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Sunday, 17 September 2006

homological algebra - An exercise in group cohomology

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. finCn(G,A). Define an operator T on f, Tf(g1,g2,cdots,gn)=g1g2ldotsgnf(g1n,g1n1,ldots,g11). It is clear that T2f=f. It is also not too hard to show T(df)=(1)n+1d(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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