4 Pure preimages and common finite observations
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The common-completion phenomenon is not confined to one selector construction. It follows from a general correspondence between rational subspaces and pure intermediate groups.
4.1 The preimage construction
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Keep \(D\subseteq P\) and \(V=P/D\) from Chapter 2. For a rational vector subspace \(W\le V\), define
This is an actual subgroup of \(P\) containing \(D\). Its quotient by \(D\) is canonically \(W\). It is dense because it contains the dense subgroup \(D\).
The construction separates the ambient coordinate space from the admissible realization. The quotient \(V\) records infinite coordinate tails modulo finite changes. A rational subspace \(W\) selects which such tails are allowed. The preimage restores all finite-support corrections.
Definition 4.1: Purity
A subgroup \(H\) of an abelian group \(M\) is pure when
Equivalently, if an equation \(my=h\) with \(h\in H\) has a solution in \(M\), then it has a solution in \(H\).
Purity is a standard algebraic notion [3]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.
Theorem 4.2: Pure-preimage correspondence
The map
is an inclusion-preserving bijection between rational subspaces of \(V\) and pure subgroups \(H\) of \(P\) containing \(D\).
Proof
First let \(W\) be rational and suppose \(h\in H_W\cap mP\). Because \(W\) is closed under division by \(m\), choose \(x\in H_W\) with \(q(x)=q(h)/m\). Then \(d=h-mx\) belongs to \(D\). At a prime dividing \(m\), the coordinate of \(h\) is zero because \(h\in mP\), and that of \(mx\) is also zero. Thus \(d_p=0\) whenever \(p\mid m\).
Define \(e_p=m^{-1}d_p\) at every other prime and \(e_p=0\) at primes dividing \(m\). The vector \(e\) has finite support and satisfies \(me=d\). Hence \(h=m(x+e)\) with \(x+e\in H_W\). This proves purity.
Conversely, let \(D\subseteq H\subseteq P\) be pure. Its image \(q(H)\) is an additive subgroup of \(V\). Given \(h\in H\) and \(m\ge1\), let \(d\in D\) agree with \(h\) on the primes dividing \(m\) and vanish elsewhere. Then \(h-d\in mP\cap H\). Purity gives \(h-d=mx\) for some \(x\in H\), so \(q(x)=q(h)/m\). Thus \(q(H)\) is closed under rational scalars. Since \(D\subseteq H\), the equality \(H=q^{-1}(q(H))\) holds. This proves that the constructions are inverse. Both preserve inclusion.
The bottom object is \(H_{\{0\}}=D\), and the top object is \(H_V=P\). Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with \(D\) serving as the bottom of the interval of intermediate groups.
4.2 Root conditions do not depend on the subspace
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Theorem 4.3: Ambient root criterion
For every rational subspace \(W\), every \(h\in H_W\), and every \(m\ge1\),
Proof
Multiplication by \(m\) on \(P\) is zero at primes dividing \(m\) and invertible on every other coordinate. Thus the right-hand condition is equivalent to \(h\in mP\). Apply purity from Theorem 4.2.
The condition is independent of \(W\), but the set of admissible \(h\) is not. This distinction is the reason identical quotient laws can coexist with different original groups.
Theorem 4.4: One finite-quotient system for every preimage
For every rational subspace \(W\) and every \(m\ge1\), the coordinate map induces an isomorphism
For \(m\mid n\), the quotient transition map is the coordinate restriction from primes dividing \(n\) to primes dividing \(m\). These quotients are cofinal among all finite quotients, and
Proof
The coordinate map onto the displayed finite product is surjective already on \(D\). Its kernel is \(mH_W\) by Theorem 4.3. The transition assertion follows by evaluating the same coordinates.
If \(H_W\to F\) is a homomorphism to a finite group, then the image is abelian and is killed by \(\lvert F\rvert\). Its kernel therefore contains \(\lvert F\rvert H_W\). Hence the displayed kernels are cofinal. Their inverse limit is the product of all prime coordinates, which is \(P\). Under this identification the completion map is the original inclusion \(H_W\hookrightarrow P\).
There is a stronger natural statement. It compares maps to every finite target rather than merely comparing the quotient groups.
Proposition 4.5: Finite-target equivalence
For every finite group \(F\), restriction along \(D\hookrightarrow H_W\) gives a natural bijection
Proof
Put \(m=\lvert F\rvert\). Every \(h\in H_W\) agrees modulo \(mH_W\) with an element of \(D\), namely its finite coordinate vector on primes dividing \(m\). Every homomorphism to \(F\) kills \(mH_W\). Thus two maps agreeing on \(D\) agree everywhere.
Conversely, a map \(D\to F\) kills every prime factor whose order does not divide \(m\). It factors through the finite coordinate product on primes dividing \(m\). Composing that finite-coordinate rule with the coordinate projection of \(H_W\) extends it. The extension is unique by the first part. Postcomposition with a map of finite groups commutes with restriction, giving naturality.
The argument never subtracts maps to a nonabelian group. The assertion is a bijection of sets of homomorphisms, and naturality is with respect to postcomposition.
4.3 Countability, rank, and nonsplitting
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Proposition 4.6: Structure of an ambient preimage
For every rational subspace \(W\),
The group \(H_W\) is reduced and residually finite. It is countable exactly when \(W\) has finite or countably infinite dimension over \({\mathbb Q}\). If \(W\ne0\), the sequence
does not split.
Proof
The torsion equality follows from \(D\subseteq H_W\subseteq P\) and Theorem 2.1. The quotient assertion is the first isomorphism theorem. Reducedness follows from reducedness of \(P\). A nonzero element has a nonzero finite coordinate, which detects it and proves residual finiteness.
A rational vector space on a finite or countable basis is countable, since each vector has a finite rational linear expression. Each fiber of \(H_W\to W\) is a coset of the countable group \(D\). Thus \(H_W\) is countable in this case. Conversely, a quotient of a countable group is countable and has at most a countable basis.
A splitting would embed the nonzero divisible group \(W\) into \(P\). Theorem 2.6 excludes every such subgroup.
The rank of the torsion-free quotient is exactly \(\dim_{{\mathbb Q}}W\). This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it admits an especially sharp classification and computability theorem.
4.4 What has been fixed and what still varies
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The entire product \(P\), all finite quotient objects, their transition maps, and the maps to arbitrary finite targets are fixed throughout the construction. The subspace \(W\) remains free. This is an ambient realization theorem, not a method for discovering \(W\) from those observations.
The preimage formula also identifies the correct role of completion. Every \(H_W\) contains the same dense torsion subgroup. Its completion admits every compatible coordinate tuple. Its original image admits only those tuples whose tail class lies in \(W\). The missing information is an admissibility condition on the quotient, not a defect in the finite coordinates themselves.