The profinite completion of the integers, the p-adic integers, and Prüfer p-groups

Jordan Bell
December 3, 2017

1 Topological rings and inverse systems

By a topological ring we mean a ring X with a Hausdorff topology such that (x,y)x+y,x-x,(x,y)xy are continuous. A morphism of topological rings is a continuous homomorphism of rings. An inverse system of topological rings is a family of topological rings Xi and a family of morphisms πi,j:XiXj for i,jI with ij, such that when ijk,

πi,k=πj,kπi,j.

If Y is a topological ring, we say that a family of morphisms ψi:YXi is compatible with the inverse system if, whenever ij,

πi,jψi=ψj.

A topological ring X and a compatible family of morphisms πi:XXi is said to be an inverse limit of the inverse system if whenever Y is a topological ring and ψi:YXi is a compatible family of morphisms, there is a unique morphism ψ:YX such that for all i,

πiψ=ψi.

If (X,πi),(Y,ψi) are inverse limits of an inverse system, one checks that there is a unique isomorphism ψ:XY such that ψiψ=πi for all i.11 1 Luis Ribes and Pavel Zalesskii, Profinite Groups, second ed., Chapter 1, “Inverse and Direct Limits”, p. 2, Proposition 1.1.1 (b). If at least one inverse limit exists for an inverse system, we permit ourselves to speak about the inverse limit of the inverse system.

For showing that the inverse limit of an inverse system exists and for establishing properties of the inverse limit, rather than stating that it is an object satisfying a universal property we can construct it in the following way. Let X be those xiIXi such that for ij,

πi,j(xi)=xj.

It is straightforward to check that X is a subring of iIXi and that with the subspace topology inherited from the direct product it is a topological ring. We define πi:XXi by πi=piι, where ι:XiXi is the inclusion map and pi:jIXjXi is the projection map. One checks that the morphisms πi are compatible with the inverse system, and then that X together with this family of morphisms is an inverse limit of the inverse system.22 2 Luis Ribes and Pavel Zalesskii, Profinite Groups, second ed., Chapter 1, “Inverse and Direct Limits”, p. 2, Proposition 1.1.1 (a). This establishes that the inverse system has an inverse limit. Furthermore, one proves that X is a closed subset of iIXi.33 3 Luis Ribes and Pavel Zalesskii, Profinite Groups, second ed., Chapter 1, “Inverse and Direct Limits”, p. 3, Lemma 1.1.2. This lets us deduce properties of the inverse limit from weakly hereditary properties of the direct product.

2 Profinite rings

A profinite ring is a topological ring that is the inverse limit of an inverse system of finite topological rings; since we demand that topological rings be Hausdorff, being finite implies having the discrete topology. Suppose that Xi with morphisms πi,j, ij,i,ji, are an inverse system of finite topological rings. Because Xi is finite it is compact, so the direct product iIXi is compact. As the inverse limit X of this inverse system is a closed subset of the direct product, X is a compact topological space.

A topological space is called totally disconnected if a subset being connected implies that the subset contains at most one point .In other words, a topological space is totally disconnected if its connected components are all the singletons. One checks that a discrete topological space is totally disconnected, and that a product of totally disconnected spaces is totally disconnected, and that being totally disconnected is hereditary.44 4 Stephen Willard, General Topology, p. 210, §29. Therefore, a profinite ring is compact and totally disconnected.55 5 In fact, a totally disconnected compact group must be an inverse limit of finite discrete groups: Markus Stroppel, Locally Compact Groups, p. 172.

3 Profinite completion of the integers

With the discrete topology, /n is a topological ring. For m|n, we take ϕn,m:/n/m to be the projection map. The topological rings /n and the morphisms ϕn,m are an inverse system in the category of topological rings (ordering the indices by nm when m|n), and we denote the inverse limit by ^, with morphisms ϕn:^/n satisfying

ϕn,mϕn=ϕm,m|n,

called the profinite completion of Z. ^ is a profinite ring, hence it is compact and totally disconnected, and because the inverse system consists of countably many metrizable limitands, the direct product n=1/n and thus the inverse limit is metrizable.

Let ψn:/n be the projection map. For m|n,

ϕn,mψn=ψm.

Namely, the morphisms ψn are compatible with the inverse system. For example,

ϕ15,3ψ15(22)=ϕ15,3(7+(15))=1+(3)=ψ3(22).

Hence there is a unique morphism ψ:^ such that for all n1,

ϕnψ=ψn.

If a,b and ab, there is some n such that ab(modn), that is, ψn(a)ψn(b). It must then be that ψ(a)ψ(b). Therefore, ψ is one-to-one.

Because ^ is compact and metrizable it is separable. We prove that the image of in its profinite completion is dense, which explicitly displays a countable dense subset.66 6 Brian Osserman, Inverse limits and profinite groups, https://www.math.ucdavis.edu/~osserman/classes/250C/notes/profinite.pdf

Theorem 1.

ψ() is a dense subset of Z^.

Proof.

Let U be a nonempty subset of ^. ^ has the subspace topology inherited from the direct product n=1/n, so there are open sets Vn in /n, where there are only finitely many n such that Vn/n, such that for V=n=1Vn, the set ^V is nonempty and is contained in U. To prove that ψ() is dense in ^ it will suffice to prove that there is some a such that ψ(a)^VU.

