All Questions
Tagged with sheaf-theory etale-cohomology
40 questions
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
0
votes
0
answers
133
views
Higher direct images of locally constant etale sheaf under smooth proper map locally constant
Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...
- 6,038
3
votes
1
answer
260
views
Etale cohomology of relative elliptic curve
Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...
- 2,907
4
votes
0
answers
216
views
When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
- 6,038
0
votes
0
answers
156
views
A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,038
3
votes
1
answer
550
views
Characterization of étale locally constant sheaves over a normal scheme
I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...
- 6,038
1
vote
1
answer
176
views
Field of fractions of etale stalk of Dedekind domain (Example from Milne's LEC)
Let $X=\operatorname{Spec}(A)$ be an affine Dedekind domain with field of fractions $K$. Let $\widetilde{A}$ be the integral closure of $A$ in separable closure $ K^{\text{sep}}$. A closed point $x$ ...
- 6,038
2
votes
1
answer
290
views
Calculate stalk of etale derived pushforward sheaf (Milne's LEC)
Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale ...
- 6,038
5
votes
0
answers
220
views
Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
- 605
2
votes
0
answers
114
views
Two natural morphisms of sheaves with the same source and target; do they agree?
Suppose we have a diagram
$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V @VV c V\\
C @>>d> D @>e>> E \\
@VfVV @VVgV @VVhV \\
F @>>i> G @>>j> H
\end{CD}...
- 125
3
votes
1
answer
249
views
Sheafifcation for the étale site
Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.
For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\...
- 481
2
votes
0
answers
167
views
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 ...
- 617
2
votes
1
answer
383
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 ...
- 6,038
8
votes
0
answers
681
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
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
- 4,418
8
votes
2
answers
532
views
A very elementary question on the definition of sheaf on a site
I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U_{i} \...
- 1,013
1
vote
0
answers
174
views
Galoisian perspective on local system tamely ramified along a smooth divisor
This question is about (1.7.8) and (1.7.11) in Deligne’s Weil II paper.
Let $X$ be a regular scheme and $D\subset X$ a smooth principal divisor cut out by the function $t$. Let $\mathcal F$ be a ...
- 1,217
5
votes
1
answer
540
views
Big etale topos vs small etale topos
Are they equivalent?
That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If ...
- 4,164
7
votes
2
answers
627
views
Vanishing of higher direct image of finite morphisms relative to the fppf topology
Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
- 345
6
votes
1
answer
281
views
Cokernel of map of étale sheaves
Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
- 115
1
vote
0
answers
180
views
Weaker version of smooth base change for étale sheaves
Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...
- 11
2
votes
0
answers
193
views
Usage of Leray spectral sequence
Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
- 1,428
2
votes
0
answers
72
views
Support of étale sheaves
Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...
user120812
4
votes
0
answers
432
views
Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
- 273
7
votes
2
answers
839
views
What is the need for torsion in the definition of lisse sheaves?
I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
- 5,670
1
vote
0
answers
101
views
How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
13
votes
2
answers
3k
views
Interpreting $f^*f_*$
For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...
- 3,555
2
votes
0
answers
175
views
Does mapping cylinder category have enough injectives?
Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \...
- 945
1
vote
0
answers
238
views
why is the local field Q_l not an etale sheaf over a scheme X?
I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?
- 11
5
votes
0
answers
336
views
Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?
For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
- 16.9k
1
vote
0
answers
249
views
On inverse images with respect to Zariski-etale topology.
For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
- 16.9k
2
votes
0
answers
358
views
What are the easiest cases of base change (for sheaves on sites)?
I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
- 16.9k
2
votes
0
answers
414
views
Do inverse images respect flabby sheaves?
Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
- 16.9k
1
vote
0
answers
236
views
Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?
For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions ...
- 16.9k
8
votes
1
answer
1k
views
Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?
This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.
For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $...
- 16.9k
1
vote
0
answers
333
views
Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?
In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....
- 945
3
votes
1
answer
463
views
For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?
Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
- 9,497
4
votes
2
answers
570
views
If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?
In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension ...
- 16.9k
18
votes
2
answers
4k
views
Locally constant sheaves for the étale topology, lack of intuition about "étale-localness"
I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
- 291
1
vote
0
answers
424
views
Another stupid question on l-adic sheaves: does a generically zero constructible sheaf vanish on an open subvariety?
Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-adic or $\mathbb{Q}_l$-adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes ...
- 16.9k
6
votes
2
answers
790
views
Do quotients of representable sheaves represent quotients?
Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
- 1,988