Questions tagged [etale-cohomology]

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

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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 $S$ ...
Yang's user avatar
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3 votes
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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 ...
Asvin's user avatar
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2 votes
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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 ...
Born to be proud's user avatar
3 votes
0 answers
525 views

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 ...
Ronald J. Zallman's user avatar
2 votes
1 answer
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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 ...
Adrien MORIN's user avatar
5 votes
1 answer
286 views

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\...
Adrien MORIN's user avatar
3 votes
0 answers
239 views

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, -- ...
AlexIvanov's user avatar
5 votes
1 answer
471 views

$\mathbf{A}^1$-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 ...
Xing Gu's user avatar
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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 ...
Adrien MORIN's user avatar
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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/\...
Hajime_Saito's user avatar
1 vote
1 answer
200 views

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,\...
user41282141's user avatar
3 votes
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160 views

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 ...
Mikhail Bondarko's user avatar
1 vote
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230 views

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 ...
Hajime_Saito's user avatar
1 vote
1 answer
175 views

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 ...
user267839's user avatar
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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 ...
user127776's user avatar
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2 votes
1 answer
331 views

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 ...
user267839's user avatar
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5 votes
1 answer
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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 ...
user avatar
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265 views

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. ...
user267839's user avatar
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1 vote
1 answer
618 views

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 ...
oleout's user avatar
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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 ...
oleout's user avatar
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540 views

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 ...
user avatar
5 votes
1 answer
389 views

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 ...
user avatar
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{...
user avatar
1 vote
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 ...
user127776's user avatar
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5 votes
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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 ...
Alessandro's user avatar
1 vote
1 answer
282 views

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 ...
oleout's user avatar
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8 votes
1 answer
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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 ...
Stefan Witzel's user avatar
3 votes
0 answers
496 views

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 ...
Yuan Yang's user avatar
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3 votes
0 answers
182 views

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 ...
Unmotivated L-function's user avatar
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 ...
Tommaso Scognamiglio's user avatar
2 votes
0 answers
191 views

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{...
curious math guy's user avatar
3 votes
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^{\...
user267839's user avatar
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16 votes
1 answer
1k 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. ...
user avatar
3 votes
2 answers
369 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 ...
Bradley04's user avatar
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8 votes
2 answers
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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 ...
Bradley04's user avatar
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1 vote
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}^{...
user127776's user avatar
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3 votes
0 answers
311 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 ...
Pulcinella's user avatar
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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^{...
curious math guy's user avatar
13 votes
1 answer
733 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 ...
user avatar
45 votes
3 answers
5k views

"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 ...
Libli's user avatar
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5 votes
0 answers
166 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 ...
user127776's user avatar
  • 5,831
2 votes
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 ...
oleout's user avatar
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1 vote
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 $\mathbf{P}^2_{\mathbf{F}_q}$?
hennlu's user avatar
  • 323
9 votes
1 answer
744 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 ...
Harry Gindi's user avatar
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2 votes
0 answers
168 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 ...
Asvin's user avatar
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9 votes
1 answer
1k 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 ...
RumDiary's user avatar
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4 votes
0 answers
319 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{...
TCiur's user avatar
  • 469
6 votes
2 answers
1k 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/...
curious math guy's user avatar
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 ...
Stabilo's user avatar
  • 1,479
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 ...
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