Euler characteristics and compact p-adic Lie groups

TL;DR

This paper investigates the conditions under which Euler characteristics are well-defined for modules over Iwasawa algebras associated with compact p-adic Lie groups, establishing key criteria related to group structure and module properties.

Contribution

It characterizes when Euler characteristics are well-defined for modules over Iwasawa algebras, linking this to the group's finiteness and nilpotency, and explores properties of pseudo-null modules.

Findings

Euler characteristic is well-defined iff the group is finite-by-nilpotent.

Pseudo-null modules have trivial Euler characteristic under certain conditions.

Results extend to Akashi series for related modules.

Abstract

We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0th homology group is finite if and only if the relevant compact p-adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules. We also prove some analogous results for the Akashi series of Coates et al.