p-local module

Let GG be a group and MM an abelian group with a GG-action, i.e. a G\mathbb{Z}G-module where G\mathbb{Z}G is the group ring of GG. Let TT be a set of prime numbers; we say that MM is a TT-local module if for all xGx\in G, the element (1+x+x 2++x n1)G(1 + x + x^2 + \cdots + x^{n-1})\in \mathbb{Z}G acts invertibly on MM for all nn not divisible by any primes in TT. If T={p}T=\{p\} we speak of a pp-local module.

Note that if GG acts trivially on MM, this is equivalent to asking that MM is a TT-local abelian group. In general, it is strictly stronger.

The notion can also be characterized as a module over the localization of the (noncommutative!) ring G\mathbb{Z}G at the set of elements 1+x+x 2++x n11 + x + x^2 + \cdots + x^{n-1}.

Last revised on June 25, 2014 at 04:54:39. See the history of this page for a list of all contributions to it.