Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
704
questions
2
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153
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Nearby cycle is tamely ramified?
Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
3
votes
0
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162
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Smooth proper varieties over the integers that are not toric
Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
2
votes
1
answer
144
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Should the identity labelled by red line be $\overline{f(Z)}=X$?
The above picture is from Milne's Etale Cohomology.
Suppose $A=\Bbb Z, \mathfrak q=(2T+3)$, consider $Z=\operatorname{Spec} \Bbb Z[T]/(2T+3)\to \operatorname{Spec} \Bbb Z[T]\to\operatorname{Spec} \Bbb ...
3
votes
0
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525
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Most general form of Poincaré duality in étale cohomology
I am interested in Poincaré duality from the point of view of Grothendieck's 6-functor formalism. I am predominantly interested in the proof that Poincaré duality holds in étale cohomology from this ...
2
votes
1
answer
144
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Extending the domain of the yoneda embedding map from étale schemes to the small étale topos so that it is still fully faithful
Let $X$ be a scheme. For $Y$ a scheme over $X$, the representable presheaf $h_Y : U\mapsto \mathrm{Hom}_X(U,Y)$ on the small étale site $X_{et}$ is actually a sheaf, and by the Yoneda lemma the ...
5
votes
1
answer
286
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On realizing a topos of sheaves as a topos of equivariant sheaves
This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...
3
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0
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239
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Ind-etale vs weakly etale
In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5):
-- any ind-etale map is weakly etale,
-- ...
5
votes
1
answer
471
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$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$
The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first ...
2
votes
0
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157
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Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules
This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...
1
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0
answers
126
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understanding higher direct images of $\mathbb{G}_m$ for a finite Galois map
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$, and let $\mu_r$ denote the group of $r$-th roots of unity, and moreover suppose $\mu_r$ (algebraically) acts on $X$ freely. Then $Y:= X/\...
1
vote
1
answer
200
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Lowest weight of compactly supported cohomology with coefficients
Let $X_0/\mathbb F_q$ be a variety, and let $\mathcal F$ be a Weil sheaf on $X := (X_0)_{\overline{\mathbb{F}_q}}$ that is pure of weight $n$. If $j < n$, does the weight $j$ piece of $H^i_c(X,\...
3
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0
answers
160
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When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?
Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...
1
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0
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230
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Higher direct images of $\mathbb{G}_m$ under a projective bundle
Let $X$ be a smooth projective rational variety over $\mathbb{C}$, and let $\pi:Y\rightarrow X$ be a principal projective bundle with fibers isomorphic to $SL(n,\mathbb{C})/P$, where $P$ is a ...
1
vote
1
answer
175
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Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl )
I have question about a statement from Etale Cohomology and the Weil Conjecture by Freitag, Kiehl
at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15
and ...
2
votes
0
answers
112
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The cohomology groups corresponding to a modified global sections functor
Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...
2
votes
1
answer
331
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Some facts about sheafification functor on étale site
I'm studying the book Etale cohomology and the Weil conjecture by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is ...
5
votes
1
answer
512
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Ordinary abelian varieties and Frobenius eigenvalues
Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We ...
0
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0
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265
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Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the Weil Conjecture by Freitag/Kiehl)
I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15.
It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. ...
1
vote
1
answer
618
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Cohomology with coefficients in $\mu_\infty$
I'm encountering a lot of problems when dealing with the root of unity sheaf $\mu_\infty := \mathrm{colim}_n\mu_n$.
Let $X$ be a smooth geometrically integral variety over a number field $k$. Although ...
3
votes
0
answers
149
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Obtaining an exact sequence of Galois modules via derived functors
This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there.
Let ...
3
votes
1
answer
540
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Subrings of Chow rings
Let $X$ be a smooth projective variety over $\mathbf{F}_p$, call $\overline{X}$ the base change to $\overline{\mathbf{F}}_p$, and denote by $F$ the base change to $\overline{X}$ of the absolute ...
5
votes
1
answer
389
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On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
8
votes
1
answer
415
views
Minimal vs characteristic polynomial of geometric Frobenius
Assume $X$ is a smooth projective variety over $\overline{\mathbf{F}}_p$ and fix a prime $\ell\neq p$.
Let $F_i$ be the geometric Frobenius on $\ell$-adic cohomology
$$H^i_{\rm ét}(X,\overline{\mathbf{...
1
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0
answers
211
views
Cohomological dimension of continuous étale cohomology of finitely generated fields
Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.
In the discussion right after [K, Lemma 2.3], the ...
5
votes
0
answers
500
views
Generalization of Weil Conjectures
is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...
1
vote
1
answer
282
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A question about a truncated object
I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:
For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
8
votes
1
answer
2k
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Cohomology of Grothendieck topology
My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...
3
votes
0
answers
496
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Etale cohomology of a nodal (cuspidal) curve
Let $k$ be a separably closed field, and $X/k$ be a curve (not necessarily complete) with a single singularity, a simple node $x$. Suppose $\ell$ is a prime number invertible in $k$, how do we compute ...
3
votes
0
answers
182
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Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
6
votes
1
answer
453
views
Irreducible components of an algebraic stack
Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $ k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of ...
2
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0
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191
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Galois-action on spectral sequence
Let $X_\bullet\to S$ be a proper surjective hypercover of a $k$-scheme by smooth proper $k$-schemes. This gives a proper surjective hypercover $X'_\bullet\to S_{\bar{k}}$ where $X'_n:=X_n\times_k \bar{...
3
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0
answers
557
views
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^{\...
16
votes
1
answer
1k
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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.
...
3
votes
2
answers
369
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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 ...
8
votes
2
answers
2k
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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 ...
1
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0
answers
105
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}^{...
3
votes
0
answers
311
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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 ...
4
votes
1
answer
242
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^{...
13
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1
answer
733
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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 ...
45
votes
3
answers
5k
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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 ...
5
votes
0
answers
166
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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 ...
2
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0
answers
158
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 ...
1
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0
answers
194
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 $\mathbf{P}^2_{\mathbf{F}_q}$?
9
votes
1
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744
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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 ...
2
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0
answers
168
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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 ...
9
votes
1
answer
1k
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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 ...
4
votes
0
answers
319
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$\ell$-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{...
6
votes
2
answers
1k
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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/...
3
votes
0
answers
432
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 ...
2
votes
0
answers
150
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 ...