# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Kunneth formula for cohomology

Why I can use Kunneth formula to say that $H^{*}(\mathbb{C}P^{\infty} \times \mathbb{C}P^{\infty})= \mathbb{C}[x_{1}] \otimes \mathbb{C}[x_{2}]$?
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### 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! :)
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### 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 ...