A cohomological support, $\operatorname{Supp}^*_{\mathcal A}(M)$,
is defined for finitely generated modules $M$ over a left noetherian
ring $R$, with respect to a ring $\mathcal A$ of central cohomology
operations on the derived category of $R$-modules. It is proved that
if the $\mathcal A$-module $\operatorname{Ext}^*_R(M,M)$ is noetherian
and $\operatorname{Ext}^*_R(M,R)=0$ for $i\gg0$, then every closed
subset of $\operatorname{Supp}^*_{\mathcal A}(M)$ is the support of
some finitely generated $R$-module. This theorem specializes to known
realizability results for varieties of modules over group algebras, over
local complete intersections, and over finite dimensional algebras over
a field. The theorem is also used to produce large families of finitely
generated modules of finite projective dimension over commutative local
noetherian rings.