All Questions
Tagged with homological-algebra sheaf-cohomology
31 questions
62
votes
8
answers
14k
views
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
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
27
votes
0
answers
468
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
17
votes
2
answers
2k
views
Is there a complex which computes Cech cohomology?
Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open ...
- 1,761
16
votes
1
answer
2k
views
Where am I suppose to actually learn how to compute hypercohomology?
I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with ...
- 1,716
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
9
votes
1
answer
2k
views
Sheaf cohomology with compact supports (and Verdier duality?)
Consider a manifold and a complex where cochains are sections of vector bundles and coboundary maps are differential operators, which are locally exact except in lowest degree (think de Rham complex). ...
- 21.5k
8
votes
1
answer
622
views
Can one determine the trace map for a nonsingular projective variety explicitly?
I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the ...
- 615
6
votes
1
answer
478
views
Unbounded acyclic resolutions
Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
- 28.7k
6
votes
1
answer
441
views
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
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 $$U_A:=\cap_{...
- 21.8k
6
votes
1
answer
385
views
Elementary proof of the exactness of Čech complex associated to a hypercovering ("Illusie's Conjecture")
Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
- 3,312
5
votes
0
answers
380
views
Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
- 6,025
4
votes
4
answers
695
views
Canonical product in sheaf cohomology
EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.8k
4
votes
1
answer
244
views
Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$
Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...
- 790
4
votes
1
answer
291
views
Exactness of $j_!$ in abelian category recollement
Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...
- 6,025
4
votes
0
answers
137
views
Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$
Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$.
I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q \...
- 371
3
votes
2
answers
371
views
Extension between vector bundles inducing non-zero map on cohomology
Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i(...
- 7,377
3
votes
1
answer
458
views
Help understand a calculation involving RHom of sheaves on manifolds
I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
arxiv.org/pdf/1005.1517v4....
- 1,893
3
votes
1
answer
515
views
For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 9,019
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
3
votes
0
answers
386
views
Spectral sequence from resolution of condensed abelian groups
I am watching Scholze's and Clausen's masterclass on Condensed Mathematics and I don't understand or can find any references on something they said.
You have a resolution
$$ \dots \to \mathbb{Z}[\...
3
votes
0
answers
154
views
$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$
Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$
Let $C = \mathbb{P}^1$.
By Kunneth $H^2(\...
- 3,968
2
votes
1
answer
210
views
Making a map in sheaf cohomology involving a theta characteristic explicit
Motivation:
For a given rank 2 vector bundle we want to know how many theta-characteristic valued twisted endomorphisms it has.
Setting:
Let $C$ be a smooth algebraic curve over a field of ...
2
votes
0
answers
156
views
Are flasque sheaves exactly the retracts of "canonically" flasque sheaves?
Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...
2
votes
0
answers
136
views
A infinity structure on Yoneda Ext group
I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
- 213
2
votes
0
answers
669
views
Čech-Alexander complex in computing (crystalline/prismatic) cohomology
I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method ...
- 21
1
vote
1
answer
634
views
Inception of modern view of Sheaf Cohomology in Mathematical Literature
From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
- 37
1
vote
0
answers
199
views
Künneth formula for local cohomology with support
In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
- 473
0
votes
1
answer
156
views
The Existence of Pure Resolutions, Given a Degree Sequence?
I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
user24421
0
votes
0
answers
308
views
Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
user81684