| Version 2 |
Version 1 |
| Let $G$ be a group and let $M$ be a (left) $G$-module. The |
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 |
$0^{th}$ \emph{cohomology group} of the $G$-module $M$ is |
| $$H^0(G,M)=\{m\in M\colon \forall \sigma \in G,\ \sigma m=m\}$$ |
$$H^0(G,M)=\{m\in M\colon \forall \sigma \in G,\ \sigma m=m\}$$ |
| which is the set of elements of $M$ which are $G$-invariant, also |
which is the set of elements of $M$ which are $G$-invariant, also |
| denoted by $M^G$. |
denoted by $M^G$. |
| A map $\phi\colon G\to M$ is said to be a \emph{crossed |
A map $\phi\colon G\to M$ is said to be a \emph{crossed |
| homomorphism} (or \emph{1-cocycle}) if |
homomorphism} (or \emph{1-cocycle}) if |
| $$\phi(\alpha\beta)=\phi(\alpha)+\alpha\phi(\beta)$$ |
$$\phi(\alpha\beta)=\phi(\alpha)+\alpha\phi(\beta)$$ |
| for all $\alpha,\beta \in G$. If we fix $m\in M$, the map |
for all $\alpha,\beta \in G$. If we fix $m\in M$, the map |
| $\rho\colon G\to M$ defined by |
$\rho\colon G\to M$ defined by |
| $$\rho(\alpha)=\alpha m-m$$ |
$$\rho(\alpha)=\alpha m-m$$ |
| is clearly a crossed homomorphism, said to be \emph{principal} (or |
is clearly a crossed homomorphism, said to be \emph{principal} (or |
| \emph{1-coboundary}). We define the following groups: |
\emph{1-coboundary}). We define the following groups: |
| \begin{eqnarray} |
\begin{eqnarray} |
| \nonumber Z^1(G,M)&=&\{\phi\colon G\to M\colon \phi \text{ is a 1-cocycle}\}\\ |
\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 |
\nonumber B^1(G,M)&=&\{\rho\colon G\to M\colon \rho \text{ is a |
| 1-coboundary}\} |
1-coboundary}\} |
| \end{eqnarray} |
\end{eqnarray} |
| Finally, the $1^{st}$ \emph{cohomology group} of the $G$-module |
Finally, the $1^{st}$ \emph{cohomology group} of the $G$-module |
| $M$ is defined to be the quotient group: |
$M$ is defined to be the quotient group: |
| $$H^1(G,M)=Z^1(G,M)/B^1(G,M)$$ |
$$H^1(G,M)=Z^1(G,M)/B^1(G,M)$$ |
| Similarly, there are cohomology groups $H^n(G,M)$ for $n>1$ but |
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 |
the first two are the ones which are most frequently found in the |
| literature. |
literature. |
| The following proposition is very useful when trying to compute |
The following proposition is very useful when trying to compute |
| cohomology groups: |
cohomology groups: |
| \begin{prop} |
\begin{prop} |
| Let $G$ be a group and let $A,B,C$ be $G$-modules related by an |
Let $G$ be a group and let $A,B,C$ be $G$-modules related by an |
| exact sequence: |
exact sequence: |
| $$0\to A\to B\to C\to 0$$ |
$$0\to A\to B\to C\to 0$$ |
| Then there is a long exact sequence in cohomology: |
Then there is a long exact sequence in cohomology: |
| $$0\to H^0(G,A)\to H^0(G,B)\to H^0(G,C)\to H^1(G,A)\to H^1(G,B)\to |
$$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)$$ |
H^1(G,C)$$ |
| \end{prop} |
\end{prop} |
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{serre} J.P. Serre, {\em Galois Cohomology}, |
\bibitem{serre} J.P. Serre, {\em Galois Cohomology}, |
| Springer-Verlag, New York. |
Springer-Verlag, New York. |
| \bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course |
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course |
| notes}{http://www.jmilne.org/math/CourseNotes/math679.html}. |
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. |
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986. |
| \end{thebibliography} |
\end{thebibliography} |
