  We show that the cosupport of a commutative noetherian ring is precisely the
set of primes appearing in a minimal pure-injective resolution of the ring. As
an application of this, we prove that every countable commutative noetherian
ring has full cosupport. We also settle the comparison of cosupport and support
of finitely generated modules over any commutative noetherian ring of finite
Krull dimension. Finally, we give an example showing that the cosupport of a
finitely generated module need not be a closed subset of
$\operatorname{Spec}(R)$, providing a negative answer to a question of
Sather-Wagstaff and Wicklein.
