Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is $$ \begin{array}[t]{lll} 1 \to & H^1(G/H, A^{G/H}) \xrightarrow{\inf} H^1(G,A) \xrightarrow{\mathrm{res}} H^1(H, A)^{G/H} \xrightarrow{\mathrm{tr}} & \\[1ex] & \to H^2(G/H,A^H) \xrightarrow{\inf} \mathrm{Ker}(H^2(G, A) \to H^2(H,A)) \xrightarrow{\rho} \\[1ex] & \xrightarrow{\rho} H^1(G/H, H^1(H, A)) \xrightarrow{\mathrm{tr}} H^3(G/H, A^H) &\\ \end{array} $$

I am interested in the morphism $\rho$ described explicitly in terms of cocycles (as in the standard elementary definition of Galois cohomology; these are called "inhomogeneous" cocycles in the book of Neukirch, Schmidt and Wingberg). I realize that this is just an exercise consisting of writing down the spectral sequence of the double complex and unwinding the definitions, but I wonder if this has already been done somewhere in an easily quotable form.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

Yes, the morhpisms have been described in detail in the paper "A seven-term exact sequence for the cohomology of a group extension" by Dekimpe, Hartl and Wauters: see http://arxiv.org/pdf/1103.4052.pdf.
Section $6$ gives details for $\rho$ (the sequence is slightly different, but applies more or less to your situation). This morphism is induced by differentials in the spectral sequence, and so far there was no explicit description available, but in this paper a purely group-theoretical description is given.

share|improve this answer
    
thanks, Dietrich! –  Dima Sustretov Jul 10 '13 at 20:19
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.