# What is the nature of the zero locus of a section of a coherent sheaf?

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.

• Even if F is torsion-free, the zero locus of a section need not be closed. Example: X=plane, F=ideal sheaf of the origin, f=linear function (vanishing at the origin). I doubt you can say more than constructibility. Jan 24, 2011 at 20:07
• Neat example, t3suji ! Jan 24, 2011 at 20:57
• Nice example indeed, t3suji! I guess the local go-to example of a non-fat but torsion-free module would have been a good place to start :) Jan 24, 2011 at 21:32
• This is an old post, but is it known that these sets are constructible (in the scheme-theoretic setting)? A reference would be great.
– DKS
Mar 26, 2021 at 19:23