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group cohomology
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(Definition)
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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: \begin{eqnarray} \nonumber Z^1(G,M)&=&\{\phi: G\to M\colon \phi \text{ is a 1-cocycle}\}\\ \nonumber B^1(G,M)&=&\{\rho: G\to M\colon \rho \text{ is a
1-coboundary}\} \end{eqnarray}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 $d_n\colon C^n(G,M) \to C^{n+1}(G,M)$ : \begin{eqnarray} \nonumber d_n(\phi)(g_1,...,g_{n+1})&=&g_1\cdot \phi(g_2,...,g_{n+1})\\ \nonumber &+& \sum_{i=1}^n(-1)^i\phi(g_1,...,g_{i-1},g_ig_{i+1},g_{i+2}, ...,g_{n+1})\\ \nonumber &+& (-1)^{n+1}\phi(g_1,...,g_n)
\end{eqnarray}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 $$H^n(G,M)=Z^n(G,M)/B^n(G,M)$$
- 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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"group cohomology" is owned by alozano.
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Cross-references: coefficients, homomorphism, additive group, long exact sequence in cohomology, exact sequence, useful, proposition, quotient group, fix, map, elements, cohomology group, group
There are 16 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 16853 times total.
Classification:
| AMS MSC: | 20J06 (Group theory and generalizations :: Connections with homological algebra and category theory :: Cohomology of groups) |
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Pending Errata and Addenda
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