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.

learn more… | top users | synonyms

22
votes
5answers
6k views

Textbook for Etale Cohomology

What is the best textbook (or book) for studying Etale cohomology?
22
votes
2answers
4k 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 ...
8
votes
1answer
3k 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.
17
votes
2answers
1k views

Etale cohomology with coefficients in the integers

Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\...
14
votes
2answers
883 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 ...
8
votes
2answers
1k views

locally constant constructible sheaves and finite etale coverings

Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way. We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
19
votes
1answer
1k views

Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
8
votes
1answer
860 views

Semisimplicity of Frobenius operation on etale cohomology?

Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$. Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? ...
10
votes
0answers
323 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
4
votes
1answer
817 views

Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology? Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$. What if $G = \...
4
votes
1answer
831 views

The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...
4
votes
2answers
524 views

Twisted forms and $\check{H}^1$

I am reading Milne's Étale cohomology, III.4. A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in ...
4
votes
1answer
387 views

Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves

I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...
4
votes
1answer
729 views

A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field

If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH_{et}(X,\mathbb{Z}/...
3
votes
2answers
196 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 ...
2
votes
1answer
156 views

Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...
6
votes
1answer
441 views

Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$?

Let $F$ be a $p$-adic field. Let $(G_{F}, W_{F}, I_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$. Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne p$...
2
votes
1answer
320 views

commutative diagram with Yoneda pairing, Weil pairing and edge morphism

Why does the following diagram commute?$\require{AMScd}$ \begin{CD} H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\ H^1(...