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I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ \xi_\sigma=m^{\sigma}-m$ for an $m \in M$. Then he says the values are in P. Clearly I am not understanding something here. Can someone clarify this for me?


edit1: I should have given the context (thank you Jyrki). So we start with an exact sequence of G-modules $$0 \rightarrow\ P\ \xrightarrow{\phi}\ M \xrightarrow{\psi}\ N \rightarrow 0.$$ Then there is a long exact sequence $$0 \rightarrow H^0(G,P) \rightarrow H^0(G,M) \rightarrow H^0(G,N) \ \xrightarrow{\delta} H^1(G,P) \rightarrow ...$$ The connecting homomorphism $\delta$ is defined als follows: Let $n \in H^0(G,N)$. Choose an $m$ such that $\psi(m)=n$ and define a cochain $\xi \in C^1(G,M)$ by $$\xi_\sigma=m^\sigma-m.$$ Then the values of $\xi$ are in $P$...

edit2: I think I see what the author means. From the definitions it follows that $\psi(m^\sigma-m)=\psi(m)^\sigma-n=0$. From the fact that $Im(\phi)=ker(\psi)$ and the injectivity of $\phi$ we can consider $m^\sigma-m$ as an element of $P$.

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It could not hurt to give a little bit of context. Like what are $G,M,\delta,P$. Many of us have a copy of Silverman's tome, but some have it in the office/home and read about your question at home/office :-) – Jyrki Lahtonen Oct 24 '11 at 20:00

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