Cartan's coherence theorem for ideal sheaves

Remarks

The question is local, and one may assume that the germ of $A$ at a point is irreducible. The local parametrization theorem presents $A$ as a branched covering of a domain in $\C^d$, and from this one writes down explicit holomorphic functions that generate $\mathcal{I}_A$ near the point. Coherence then follows from the Oka coherence theorem, since $\mathcal{I}_A$ is a subsheaf of finite type of the coherent sheaf $\mathcal{O}_\Omega$.

Example: for $A = \set{z_1 = z_2 = 0} \subseteq \C^3$, the sheaf $\mathcal{I}_A$ is generated by $z_1$ and $z_2$ at every point. For the cusp $A = \set{z_1^2 = z_2^3}$, it is generated by $z_1^2 - z_2^3$. In contrast, the sheaf generated by $z_1^2$ has the same zero set $\set{z_1 = 0}$ but is strictly smaller than the ideal sheaf of that set.

Going deeper

For a student

The theorem gives every analytic set $A$ a canonical structure of reduced complex space, with structure sheaf $\mathcal{O}_\Omega / \mathcal{I}_A$ restricted to $A$. By the Rückert Nullstellensatz, the nilradical of a local model $\mathcal{O}_D / \mathcal{I}$ is $\mathcal{I}_Z / \mathcal{I}$, where $Z$ is the zero set of $\mathcal{I}$. So the theorem shows that the nilradical of the structure sheaf of any complex space is coherent, and every complex space has a reduction. Together with Oka's theorem it is one of the two pillars of the theory of coherent analytic sheaves.