All Questions
Tagged with etale-cohomology constructible-sheaves
15 questions
3
votes
1
answer
205
views
Inertia Action on Kummer Sheaves
In 7.0.2 of Katz's book "Gauss Sums, Kloosterman Sums, and Monodromy Groups", Katz states the following (when $x=0$).
Let $\chi:\mathbb{F}_q^\times\to\mathbb{Q}_\ell^\times$ be a ...
- 65
2
votes
0
answers
80
views
Lift of nearby cycles functor
Let $S$ be the spectrum of a Henselian discrete valuation ring (called a Henselian trait). Let $f:X\to S$ be a finite type, separated morphism of schemes. Let $\eta\in S$ be the generic point. Let $s\...
- 615
2
votes
1
answer
200
views
Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.
Let $f:X = \text{...
- 6,018
2
votes
0
answers
227
views
"Simple Limit Argument" in Freitag's and Kiehl's Etale Cohomology
I have a question about an argument used in Freitag's and Kiehl's Etale Cohomology and the Weil Conjecture in the proof of:
4.4 Lemma. (p 41) Every sheaf $F$ representable by an étale scheme $U \to X$,...
- 6,018
2
votes
0
answers
206
views
Stratified sites/topoi and constructible sheaves
Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like ...
- 1,516
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
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
2
votes
1
answer
329
views
Help with $\mathbf{Q}_{\ell}$ sheaves
Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (not a geometric generic point) and ...
- 153
7
votes
1
answer
996
views
Generalized Behrend version for Grothendieck-Lefschetz trace formula
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$,
$$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
- 455
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
4
votes
0
answers
253
views
How to compute the first etale cohomology of a constructible torsion-free sheaf?
I am interested in the following example!
Let $k$ be a field, let $X_0$ be the scheme $\mathrm{Spec}R$ with $R_0=k[x,y]/(xy)$, let $R$ be the strict Hensilian localalisation of $R_0$ at the origin ...
- 997
2
votes
1
answer
505
views
Higher direct image of locally constant torsion sheaf (étale cohomology)
Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...
- 155
0
votes
0
answers
188
views
constructibility for pushforward
Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
- 3,472
2
votes
0
answers
102
views
Continuity of constructible derived category
Let $X_0$ be a variety over $\mathbb F_q$. Denote by $X_n$ its basechange to $\mathbb F_{q^n}$ and let $X=\lim X_n$ be its basechange to the algebraic closure $\overline{\mathbb F}_q$.
Let $D^b_c(X_n,...
- 13.2k
6
votes
0
answers
466
views
Purity and six operations?
The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...
- 13.2k