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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)].

16S34 Group rings
22E35 Analysis on \(p\)-adic Lie groups
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