# Questions tagged [etale-cohomology]

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

**5**

votes

**1**answer

237 views

### How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...

**10**

votes

**1**answer

378 views

### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

**6**

votes

**1**answer

325 views

### Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...

**5**

votes

**0**answers

128 views

### Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?

Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...

**6**

votes

**1**answer

426 views

### Quaternion algebra actions on $\ell$-adic cohomology

Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
...

**7**

votes

**0**answers

120 views

### Invariants of etale topological type that are not homotopy invariants

Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...

**5**

votes

**0**answers

164 views

### Cohomology groups on small fppf site and small etale site are not the same

Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?

**3**

votes

**0**answers

100 views

### Etale cohomology of projective spaces in the rigid analytic setting

Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...

**4**

votes

**1**answer

179 views

### Henselianizations over countable index sets

Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...

**3**

votes

**1**answer

97 views

### Etale homotopy type of non-unibranch scheme over $\mathbb{C}$

In these notes the following theorem is stated, among other things.
Let $X$ be a pointed connected geometrically unibranch scheme over
$\mathbb{C}$. Then Artin-Mazur etale homotopy type of $X$ is ...

**2**

votes

**0**answers

101 views

### Fundamental Group of small Zariski open set

Let $Y$ be an integral affine variety over $\mathbb{C}$ and $K$ be its function field. How to find a sufficiently small Zariski open set of $Y$ such that it is isomorphic to $K(\pi,1)$? Here $\pi$ is ...

**2**

votes

**0**answers

154 views

### Can etale-analytic comparison hold when etale-Cech comparison doesn't?

Assume we have a scheme over $\mathbb{C}$ and a constructible sheaf on $X$. We have a natural morphism from etale cohomology to derived functor cohomology in complex-analytic topology
$$
H_{et}(X, F)\...

**1**

vote

**0**answers

78 views

### Etale-analytic comparison without elementary fibrations

A theorem due to Artin states that for a smooth scheme $X$ of finite type over $\mathbb{C}$ and a locally constant constructible sheaf $F$ we have an isomorphism
$$
H^*_{et}(X, F)\approx H^*(X(\...

**3**

votes

**1**answer

137 views

### Direct limit of strict henselizations

Assume we have a map $A \rightarrow A'$ of strictly henselian local rings, such that the induced map between spectra $S'\rightarrow S$ is essentially smooth. Is is true that $S'$ is a direct limit of ...

**3**

votes

**0**answers

135 views

### Finiteness of $H^2(X,\mu_n)$

Let $X$ be a proper curve over $k$ (algebraically closed) of characteristic $p>0$.
When is $H_{fl}^2(X,\mu_n)$ is a finite group?
It's true when $X$ is smooth but are there any more general ...

**0**

votes

**0**answers

98 views

### When is the $p$-power on etale sheaves $\mathbb{G}_m$ injective

Let $X$ be a proper scheme over $k$ where the characteristic of $k$ is $p>0$.
Consider the etale sheaves $\mathbb{G}_m$ over $X$ and consider the $p$-th power map from $\mathbb{G}_m \to \mathbb{G}...

**4**

votes

**0**answers

106 views

### Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...

**1**

vote

**0**answers

142 views

### Complex varieties which are not homotopic in complex-analytic topology but have the same etale homotopy types

Do there exist smooth quasi-projective complex varieties $X_1$, $X_2$ such that $X_1(\mathbb{C})$ is not homotopy equivalent to $X_2(\mathbb{C})$ but their etale homotopy types coincide?

**3**

votes

**1**answer

139 views

### Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...

**6**

votes

**1**answer

190 views

### Frobenius eigenvalues algebraic numbers

Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...

**6**

votes

**1**answer

154 views

### The weight filtration on etale cohomology and Berkovich analytic geometry

If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also ...

**1**

vote

**0**answers

91 views

### Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...

**4**

votes

**1**answer

307 views

### etale higher direct image sheaf

Let $f:X\rightarrow Y$ be a smooth morphism of schemes such that all the fibres (for geometric points) are affine spaces. Let $F$ be a coherent sheaf on $X$. Is $R^i_{et}~f_*F=0~~~\forall i>0$? ...

**6**

votes

**1**answer

248 views

### Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...

