PlanetMath: group cohomology

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group cohomology

(Definition)

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

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

which is the set of elements of

which are

-invariant, also denoted by

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

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

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

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

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

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

Finally, the $1^{st}$ cohomology group of the

-module

is defined to be the quotient group:

$\displaystyle H^1(G,M)=Z^1(G,M)/B^1(G,M)$

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

Proposition 1 Let be a group and let be -modules related by an exact sequence:

$\displaystyle 0\to A\to B\to C\to 0$

Then there is a long exact sequence in cohomology:

$\displaystyle 0\to H^0(G,A)\to H^0(G,B)\to H^0(G,C)\to H^1(G,A)\to H^1(G,B)\to H^1(G,C)\to \ldots$

In general, the cohomology groups can be defined as follows:

Definition 1 Define and for $n\geq 1$ define the additive group:

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

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

$\displaystyle d_n(\phi)(g_1,...,g_{n+1})$	$\displaystyle =$	$\displaystyle g_1\cdot \phi(g_2,...,g_{n+1})$
	$\displaystyle +$	$\displaystyle \sum_{i=1}^n(-1)^i\phi(g_1,...,g_{i-1},g_ig_{i+1},g_{i+2}, ...,g_{n+1})$
	$\displaystyle +$	$\displaystyle (-1)^{n+1}\phi(g_1,...,g_n)$

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

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

$\displaystyle H^n(G,M)=Z^n(G,M)/B^n(G,M)$

Bibliography

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

"group cohomology" is owned by alozano.

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