Coherent analytic sheaf

Remarks

Example: the sheaf $\mathcal{O}_X^p$ is coherent for every complex space $X$; for $X = \C^n$ this is the Oka coherence theorem, and the general case follows from it. Given it, a sheaf is coherent if and only if it has, locally, a presentation $\mathcal{O}_U^q \to \mathcal{O}_U^p \to \mathcal{F}|_U \to 0$.

Non-example: let $X = \C$, $U = \C \setminus \set{0}$, and let $\mathcal{F}$ be the sheaf of holomorphic functions on $U$ extended by zero to $\C$. Its stalk at $0$ is $0$ and its stalks at other points are $\mathcal{O}_{\C,x}$. A section of $\mathcal{F}$ on a connected open set $V \ni 0$ is a holomorphic function on $V \setminus \set{0}$ that vanishes near $0$, so it is zero by the identity theorem. So no finite set of sections over a neighbourhood of $0$ generates the nonzero stalks near $0$, and clause 1 fails.