6
votes
1answer
228 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
6
votes
0answers
166 views

The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...
6
votes
1answer
188 views

'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization of $\mathbb{Q}[t]$?

For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain ...
1
vote
0answers
187 views

'etale topology

Could you recommend me please some basic, self-contained books on 'etale topology. I read Yoshida's article "local class field theory via lubin-tate theory" and some people said that it is somehow ...
0
votes
1answer
296 views

The fibres of smooth projective families over all geometric points have isomorphic cohomology; are these isomorphisms 'functorial'?

Let $p:P\to S$ (and $p':P\to S$) be proper smooth morphisms of 'nice' schemes (one may assume that $S$ is a complex variety). It is well-known that the fibres of $p$ (and $p'$) over all geometric ...
7
votes
1answer
509 views

Points in sites (etale, fppf, … )

I asked a part of this in an earlier question, but that part of my question didn't receive precedence. Etale site is useful - examples of using the small fppf site? Let $X$ be a scheme (assume it ...
16
votes
1answer
820 views

Etale site is useful - examples of using the small fppf site?

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here: Points in sites (etale, fppf, ... ) There, ...
3
votes
2answers
532 views

Weights for etale cohomology: why does Deligne's definition work?

For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ ...
5
votes
1answer
352 views

The main idea in the proof of Artin's vanishing

Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees ...
0
votes
1answer
335 views

Lefschetz fixed point formula: an 'easy' proof; cohomology with compact support

I have two questions concerning the LFPF (for etale cohomology). Is there an easy 'explanation' of this statements (that could be understood by students)? In particular, I would like to get away ...
16
votes
1answer
910 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 ...
2
votes
1answer
443 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 ...
3
votes
1answer
529 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 $Z\to X$. ...
5
votes
0answers
311 views

Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?

For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' ...
1
vote
1answer
235 views

The cohomology of a $G_m$-bundle

Let $X$ be a smooth variety over an algebraically closed field (whose characteristic could be positive), $Y\to X$ is a $G_m$-bundle ($G_m=\mathbb{A}\setminus \{0\}$). Then I want to have a long exact ...
15
votes
5answers
4k views

Textbook for Etale Cohomology

What is the best textbook (or book) for studying Etale cohomology?
6
votes
1answer
438 views

Basic properties of Nisnevich cohomology; $l'$-topology?

I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...
2
votes
0answers
182 views

The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...
0
votes
0answers
213 views

Ordered Cech(-like) complexes that compute etale cohomology (of fields!)

It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...
1
vote
0answers
189 views

On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
2
votes
0answers
299 views

What are the easiest cases of base change (for sheaves on sites)?

I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
2
votes
0answers
236 views

Do inverse images respect flabby sheaves?

Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when ...
8
votes
0answers
340 views

Comparison of etale and formal etale cohomologies for l=p

Let $K$ be a finite extension of $\mathbb{Q} _p$ with a field of integers $\mathcal{O} _K$. Let $X$ be a semistable proper scheme over $\mathcal{O} _K$, and $\mathcal{X}$ the associated p-adic formal ...
4
votes
1answer
294 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, ...
0
votes
0answers
174 views

On 'special properties' of various 'sheaf image' functors for a local complete intersection morphism

Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived ...
3
votes
0answers
198 views

On (the cohomology of) Hensel pairs

I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$. I would like the following facts to be true (and to make sense!:)). a.) The motivic ...
2
votes
1answer
467 views

Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?

This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly. For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf ...
14
votes
1answer
962 views

Serre and Tate's conjectures on étale cohomology

In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures. Suppose that $X$ is a smooth proper ...
2
votes
0answers
192 views

structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety

Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
1
vote
0answers
138 views

When inverse image is conservative; a reference or a generalization?

I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
4
votes
0answers
314 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 ...
0
votes
2answers
501 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 ...
2
votes
1answer
363 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 ...
7
votes
0answers
437 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 ...
6
votes
2answers
973 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 ...
7
votes
1answer
677 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 ...
7
votes
4answers
963 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 $g$ ...