Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
574
questions
1
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0answers
143 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^{\...
11
votes
1answer
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.
...
3
votes
2answers
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 ...
6
votes
2answers
1k views
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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0answers
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}^{...
3
votes
0answers
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 ...
2
votes
1answer
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^{...
11
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1answer
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 ...
41
votes
3answers
3k 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 ...
5
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0answers
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 ...
2
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0answers
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 ...
1
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0answers
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}$?
7
votes
1answer
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 ...
2
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0answers
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 ...
8
votes
1answer
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 ...
3
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0answers
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{...
5
votes
2answers
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/...
3
votes
0answers
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 ...
2
votes
0answers
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 ...
6
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0answers
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 ...
7
votes
2answers
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} \...
1
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0answers
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 ...
3
votes
0answers
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 ...
15
votes
0answers
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 ...
3
votes
0answers
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 ...
5
votes
0answers
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 ...
1
vote
1answer
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)...
4
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0answers
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 $x\in C$...
4
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0answers
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],\...
6
votes
1answer
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{...
5
votes
1answer
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 ...
3
votes
0answers
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 ...
4
votes
0answers
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 ...
6
votes
0answers
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 ...
3
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0answers
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, ...
3
votes
0answers
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-...
2
votes
0answers
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 \...
1
vote
0answers
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-...
10
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0answers
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). ...
2
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0answers
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}...
4
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1answer
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 ...
3
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0answers
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 ...
1
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0answers
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 ...
5
votes
1answer
510 views
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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0answers
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). ...
9
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0answers
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 ...
4
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0answers
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.
...
4
votes
0answers
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}\...
4
votes
1answer
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 ...
3
votes
1answer
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....
