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On the structure theory of the Iwasawa algebra of a $$p$$-adic Lie group. (English) Zbl 1049.16016
Summary: This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $$\Lambda$$ of a $$p$$-adic analytic group $$G$$. For $$G$$ without any $$p$$-torsion element we prove that $$\Lambda$$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $$\Lambda$$-module. This is classical when $$G=\mathbb{Z}^k_p$$ for some integer $$k\geq 1$$, but was previously unknown in the non-commutative case. Then the category of $$\Lambda$$-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the $$\mathbb{Z}_p$$-torsion part of a finitely generated $$\Lambda$$-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of Abelian varieties are published elsewhere [cf. Compos. Math. 138, No. 1, 1-54 (2003; Zbl 1039.11073)].

##### MSC:
 16S34 Group rings 22E35 Analysis on $$p$$-adic Lie groups
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