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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.
Reviewer: Xu Fei (Münster)

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