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Krull dimension of Iwasawa algebras Konstantin Ardakov

Summary: Krull dimension of Iwasawa algebras
Konstantin Ardakov
1 Introduction
Let G be a compact p-adic Lie group. In the recent years, there has been an
increased amount of interest in completed group algebras (Iwasawa algebras)
G = Z p [[G]] := lim
N/oGZ p [G=N ];
for example, because of their connections with number theory and arithmetic
geometry; see the paper by Coates, Schneider and Sujatha ([4]) for more details.
When G is a uniform pro-p group, G is a concrete example of a complete
local Noetherian ring (noncommutative, in general) with good homological prop-
erties: it is known that G has nite global dimension and is an Auslander regu-
lar ring. Thus, G falls into the class of rings studied by Brown, Hajarnavis and
MacEacharn in [1]. There they consider various properties of Noetherian rings
R of nite global dimension, including the Krull(-Gabriel-Rentschler) dimension
K(R) - a module-theoretic dimension which measures how far R is from being
Artinian. They also posed the following question:
Question ([1], Section 5). Let R be a local right Noetherian ring, whose
Jacobson radical satis es the Artin-Rees property. Is the Krull dimension of R
always equal to the global dimension of R?


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


Collections: Mathematics