All Questions
Tagged with sheaf-cohomology cohomology
15 questions with no upvoted or accepted answers
6
votes
0
answers
224
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
5
votes
0
answers
370
views
Continuity property for Čech cohomology
Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
- 275
4
votes
0
answers
211
views
Does cohomology and base change hold if supported at a point?
I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and ...
- 1,104
4
votes
0
answers
343
views
Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
- 1,716
4
votes
0
answers
177
views
Reference request: local cohomology in disjoint union
I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
- 41
3
votes
0
answers
256
views
Convolution of sheaves on R
I am trying to understand a basic computation of convolution. Throughout, $R$ is the real line as a topological group and $k$ is some base field. I would like to understand the computation of the ...
3
votes
0
answers
130
views
Action of automorphisms on cohomology with supports
Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
- 525
2
votes
0
answers
347
views
l-adic cohomology and perverse sheaves
Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
- 3,472
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
1
vote
0
answers
120
views
A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,367
1
vote
0
answers
295
views
Exterior product of Euler Exact Sequence
Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...
- 131
1
vote
0
answers
217
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
135
views
Base change of cohomology when the cohomology is a torsion
Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
- 51
1
vote
0
answers
163
views
Explicit map between $\check{H}^1(M,\underline{\mathbb{R}})$ and $H^1(M,\mathbb{R})$
Is there a way to construct an explicit isomorphism between Cech cohomology and singular cohomology on a smooth manifold for degree 1? If yes can this be extended to higher degee?
- 789
1
vote
0
answers
152
views
What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
- 435