Completions of filtered groups, rings and modules. Applications to m-adic topologies

Abstract

Let G be a filtered abelian group (i.e. a filtered ℤ-module, see § 1), and let (G _n ) be its filtration. We can define a mapping

$$ v:G \to \mathbb{N} \cup \{ \infty \} $$

in the following way: $v(x) = \sup \{ n \in \mathbb{N}\} |x \in {G_n}$. It is clear that ν(x) = ∞ if and only if $x \in \cap {G_n},i.e.x \in \mathop 0\limits^ -$ (lemma 1.1). The mapping allows us to define a pseudometric in G: let

$$ d:G \times G \to G $$

be the mapping defined by d(x, y) = e^-ν(x-y) (we agree that e^-∞ = 0). Then it is easy to see that

$$ d(x,y) \leqslant \sup \{ d(x,z),d(y,z)\} $$

and that d defines in G the topology induced by the filtration (G _n ).