For questions on the étale cohomology groups of an algebraic variety or scheme, algebraic analogues of the usual cohomology groups with finite coefficients of a topological space.

# Questions tagged [etale-cohomology]

260 questions

**110**

votes

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### Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture for the Fermat variety $X_m^r$, defined by the…

Alex

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votes

**1**answer

### Has SGA 4½ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it, as this seemed like the best way to get the news out.

Daniel Miller

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**23**

votes

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### What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the other hand $\varprojlim…

user10676

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### In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7):
The goal of p-adic Hodge theory is to identify and study various “good” classes of
$p$-adic representations of $G_K$ for p-adic fields $K$,…

Joachim

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**20**

votes

**2**answers

### Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be essentially nonexistent.
I think that I understand…

user101616

**18**

votes

**1**answer

### Why étale?

Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions and theories such as étale cohomology, a Galois…

user554397

**15**

votes

**1**answer

### Relationship between Galois cohomology and etale cohomology.

Why is étale cohomology a natural generalization of Galois cohomology ?
I would like to inform you that I have a few quite sufficient prerequisites Galois cohomology and its application to solve the $90$ - th problem of Hilbert . So I can…

Albert007

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votes

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### The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continuous map $f\colon (X,x)\rightarrow (Y,y)$ induces a homomorphism $f_* \colon \pi_1 (X,x)\rightarrow \pi_1…

Alex Saad

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### What is the intuition behind the Fontaine-Mazur Conjecture?

The Fontaine-Mazur conjecture (over $\textbf{Q}$ for simplicity) says that a (continuous irreducible) Galois representation
$$
\rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to GL_n(\overline{\textbf{Q}}_\ell)
$$
"comes from geometry," that is,…

Miles Lake

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### Understanding etale cohomology versus ordinary sheaves

I am a physicist trying to understand etale cohomology from Shafaverich, and I would like to check a misunderstanding, undoubtedly.
When defining etale cohomology, it seems it is sheaf cohomology in the sense of right-derived functors, but with the…

JPhy

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**10**

votes

**1**answer

### Étale cohomology of projective space

I have some very basic question about étale cohomology.
Namely I would like to compute the étale cohomology of of the projective space over the algebraic closure of $\mathbb F_q$ along with its Frobenius operation:
$$H^i(\mathbb P^n_{\mathbb…

Jan

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**10**

votes

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### Lack of "good covers" in the étale topology

Disclaimer: This question might be terribly naive and almost certainly reflects my own ignorance.
If $X$ is a topological space admitting a finite triangulation, then it admits a "good covering," i.e. an open covering by contractible sets $U_i$ such…

Keenan Kidwell

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**9**

votes

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### Comparison between etale and singular cohomology for a singular variety

In his Lectures on Etale Cohomology Milne proves in Theorem 21.1, that for all $r\geq 0$
$$
H^r_{\acute{e}t}(X,\Lambda)\cong H^r(X_{cx},\Lambda)
$$
with $X$ a nonsingular $\mathbb{C}$-variety and $\Lambda$ a finite abelian group. The right-hand-side…

Ronald Bernard

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### Kummer sequence etale topology

Consider the category $C=Sch/S$ of schemes over $S$ and let $n \in \Gamma(S,\mathcal{O}_S)^{*}$. It is possible to show that $$0 \rightarrow \mu_{n,S} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$$
is exact in the etale topology,…

Tommaso Scognamiglio

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votes

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### Computing étale cohomology group $H^1( \text{Spec}(k), \mu_n)$ and $H^1( \text{Spec}(k), \underline{\Bbb{Z}/\mathord{n \Bbb{Z}}})$

I am starting to learn about étale cohomology and would like to compute a simple example. Let $k$ be a field with a fixed separable extension $k^s.$ I want to compute $H^1( \operatorname{Spec}(k), \mu_n)$ (I am speaking about the étale cohomology…

proofromthebook

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