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LOCALISATION AT AUGMENTATION IDEALS IN IWASAWA KONSTANTIN ARDAKOV
 

Summary: LOCALISATION AT AUGMENTATION IDEALS IN IWASAWA
ALGEBRAS
KONSTANTIN ARDAKOV
Abstract. Let G be a compact p-adic analytic group and let G be its
completed group algebra with coefficient ring the p-adic integers Zp. We
show that the augmentation ideal in G of a closed normal subgroup H of
G is localisable if and only if H is finite-by-nilpotent, answering a question
of Sujatha. The localisations are shown to be Auslander-regular rings with
Krull and global dimensions equal to dim H. It is also shown that the minimal
prime ideals and the prime radical of the Fp-version G of G are controlled
by + , where + is the largest finite normal subgroup of G. Finally, we
prove a conjecture of Ardakov and Brown[1].
1. Introduction
1.1. Iwasawa algebras. Let G be a compact p-adic analytic group and let G
and G denote the completed group algebras of G with coefficients in Zp and Fp,
respectively. Otherwise known as Iwasawa algebras, these rings were first defined
by Lazard [16] and have been the focus of increasing attention in recent years,
primarily because of their connections to number theory and arithmetic geometry.
We refer the reader to [11] and [12] for more information about these connections.
1.2. Iwasawa algebras also form a natural class of Noetherian algebras, analogous

  

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

 

Collections: Mathematics