All Questions
Tagged with cohomology sheaf-cohomology
46
questions
3
votes
1
answer
301
views
Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,452
4
votes
1
answer
369
views
Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
- 498
1
vote
0
answers
116
views
A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,301
4
votes
4
answers
616
views
Canonical product in sheaf cohomology
EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.3k
1
vote
0
answers
230
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
2
votes
1
answer
249
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
- 7,313
2
votes
1
answer
188
views
On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 553
3
votes
1
answer
334
views
Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 553
1
vote
1
answer
225
views
A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 553
1
vote
0
answers
192
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
129
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
5
votes
1
answer
596
views
First Chern class of torsion sheaves
Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...
- 113
8
votes
1
answer
2k
views
Cohomology of Grothendieck topology
My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...
- 2,050
5
votes
1
answer
200
views
Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
- 191
4
votes
0
answers
346
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(\...
- 265