group cohomology

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:

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

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 $d_{n}\colon C^{n}(G,M)\to C^{n+1}(G,M)$:

 $\displaystyle d_{n}(\phi)(g_{1},...,g_{n+1})$ $\displaystyle=$ $\displaystyle g_{1}\cdot\phi(g_{2},...,g_{n+1})$ $\displaystyle+$ $\displaystyle\sum_{i=1}^{n}(-1)^{i}\phi(g_{1},...,g_{i-1},g_{i}g_{i+1},g_{i+2}% ,...,g_{n+1})$ $\displaystyle+$ $\displaystyle(-1)^{n+1}\phi(g_{1},...,g_{n})$

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

 $H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M)$

References

• 1 J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
• 2 James Milne, .
• 3 Joseph H. Silverman, . Springer-Verlag, New York, 1986.
 Title group cohomology Canonical name GroupCohomology Date of creation 2013-03-22 13:50:07 Last modified on 2013-03-22 13:50:07 Owner alozano (2414) Last modified by alozano (2414) Numerical id 11 Author alozano (2414) Entry type Definition Classification msc 20J06 Synonym cohomology Related topic SelmerGroup Related topic CohomologyGroupTheorem Related topic ProofOfCohomologyGroupTheorem Related topic OmegaSpectrum Related topic NaturalEquivalenceOfC_GAndC_MCategories Defines group cohomology Defines coboundary Defines cocycle Defines crossed homomorphism