The Stacks project

Proof. The question is local on $X$. Hence we may assume that $X = \mathop{\mathrm{Spec}}(A)$, and $\mathcal{L} = \mathcal{O}_ X$. After shrinking further we may assume that $s = a/b$ with $a, b \in A$ both nonzerodivisors in $A$. Set $I = \{ x \in A \mid x(a/b) \in A\} $.

To show that $\mathcal{I}$ is quasi-coherent we have to show that $I_ f = \{ x \in A_ f \mid x(a/b) \in A_ f\} $ for every $f \in A$. If $c/f^ n \in A_ f$, $(c/f^ n)(a/b) \in A_ f$, then we see that $f^ mc(a/b) \in A$ for some $m$, hence $c/f^ n \in I_ f$. Conversely it is easy to see that $I_ f$ is contained in $\{ x \in A_ f \mid x(a/b) \in A_ f\} $. This proves quasi-coherence.

Let us prove the final statement. It is clear that $(b) \subset I$. Hence $V(I) \subset V(b)$ is a nowhere dense subset as $b$ is a nonzerodivisor. Thus the cokernel of $1$ is supported in a nowhere dense closed set. The same argument works for the cokernel of $s$ since $s(b) = (a) \subset sI \subset A$. $\square$