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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 ...
Saal Hardali's user avatar
  • 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 ...
C. Dubussy's user avatar
  • 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): ...
David Zureick-Brown's user avatar
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 ...
algori's user avatar
  • 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^...
cll's user avatar
  • 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?
Louis A's user avatar
  • 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 ...
Hyperion's user avatar
  • 213
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_{...
asv's user avatar
  • 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 ...
Gabriel's user avatar
  • 721
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 ...
algebrachallenged's user avatar
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 ...
Gabriel's user avatar
  • 721
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 ...
Sergey Guminov's user avatar
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 ...
Louis A's user avatar
  • 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}...
Wilhelm's user avatar
  • 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 ...
Gabriel's user avatar
  • 721
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 ...
Louis A's user avatar
  • 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 ...
Louis A's user avatar
  • 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 ...
algori's user avatar
  • 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 ...
XT Chen's user avatar
  • 1,168