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

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