Serge Lang ANT Theorem 7 p. 148

Question

Let $\mathfrak c, \mathfrak f$ be admissible cycles for an extension of number fields $K/k$ with $\mathfrak f$ dividing $\mathfrak c$. If you're not familiar with the terminology, a cycle $\mathfrak c$ is admissible if any $1 + \mathfrak p_v^{ord_v(\mathfrak c)}$ is contained in the group of local norms for all finite $v$ dividing $\mathfrak c$, and $ord_v(\mathfrak c) = 1$ whenever $v$ is a real place which has a complex place lying over it. There is a natural inclusion $$I(\mathfrak c) \subseteq I(\mathfrak f)$$ as well as inclusions $P_{\mathfrak c} \subseteq P_{\mathfrak f}, \mathfrak N(\mathfrak c) \subseteq \mathfrak N(\mathfrak f)$ (where $I(\mathfrak c)$ is the group of fractional ideals relatively prime to $\mathfrak c$, $P_{\mathfrak c}$ is the group of principal fractional ideals $(x)$ where $x \equiv 1 \mod^{\ast} \mathfrak c$, and $\mathfrak N(\mathfrak c)$ is the group of norms of fractional ideals of $K$ which are relatively prime to the places lying over those dividing $\mathfrak c$. Serge Lang uses the approximation theorem to prove that $$ P_{\mathfrak f} \mathfrak N(\mathfrak f) \cap I(\mathfrak c) = P_{\mathfrak c} \mathfrak N(\mathfrak c)$$ which shows that the inclusion above induces an injection $$I(\mathfrak c)/ P_{\mathfrak c} \mathfrak N(\mathfrak c) \rightarrow I(\mathfrak f)/P_{\mathfrak f} \mathfrak N(\mathfrak f)$$ However, he claims this is an isomorphism. Why is it surjective?

Answer 1

This is again an approximation theorem argument. Given an ideal $\frak{a}$ in $I(\frak{f})$, we can find some $\gamma \in k^\times$ such that $v(\gamma)=0$ if $v|\frak{f}$, and $v(\gamma)=-v(\frak{a})$ if $v|\frak{c}$, $v\not| \frak{f}$. Then $\gamma{\frak{a}} \in I(\frak{c})$. So in fact $I({\frak{c}}) \rightarrow I({\frak{f}})/P_{\frak{f}}$ is already surjective. Compare this with $J_{\frak{c}}/k_{\frak{c}}\simeq J/k^*$ for any $\frak{c}$.