On the failure of pseudo-nullity of Iwasawa modules

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 λ-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 μ = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in pro-cyclic extensions. These sequences are derived in an appendix by the second author.