We consider the family of CM-fields which are pro-p p-adic Lie extensions of
number fields of dimension at least two, which contain the cyclotomic
Z_p-extension, and which are ramified at only finitely many primes. We show
that the Galois groups of the maximal unramified abelian pro-p extensions of
these fields are not always pseudo-null as Iwasawa modules for the Iwasawa
algebras of the given p-adic Lie groups. The proof uses Kida's formula for the
growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact,
we give a new proof of Kida's formula which includes a slight weakening of the
usual assumption that mu is trivial. This proof uses certain exact sequences
involving Iwasawa modules in procyclic extensions. These sequences are derived
in an appendix by the second author.Comment: 26 page
