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.

Title	group cohomology
Canonical name	GroupCohomology
Date of creation	2013-03-22 13:50:07
Last modified on	2013-03-22 13:50:07
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	11
Author	alozano (2414)
Entry type	Definition
Classification	msc 20J06
Synonym	cohomology
Related topic	SelmerGroup
Related topic	CohomologyGroupTheorem
Related topic	ProofOfCohomologyGroupTheorem
Related topic	OmegaSpectrum
Related topic	NaturalEquivalenceOfC_GAndC_MCategories
Defines	group cohomology
Defines	coboundary
Defines	cocycle
Defines	crossed homomorphism