Long Exact Sequence in Group Cohomology Proof

Question

How do you prove this? I have looked all around and am unable to find a real, in-depth proof.

Answer 1

Consider a projective resolution $$\cdots \to P_i \to P_{i-1} \to \cdots \to P_0 \twoheadrightarrow \mathbb{Z}$$ Let \begin{eqnarray*} A^i&=&{\rm Hom}_G(P_i,A),\\ B^i&=&{\rm Hom}_G(P_i,B),\\ C^i&=&{\rm Hom}_G(P_i,C). \end{eqnarray*}

Then form the diagram:

\begin{eqnarray*} \begin{array}{cccccccc} &&\vdots &&\vdots&&\vdots&&\\ &&\downarrow d&&\downarrow d&&\downarrow d\\ 0&\to&A^i&\to& B^i&\to& C^i&\to&0\\ &&\downarrow d&&\downarrow d&&\downarrow d\\ 0&\to&A^{i+1}&\to& B^{i+1}&\to& C^{i+1}&\to&0\\ &&\downarrow d&&\downarrow d&&\downarrow d\\ &&\vdots &&\vdots&&\vdots&&\\ \end{array} \end{eqnarray*} where $d$ is precomposition with the relevant map from the resolution and the horizontal maps are post-composition with the maps $A\to B$ and $B\to C$.

${\bf Step\,\, 1}$ You need to check that this diagram commutes and then use the projective property of the $P_i$ to show that the rows are exact.

Once you have done that, define the Snake map on cocycles in $C^i$ as follows:

(1) Take a preimage of your cocycle in $B^i$ (as in the film clip).

(2) Apply $d$.

(3) Take a preimage in $A^{i+1}$.

${\bf Step\,\, 2}$ Check that each of those moves was possible, and that the end result is well defined up to coboundaries in $A^{i}$. Also make sure that you end up with a cocycle.

${\bf Step\,\, 3}$ Check that the resulting sequence of maps on homology (induced from the horizontal arrows) is exact.

There are a lot of little things to check at each stage, but you should find it really satisfying. If you get stuck on anything specific, drop a comment.