Oka coherence theorem

Remarks

The proof is by induction on $n$. By the Weierstrass preparation theorem, after a linear change of coordinates one may take the $f_i$ to be Weierstrass polynomials in $z_n$ with coefficients holomorphic in $z' = (z_1, \dots, z_{n-1})$. With the Weierstrass division theorem, relations among polynomials in $z_n$ of bounded degree are then reduced to relations over $\mathcal{O}_{\C^{n-1}}$, where the induction hypothesis applies.

Example: on $\C^2$ the relations of $f_1 = z_1$ and $f_2 = z_2$ are generated at every point by the single relation $(z_2, -z_1)$.

A consequence is that the structure sheaf $\mathcal{O}_X$ of every complex analytic space is coherent. Indeed, locally it is a quotient of $\mathcal{O}_\Omega$ by an ideal sheaf generated by finitely many sections. The real analogue fails: the sheaf of real analytic functions on $\R^3$ has non-coherent ideal sheaves, such as the ideal of germs vanishing on Cartan's umbrella $\set{x^3 = z(x^2 + y^2)}$.

Going deeper

For a student

Coherence is what makes sheaf theory effective in several complex variables. Kernels, images and cokernels of maps between finite sums of $\mathcal{O}$ are coherent, so one can apply Cartan's theorems A and B and the finiteness theorems to them. Oka's theorem is the base case, and Cartan's coherence of ideal sheaves of analytic sets and Oka's coherence of the normalization sheaf build on it.