Specialization method in Krull dimension two and Euler system theory over normal deformation rings

Abstract

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 $${\mathcal {O}}[[x_1,\ldots ,x_d]]$$ , where $${\mathcal {O}}$$ is the ring of integers of a finite extension of the field of p-adic integers $${\mathbb {Q}}_p$$ . The specialization method is a technique that recovers the information on the characteristic ideal $${\text {char}}_R (M)$$ from $${\text {char}}_{R/I}(M/IM)$$ , 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 Ochiai (Nagoya Math J 218:125–173, 2015) and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in Ochiai (Compos Math 142:1157–1200, 2006).