This tag is for questions relating to cohomology groups and cochain complexes.
8
votes
0answers
204 views
Why do universal $\delta$-functors annihilate injectives?
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
6
votes
0answers
82 views
When does a cohomology theory have a ring structure?
I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
6
votes
0answers
81 views
Computing a specific direct limit
Suppose we have the following sequence of $\mathbb{Z}$-modules
$$G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, \cdots,$$
where each ...
5
votes
0answers
51 views
Injective Resolutions in $\mathfrak{Ab}(X)$
Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
5
votes
0answers
35 views
“Inverse problem” for Brauer groups
This question is really just a curiosity, but I'm really interested in the answer.
Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
5
votes
0answers
113 views
Why are injective $\mathscr{O}$-modules flasque?
Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
4
votes
0answers
55 views
Example where Čech and derived functor cohomologies don't agree.
I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces.
Is there a simple example of a topological space ...
4
votes
0answers
116 views
de Rham Cohomology of Non-Flat Bundle
Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$.
If $E$ ...
4
votes
0answers
71 views
Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions
I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$
$0 ...
4
votes
0answers
60 views
Explicit quasi-inverse of Künneth-isomorphism?
With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that
\begin{align*}
A_X \otimes A_Y &\to A_{X \times Y}, \\
(\omega,\eta) &\mapsto {\rm ...
4
votes
0answers
105 views
Cohomology of fiber bundle with a section
Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map
$$
...
4
votes
0answers
137 views
Composition of derived functors and comparison between hypercohomology and sheaf cohomology
I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book:
...
4
votes
0answers
276 views
De Rham cohomology for non-compact manifolds
Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's ...
4
votes
0answers
121 views
What is a high level reason for the fecundity of (co)homological algebra?
A colleague once disparaged his own research to me by saying that it didn't involve any sort of cohomology.
It does, in fact, seem like homological ideas appear across disciplines...and are ...
3
votes
0answers
31 views
Intuitive definition of Čech cohomology for compact surfaces
Let $X$ be a smooth compact $k$-surface in $\mathbb R^n$ without boundary. Today on my lection lecturer introduced Čech cohomology as follows (not like in Wikipedia): let $\mathcal U$ be a finite open ...
3
votes
0answers
85 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
0answers
65 views
A few questions about nonabelian cohomology of finite groups.
I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
3
votes
0answers
62 views
Twisted de Rham Cohomology
Let $M$ be a smooth manifold and $H$ a closed odd-degree form. Then $(\Omega^{\bullet}(M), d_H)$ defines a complex where $d_H := d + H\wedge$. The cohomology of this complex is called twisted de Rham ...
3
votes
0answers
87 views
Cohomology of a tensor product
Let $k$ be a field of characteristic $p$ and $V$ be a $k^p$ vector space.
Denote by $k_s$ the separable closure of $k$ and set $G_k := Gal(k_s|k)$.
Prove that
$$
H^0(G_k, V \otimes_{k^p} k_s^p) = V ...
3
votes
0answers
136 views
short exact sequence - split, as a semidirect product, with some cohomology
I've looked at several s.e.s. examples and I feel I am quite close but here is a question I am still a little confused on.
Let $E$ be a group and $A$ an abelian normal subgroup s.t. have an exact ...
3
votes
0answers
110 views
cohomology isomorphism
Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$,
$$H^{i}(X-A;\mathbb ...
3
votes
0answers
144 views
The first cohomology of group
I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
3
votes
0answers
67 views
Examples showing the usefulness of derived categories
What are examples that show that derived categories really makes things easier/more transparent/have a real use?
3
votes
0answers
75 views
Technical question about Brauer groups and smoothness
For a variety $X$ over a field $k$, we define $Br(X) = H^2_{et}(X,\mathbb{G}_m)$.
Suppose $X$ is a smooth variety (finite type, separated) over an algebraically closed field $k$ together with a ...
3
votes
0answers
156 views
group cohomology with coefficient in an induced module
We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second ...
2
votes
0answers
39 views
Simplicial cup product on torus
I'm trying to compute the simplicial cup product on the torus (using $\Delta$-complexes) but running into a problem: each way I draw the fundamental polygon I get different answers! When I draw it as ...
2
votes
0answers
57 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
2
votes
0answers
42 views
$f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$ using simplicial chains to define cochains
Let $f \colon X \to Y$ be a continuous map between topological spaces $X$ and $Y$, $f_\ast$ be the induced homomorphism of singular chains $C_k^s(X;G)$, $C_k^s(Y;G)$ and $f^\ast$ be the induced ...
2
votes
0answers
70 views
Vanishing of local cohomology of constructible sheaves
Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.
Is there an analogous statement for ...
2
votes
0answers
37 views
Why do the two topologies on a Galois group coincide?
In the following one is referred to the book.
At this page, the author defines Krell topology on a Galois group(not necessarily finite); at the 22-th page of the same book, the quthor defines then the ...
