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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}...
Wilhelm's user avatar
  • 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 ...
user267839's user avatar
  • 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}...
kindasorta's user avatar
  • 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 ...
user267839's user avatar
  • 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} \...
user267839's user avatar
  • 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 ...
user267839's user avatar
  • 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 $x$ ...
user267839's user avatar
  • 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 ...
user267839's user avatar
  • 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 \...
C.D.'s user avatar
  • 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}...
user333154's user avatar
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}}\...
H U's user avatar
  • 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 ...
Adrien MORIN's user avatar
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 ...
user267839's user avatar
  • 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 ...
curious math guy's user avatar
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} \...
gualterio's user avatar
  • 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 ...
Tomo's user avatar
  • 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 ...
Kim's user avatar
  • 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 ...
grok's user avatar
  • 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 $T\to p_*p^*T$ ...
HuynA's user avatar
  • 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 : ...
Michele's user avatar
  • 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 $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
User0829's user avatar
  • 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 ...
user avatar
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 ...
Student's user avatar
  • 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 ...
Filippo Alberto Edoardo's user avatar
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}\...
Mikhail Bondarko's user avatar
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 ...
LMN's user avatar
  • 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 \...
Hiro's user avatar
  • 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?
John's user avatar
  • 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' (...
Mikhail Bondarko's user avatar
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 ...
Mikhail Bondarko's user avatar
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 ...
Mikhail Bondarko's user avatar
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*...
Mikhail Bondarko's user avatar
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 ...
Mikhail Bondarko's user avatar
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 $...
Mikhail Bondarko's user avatar
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....
Hiro's user avatar
  • 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 ...
Ariyan Javanpeykar's user avatar
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 ...
Mikhail Bondarko's user avatar
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 ...
Lorenzo's user avatar
  • 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 ...
Mikhail Bondarko's user avatar
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 ...
Rebecca Bellovin's user avatar