All Questions
Tagged with etale-cohomology derived-categories
16
questions
2
votes
1
answer
190
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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{...
- 5,780
3
votes
0
answers
169
views
Relations between some categories of étale sheaves
I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..
Let $X$ be a scheme over a number field $k$. Feel free to add ...
- 1,125
1
vote
1
answer
211
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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
151
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Obtaining an exact sequence of Galois modules via derived functors
This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there.
Let ...
- 885
1
vote
1
answer
291
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A question about a truncated object
I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:
For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
- 885
4
votes
1
answer
645
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"Universal coefficent theorem" for pro-étale cohomology
In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have
$$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)...
- 4,286
10
votes
1
answer
1k
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Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 27.9k
1
vote
0
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143
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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
3
votes
1
answer
445
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Spectral sequence for tensor product of complexes
Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.
I want to compute the hypercohomology:
$$\mathbb{H}^*(X_{\rm ét}, K^{\bullet}\otimes^L_{\...
user113452
1
vote
0
answers
245
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Stalks of derived tensor product (in the Kunneth formula)
So, essentially here's what I'm curious about. Suppose that $k$ is a (separably closed/algebraic closed) field $X_i,Y_i/k$ are finite type and $f_i:X_i\to Y_i$ are $k$-maps (all of this for $i=1,2$). ...
- 833
1
vote
0
answers
189
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Verdier duality on excellent schemes
Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension.
In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
- 3,452
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,...
- 13k
12
votes
1
answer
1k
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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
2
votes
1
answer
437
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Equivalent forms of the proper base change isomorphism
$\DeclareMathOperator{\Nat}{Nat}$In a current project, I am trying to "commute" $!$ and $*$ functors that are both upper or both lower. (Sheaf-theoretic context: constructible étale sheaves.) ...
- 7,213
3
votes
1
answer
448
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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 ...
- 9,447