2
votes
0answers
102 views
Question on $\operatorname{res}^G_U \circ \operatorname{cor}^U_G = N_{G/U}$
This is probably a very basic question, but I can't wrap my head around it.
Given a normal open subgroup $U$ of a profinite group $G$ and a $G$-Module $A$ we have the following equation in group ...
2
votes
0answers
66 views
Computing Derived Pullback on the Complement
Let $X$ be a scheme and $\iota: Z\hookrightarrow X$ the embedding of a closed subscheme $Z$; let $j: U\hookrightarrow X$ be the open complement. Suppose $\mathcal{F}$ is a coherent sheaf on $X$.
...
2
votes
0answers
265 views
Homology and cohomology: why does Poincaré duality fail for domains with boundary?
Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic.
For domains with boundary, it's easy to construct examples where ...
2
votes
0answers
123 views
On a canonical homomorphism in cohomology
When $K$ is a simplicial complex, the dual complex $C^*(K)$ to the chain complex $C_*(K)$ has a concrete interpretation: an element in $C^n(K)$ is given by assigning an integer to every oriented ...
2
votes
0answers
24 views
How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?
$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...
2
votes
0answers
227 views
Computing the cohomology ring of $\Sigma_2$
This question is from an old exam. I was completely lost on it and not sure where to start- hoping for even a point in the right direction.
Let $\Sigma_2$ be the genus 2 surface and ...
1
vote
0answers
20 views
Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$
What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
1
vote
0answers
19 views
Chern Character Isomorphism for non-CW complexes
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
1
vote
0answers
46 views
Explanation of notations
I was reading Binary icosahedral group in Wikipedia. The author uses $<2,3,5>$ and $(2,3,5)$ to denote the groups. And the Coxeter group of type $H_4$ is denoted by $[3,3,5]$. Could anyone ...
1
vote
0answers
30 views
Dolbeault cohomology of $S^{2n-1} \times S^1$
Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}}(X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$. I don't know how to do this but if we use Kunnet formula we have that ...
1
vote
0answers
43 views
Cohomology and 1-forms with compact support
I'm, having troubles with the following
Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$.
I ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
1
vote
0answers
45 views
Cohomology of a chain complex
I know that one can define a chain complex for a CW complex X by taking the chain groups $C_n(X)$ as the free group generated by the $n$-cells, $C_n(X;\mathbb{Z}) = \mathbb{Z}\langle ...
1
vote
0answers
17 views
Sending the Poincaré dual class of a point in a smooth manifold to 1
I have been given the following problem:
For a compact oriented $n$-dimensional manifold, use a nowhere zero $n$-fold $\omega\in\Omega^n(M)$ to define a linear map
\begin{equation}
[M]: H^n(M)\to ...
1
vote
0answers
26 views
Relation between the pull back and the pull forward map on the (co)homology groups
So let $f:X\rightarrow Y$ be a continuous map. Is there any relation between
$$f_{*}:H_{k}(X)\rightarrow H_{k}(Y)$$
and
$$f^{*}:H^{k}(Y)\rightarrow H^{k}(X).$$
1
vote
0answers
61 views
De Rham cohomology of $\mathbb R^3$ without lines and a circumference
I am trying to calculate De Rham cohomology of the following spaces:
$X=\mathbb R^3\setminus r$ where $r$ is a line;
$Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
1
vote
0answers
38 views
Equivariant localization and integration equivariant forms
I have two problems:
Let it $\Omega^{*}_{G}:=(\mathbb{C}[\mathfrak{g}]\otimes\Omega^{*}(M))^{G}$ be the complex of equivariant differential forms on a differential manifold $M$ (in which acts a Lie ...
1
vote
0answers
37 views
Equivariant Mayer Vietoris and Borel localization
We have this theorem:
Let $U$, $V$ two open sets of manifold $M$, ($U \cup V = M$). If they are $G$-stable the induced sequence in cohomology
$$ \cdots \rightarrow H^{k}_{G}(U \cup V) \rightarrow ...
1
vote
0answers
83 views
group action - compact complex torus with $H^2_{DR}(X,C) = 0$ (de Rham cohomology)
If $\mathbb{Z}$ acts on $\mathbb{C}^n \backslash \{0\}$ by $(m,z) \mapsto 2^m\,z$, I need to show that $H^2_{DR}(X=(\mathbb{C}^n \backslash \{0\})/\mathbb{Z},\mathbb{C})=0$.
I start with showing it ...
1
vote
0answers
22 views
Generators of $H^k(X,\mathbb{Q})$ and $H^k(X,\mathbb{Z})/torsion$.
Let $X$ be a compact smooth manifold. Given a $\mathbb{Q}$-basis$e_1,\dots,e_n$ of the vector space $H^k(X,\mathbb{Q})$.
Is it true that there exist rational numbers $a_1,\dots,a_n$ such that
$$
...
