nLab completion of a group


Not to be confused with group completion.






(RR-complete group)
For RR a solid ring, a group GG is called RR-complete if it has a central series

G=G 0G 1G k=1 G \,=\, G_0 \supset G_1 \supset \cdots \supset G_k \,=\, 1

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

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


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

G R^limGCGrp /RCompGrp G/C 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 RR-complete groups under GG.

More generally:


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

(1)1NG^G1 1 \to N \longrightarrow \widehat{G} \longrightarrow G \to 1

is fiberwise or relative RR-complete if NN is so (in the sense of Def. ).


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

1 N G^ G 1 = 1 N R^ G^ R^ G G 1 \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)


Special cases


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


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

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


Original articles:

Further developments:

Last revised on July 16, 2022 at 21:21:09. See the history of this page for a list of all contributions to it.