Lemma 6.8.28DDA48CD81CB

Proof. The set of points $z$ with $\mathcal I_ z=0$ is open: near such a point, choose sections $g_1,\ldots ,g_ r$ generating all stalks of $\mathcal I$ on a neighborhood; their germs at $z$ vanish, so they vanish on a neighborhood $U'$ of $z$, and $\mathcal I$ has zero stalks on $U'$. The set of points with $\mathcal I_ z\neq 0$ is open as well: a section with nonzero germ at $z$, defined on a connected neighborhood of $z$, has nonzero germs at all points of this neighborhood, by the identity theorem. Since $W$ is connected, one of the two sets is empty. $\square$