Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
5
votes
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
$$
H_{et}(X, F)\...
1
vote
0answers
78 views
Etale-analytic comparison without elementary fibrations
A theorem due to Artin states that for a smooth scheme $X$ of finite type over $\mathbb{C}$ and a locally constant constructible sheaf $F$ we have an isomorphism
$$
H^*_{et}(X, F)\approx H^*(X(\...
3
votes
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
$$ \eta : ...
2
votes
0answers
133 views
Lefschetz trace formula over truncated Witt ring
Let $k$ be a finite field, $W_n(k)$ be its $n$-th truncated Witt ring. We have a Frobenius on $W_n(\bar{k})$ whose fixed point is exactly $W_n(k)$. Let $X$ be a finite type separated scheme over $W_n(...
8
votes
1answer
307 views
Picard group and reduced schemes
If $A$ is a ring, then we know that $Pic(A)=Pic(A_{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $Pic(X)=H^{1}_{et}(X,\mathbb{G}_m)$ and étale cohomology doesn'...
1
vote
0answers
87 views
Special formal lifts of smooth algebras
Let $A$ be a smooth algebra over $k$ a finite field.
Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$.
Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...
0
votes
0answers
67 views
Twisted sheaves on tower of $\mathbb{P}^n$
Take the projective space $\mathbb{P}^n$ over a ring $W$.
We call $\mathcal{O}(q)$ the usual twisted line bundle.
Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by
$$[x_0,\ldots, x_n]\...
3
votes
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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 ...
