group cohomology

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group cohomology

Let $G$ be a group and let $M$ be a (left) $G$ -module. The $0^{{th}}$ cohomology group of the $G$ -module $M$ is

H^{0}(G,M)=\{m\in M:\forall\sigma\in G,\ \sigma m=m\}

which is the set of elements of $M$ which are $G$ -invariant, also denoted by $M^{G}$ .

A map $\phi\colon G\to M$ is said to be a crossed homomorphism (or 1-cocycle) if

\phi(\alpha\beta)=\phi(\alpha)+\alpha\phi(\beta)

for all $\alpha,\beta\in G$ . If we fix $m\in M$ , the map $\rho\colon G\to M$ defined by

\rho(\alpha)=\alpha m-m

is clearly a crossed homomorphism, said to be principal (or 1-coboundary). We define the following groups:

	$\displaystyle Z^{1}(G,M)$	$\displaystyle=$	$\displaystyle\{\phi:G\to M\colon\phi\text{ is a 1-cocycle}\}$
	$\displaystyle B^{1}(G,M)$	$\displaystyle=$	$\displaystyle\{\rho:G\to M\colon\rho\text{ is a 1-coboundary}\}$

Finally, the $1^{{st}}$ cohomology group of the $G$ -module $M$ is defined to be the quotient group:

H^{1}(G,M)=Z^{1}(G,M)/B^{1}(G,M)

The following proposition is very useful when trying to compute cohomology groups:

Proposition 1.

Let $G$ be a group and let $A,B,C$ be $G$ -modules related by an exact sequence:

0\to A\to B\to C\to 0

Then there is a long exact sequence in cohomology:

0\to H^{0}(G,A)\to H^{0}(G,B)\to H^{0}(G,C)\to H^{1}(G,A)\to H^{1}(G,B)\to H^{% 1}(G,C)\to\ldots

In general, the cohomology groups $H^{n}(G,M)$ can be defined as follows:

Definition 1.

Define $C^{0}(G,M)=M$ and for $n\geq 1$ define the additive group:

C^{n}(G,M)=\{\phi\colon G^{n}\to M\}

The elements of $C^{n}(G,M)$ are called $n$ -cochains. Also, for $n\geq 0$ define the $n^{{th}}$ coboundary homomorphism $d_{n}\colon C^{n}(G,M)\to C^{{n+1}}(G,M)$ :

$\displaystyle d_{n}(\phi)(g_{1},...,g_{{n+1}})$	$\displaystyle=$	$\displaystyle g_{1}\cdot\phi(g_{2},...,g_{{n+1}})$
	$\displaystyle+$	$\displaystyle\sum_{{i=1}}^{n}(-1)^{i}\phi(g_{1},...,g_{{i-1}},g_{i}g_{{i+1}},g% _{{i+2}},...,g_{{n+1}})$
	$\displaystyle+$	$\displaystyle(-1)^{{n+1}}\phi(g_{1},...,g_{n})$

Let $Z^{n}(G,M)=\operatorname{ker}d_{n}$ for $n\geq 0$ , the set of $n$ -cocyles. Also, let $B^{0}(G,M)=1$ and for $n\geq 1$ let $B^{n}(G,M)=\operatorname{image}d_{{n-1}}$ , the set of $n$ -coboundaries.

Finally we define the $n^{{th}}$ -cohomology group of $G$ with coefficients in $M$ to be

H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M)

References

1 J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
2 James Milne, Elliptic Curves.
3 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

Defines:

group cohomology, coboundary, cocycle, crossed homomorphism

Keywords:

cohomology, coboundary, cocycle

Synonym:

cohomology

Type of Math Object:

Definition

Major Section:

Reference

Mathematics Subject Classification

20J06 Cohomology of groups

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group cohomology

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group cohomology

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group cohomology

Proposition 1.

Definition 1.

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