This tag is for questions relating to cohomology groups and cochain complexes.

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3
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1answer
55 views

On which groups can homomorphisms be defined subgroup-locally?

Let $G$ be a group and suppose that $G$ is the set-theoretic union of a collection $\mathcal U$ of subgroups. I will call such a collection a "covering" of $G$. Suppose that for each $L \in \mathcal ...
3
votes
1answer
42 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, ...
3
votes
1answer
123 views

Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
3
votes
1answer
60 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
3
votes
2answers
138 views

de Rham cohomology of $\mathbb R^2 \setminus \mathbb Z^2 $

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
3
votes
1answer
92 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
3
votes
1answer
27 views

Identity in cohomology

Let $N^{4k+1}$ be a compact oriented manifold with boundary $i:M^{4k} \hookrightarrow N$. Suppose $c \in H^{4k}(N,A)$ for some abelian group $A$. I have to prove that $ \langle i^*(c), [M] \rangle =0 ...
3
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1answer
150 views

Homology of symplectic manifolds

Could you show me some example of compact symplectic 4-manifold $M$ with the torsion in $H_{2}(M;\mathbb{Z})$
3
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1answer
83 views

Algebraic Topology, Homology - Cohomology

This problem made me crazy, so please take a look at it: "$X=S^1 \cup_\phi D^2$" where $\phi$ is a degree $g$ map. (that means $X$ is obtained by glueing a disk to $S^1$ with a degree $g$ map) $H_n(X, ...
3
votes
1answer
170 views

Poincaré duality

Let $X$ be a a compact oriented manifold of dimension $n$. Assume that its (co)homologies have no torsion. Then Poincaré duality says that $$ H^{k}(X,\mathbb{Z})\cong H_{n-k}(X,\mathbb{Z}) $$ holds ...
3
votes
1answer
83 views

Geometric genus of a (possibly non-complete) intersection in P^n

Let $Y\subseteq \mathbb{P}^n$ be the zero locus of $f_1,...,f_k$ of degree $d_1,...,d_k$, and put $d=\sum d_i$. If $Y$ is nonsingular and a complete intersection, then ...
3
votes
1answer
170 views

Hartshorne Exercise III.2.1(a)

Show that $H^1(\mathbf{A}^1_k, \mathbf{Z}_U) \neq 0$ for $U = \mathbf{A}^1_k \setminus \{P,Q\}$, $k$ infinite field. Is it really neccessary that $P \neq Q$? My proof is as follows: Take the long ...
3
votes
1answer
94 views

cohomology of the additive group of imperfect field

In Springer's Encyclopaedia of Mathematics> Galois Cohomology, it is mentioned that For an imperfect field $k$, $H^1(k,\mathbb{G}_a)\neq 0$ in general. I'm looking for such an example or a ...
3
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0answers
35 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
3
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0answers
54 views

Geometric Interepretation of $\mathbb{G}_a$-torsors

Let's fixed a locally ringed space $(X,\mathcal{O}_X)$ (although, this should apply to any ringed topos, but I haven't thought that through). In fact, if it's helpful, you can assume that $X$ is a ...
3
votes
0answers
59 views

De Rham cohomology of the pointed plane

i try to work out some examples for de Rham cohomology, but i have some problems: I want to figure out what $H^k(\mathbb{R}^2\setminus\{0\})$ is and want to generalize this to arbitrary finite points ...
3
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0answers
44 views

When is a graded ring the cohomology ring of a CW-complex?

Let $A^*$ be a graded-commutative ring with $A^n = 0$ for sufficiently large $n$ and each $A^n$ finitely generated. When does there exist a finite CW-complex $X$ with $H^*(X) \cong A$ as graded rings? ...
3
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0answers
57 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
3
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0answers
41 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
3
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1answer
56 views

Why do we have that Hom is an exact functor in the situation described below?

We are given a finite $p$-group $G$ and a finite $G$-module $M$ such that $pM=0$ (therefore $M$ is in particular a $\mathbb{F}_p$-vector space). In addition we have an arbitrary $G$-module $N$ which ...
3
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0answers
51 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
3
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0answers
45 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
3
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0answers
38 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...
3
votes
1answer
38 views

Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
3
votes
0answers
36 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
3
votes
1answer
76 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
3
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0answers
112 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
3
votes
0answers
75 views

Tangent space of a moduli space.

Let $X$ be a compact Riemann surface with genus $2$ and $M^2$ the moduli space of stable principal $SL(2)$-bundles of rank $r$. We know that $M^2$ is a complex projective variety of dimention ...
3
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0answers
63 views

Homotopy operator for the Gysin sequnce

Let's consider some sphere bundle $π:E ↦ M$ with fiber $S^{r}$. What is homotopy operators in case of the Gysin sequence? $$ \ldots \rightarrow H_{dR}^p(B) \xrightarrow{\wedge e} H_{dR}^{p+r+1}(B) ...
3
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0answers
58 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
3
votes
0answers
78 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
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0answers
93 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
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0answers
152 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
1answer
108 views

Actions of automorphisms in cohomology

Let X be a smooth, projective variety over a field $k \hookrightarrow \mathbb{C}$ and let $g$ be an automorphism of $X$ of finite order. Consider the induced automorphism on the singular cohomology ...
3
votes
0answers
124 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 ...
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0answers
48 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 the Krull topology on a Galois group (not necessarily finite); on the 22nd page of the same book, the author defines the ...
3
votes
0answers
178 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
127 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
158 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
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0answers
94 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
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0answers
83 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
200 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
2answers
142 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
2
votes
1answer
80 views

Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$ h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10 $$ Do you know why this is the case? In ...
2
votes
2answers
219 views

Serre duality for curves, the other statement.

Here's a question from someone who's just found out what Serre duality (in the case of curves) is. It occurs to me that the popular statement which can also be interpreted as the Riemann-Roch theorem ...
2
votes
2answers
383 views

Brouwer's fixed point theorem for free?

I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books. One part is standard: Suppose there is an $f:D^n \rightarrow D^n$ with no fixed ...
2
votes
3answers
438 views

Acyclic vs Exact

I have a question about the words "acyclic" and "exact." Why does Brown use the term "acyclic" instead of "exact" in his book Cohomology of Groups? It seems to me that these two terms exactly ...
2
votes
1answer
276 views

Kunneth formula for cohomology

Why I can use Kunneth formula to say that $H^{*}(\mathbb{C}P^{\infty} \times \mathbb{C}P^{\infty})= \mathbb{C}[x_{1}] \otimes \mathbb{C}[x_{2}]$?
2
votes
3answers
128 views

What does $p+q=k$ mean in the index of summation?

I need help solving something I don't understand. OK so the problem is this: $$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$ What does the $\;p+q=k\;$ mean? Thank you anybody that helps! :)
2
votes
1answer
204 views

Top cohomology detecting compactness

Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold ...