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'group cohomology'
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| Title of object: |
group cohomology |
| Canonical Name: |
GroupCohomology |
| Type: |
Definition |
| Created on: |
2003-08-08 16:36:18 |
| Modified on: |
2003-08-08 16:36:18 |
| Classification: |
msc:20J06 |
| Keywords: |
cohomology, coboundary, cocycle |
| Defines: |
group cohomology, coboundary, cocycle |
| Synonyms: |
group cohomology=cohomology |
Preamble:
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Content:
Let $G$ be a group and let $M$ be a (left) $G$-module. The
$0^{th}$ \emph{cohomology group} of the $G$-module $M$ is
$$H^0(G,M)=\{m\in M\colon \forall \sigma \in G,\ \sigma m=m\}$$
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 \emph{crossed
homomorphism} (or \emph{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 \emph{principal} (or
\emph{1-coboundary}). We define the following groups:
\begin{eqnarray}
\nonumber Z^1(G,M)&=&\{\phi\colon G\to M\colon \phi \text{ is a 1-cocycle}\}\\
\nonumber B^1(G,M)&=&\{\rho\colon G\to M\colon \rho \text{ is a
1-coboundary}\}
\end{eqnarray}
Finally, the $1^{st}$ \emph{cohomology group} of the $G$-module
$M$ is defined to be the quotient group:
$$H^1(G,M)=Z^1(G,M)/B^1(G,M)$$
Similarly, there are cohomology groups $H^n(G,M)$ for $n>1$ but
the first two are the ones which are most frequently found in the
literature.
The following proposition is very useful when trying to compute
cohomology groups:
\begin{prop}
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:
$$0\to H^0(G,A)\to H^0(G,B)\to H^0(G,C)\to H^1(G,A)\to H^1(G,B)\to
H^1(G,C)$$
\end{prop}
\begin{thebibliography}{9}
\bibitem{serre} J.P. Serre, {\em Galois Cohomology},
Springer-Verlag, New York.
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course
notes}{http://www.jmilne.org/math/CourseNotes/math679.html}.
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\end{thebibliography} |
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