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profinite completion of a group

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Definition

Profinite completion of a group

The profinite completion G^\hat{G} of a (discrete) group GG is the limit (in the category of topological groups) over the diagram with objects all the finite quotient groups G/N finG/N_{fin} where N finN_{fin} is a normal subgroup of GG with finite index, and morphisms induced from the lattice of subgroups of GG.

Note that the profinite completion actually is a profinite group, and there is a canonical homomorphism GG^G \to \hat{G}.

More formally, we note that for any group GG, the family of its normal finite index subgroups forms a cofiltered category under inclusion. (Denote it by Ω G\Omega_G.) The assignment of G/NG/N to NN gives a functor from Ω G op\Omega_G^{op} to the category of finite groups. It is thus a profinite group in the sense given in that entry, i.e. a pro-object in the category of finite groups. This pro-object is the profinite completion of GG.

The above topological version of this, with which we started, is obtained by means of the equivalence between the category of pro-(finite groups) and that of the groups internal to profinite spaces that is by taking the limit in the category of topological groups of the diagram of (discrete) finite groups that the above construction gives one.

Remark

The inclusion functor incinc from the category, FinGrpsFinGrps, of finite groups, into that of groups does not have a left adjoint. It does have a left pro-adjoint, that is to say, it induces a functor on procategories

proinc:proFinGrpsproGrps pro\!-\!inc : pro\!-\! FinGrps\to pro\!-\Grps

and that functor does have a left adjoint. If we restrict that ‘pro-adjoint’ to the subcategory of proFinGrpspro\!-\! FinGrps given by the ‘constant’ pro-objects, then the result is the pro-finite completion construction that is given above. Because of this, if we think of the natural functor CproCC\to pro\!-\! C to be an inclusion, i.e. think of an object as a pro-object indexed by the one arrow category, we can give a universal property for the pro-finite completion of a group GG. This universal property gives a universal cone from GG to finite groups, and just encodes the obvious fact that any homomorphism from GG to a finite group factors through one of its finite quotient groups. If we write G^\hat{G} for the pro-finite completion, the universal cone is a map GG^G\to \hat{G} in proGrpspro\!-\! Grps.

Pro-𝒞\mathcal{C} completion of a group

Let 𝒞\mathcal{C} be any class of finite groups that is closed under the formation of subgroups, homomorphic images and group extensions.

Definition

A pro-𝒞\mathcal{C} group is an inverse limit of an inverse system of groups in the class 𝒞\mathcal{C} or alternatively a pro-object in the full subcategory 𝒞\mathcal{C} determined by the class 𝒞\mathcal{C}.

The subcategory of proFinGrpspro\!-\!FinGrps consisting of the pro-𝒞\mathcal{C} groups and the continuous homomorphisms between them will be denoted pro𝒞pro\!-\!\mathcal{C}.

This notation now has two definitions, but, as the corresponding categories are equivalent, this causes no problem.

The categories of the form pro𝒞pro\!-\!\mathcal{C} form varieties in profFinGrpsprof-FinGrps. Recall that a variety in any algebraic context means a subcategory of ‘algebras’ closed under products, subobjects and quotients. We note the condition on 𝒞\mathcal{C} implies the closure of 𝒞\mathcal{C} under finite products, so 𝒞\mathcal{C} is what is called a pseudovariety. The category proFinGrpspro\!-\!FinGrps is monadic over the category of spaces. This means that free objects exist in all the pro𝒞pro\!-\!\mathcal{C}. A good reference for this is Gildenhuys and Kennison, (1971), see below.

Examples

Example

Consider the profinite completion of the fundamental group of an complex projective variety XX. Since XX has an underlying topological space, its fundamental group of loops π 1 top(X)\pi_1^{top}(X) can be defined in the usual way. But one can also define the algebraic fundamental group π 1 alg(X)\pi_1^{alg}(X). This is a profinite group, which is isomorphic to the profinite completion of π 1 top(X)\pi_1^{top}(X).

Example

The profinite completion of the integers is

^lim n/n. \widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_n \mathbb{Z}/n\mathbb{Z} \,.

This is isomorphic to the product of the p-adic integers for all pp

^p p. \widehat{\mathbb{Z}} \simeq \underset{p}{\prod} \mathbb{Z}_p \,.

For more on this see at p-adic integers, at adele and idele.

Profinite completion of spaces

(Beware there are two possible interpretations of this term. One is handled in the section above, being profinite completion of the homotopy type of a space. The entry linked to here treats another more purely topological concept.)

Equational or monadic completions

Profinite completion of groups is a special case of a general process that ‘completes’ a category together with a ‘forgetful functor’ to some ‘base’ category, replacing it by a category which is equational/monadic over the base.

  • D. Gildenhuys and J. Kennison, Equational completions, model induced triples and pro-objects, J. Pure Applied Alg., 4, (1971), 317–346.

Pro-finite completions of homotopy types.

Artin and Mazur in their lecture note on étale homotopy introduced a process of profinite completion, generalising that for groups in as much as the profinite completion of an Eilenberg-Mac Lane space having GG as fundamental group has the profinite completion og GG as its fundamental group. (WARNING: This needs a bit more detail to make it true! so this part of the entry needs more work.)

References

  • L. Ribes and P. Zalesskii, 2000, Profinite groups , volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge , Springer-Verlag, Berlin.

  • J. Dixon, M. du Sautoy, A. Mann and D. Segal, 1999, Analytic pro-p groups, volume 61 of Cambridge Studies in Advanced Mathematics , Cambridge Univ. Press.

Revised on April 18, 2014 04:23:05 by Urs Schreiber (89.204.155.204)