Germ of an analytic set

Remarks

This is the notion of germ applied to analytic sets. Inclusion, finite union and finite intersection of germs are defined through representatives on a common neighbourhood. To a germ $A_a$ one attaches the ideal $\mathcal{I}(A_a)$ of all holomorphic germs $f_a \in \mathcal{O}_{n,a}$ that have a representative vanishing on a representative of $A_a$; it is a radical ideal. Conversely, an ideal $I \subseteq \mathcal{O}_{n,a}$ with generators $f_{1,a}, \dots, f_{k,a}$ defines the germ $V(I)$ of the common zero set of representatives $f_1, \dots, f_k$. Example: the sets $\set{z_1 z_2 = 0}$ and $\set{z_1 z_2 (z_1 - 1) = 0}$ in $\C^2$ have the same germ at $0$, since they agree on the ball of radius $1$ about $0$. Non-example: the sets $\set{z_1 = 0}$ and $\set{z_1 = z_2^2}$ have different germs at $0$. Indeed, every neighbourhood of $0$ contains a point $(0, t)$ with $t \ne 0$, which lies in the first set but not the second. The germ of $A$ at a point $a \notin A$ is the empty germ.