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In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue.  It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.

Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense.  In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).

This post serves to resolve some questions posed at the end of a previous article, Unit-Prime Dichotomy for Complex Subrings, in which we began to look at the tension between the relative sizes of the units group and the spectrum of  a complex subring (with unity).  As a disclaimer, I assume from this point familiarity with the topics presented therein.  To begin, we recall a Theorem from the prequel (Theorem 4):

Theorem: Let $R \subset \mathbb{C}$ be a subring (with unity), and suppose that $R$ has finite spectrum.  Then $R^\times$ is dense in $\mathrm{cl}(\mathrm{Frac}(R))$, the topological closure of the fraction field of $R$.

In particular — assuming a finite spectrum — it follows that the rank of $R^\times$ (considered as an abelian group) is bounded below by two. Unfortunately, this is as far as our previous methods will take us, even when $R \not\subset \mathbb{R}$ (see the Exercises).

The primary objective of this post is an extension to this result.  Specifically, we would like to capture in a purely algebraic way (e.g. without mention of the topology on $\mathbb{C}$) the fact that $R^\times$ becomes quite large in certain rings with finite spectrum.  After the fold we’ll accomplish exactly that, with the following Theorem and its generalizations to a wide class of commutative rings:

Theorem 1: Suppose that $R \subset \mathbb{C}$ is a subring (with unity).  If $R$ has finite spectrum, then $R^\times$ has infinite rank as an abelian group.

Throughout, we take $R \subset \mathbb{C}$ as a complex subring (with unity).  In this article, we’ll be interested in natural analogues of Euclid’s proof of the infinitude of the primes (i.e. the case $R = \mathbb{Z})$.  In particular, we’ll show that Euclid’s proof (and generalizations to this method) can be recast as a relationship between the size of the unit group $R^\times$ of $R$ and $\# \mathrm{Spec}(R)$, the number of prime ideals in$R$. (Here, $\mathrm{Spec}(R)$ denotes the spectrum of $R$.)

For the sake of explicit analogy, we include a (needlessly abstracted) version of Euclid’s result now:

Theorem 1 (Euclid):  The integers have infinite spectrum.

Proof: If not, let $\mathfrak{p}_1,\ldots, \mathfrak{p}_m$ exhaust the list of prime ideals.  As the integers form a PID, (in fact, they form a Euclidean domain), we may associate to each given ideal a prime generator $p_i$ such that $(p_i) = \mathfrak{p}_i$.  Let $N:= \prod p_i$; as $N+1$ is not a unit (replacing $N$ with $kN \gg 0$ if necessary), it admits a prime factor $p$ which equals some $p_i$.  But then $p$ divides both $N$ and $N+1$, whence $1$ as well.  This is a contradiction, and our result follows. $\square$

It is worth remarking that the “PID-ness” of the integers is not needed in the derivation of Theorem 1.  Indeed, if $\mathfrak{p}_i$ were not principal, we may take instead $p_i$ to be any non-zero element of $\mathfrak{p}_i$.  Thus, we see that the PID-ness of $\mathbb{Z}$ is not the crucial property of the integers (viewed as a subfield of $\mathbb{C}$) upon which our proof stands.  Rather — as we’ll see after the fold — Euclid’s proof frames the infinitude of primes as a consequence of the finiteness of the units group!