Take n0 such that for n>n0, Vn=/n. (In this proof by we mean the usual order on the positive integers, not nm when m|n.) Because ^V is nonempty, there is some x^V. Let N=lcm(1,2,,n0) and let aψN-1(ϕN(x)). For 1nn0, n|N and

ϕn(ψ(a)) =(ϕN,nϕN)(ψ(a))
=(ϕN,nϕNψ)(a)
=(ϕN,nψN)(a)
=ϕN,n(ψN(a))
=ϕN,n(ϕN(x))
=(ϕN,nϕN)(x)
=ϕn(x).

Hence ϕn(ψ(a))Vn, and so ψ(a)V. ψ:^ so ψ(a)^. Therefore, ψ(a)^V, which proves the claim. ∎

4 p-adic integers

Let p be a prime. /pn with the discrete topology is a topological ring. For nm, let πn,m:/pn/pm be the projection map. For example, with p=3,

π3,2(15+(33))=6+(32).

The topological rings /pn and the morphisms πn,m are an inverse system in the category of topological rings. The inverse limit of this inverse system is a topological ring denoted by p, together with morphisms πn:p/pn such that

πn,mπn=πm

for nm. We call p the ring of p-adic integers. It is compact and totally disconnected. Furthermore, because each limitand /pn is metrizable by the discrete metric, the countable direct product n=1/pn is metrizable, and therefore so is p.

Let χn:/pn be the projection maps. For nm,

πn,mχn=χm.

Namely, the morphisms χn are compatible with the inverse system, and therefore there is a unique morphism χ:p such that for all n1,

πnχ=χn.

If a,b and ab, there is some n such that ab(modpn), so that χn(a)χn(b), whence χ(a)χ(b). This shows that χ:p is one-to-one. Furthermore, like how the image of in ^ is dense, the image of in p is dense.

Theorem 2.

χ() is a dense subset of Zp.

5 The Chinese remainder theorem

For n a positive integer, let vp(n) denote the highest power of the prime p that divides n. For example, v3(45)=2 and v3(11)=0. The Chinese remainder theorem states that

/np/pvp(n).

Then, supposing that the following steps are correct,77 7 See Paul Garrett, The ur-solenoid and the adeles, http://www.math.umn.edu/~garrett/m/mfms/notes/04_ur_solenoid.pdf

^=limn/nlimnp/pvp(n)plimν/pνpp

as topological rings.

6 Direct systems

A direct system of abelian groups is a family of abelian groups Ai and a family of group homomorphisms ϕi,j:AiAj for i,jI with ij, such that when ijk,

ϕi,k=ϕj,kϕi,j.

If A is an abelian group, we say that a family of group homomorphisms ψi:AiA is compatible with the direct system if, whenever ij,

ψjϕi,j=ψi.

An abelian group A and a compatible family of group homomorphisms ϕi:AiA is said to be a direct limit of the direct system if whenever B is an abelian group and ψi:AiB is a compatible family of group homomorphisms, there is a unique group homomorphism ψ:AB such that for all i,

ψϕi=ψi.

It can be proved that a direct system of abelian groups has a direct limit, and that if if (A,ϕi),(B,ψi) are direct limits of a direct system, then there is a unique group isomorphism ψ:AB such that ψi=ψϕi for all i.88 8 Luis Ribes and Pavel Zalesskii, Profinite Groups, second ed., Chapter 1, “Inverse and Direct Limits”, p. 15, Proposition 1.2.1. We permit ourselves to speak about the direct limit of the direct system.

7 Pontryagin duality

A morphism of a locally compact abelian group is a continuous group homomorphism. Let S1={z:|z|=1}. The Pontryagin dual of a locally compact abelian group G is the collection of morphisms GS1, where we define ϕ1ϕ2 by (ϕ1ϕ2)(x)=ϕ1(x)ϕ2(x).

It is a fact that if Gi is an inverse system of compact abelian groups with surjective morphisms GiGj for ij, then the Pontryagin dual of the inverse limit is isomorphic to the direct limit of the Pontryagin duals of the Gi, and that the direct limit is equal to the union of the images of the Pontryagin duals.99 9 Karl H. Hofmann and Sidney A. Morris, The Structure of Compact Groups, 2nd revised and augmented edition, p. 24, Proposition 1.36.

The Pontryagin dual of the compact abelian group /N is isomorphic to the discrete abelian group /N. (The discrete topology on a finite abelian group is compact.) The dual of the inverse system of projections πn,m:/pn/pm, nm, is the direct system of inclusion maps im,n:/pm/pn, mn, and the direct limit of this direct system is a discrete abelian group denoted by (p), called the Prüfer p-group, with morphisms in:/pn(p) and which satisfies

(p)=n+in(/pn).

8 Solenoids

For n0, let πn:/pn be the projection map, and give /pn the final topology induced by this map, with which /pn is a compact abelian group. For nm, let

φn,m:/pn/pm

be the projection map. The following diagram commutes:

It is immediate that the compact abelian groups /pn and the morphisms φn,m, nm, are an inverse system. We call the inverse limit of this sytem the p-adic solenoid, denoted 𝕋p, with morphisms φn:𝕋p/pn.1010 10 There are few books that present the p-adic solenoid. Two are Alain M. Robert, A Course in p-adic Analysis, p. 54, Appendix to Chapter 1, and Karl A. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups, p. 589, Example 14.4. 𝕋p is a compact abelian group.

One proves that each morphism φn:𝕋p/pn is onto. We now relate the p-adic solenoid to the p-adic integers.1111 11 Alain M. Robert, A Course in p-adic Analysis, p. 55, Appendix A.1. implies that /pn=/pn/pn. We model p as a subset of the direct product /pn and model 𝕋p as a subset of the direct product /pn, and thus the statement of the following theorem makes sense.

Theorem 3.

kerφn=pnp.