  Let $k$ be a field, with absolute Galois group $\Gamma$. Let $A/k$ be a
finite \'etale group scheme of multiplicative type, i.e. a discrete
$\Gamma$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a
cohomology class. We show that there exists a countable set $I$, and a familiy
$(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that
the following holds. For any field extension $l/k$, the restriction of $x$
vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an
$l$-point. We moreover show that the $X_i$'s can be made into an ind-variety.
In the case $n=2$, we note that one variety is enough.

Algebraic Geometry