# Questions tagged [etale-cohomology]

for questions about etale cohomology of schemes, including foundational material and applications.

574
questions

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### fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map
let $\phi: R \to S$). Then the so called Amitsur complex
$R \to S^{\...

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**1**answer

776 views

### Proof of main theorems in étale cohomology theory

(In this question, $p$ can be $0$.)
I'm curious if theorems on étale cohomology can be proved by easier way.
For example, proper base change theorem. This theorem can be stated as the following way.
...

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279 views

### Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \, \mathbb{Q}_{ \ell } (r) )^G $?

Let $ k $ be a field and let $ X $ be a smooth projective variety over $ k $ of dimension $ d $.
We denote by $ \overline{X} = X \times_k \overline{k} \ $ the base change of $ X $ to the algebraic ...

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### The Mumford-Tate conjecture

The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...

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79 views

### Divisible elements in the cohomology of Milnor $K$-theory

As a consequence of the strong Tate conjecture over finite fields one can deduce (see here proposition 8.20) for every smooth variety $X$ over a finite field:
$$H_{cont}^i(X,\mathbb{Q}_l(n))=H_{Zar}^{...

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181 views

### Cohomological dimension for stacks

If $X$ is a scheme (maybe with conditions), I'm pretty sure that the ($\ell$-adic/de Rham) rational cohomology $H^*(X,\mathcal{F}$) of an $\ell$-adic sheaf/holonomic $D$-module $\mathcal{F}$ vanishes ...

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154 views

### Semisimplicity of the étale cohomology mod $p$

Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...

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511 views

### Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?

Let $K$ be a henselian valuation field with residue field $k$, then the decomposition group surjects onto Galois group of the residue field, with kernel the inertia subgroup, namely we have short ...

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### “Cute” applications of the étale fundamental group

When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...

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144 views

### Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra

By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...

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107 views

### Interpretation of some maps involving cohomology groups

I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...

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104 views

### What is the étale fundamental group of projective spaces over finite fields?

Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?

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318 views

### Giraud's proper base change for Gerbes - Elimination of Noetherian hypotheses

I was looking through Giraud's book Cohomologie Non-abelienne, and there is a very nice theorem that Giraud proves in the Noetherian case (Cohomologie Non-Abelienne VII.2.2):
Let $f:X\to Y$ be a ...

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101 views

### Computing monodromy groups of curves over function fields

Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...

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614 views

### On the definition of the etale site of an adic space

I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...

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200 views

### $\ell$-adic cohomology of a quotient by group action

Suppose $Y \to Y/G$ is the Galois cover induced from a finite group $G$ acting on a scheme $Y$ and that this is indeed a Galois cover with $Y/G$ a scheme. In my case $Y$ is the Drinfeld curve $\mathrm{...

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532 views

### Cohomology of resolution of singularity

If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see https://mathoverflow.net/q/...

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340 views

### Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole

Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by the two affine ...

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108 views

### Rigid \'etale cohohomology of flag variety minus its rational points e.g $p$-adic Drinfeld half plane

Let $Fl=G/B$ over $\mathbb Q_p$ be the flag variety of a quasi-split reductive group $G$ over $\mathbb Q_p$, then $X=Fl-Fl(Q_p)$ shall exist as a rigid analytic variety over $\mathbb Q_p$, how to ...

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254 views

### Stalks of limit sheaves

Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...

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486 views

### A very elementary question on the definition of sheaf on a site

I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U_{i} \...

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107 views

### Higher pushforwards of roots of unity

Suppose $X,Y$ are smooth projective varieties over a field $k$ with $Y$ one-dimensional, connected, and normal, and $f\colon X\to Y$ a proper, flat, surjective morphism so that its fibers are ...

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236 views

### Absolute purity for intersection cohomology

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then
$$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$
where $(\star)$ is true when $i$ is in addition regular.
Here ...

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222 views

### Applications of the Weight Monodromy conjecture

I think of the Weight Monodromy conjecture as an analogue of the Weil conjectures in the case of bad reduction. The Weil conjectures of course have lots of applications, from point counting to ...

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152 views

### Braverman-Gaitsgory definition of local acyclicity

In the Appendix B of the published version of ‘Geometric Eisenstein Series’ (but not found in the arXiv version), Braverman and Gaitsgory compare their notion of local acyclicity with Deligne’s. I ...

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162 views

### Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...

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409 views

### Fpqc-locally constant if and only if étale-locally constant?

Also in SE.
Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...

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121 views

### Sato-Tate over function fields

Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.
Over any point $x\in C$...

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157 views

### nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...

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267 views

### Etale fundamental group of an order in a number field

Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{...

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325 views

### What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?

(I asked it first in MathStackExchange but I haven't get an answer yet)
Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms.
For unramified ...

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147 views

### A Künneth formula for relative fiber products

There is a Künneth formula for the cohomology of a product of spaces $X\times Y$ in quite a lot of generality.
Is there a Künneth formula for relative fiber products $X\times_S Y$? The case I am most ...

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151 views

### Gysin map and $B\mathbf{G}_m$, confusion

Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object. In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've ...

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196 views

### Computing Hodge numbers by point counting

In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...

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212 views

### Etale cohomology and Kummer theory

If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...

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172 views

### Étale homotopy equivalent varieties are deformation equivalent

Let $k$ be an algebraically closed field of characteristic $p>0$.
Let $V_1$ and $V_2$ be étale simply-connected smooth proper varieties over $k$. Assume there is an isomorphism between the prime-to-...

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116 views

### Surjectivity of Étale cohomology

I've come across https://mathoverflow.net/q/172523, which says that if $D\subset Y$ is a contractible divisor, then we have a surjection
$$H^n(Y,\mathbb{C})\rightarrow H^n(D,\mathbb{C})$$
for $n\geq \...

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121 views

### To see that the fundamental class of a local complete intersection is independent of choice of regular sequence

In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...

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347 views

### How do I produce a basis of cohomology?

Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...

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167 views

### What unramified Galois representations come from geometry?

I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...

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166 views

### Suspension Theorem in $\mathbb{A}^1$-homotopy

In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...

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193 views

### $l$-adic Galois representations factor through a common finite quotient

Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...

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95 views

### Essential Image of the Étale Homotopy type

For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...

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### Homology of the étale homotopy type

$\DeclareMathOperator\Et{Et}$Let $X$ be a scheme and denote by $\Et(X)$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $A$, we have
$$H^n(\Et(X),...

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158 views

### $p$-adic Galois representation and Étale homology

Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...

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244 views

### How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?

In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...

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253 views

### Explicit linear object underlying $l$-adic cohomology for almost all $l$

If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...

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273 views

### Étale cohomology and normalization?

I have an argument, which I wonder if someone could check:
Let $X$ be an irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\...

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**1**answer

195 views

### Pushout of schemes and étale cohomology

Let $k$ be an algebraically closed field and $X,Y$ two $k$-schemes. We fix a $k$-point in $X$ and in $Y$ each, which we denote by abuse of notation by $P$. Since the pushout of schemes along closed ...

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**1**answer

277 views

### Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes

Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group....