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group cohomology (Definition)

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

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

$\displaystyle \phi(\alpha\beta)=\phi(\alpha)+\alpha\phi(\beta)$
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 $ G$-module $ M$ 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 $ G$ be a group and let $ A,B,C$ be $ G$-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 $ 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:
$\displaystyle C^n(G,M)=\{\phi\colon G^n \to M\}$
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)$:
$\displaystyle d_n(\phi)(g_1,...,g_{n+1})$ $\displaystyle =$ $\displaystyle g_1\cdot \phi(g_2,...,g_{n+1})$  
  $\displaystyle +$ $\displaystyle \sum_{i=1}^n(-1)^i\phi(g_1,...,g_{i-1},g_ig_{i+1},g_{i+2}, ...,g_{n+1})$  
  $\displaystyle +$ $\displaystyle (-1)^{n+1}\phi(g_1,...,g_n)$  

Let $ Z^n(G,M)=\operatorname{ker} d_n$ for $ n\geq 0$, the set of $ 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

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

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.



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See Also: Selmer group

Other names:  cohomology
Also defines:  group cohomology, coboundary, cocycle, crossed homomorphism
Keywords:  cohomology, coboundary, cocycle

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Cross-references: coefficients, homomorphism, additive group, exact sequence, proposition, quotient group, fix, map, group
There are 26 references to this entry.

This is version 6 of group cohomology, born on 2003-08-08, modified 2004-06-04.
Object id is 4571, canonical name is GroupCohomology.
Accessed 12981 times total.

Classification:
AMS MSC20J06 (Group theory and generalizations :: Connections with homological algebra and category theory :: Cohomology of groups)

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