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The aim of this article is to establish the specialization method oncharacteristic ideals for finitely generated torsion modules over a completelocal 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-adicintegers $Q_p$. The specialization method is a technique that recovers theinformation 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. Asapplications, we prove Euler system bound over Cohen-Macaulay normal domains bycombining the main results in an earlier article of the first named author andthen we prove one of divisibilities of the Iwasawa main conjecture fortwo-variable Hida deformations generalizing the main theorem obtained in anarticle of the first named author.

Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings