All Questions
Tagged with etale-cohomology perverse-sheaves
16
questions
4
votes
0
answers
336
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 ...
- 5,565
3
votes
1
answer
500
views
Example of an intersection complex not concentrated in a single degree
I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.
I want to construct an example of an intersection ...
- 133
1
vote
0
answers
247
views
When is $\mathbb{Q}_X$ pure?
I'll ask this question in the language of mixed Hodge modules, since that's where I'm coming from, but the question has an exact analogue for mixed l-adic complexes on schemes over fields of positive ...
- 341
8
votes
0
answers
954
views
What is an example of a non-mixed $\ell$-adic sheaf?
$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
- 153k
0
votes
1
answer
230
views
cohomology of an intermediate extension of a local system
Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.
Suppose the cohomology of $H^{\bullet}(U,L)$ does not ...
- 3,452
2
votes
0
answers
344
views
l-adic cohomology and perverse sheaves
Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
- 3,452
8
votes
1
answer
737
views
$\ell$-adic monodromy theorems (over $\mathbb{C}$)
This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)
Notation
$H$ denotes étale cohomology.
Let $...
- 5,454
8
votes
1
answer
702
views
DG enhancements of $\ell$-adic derived categories
This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...
- 1,595
4
votes
2
answers
1k
views
Pullbacks of intermediate/middle extensions and Gabber's purity theorem
I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of a pure perverse ...
- 1,828
12
votes
1
answer
1k
views
On the derived category of constructible étale sheaves
The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...
- 15.5k
1
vote
1
answer
237
views
How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?
Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...
- 16.7k
5
votes
1
answer
525
views
Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves
I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...
- 16.7k
5
votes
0
answers
729
views
Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
- 16.7k
10
votes
1
answer
1k
views
Bad behaviour of perverse sheaves over 'general' bases?
Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...
- 16.7k
15
votes
1
answer
7k
views
A nice explanation of what is a smooth (l-adic) sheaf?
I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
- 16.7k