= First Galois cohomology unramified outside a finite set
{title2=$H^1_S(K,M)$}
For a finite Galois module $M$ unramified outside a finite set $S$, define $H^1_S(K,M)$ as the classes in <Galois cohomology> whose restriction to inertia is zero at every finite prime outside $S$. This means zero as a cohomology class, rather than zero for every possible representative. Passing to a finite Galois extension on which $M$ is constant reduces the finiteness problem to characters with bounded ramification and a finite restriction kernel.
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