Journal of Algebraic Geometry

Author(s): Konstantin Ardakov; Simon Wadsley
Journal: J. Algebraic Geom. 15 (2006), 339-377.
Posted: June 7, 2005
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Abstract: Let

be a compact

-adic analytic group with no elements of order

. We provide a formula for the characteristic element (J. Coates, et. al., The main conjecture for elliptic curves without complex multiplication, preprint) of any finitely generated

-torsion module

over the Iwasawa algebra $\Lambda_G$ of

in terms of twisted $\mu$ -invariants of

, which are defined using the Euler characteristics of

and its twists. A version of the Artin formalism is proved for these characteristic elements. We characterize those groups having the property that every finitely generated pseudo-null

-torsion module has trivial characteristic element as the

-nilpotent groups. It is also shown that these are precisely the groups which have the property that every finitely generated

-torsion module has integral Euler characteristic. Under a slightly weaker condition on

we decompose the completed group algebra $\Omega_G$ of

with coefficients in $\mathbb{F} _p$ into blocks and show that each block is prime; this generalizes a result of Ardakov and Brown (Primeness, semiprimeness and localisation in Iwasawa Algebras, submitted). We obtain a generalization of a result of Osima (On primary decomposable group rings, Proc. Phy-Math. Soc. Japan (3) 24 (1942) 1-9), characterizing the groups

which have the property that every block of $\Omega_G$ is local. Finally, we compute the ranks of the

group of $\Omega_G$ and of its classical ring of quotients $Q(\Omega_G)$ whenever the latter is semisimple.

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Konstantin Ardakov
Affiliation: Christ's College, University of Cambridge, Cambridge CB2 3BU, United Kingdom
Email: K.Ardakov@dpmms.cam.ac.uk

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