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4
votes
1answer
383 views

Exact sequence in étale cohomology related with proper birational morphism

Let f:X→Y be proper birational morphism between two quasi-projective varieties over an algebraically closed field $k$. I am particularly interested in the case where the characteristic of $k$ is ...
4
votes
0answers
332 views

Étale cohomology of linear groups

This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of ...
4
votes
2answers
494 views

Relationship between topological cohomology and $\ell$-adic cohomology

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope ...
7
votes
1answer
2k views

A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
0
votes
2answers
515 views

Poincaré duality for smooth projective varieties over finite fields

What is exacly the statement of Poincaré duality for smooth projective varieties over finite fields and twisted constant $\mathbf{Z}_\ell$ sheaves? Where can I find a proof? By twisted constant ...
9
votes
2answers
574 views

Cohomological dimension-doubling

I'm sure this is a question which has been asked many times, if not necessarily on this site: Why does a (smooth, projective) scheme over a field, with dimension d, behave as though it were a ...
6
votes
3answers
804 views

What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure?

I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples. For example, it's a ...
10
votes
1answer
1k views

Leray-Hirsch principle for étale cohomology

Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. ...
14
votes
1answer
1k views

Are all Galois cohomology groups also étale cohomology groups?

Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology). By Grothendieck's Galois Theory, we have the isomorphism ...
2
votes
1answer
371 views

Reference for comparison of complex and étale cohomology

Hello everybody! I am looking for a nice DETAIL account of the comparison of étale cohomology and complex cohomology, an alternative reference instead of SGA. Especially on this stuff of "Artin ...
12
votes
1answer
854 views

Analogue of Shimura curves in the symplectic case?

My understanding is this: one can attach 2-d Galois representations to classical modular eigenforms because one can look in the etale cohomology of modular curves. For Hilbert modular forms the naive ...
7
votes
0answers
525 views

Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?

Let me recall two theorems : Let $K$ be a field, $\overline{K}$ be a separable closure of $K$ with absolute Galois group $G_K:=Gal(\overline{K}/K)$, and let $\ell$ be a prime that is ...
4
votes
1answer
646 views

2d Weil conjecture

Does there exist a two variable analogue of the Weil conjecture? What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points ...
15
votes
1answer
469 views

Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?

More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...
14
votes
6answers
1k views

weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come from Calabi-Yaus?

A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a ...
12
votes
1answer
914 views

Why does the naive definition of compactly supported étale cohomology give the wrong answer?

Illusie's article about étale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not ...
17
votes
1answer
1k views

Cohomology of sheaves in different Grothendieck topologies

Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
20
votes
2answers
3k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
6
votes
0answers
938 views

An example computation of etale cohomology

(edited for clarity) In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
3
votes
1answer
1k views

etale fundamental group and etale cohomology of curves

Given a curve $C$. Is there any relation between the etale fundamental group $\pi_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ? For example, if $C$ is a complex ...
6
votes
2answers
1k views

Applications of étale cohomology

It is well-known that étale cohomology is used in the proof of Weil conjectures and that SGA 4.5 is devoted to it. Also it seems(from a brief perusal of Milne's notes) that it is a kind of Galois ...
11
votes
1answer
819 views

Motivation for the étale topology over other possibilities

In the search for a Weil cohomology theory $H$ over a field $K$ (with $\text{char}(K)=0$) for varieties in characteristic $p$, a classical argument by Serre shows that the coefficient field cannot be ...
7
votes
1answer
708 views

étale cohomology with G_m coefficients

Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are ...
6
votes
1answer
751 views

Flat cohomology and Picard groups

Let $(R,m)$ be a local complete intersection of dimension $3$. Let $X=Spec(R)$ and $U=Spec(R) -\{m\}$ be the punctured spectrum of $R$. I am trying to understand the following comment by Gabber (see ...
21
votes
3answers
603 views

A hypersurface with many points

Ok, it's time for me to ask my first question on MO. Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...
14
votes
4answers
3k views

Etale cohomology and l-adic Tate modules

$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$ Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
10
votes
2answers
449 views

intuition about the “section after base-change” for flat descent and exactness of the Amitsur complex

Suppose $A \rightarrow B$ is a faithfully flat map of rings. Then the Amitsur complex is exact: $0 \rightarrow A \rightarrow B \rightarrow B \otimes_A B \rightarrow \dots$ (the second map is $id ...
7
votes
2answers
427 views

When can cohomology be calculated on the coarse moduli space?

Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that ...
49
votes
4answers
6k views

Etale cohomology — Why study it?

I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...
6
votes
2answers
559 views

Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
2
votes
3answers
1k views

Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in : Algebraic Geometry sources: Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
20
votes
7answers
2k views

Etale covers of the affine line

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
8
votes
4answers
976 views

cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
2
votes
2answers
663 views

roots of analytic functions

Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
8
votes
4answers
1k views

Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with q elements says that: the eigenvalues of ...