  Let $K$ be a number field and $S$ a finite set of places of $K$. We study the
kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S}
H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$,
into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$,
which is always an isomorphism in the wild case. The tame case is much more
mysterious. Our main result is that given a finite $X$ coprime to $p$, there
exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X}
\stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X}
\stackrel{\simeq}{\twoheadleftarrow} \CyB_X \hookleftarrow \Sha_X$. In
particular, we show that in the tame case $\Sha_Y$ can {\it increase} with
increasing $Y$. This is in contrast with the wild case where $\Sha_Y$ is
nonincreasing in size with increasing $Y$.
