Reduced complex space

Remarks

Example: the local model in $\C^2$ given by $f = z_1 z_2$ is reduced. Its stalk at $0$ is $\C\{z_1, z_2\}/(z_1 z_2)$, and if $g^k \in (z_1 z_2)$ then $z_1 \mid g^k$ and $z_2 \mid g^k$ in the factorial ring $\C\{z_1, z_2\}$, so $z_1 \mid g$ and $z_2 \mid g$, hence $z_1 z_2 \mid g$. At the other points the stalk is a ring of convergent power series, which is reduced.

Non-example: the local model in $\C$ given by $f = z^2$ is not reduced, since the class of $z$ in $\C\{z\}/(z^2)$ is nonzero and has square $0$.

Equivalently, $X$ is reduced if every section of $\mathcal{O}_X$ whose value vanishes at every point of its domain is zero. For any complex space $X$, the quotient of $\mathcal{O}_X$ by its sheaf of nilpotent germs defines a reduced complex space $X_{\mathrm{red}}$ on the same topological space.