the derived tensor product ...

14

Not a textbook, but a free PDF by J.S. Milne, http://www.jmilne.org/math/CourseNotes/LEC.pdf, pretty good IMHO.

14

The following result follows from Tate-Honda theory
Let $A$ be an abelian variety over a finite field $k$, and let $f_A$ be the characteristic polynomial of $A$. Then $A$ is isogenous to a power of a simple abelian variety if and only if $f_A$ is a power of an irreducible polynomial.
I can't find a set of online notes which contains this statement. ...

answered Feb 15 '12 at 15:09

David E Speyer

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13

More generally, if $X$ is proper over an algebraically closed field, then $H^1(X,\mathbb Z)$
is isomorphic to the cocharacter module of the maximal torus of the Picard variety
$Hom(\mathbb G_m,Pic^0)$.

13

As already pointed out, the Hodge numbers may go up under reduction modulo $p$. On the other hand, let me also point out that the situation can be controlled:
1.) For all $p$, where $\overline{X}_p$ is smooth, the $\ell$-adic Betti numbers of $X$ and $\overline{X}_p$ are the same.
2.) Now, by the universal coefficient formula relating crystalline and ...

13

Sure thing! There's an equivalence of categories between cosheaves valued in profinite abelian groups and sheaves valued in torsion abelian groups, by Pontryagin duality one could say. So you can directly define the first etale homology of Z_ell and it will give you the l-adic Tate module. But it's formally the same as defining the l-adic Tate module to ...

13

As you said, it is a generalization of Cech theory. The standard example to understand first is a divisor $D=\bigcup D_i$ with simple normal crossings. A resolution of singularities is obtained by simply taking a disjoint union of components $X_0= \coprod D_i$. Let $\pi_0:X_0\to D$ be the obvious map. The cohomology of $X_0$ with your favourite coefficients ...

13

You probably meant to assume $R$ and $S$ are noetherian. The answer is "no" to the initial hypergeneral part of the question. EDIT: In the 2nd half (below the long line), I now give a proof of an affirmative answer to the added part involving maps of affine spaces.
Counterexamples to the initial hypergeneral part can be made using 2-dimensional regular ...

13

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. However, if $E$ does have CM, then $\text{End}(E)\otimes\mathbb Q\cong\mathbb Q(\tau)$, and Frobenius lifts to an endomorphism of $E$ that is in the ring of ...

13

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 $\mathbb A^2$ be the open immersion. Then there is a Leray spectral sequence relating the etale cohomology of $\mathbb A^2 \backslash \langle 0,0\rangle$ to the ...

13

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 characteristic zero simply says that it is enough to consider $\mathbb{C}$ (via Lefschetz principle), and that there you can use Hodge theory, the Hodge index theorem in ...

13

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 intersection into a smooth toric variety then $\pi_1(X)=\{1\}$.
In particular, for instance, abelian varieties of dimension at least $3$ cannot be realized as ...

answered Mar 19 '18 at 9:02

Francesco Polizzi

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13

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 is a separated, smooth morphism whose geometric fibers are affine spaces.
Definition 2. A morphism of schemes is a étale cohomological equivalence if ...

12

With your particular choice of $R$, the $H^2$ is $0$. More generally, if $R$ is a strictly Henselian regular local ring of dimension $2$, then by the purity for the Brauer group (in this particular case it is known and due to, I believe, Grothendieck; for a proof see Grothendieck "Le groupe de Brauer II", Prop. 2.3) $H^2_{et}(R \setminus \{ \mathfrak{m} \}, \...

answered May 9 '13 at 0:50

Kestutis Cesnavicius

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12

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 nontrivial class in $H^1(K, H^1(E_{\bar K}, \mathbf{Q}_\ell)(1))$ and thus corresponds to a non-split extension of $H^1(E_{\bar K}, \mathbf{Q}_\ell)(1)$ by $\...

12

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 the Zariski covering $X = U \cup V$. (Hint: use that $k[[t]]$ is strictly henselian.) Thus the etale Cech cohomology is the usual Cech cohomology. For a ...

11

Lei Fu, Étale Cohomology Theory is also nice and has not been mentioned yet.
And the lecture notes of Alexander Schmidt: http://theorics.yichuanshen.de/etale-kohomologie/ (unfortunately in German)

11

Dear Kestutis,
your question is integral in two ways: first of all, you would like to consider schemes over a whole DVR instead of the generic fiber only. Secondly, you would like to have a comparison theorem between two $\mathcal{O}_\overline{K}$-modules and not only between $K$-vector spaces. The two are tightly connected since $H_\text{dR}(\mathcal{X})\...

answered May 30 '12 at 2:00

Filippo Alberto Edoardo

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