The etale-cohomology tag has no usage guidance.

**17**

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### Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?

As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...

**5**

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**1**answer

434 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 ...

**15**

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**2**answers

694 views

### Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...

**2**

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**1**answer

273 views

### Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...

**6**

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**1**answer

390 views

### motivic t-structure and realisations

Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $
\mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...

**0**

votes

**1**answer

376 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 ...

**5**

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**1**answer

402 views

### Is there a “universal” cohomology theory for varieties over p-adic fields?

Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, ...

**6**

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**1**answer

336 views

### What is the importance of the conjectural semi-simplicity of the action of the Frobenius on the etale cohomology of a variety over a finite field ?

It is conjectured that the action of the Frobenius acting on the etale cohomology of an algebraic variety over a finite field is semisimple.
A first approximation of my question is :
What is the ...

**18**

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**1**answer

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 ...

**5**

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**1**answer

653 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 $X_0$)? ...

**7**

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**1**answer

393 views

### Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes?

Let $X$ be a smooth variety of finite type over a number field $k$. Let $\overline{X} = X \times_{k} \overline{k}$, and let $\ell$ be a prime. It's well known that if $X$ is proper, then the ...

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**0**answers

141 views

### Does mapping cylinder category have enough injectives?

Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to ...

**1**

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**1**answer

217 views

### Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?
In other words, is the following statement true?
If it is true, then, how can one prove it?
...

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**0**answers

197 views

### why is the local field Q_l not an etale sheaf over a scheme X?

I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?

**4**

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**1**answer

416 views

### Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried ...

**3**

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379 views

### Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...

**6**

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**0**answers

299 views

### Formality of algebraic varieties via l-adic cohomology?

The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, ...

**4**

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**1**answer

512 views

### etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...

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**1**answer

745 views

### Long exact sequence of cohomology and fibration

In an article I'm reading, the author is stating :
$O$ is isomorphic to the complement of a zero section of a line bundle over $X$. We have a long exact sequence of (étale) cohomology associated ...

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359 views

### classify \mu_n torsors

Recently I read in Milne's book "etale cohomology" that the set $H^1(X,\mu_n)$ ($X$ a scheme, $n$ a nature number, the cohomology is flat cohomology) can be described as the set of pairs $(L,\phi)$, ...

**4**

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**1**answer

393 views

### is generically split Azumaya algebra locally split?

Let $A$ be an Azumaya algebra over a scheme $X$ (or maybe more specifically a scheme of finite type over a field). Suppose that the restriction of $A$ to $U=X\setminus Z$ (where $Z$ is a closed set) ...

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**2**answers

208 views

### Detecting zero morphisms via an open subscheme and its complement.

In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the ...

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**1**answer

397 views

### Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...

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**1**answer

579 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 ...

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**1**answer

179 views

### How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...

**3**

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**1**answer

679 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 $Z\to X$. ...

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**0**answers

201 views

### What can one say about a smooth variety whose lower cohomology is trivial?

Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...

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766 views

### Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...

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**0**answers

625 views

### Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In
[S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70
Serre presents nine conjectures ...

**6**

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**1**answer

385 views

### How to compute the étale cohomology of the quotient of a surface by a finite group of automorphisms ?

Let $S$ be a smooth surface defined over a finite field $K$ of char. $p$. Let $G$ be a finite group of automorphisms of $S$. Let $Z\to S/G$ be the minimal resolution of the quotient of $S$ by $G$. ...

**3**

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424 views

### Can we decide if an abelian variety is simple by knowing its Zeta function ?

Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group.
Is ...

**4**

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**1**answer

272 views

### Reference wanted - etale sheaves on $X$ versus on $\overline{X}$

Hello,
Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I ...

**5**

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312 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**

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**1**answer

369 views

### vanishing of cohomology sheaves with supports and values in the multiplicative group

Let $X$ be a locally noetherian regular scheme and $Y$ be a closed subscheme of codimension $d > 0$ in every point. Why does it "immédiatement" (Grothendieck, Groupe de Brauer III, §6, p. 133 f.) ...

**3**

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195 views

### limit of étale cohomology with supports

I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology)
Let $Y \hookrightarrow X$ ...

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**1**answer

246 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 ...

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285 views

### Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?

As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on ...

**1**

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**1**answer

182 views

### combination of two duality theorems

For a variety $X$ over a finite field $k$, one combines étale Poincaré duality for $X \times_k \bar{k}$ with duality for $k$ ($H^0(k,M) \times H^1(k,M^\vee) \to H^1(k,\mathbf{Z}/n) = \mathbf{Z}/n$) to ...

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679 views

### $\ell$-adic Weil cohomology theory

I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...

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**1**answer

284 views

### Equivalent forms of the proper base change isomorphism

$\DeclareMathOperator{\Nat}{Nat}$In a current project, I am trying to "commute" $!$ and $*$ functors that are both upper or both lower. (Sheaf-theoretic context: constructible étale sheaves.) ...

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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 ...

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### project proposal: English translation of Deligne's “La conjecture de Weil : II” [closed]

First of all, I hope this "question" is appropriate here. If not, please delete it.
I would like to propose a translation project of Deligne's "La conjecture de Weil : II" ...

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**0**answers

251 views

### Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for ...

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**3**answers

581 views

### étale cohomology with values in an abelian scheme is torsion?

Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?
Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the ...

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340 views

### General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...

**3**

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323 views

### vanishing of étale cohomology groups with small support with values in an abelian scheme

Let $S/k$ be a smooth variety and $A/S$ be an abelian scheme. Let $Z \hookrightarrow S$ be a reduced closed subscheme of codimension $\geq 2$.
I want to show that in this situation, $H^i_Z(S, A) = 0$ ...

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710 views

### Etale cohomology in the $p$-adic setting

Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?
Recall that semialgebraic subsets are obtained from $p$-adic ...

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382 views

### is the presheaf of automorphisms a sheaf?

In Chapter III,$\S 4$ of Milne's Etale cohomology a correspondence between twisted forms and Cech cohomology cocycles is described.
Fix some Grothendieck topology, say, etale, and let $A$ be a ...

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### Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...