Questions tagged [l-adic-sheaves]
The l-adic-sheaves tag has no usage guidance.
27 questions
4
votes
0
answers
127
views
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 ...
- 2,871
1
vote
0
answers
180
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 ...
- 426
3
votes
1
answer
224
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
$$\left\lbrace \text{$\ell$-adic smooth sheaves over $X$} \right\rbrace \leftrightarrow \left\lbrace \text{$\ell$-adic ...
- 1,270
2
votes
0
answers
290
views
Tate's conjecture for arithmetic schemes
Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
- 5,901
2
votes
1
answer
244
views
Compatibility of Beck Chevalley condition: sheaves
Given a (not necessarily Cartesian) square of spaces
$$\require{AMScd}\begin{CD}
X @>g>> \overline{X} \\
@VVfV @VV\overline{f}V \\
Y @>\overline{g}>> \overline{Y}
\end{CD}$$
does the ...
- 5,701
3
votes
1
answer
183
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 ...
- 5,901
1
vote
1
answer
191
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 = \...
- 769
1
vote
1
answer
253
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. ...
- 769
3
votes
0
answers
88
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
225
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 ...
- 21.8k
1
vote
0
answers
154
views
$\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
209
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, ...
- 1,071
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
12
votes
1
answer
558
views
Katz's $\ell$-adic Airy sheaf
The Airy differential equation
$$y''(x)\ = \ xy(x)$$
is one of the simplest irregular differential equations (so not determined by its monodromy data, there is more structure, the Stokes data). ...
- 5,701
2
votes
1
answer
429
views
Finiteness result for higher direct image of $\ell$-adic sheaves
Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf_*\mathbf{...
- 5,701
1
vote
0
answers
111
views
Divisible elements in the cohomology of Milnor $K$-theory
As a consequence of the strong Tate conjecture over finite fields one can deduce (see here proposition 8.20) for every smooth variety $X$ over a finite field:
$$H_{cont}^i(X,\mathbb{Q}_l(n))=H_{Zar}^{...
- 5,901
1
vote
0
answers
110
views
constructible sheaves not defined over the base field
Let $k$ be a finite field, $X$ a smooth variety over $k$, and $D^b_c(X)$ the bounded derived category of complexes of $\ell$-adic sheaves with constructible cohomology sheaves. Let now $\bar{k}$ be an ...
- 11
6
votes
0
answers
265
views
A lisse mixed sheaf as an extension of pure lisse sheaves
I am trying to understand Corollary 1.8.11 in Deligne's Weil II paper. The statement is that for a normal scheme $X_0$ that is of finite type over $\mathbb{F}_q$, every lisse $\ell$-adic $\iota$-mixed ...
- 2,663
9
votes
1
answer
459
views
$\ell$-adic schematic homotopy type
In the groundbreaking paper Champs Affines (DOI), Toen constructs a generalisation of rational homotopy types which he calls schematic homotopy types. This is part of a larger
programme of a theory ...
- 1,635
1
vote
0
answers
77
views
Is a $\tau$-mixed Weil sheaf always a direct summand of a $\tau$-real sheaf?
Fix an isomorphism $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$ and consider Weil sheaves on a scheme $X$ over $\mathbf{F_q}$.
It is a theorem that a $\tau$-real sheaf $\mathscr{F}$ is $\tau$-...
- 4,164
3
votes
0
answers
319
views
What is an example of a lisse Weil sheaf which is not mixed?
Fix a base field $\mathbf{F}_q$. What is a simple example of a lisse Weil sheaf which is not $\tau$-mixed for any identification $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$?
Addendum: it ...
- 4,164
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
1
vote
0
answers
145
views
Definition for equivariant $l$-adic sheaves
What is the definition of equivariant $l$-adic or ($\mathbb{Z}_l$-) sheaves?
Suppose $G$ acts on $X$, I could pick a $G$-equivariant etale sheaf of $\mathbb{Z}/l^n$ module on $X$ for each $n$, and ...
- 677
4
votes
0
answers
177
views
IC sheaves and formal neighbourhoods
Let $X$ and $Y$ be two schemes of finite type over a finite field $\mathbb F_q$. Let $x$ (resp. $y$) be an $\mathbb F_q$-point of $X$ (resp. of $Y$).
Let now $l$ be a prime which is prime to $q$. ...
- 7,037
1
vote
0
answers
150
views
Convolution of $\ell$-adic sheaves and group homomorphisms
This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves.
Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ ...
- 329
4
votes
1
answer
290
views
Convolution of $\ell$-adic sheaves is commutative if the group is commutative
[This is a duplicate of this question on Stackexchange]
I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" ...
- 329
8
votes
0
answers
469
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k