  The aim of this article is to establish the specialization method on
characteristic ideals for finitely generated torsion modules over a complete
local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where
$O$ is the ring of integers of a finite extension of the field of p-adic
integers $Q_p$. The specialization method is a technique that recovers the
information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$,
where I varies in a certain family of nonzero principal ideals of R. As
applications, we prove Euler system bound over Cohen-Macaulay normal domains by
combining the main results in an earlier article of the first named author and
then we prove one of divisibilities of the Iwasawa main conjecture for
two-variable Hida deformations generalizing the main theorem obtained in an
article of the first named author.
