Some types of questions in algebra immediately reduce to the “countable” case. Simple example: Let A be a ring and let φ : A —> A be an automorphism. Then for every finite subset E of A there exists a countable subring A’ ⊂ A containing E such that φ induces an automorphism of A’. The proof is to let A’ be the subring of A generated by φ^n(e) for all e in E and all n ∈ Z.
Another example of this phenomenon is that any projective module is a direct sum of countably generated (projective) modules (Kaplansky’s theorem).
The technique also applies to the following problem: Let A ⊂ B be an integral extension of rings with A Noetherian. Let M be an A-module such that M ⊗_A B is flat over B. Problem: Show M is flat over A. (This is equivalent to the direct summand conjecture by a 1 page paper of Ohi.) A key case is to show M ⊗_A B = 0 implies M = 0. Picking suitable families of elements this reduces to the case where B is countably generated over A and M is a countably generated A-module.
Here are two examples involving algebraic stacks: (1) Suppose X is a quasi-compact algebraic stack with affine diagonal. I claim you can write X as a filtered limit of stacks X’ of the form X’ = [U’/R’] with U’ and R’ spectra of countable rings. I haven’t written out the details but it seems to me one can do this by just “adding elements” as above. (2) A quasi-coherent module F on a quasi-separated and quasi-compact algebraic stack is a filtered colimit of countably generated quasi-coherent modules.
I wonder if this type of argument can ever be used to bootstrap? Do some general arguments become easier if you assume all rings/modules in question are countable? Are there some _useful_ properties that hold for countable rings? Things like “the topology on Spec has a countable basis” aren’t really useful, or are they?
Before you say “No!” let me just point out that Kaplansky’s theorem is used in the proof of faithfully flat descent for projectivity of modules, so sometimes…