Here is an exercise from Serre's "local fiels" when he starts to do cohomology: Let G act on an abelian group A, f be an inhomogenous n cochain, i.e. $f\in C^n(G,A).$ Define an operator T on f, $Tf(g_1,g_2,\cdots,g_n)=g_1g_2\ldots g_n f(g_n^{-1},g_{n-1}^{-1},\ldots,g_1^{-1})$. It is clear that $T^2f=f$. It is also not too hard to show $T(df)=(-1)^{n+1}d(Tf)$. Thus f is a cocycle iff Tf is, and f is a coboundary iff Tf is. When n=1, it is straightforward to see -f is cohomologous to Tf.
Then the exercise wants us to show when n= 0,3 mod 4, f is cohomologous to Tf,
while when n=1,2 mod 4, Tf is cohomologous to -f.
Any idea will be appreciated.
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Fix the signs as Wilberd suggests in the comments, check that you get a natural automorphism of the complex computing cohomology, see that it induces in fact an automorphism of the universal $\delta$-functor $H^\bullet(G,\mathord-)$, and see what it does in degree zero. |
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