Fitting ideals of class groups in a \(\mathbb{Z}_p\)-extension. (English) Zbl 0926.11084

Cornacchia, Pietro

Acta Arith. 87, No. 1, 79-88 (1998).

The author first proves a result about the module of Iwasawa algebra, which is finitely generated as \(\mathbb{Z}_p\)-module. The result can be roughly described as follows: the cohomology groups at finite levels determine the \(\mathbb{Z}_p\)-torsion part and the characteristic ideal gives information on the \(\mathbb{Z}_p\)-free part.
As an application of this result, the author proves that the Fitting ideal of the \(\chi\)-parts of the ideal class group is equal to the fitting ideal of the \(\chi\)-parts of the group of units modulo cyclotomic units in any finite level for the cyclotomic \(\mathbb{Z}_p\)-extension of a totally real abelian number field and \(p\)-adic character \(\chi\) of this field.

Reviewer: Xu Fei (Münster)

MSC:

11R29	Class numbers, class groups, discriminants
11R23	Iwasawa theory
11R18	Cyclotomic extensions

Keywords:

\(\mathbb{Z}_p\)-extension; Tate cohomology group; ideal class group; Iwasawa algebra; Fitting ideal