nLab completion of a group

Not to be confused with group completion.

Idea

(…)

Definition

Definition

($R$-complete group)
For $R$ a solid ring, a group $G$ is called $R$-complete if it has a central series

$$G \,=\, G_0 \supset G_1 \supset \cdots \supset G_k \,=\, 1$$

(i.e. such that under commutator subgroups $[-,-]$ we have $[G, G_i] \subset G_{i+1}$)

such that each quotient group $G_j/G_{j+1}$ (which is an abelian group by the previous property) carries the structure of an $R$-module.

Proposition

(completion of a group)
Given a solid ring $R$ and any group $G$, there is a universal construction of an $R$-complete group $G \xrightarrow{\;\;} G_{\widehat{R}}$ – the $R$-completion of $G$ – given by the limit

$$G_{\widehat{R}} \;\simeq\; \underset{ \underset{ { G \to C \in } \atop { Grp^{G/}_{/R CompGrp} } }{\longleftarrow} }{\lim} \;C\;$$

over the codomain functor from the comma category of $R$-complete groups under $G$.

More generally:

Definition

(fiberwise $R$-complete group)
A short exact sequence of groups

(1)$$1 \to N \longrightarrow \widehat{G} \longrightarrow G \to 1$$

is fiberwise or relative $R$-complete if $N$ is so (in the sense of Def. ).

Proposition

(fiberwise completion of a group)
Given a solid ring $R$ and any short exact sequence of groups (1), there is a universal construction of a fiberwise $R$-complete exact sequence

$$\array{ 1 &\to& N &\longrightarrow& \widehat{G} &\longrightarrow& G &\to& 1 \\ && \big\downarrow && \big\downarrow && \big\downarrow \mathrlap{{}^{=}} \\ 1 &\to& N_{\widehat{R}} &\longrightarrow& \widehat{G}_{\widehat{R}_G} &\longrightarrow& G &\to& 1 }$$

(Bousfield & Kan 1971, §2)

Properties

Special cases

Example

(profinite completion)
For $R = \mathbb{Z}/p$ (a cyclic group of prime order) and applied to finitely generated groups, $R$-completion coincides with $p$-profinite completion.

Example

(Malcev completion)
For $R = \mathbb{Q}$ (the rational numbers) and applied to nilpotent groups, $R$-completion coincides with Malcev completion.

(e.g. Bousfield & Kan 1972, p. 99)

Related concepts

• completion of a ring

• completion of a space

  • p-completion, p-adic homotopy theory

  • rationalization, rational homotopy theory

• solid ring

References

Original articles:

• Aldridge Bousfield, Daniel Kan, §2 of: Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 6 (1971) 1006-1010 [doi:10.1090/S0002-9904-1971-12837-9, pdf]

• Aldridge Bousfield, Daniel Kan, Chapter IV of: Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 Springer (1972) [doi:10.1007/978-3-540-38117-4]

Further developments:

• Carles Casacuberta, An Descheemaeker, Relative group completions, Journal of Algebra, 285 2 (2005) 451-469 [doi:10.1016/j.jalgebra.2004.10.023]