All Questions
Tagged with sheaf-theory cohomology
11 questions with no upvoted or accepted answers
8
votes
0
answers
303
views
Examples of locally-but-not-globally (pre)sheaf toposes?
Re-reading my own recently posted question What is the total space of a stack after all? I realized that I don't know something more simple and presumably more basic.
Are there (bounded if you like) ...
- 18.4k
6
votes
0
answers
223
views
Under what generality are the compactly supported singular and sheaf cohomologies equal?
Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of ...
- 1,020
6
votes
0
answers
452
views
Yoneda product on Ext
Let $M$ be a closed manifold and consider the constant sheaf $\mathbb{R}_M$. I have heard that the Yoneda product on $\operatorname{Ext}(\mathbb{R}_M, \mathbb{R}_M)= H^*(M; \mathbb{R})$ coincides with ...
- 177
3
votes
0
answers
411
views
Two definitions of Čech cohomology
Hello,
I have found different definitions of Čech complex for sheaf $F$ od abelian groups on topological space $X$ with respect to the cover $\mathcal U$. One in Gelfand-Manin says to take product of ...
- 1,368
2
votes
0
answers
121
views
Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
2
votes
0
answers
306
views
Cech cohomology.
Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with $U_{\...
- 181
2
votes
0
answers
228
views
The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.
For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram $U\...
- 16.9k
1
vote
0
answers
215
views
Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
1
vote
0
answers
343
views
Cokernel of section of a general coherent sheaf
Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...
- 1,701
1
vote
0
answers
333
views
Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?
In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....
- 945
0
votes
0
answers
116
views
cohomology of curves
Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$
in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle.
If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
- 237