Why are stably free modules called stably free?

Being of the kind who easily get troubled by questions of semantics, I've been troubled for a while now about why "stably free" modules are called just "stably free".

It seems to me that, logically, that by applying the adverb stably to the adjective free, you would wish to specify something stronger. That a module isn't just free, but stably so. And yet, stably free is a weaker statement that free. All free modules are stably free, not all stably free modules are free.

So what's the etymology here. In what sense is the freedom of the module something stable?

As always, I look forward to your responses.

Solution 1:

Okay, so I think I finally can give some background to this myself. The precise terminology (as explained by Eric Wofsey in the comments to the original post) is rather to mean "the module is free once stabilized." This in turn would seem to derive from the terminology of stably isomorphic modules, which is that two $R$-modules $P$ and $Q$ are stably isomorphic if there exists a number $n \in \mathbb{N}_0$ such that $P \oplus R^n \cong Q \oplus R^n$.

This in turn would seem to derive from the notion of stably equivalent manifolds, which was first introduced by Barry Mazur in an article in 1961. Mazur was interested in finding out when two manifolds were diffeomorphic (for a variety of purposes, one of the reasons he mentions in the paper is the Poincaré Conjecture), and so produced what he in that paper calls a "simpleminded stabilization": two real manifolds $M_1$ and $M_2$ are $k$-equivalent when there exists a natural number $k \in \mathbb{N}_0$ such that $M_1 \times \mathbb{R}^n \cong M_2 \times \mathbb{R}^n$. This notion of $k$-equivalence would in papers that followed soon be dubbed stably equivalent.

When I emailed Prof. Mazur about this, he explained to me that in his conception, the term stable here was to be read "in the same spirit" as the stable in stable homotopy theory (which predated his article), and that the process was to be thought of as analogous to taking the suspension of a space sufficiently many times. There, the term stable comes from the stable homotopy groups of spheres, which has to do with the Freudenthal suspension theorem and all that jazz.

So there we have it. The terms stable and stably have arisen to become the favoured nomenclature due to organic processes, which is why it comes across as a bit confusing. I cannot help but feel that ironically, perhaps a better term would be unstably free since in many cases, you actually need to stabilize the module by adding a few summands for it to be free.

But well, language is never perfect. I hope this little summary might be of some use to any future student who has the same confusion with the terminology as I did.