coherent analytic sheaf

Let $M$ be a complex manifold and $ℱ$ be an analytic sheaf.For $z\in M$, denote by ${ℱ}_z$ the stalk of $ℱ$ at $z$.By $𝒪$ denote the sheaf of germs of analytic functions. For a section$f$ and a point $z\in M$ denote by $f_z$ the germ of $f$ at $z$.

$ℱ$ is said to be locally finitely generated if for every $z\in M$,there is a neighbourhood $U$ of $z$, a finite number of sections$f_1,\mathrm{\dots },f_k\in \mathrm{\Gamma }(U,ℱ)$ such that for each $w\in U$,${ℱ}_w$ is a finitely generated module (as an ${𝒪}_w$-module).

Let $U$ be a neighbourhood in $M$ andSuppose that $f_1,\mathrm{\dots },f_k$ are sections in $\mathrm{\Gamma }(U,ℱ)$.Let $ℛ(f_1,\mathrm{\dots },f_k)$ be the subsheaf of ${𝒪}^k$over $U$ consisting of the germs

$$\{(g_1,\mathrm{\dots },g_k)\in {𝒪}_z^k\mid \sum _{j=1}^k{g}_j{(f_j)}_z=0\}.$$

$ℛ(f_1,\mathrm{\dots },f_k)$ is called the sheaf of relations.

Definition.

$ℱ$ is called a coherent analytic sheaf if $ℱ$is locally finitely generated and if for every open subset $U\subset M$,and $f_1,\mathrm{\dots },f_k\in \mathrm{\Gamma }(U,ℱ)$,thesheaf $ℛ(f_1,\mathrm{\dots },f_k)$ is locally finitely generated.