This is a very naive question.


Let $X$ be a complex manifold. Let $\mathcal{O}\_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic functions $U\rightarrow\mathbb{C}$.


A sheaf of $\mathcal{O}\_X$-modules $F$ is coherent if:


1. It is locally finitely generated: Ie, there is an open cover $\{U\_i\}$ of $X$ such that $F|\_{U\_i}$ admits surjections $\mathcal{O}\_{U\_i}\rightarrow F|\_{U\_i}$ for each $i$.
2. For any open $V\subset X$, and any morphism $f : \mathcal{O}\_V^s\rightarrow F|\_V$, $\text{Ker}(f)$ is a locally finitely generated sheaf on $V$ (ie, satisfies condition 1).


Then, Oka's coherence theorem states that the structure sheaf $\mathcal{O}\_X$ is coherent.


When $X$ is a scheme, this statement is almost tautological. What is it about the setting of complex manifolds that makes this theorem deep?



Why is Oka's coherence theorem a deep result?
In scheme theory applied to complex geometry one usually does not encounter coherent rings which are not noetherian as well.
However if $X$ is (for example) a Stein manifold then the ring $R = \mathcal{O}(X)$ of global holomorphic functions is typically non-noetherian, which makes remarkable the fact that $R$ is nonetheless coherent.


