In classical commutative algebra, a ring morphism $f:A\to B$ is injective/surjective iff it is so Zarsiki locally on its source. That is, iff there are elements $a_i\in A$ so that $1\in(a_i)$ and all $A[1/a_i]\to B[1/f(a_i)]$ are injective/surjective.
I was wondering how much is known about this in the world of $C^\infty$-rings? I am able to prove the analogue of the above statement for $C^\infty$-rings of the form $C^\infty X$ for $X$ a closed subset of a manifold, but I would really like to know it for all finitely generated ones.
I do suspect, however, that this will only hold for finitely generated and germ-determined ones...
Edit: if anyone knows, I would also be interested in a counterexample: an injection $f:A\to B$ of $C^\infty$-rings such that $A\{1/a\}\to B\{1/f(a)\}$ is not injective, where the localization is in the category of $C^\infty$-rings.