It seems like this particular theorem is always stated in a way that's slightly hard to interpret. Let $S$ be some finite set of primes containing all the primes of $K$ ramifying in $L/K$. Then the statement is that the map
$\psi:I^S_K\to Gal(L/K)$
admits a modulus $\mathfrak{m}$ s.t. $S(\mathfrak{m})\subset S$ and we get a factorization
$\psi:I^S_K\to I^{\mathfrak{m}}_K/i(K_{\mathfrak{m},1})\to Gal(L/K)$
I was wondering if we always know precisely what the kernels are for both maps in the factorization below? Is it dependent on $\mathfrak{m}$? I've seen
i(K_{\mathfrak{m},1})\cdot Nm_{L/K}(I^{\mathfrak{m}'}_L)
show up in some places where \mathfrak{m}' is the modulus consisting of the set of primes in $L$ lying above the primes in $\mathfrak{m}$. How does this norm thing fit in?