Lemma 31.23.9. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. Let us denote $\mathcal{I} \subset \mathcal{O}_ X$ the sheaf of ideals defined by the rule

$\mathcal{I}(V) = \{ f \in \mathcal{O}_ X(V) \mid fs \in \mathcal{L}(V)\} .$

The formula makes sense since $\mathcal{L}(V) \subset \mathcal{K}_ X(\mathcal{L})(V)$. Then $\mathcal{I}$ is a quasi-coherent sheaf of ideals and we have injective maps

$1 : \mathcal{I} \longrightarrow \mathcal{O}_ X, \quad s : \mathcal{I} \longrightarrow \mathcal{L}$

whose cokernels are supported on closed nowhere dense subsets of $X$.

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 $s(b) = (a) \subset sI \subset A$. $\square$

Comment #3768 by Laurent Moret-Bailly on

Typo on second line of proof: "$s=x/y$" should be "$s=a/b$". I think the quasicoherence property does not use the regularity of $s$.

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