The tower of profinite completions

Abstract

We show that the tower of profinite completions of a nonstrongly complete profinite group continues indefinitely.

Cohomological properties of absolute Galois groups and their profinite completion

“…Notice that the orientation makes Z p a Z p [G]− module, by defining g.x = θ(g)x. We denote this module by Z p (1).…”

“…In [1] the author presented the infinite tower of profinite completions which is defined as follows: Let G be a nonstrongly complete profinite group and α an ordinal. Denote G 0 = G.…”

Profinite Completion of Free Pro-$\mathcal{C}$ Groups

“…If α is a limit ordinal, G α = H α where H α = lim →β<α {G β , ϕ γβ }, where ϕ γβ are compatible homomorphisms which are defined for every γ < β. In [2] it is shown that all the groups in this chain are nonstrongly complete, and all the homomorphisms are injections. That makes the chain a strictly increasing chain, which never terminates.…”

“…In addition to the profinite completion we just define, which we can treat as the "first" profinite completion, a nonstrongly complete profinite group admits a series of profinite completions of higher orders, which we denote by G α , and are defined for every ordinal α. These completions of higher order were first introduced in [2] and are built as follows: Let G be a nonstrongly complete profinite group. Denote G = G 0 .…”