**0**

votes

**1**answer

102 views

### Smooth loci and formal neighborhoods

Let $R$ be a Noetherian local ring with maximal ideal $I$.
Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$
$$f_n : A/I^n \to B/I^n$$
is an ...

**5**

votes

**1**answer

215 views

### What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space.
...

**14**

votes

**2**answers

438 views

### When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?

Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}...

**1**

vote

**0**answers

125 views

### Group scheme representation from action of a group scheme on a variety?

Let f be some homogenous polynomial of d.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$
where $k$ is algebraically closed field of characteristic $p>0$.
Now $G$ is a group scheme acting on $X$.
...

**1**

vote

**0**answers

166 views

### Cohomology and base change

Let $f$ be some homogenous polynomial of $d$.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$.
Now let $R$ be a $k$-algebra.
What ...

**4**

votes

**1**answer

228 views

### Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...

**11**

votes

**1**answer

310 views

### “Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion ...

**7**

votes

**1**answer

374 views

### Vector space objects in schemes - confusion

Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...

**15**

votes

**0**answers

321 views

### Étale cohomology of varieties in positive characteristic via singular cohomology

Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \...

**2**

votes

**0**answers

131 views

### Specialization map on geometric points

Let $\mathcal{X}$ be a proper and smooth scheme over $\text{Spec}(\mathbf{Z}_p)$, and let’s call $X$ the geometric generic fiber of $\mathcal{X}$, and $X_0$ the geometric special fiber of $\mathcal{X}$...

**1**

vote

**0**answers

115 views

### Lefschetz trace formula and Frobenius elements

Let $\mathcal{X}$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z}_p)$ (with $\mathbf{Z}_p$ the $p$-adic integers).
Let’s call $X$ its geometric generic fiber $X := \mathcal{X}_{\overline{\...

**2**

votes

**0**answers

145 views

### Weaker version of smooth base change for étale sheaves

Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...

**2**

votes

**0**answers

133 views

### Lefschetz trace formula over truncated Witt ring

Let $k$ be a finite field, $W_n(k)$ be its $n$-th truncated Witt ring. We have a Frobenius on $W_n(\bar{k})$ whose fixed point is exactly $W_n(k)$. Let $X$ be a finite type separated scheme over $W_n(...

**8**

votes

**1**answer

307 views

### Picard group and reduced schemes

If $A$ is a ring, then we know that $Pic(A)=Pic(A_{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $Pic(X)=H^{1}_{et}(X,\mathbb{G}_m)$ and étale cohomology doesn'...

**1**

vote

**0**answers

87 views

### Special formal lifts of smooth algebras

Let $A$ be a smooth algebra over $k$ a finite field.
Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$.
Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...

**0**

votes

**0**answers

67 views

### Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$.
We call $\mathcal{O}(q)$ the usual twisted line bundle.
Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by
$$[x_0,\ldots, x_n]\...

**3**

votes

**0**answers

151 views

### Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.
If ...

**6**

votes

**1**answer

206 views

### Smooth algebras always lift

Let $k$ be a finite field, $A$ a smooth $k$-algebra.
Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?

**7**

votes

**0**answers

115 views

### Do residues commute with transverse base change?

Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...

**3**

votes

**1**answer

113 views

### Liftings and closed immersions

Let $A$ be a flat $\mathbf{Z}_p$-algebra, $\overline{I}\subset A/p$ an ideal.
Can we find an ideal $I\subset A$ such that
$I$ mod $p$ = $\overline{I}$
$I$ does not contain $p$.
It's harder than it ...

**4**

votes

**1**answer

229 views

### Coherent modules over complete adic rings: counterexamples

Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$.
Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...

**2**

votes

**0**answers

118 views

### Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...

**2**

votes

**0**answers

156 views

### Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...

**1**

vote

**1**answer

155 views

### Approximation of constructible abelian étale sheaves

Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$.
Suppose $X = \varprojlim X_j$, ...

**3**

votes

**1**answer

172 views

### Projective embeddings and quasi-compactness

Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding.
Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...

**4**

votes

**0**answers

135 views

### What is the map induced by Frobenius on the cohomology group $H_{fppf}^1(X, \mathbb{G}_a)$?

Let $X$ be a projective curve of degree $p$ in $\mathbb{P}^2_k$, where $k$ is a field of charcteristic $p>0$. $X$ may not be reduced. I'm trying to compute $H_{fppf}^1(X, \alpha_p)$ using the ...