  The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules
that are abelian pro-p Galois groups with ramification allowed at a maximal set
of primes over p such that the module is torsion. A main conjecture for such an
Iwasawa module describes its codimension one support in terms of a p-adic
L-function attached to the primes of ramification. In this paper, we study more
general and potentially much smaller Iwasawa modules that are quotients of
exterior powers of Iwasawa modules with ramification at a set of primes over p
by sums of exterior powers of inertia subgroups. We show that the higher
codimension support of such quotients can be measured by finite collections of
p-adic L-functions under the relevant CM main conjectures.
