# ag.algebraic geometry – Determine vanishing of a section from its zero loci

Let $$mathcal{F}$$ be a coherent sheaf on a complex algebraic variety $$X$$. In this paper (section 5), it is proved that the zero loci of a section $$s$$, denoted by $$Z(mathcal{F}, s)$$, is constructible.

Question: is it true that $$s=0$$ if and only if $$Z(mathcal{F}, s)=X$$? If so, how to prove this?

Since everything is local, we can assume that $$X=text{Spec} R$$, and then $$mathcal{F}$$ corresponds to a finitely generated $$R$$-module $$M$$. For any prime ideal $$mathfrak{p}subset R$$, let $$kappa(mathfrak{p})$$ be the residue field at $$mathfrak{p}$$. Recall the zero loci of $$sin M$$ is defined as
begin{align*} Z(M, s)={mathfrak{p}: overline{s}in Motimes_R kappa(mathfrak{p}) text{ is }0}. end{align*}