Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the *zero locus* of $f$ is the set of points $x\in X$ at which $f$ vanishes in the *fiber* ${\scr F}(x) = {\scr F}\otimes_{{\scr O}_X}k(x)$.

If ${\scr F}$ is locally free, then the zero locus of a section is an analytic set (by which I mean the zero locus of a coherent ideal sheaf). In general, this is quite false (consider a sky-scraper sheaf at a point).

Here a probably too general question:

Which sets are the zero loci of such sections?

Let's call ${\scr F}$ *torsion-free* if it is without torsion by non-zero-divisors in the structure sheaf. Equivalently, if the natural map ${\scr F}\to {\scr F}\otimes_{{\scr O}_X} {\scr M}_X$ is injective, where ${\scr M}_X$ is the sheaf of meromorphic functions on $X$.

What if ${\scr F}$ is torsion-free? Are the zero loci analytic in this generality?

Though I've phrased the problem in the context of rigid spaces, there are obvious analogues for schemes and complex analytic spaces. I'd welcome comments in any of these contexts.