Realization Calculus: twin edition

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1 \chapter{Pure preimages and common finite observations}
2 \label{ch:purity}
3 \chapterintro{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 
5 \section{The preimage construction}
6 Keep $D\subseteq P$ and $V=P/D$ from Chapter~\ref{ch:ambient}. For a rational vector subspace $W\le V$, define
7 \[
8 H_W=q^{-1}(W).
9 \]
10 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 $D$.
11 
12 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.
13 
14 \begin{definition}[Purity]
15 A subgroup $H$ of an abelian group $M$ is pure when
16 \[
17 H\cap mM=mH\qquad(m\ge1).
18 \]
19 Equivalently, if an equation $my=h$ with $h\in H$ has a solution in $M$, then it has a solution in $H$.
20 \end{definition}
21 Purity is a standard algebraic notion \cite{Fuchs}. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.
22 
23 \begin{theorem}[Pure-preimage correspondence]
24 \label{thm:purecorrespondence}
25 The map
26 \[
27 W\longmapsto H_W=q^{-1}(W)
28 \]
29 is an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.
30 \end{theorem}
31 \begin{proof}
32 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 $p\mid m$.
33 
34 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.
35 
36 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.
37 \end{proof}
38 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.
39 
40 \section{Root conditions do not depend on the subspace}
41 \begin{theorem}[Ambient root criterion]
42 \label{thm:ambientroot}
43 For every rational subspace $W$, every $h\in H_W$, and every $m\ge1$,
44 \[
45 h\in mH_W\quad\Longleftrightarrow\quad
46 h_p=0\text{ for all primes }p\mid m.
47 \]
48 \end{theorem}
49 \begin{proof}
50 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~\ref{thm:purecorrespondence}.
51 \end{proof}
52 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.
53 
54 \begin{theorem}[One finite-quotient system for every preimage]
55 \label{thm:generalquotients}
56 For every rational subspace $W$ and every $m\ge1$, the coordinate map induces an isomorphism
57 \[
58 H_W/mH_W\ \cong\ \prod_{p\mid m}C_p.
59 \]
60 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
61 \[
62 \widehat{H_W}\cong P.
63 \]
64 \end{theorem}
65 \begin{proof}
66 The coordinate map onto the displayed finite product is surjective already on $D$. Its kernel is $mH_W$ by Theorem~\ref{thm:ambientroot}. The transition assertion follows by evaluating the same coordinates.
67 
68 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 $H_W\hookrightarrow P$.
69 \end{proof}
70 There is a stronger natural statement. It compares maps to every finite target rather than merely comparing the quotient groups.
71 
72 \begin{proposition}[Finite-target equivalence]
73 \label{thm:finitehom}
74 For every finite group $F$, restriction along $D\hookrightarrow H_W$ gives a natural bijection
75 \[
76 \Hom(H_W,F)\longrightarrow\Hom(D,F).
77 \]
78 \end{proposition}
79 \begin{proof}
80 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.
81 
82 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.
83 \end{proof}
84 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.
85 
86 \section{Countability, rank, and nonsplitting}
87 \begin{proposition}[Structure of an ambient preimage]
88 \label{thm:HWstructure}
89 For every rational subspace $W$,
90 \[
91 \Tor(H_W)=D,\qquad H_W/D\cong W.
92 \]
93 The group $H_W$ is reduced and residually finite. It is countable exactly when $W$ has finite or countably infinite dimension over $\Q$. If $W\ne0$, the sequence
94 \[
95 0\longrightarrow D\longrightarrow H_W\longrightarrow W\longrightarrow0
96 \]
97 does not split.
98 \end{proposition}
99 \begin{proof}
100 The torsion equality follows from $D\subseteq H_W\subseteq P$ and Theorem~\ref{thm:torP}. 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.
101 
102 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.
103 
104 A splitting would embed the nonzero divisible group $W$ into $P$. Theorem~\ref{thm:Vrational} excludes every such subgroup.
105 \end{proof}
106 The rank of the torsion-free quotient is exactly $\dim_{\Q}W$. 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.
107 
108 \section{What has been fixed and what still varies}
109 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.
110 
111 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.

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