Coherent module sheaves on complex manifolds

Let $(X,{\mathcal{O}}_X)$ be a ringed space, so $X$ is a topological space and ${\mathcal{O}}_X$ is a sheaf of commutative rings. An ${\mathcal{O}}_X$-module sheaf $\mathcal{F}$ is locally of finite type if for every $x\in X$ there are an open neighborhood $U$ of $x$, a finite integer $r\ge 0$, and sections $s_1,\dots ,s_r\in \mathcal{F}(U)$ such that their germs generate ${\mathcal{F}}_y$ as an ${\mathcal{O}}_{X,y}$-module for every $y\in U$.

For an open $V\subseteq X$, a finite integer $p\ge 0$, and an ${\mathcal{O}}_X$-module morphism $\varphi :{\mathcal{O}}_V^{\oplus p}\to \mathcal{F}{\mathrm{\mid }}_V$, its relation sheaf is the kernel sheaf $\ker \varphi$. The module sheaf $\mathcal{F}$ is coherent if it is locally of finite type and, for every such $V$, $p$, and $\varphi$, the relation sheaf $\ker \varphi$ is locally of finite type. Here ${\mathcal{O}}_V^{\oplus 0}$ is the zero module sheaf.

In particular, for a finite tuple $f_1,\dots ,f_p\in {\mathcal{O}}_X(U)$, its holomorphic relation sheaf is the kernel of the morphism ${\mathcal{O}}_U^{\oplus p}\to {\mathcal{O}}_U$ that sends $(a_1,\dots ,a_p)$ to ${\sum }_{j=1}^p{a}_j{f}_j$. The structure sheaf is coherent when it is coherent as a module over itself. If $M$ is a complex manifold, an analytic coherent module on $M$ means a coherent module sheaf over its sheaf ${\mathcal{O}}_M$ of holomorphic functions.

Ringed space and structure sheaf

Module sheaf on a ringed space

Kernel sheaf

Stalk of a presheaf

Complex manifold and holomorphic vector bundle conventions

Demailly, Chapter II §3.2.1, Definitions (3.8) and (3.11)

Wehler, §4.2, Definitions 4.25–4.26