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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 ...


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 ...


19

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 $\Bbb R$. For example, the field $\Bbb Q(i)$ is totally imaginary, while $\Bbb Q$ and $\Bbb Q(\sqrt{3})$ are not. We fix an algebraic closure $\overline K$ of $...


18

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 $...


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

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 ...


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

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 ...


14

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 ...


14

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 ...


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

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

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 ...


13

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 description of the Thom isomorphism, hence, in the description/construction of Gysin maps, trace maps, and so forth. A natural way to make this transparent ...


13

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 given by the Gabber-Fujiwara theorem: see Corollary 6.6.4 in [1]. Let us consider the following general setup: let $(A, I)$ be a henselian couple with $A$ ...


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 ...


12

I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth varieties over the complex numbers and singular cohomology (theorem of Andreotti-Frankel) follows from the fact that Morse theory shows that the variety is homotopy equivalent ...


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} \}, \...


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

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 $X$ be a normal noetherian scheme. We'll show the higher etale cohomology with coefficients in any flat $\mathbf{Z}$-module $M$ is torsion, and hence vanishes when $...


11

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" with "geometrically unibranch.") To see this, recall the following presumably classical fact. Lemma: If $X$ is a normal integral scheme with generic point $j:\...


11

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 forms for $SL_2(\mathbb{Z})$. Blasius shows in the article "Modular forms and abelian varieties" (Séminaire de Théorie des Nombres, Paris, 1989–90, 23–29,...


11

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 numbers you can make it single valued by picking one branch. Riemann had a vastly better idea: there is a two-sheeted covering surface for the complex plane (...


11

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 $\pi_1(X)\text{ -}\operatorname{Set}_f$ of finite sets with a continuous $\pi_1(X)$-action. Under this correspondence, the $Y \to X$ finite étale with $Y$ connected correspond to the ...


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