All Questions
19 questions
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
20
votes
5
answers
2k
views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
- 10.5k
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
11
votes
1
answer
866
views
Serre spectral sequence for de Rham cohomology
Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...
- 2,305
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
7
votes
1
answer
353
views
Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 173
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
5
votes
0
answers
290
views
About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
- 711
5
votes
0
answers
113
views
How "commutative" are Cech cochains of a sheaf of commutative (dg) algebras?
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
4
votes
2
answers
228
views
Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?
Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
- 711
2
votes
1
answer
242
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,071
2
votes
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
2
votes
0
answers
372
views
How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
- 711
1
vote
2
answers
1k
views
Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
1
vote
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
1
vote
0
answers
58
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.5k
1
vote
0
answers
213
views
Zero in colimit of sheaves category
This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...
- 1,168