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-where O is the ring of integers of a finite extension of the field of p-adic integers Qp. The specialization method is a technique that recovers the information on the characteristic ideal charR(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 [31] and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in [28].

An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case

“…It is also essential in the proof of the Bertini-type theorem for reduced hyperplane quotients of complete local rings of characteristic p > 0. This result is proved in [7]. Finally, we mention that there is a version of Theorem 1.1 for an affine domain over a perfect field (see [8,Theorem 4.2.2]).…”