Questions tagged [l-adic-sheaves]
The l-adic-sheaves tag has no usage guidance.
27
questions
4
votes
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Introduction to the theory of $D$-modules and the role of the characteristic cycle
I am seeking recommendations for a concise introduction to the theory of $D$-modules suitable for an algebraic geometer. Specifically, I am interested in understanding:
The role of the characteristic ...
- 2,811
1
vote
0
answers
176
views
Moduli stack of l-adic sheaves?
Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors.
Let $\ell$ be a prime ...
- 341
2
votes
1
answer
204
views
Does base change respect Galois correspondence between $\ell$-adic sheaves and representations of the fundamental étale group?
It is known that for $X$ a connected scheme there is an equivalence of categories
$$\left\lbrace \text{$\ell$-adic smooth sheaves over $X$} \right\rbrace \leftrightarrow \left\lbrace \text{$\ell$-adic ...
- 1,125
2
votes
0
answers
258
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Tate's conjecture for arithmetic schemes
Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
- 5,861
2
votes
1
answer
233
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Compatibility of Beck Chevalley condition: sheaves
Given a (not necessarily Cartesian) square of spaces
$$\require{AMScd}\begin{CD}
X @>g>> \overline{X} \\
@VVfV @VV\overline{f}V \\
Y @>\overline{g}>> \overline{Y}
\end{CD}$$
does the ...
- 5,565
3
votes
1
answer
176
views
$l$-adic cohomology of hyperplane arrangements
Consider an arrangement of hyperplanes given by the faces of a simplex. Let's consider it as a scheme (a non-regular scheme) and let's also work over a finite field. Has the rational $l$-adic ...
- 5,861
1
vote
1
answer
173
views
Reference for localization distinguished triangles in the derived category of $\ell$-adic sheaves
Let us consider a variety $X$ over a field $k$ which is a finite field or an algebraic closure thereof. Let $\ell$ be a prime number different from the characteristic of $k$, and let $\Lambda = \...
- 717
1
vote
1
answer
211
views
Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology
I have recently learned about relative Poincaré duality in the book Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform by Kiehl and Weissauer (2001). The reference is section II.7. ...
- 717
3
votes
0
answers
86
views
Is there a reasonable K-grroup of Behrend’s absolutely convergent complexes?
Let $\mathfrak X$ be an algebraic stack over $\mathbb F_q$ and let $D_{\mathrm{abs}}(\mathfrak X)$ be the derived category of complexes of $\overline{\mathbb Q}_\ell$-sheaves which are absolutely ...
- 137
2
votes
1
answer
208
views
Eigenvalues of Frobenius in $l$-adic cohomology
Let $X_0$ be a smooth projective variety over a finite field $\mathbb{F}_q$. Let $X$ be the corresponding variety over the algebraic closure $\bar{\mathbb{F}}_q$. Let $Fr_q\colon X\to X$ be the ...
- 21.3k
1
vote
0
answers
142
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$\ell$-adic cohomology and finiteness of the $\mathbf{Q}_\ell$-vector space
Let $X$ be a smooth projective variety over $K$. Fix $\ell \neq \mathrm{char}(K)$. I'm looking for references describing how the absolute Galois group $G_k$ acts on $H_{et}^i(X \times_K \bar{K}, \...
- 11
2
votes
0
answers
196
views
Intermediate extensions of pure perverse sheaves (BBD 5.4.3)
I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
- 1,061
2
votes
0
answers
196
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 ...
- 1,472
12
votes
1
answer
540
views
Katz's $\ell$-adic Airy sheaf
The Airy differential equation
$$y''(x)\ = \ xy(x)$$
is one of the simplest irregular differential equations (so not determined by its monodromy data, there is more structure, the Stokes data). ...
- 5,565
2
votes
1
answer
398
views
Finiteness result for higher direct image of $\ell$-adic sheaves
Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf_*\mathbf{...
- 5,565