# coherent analytic sheaf

Let $M$ be a complex manifold and $\mathcal{F}$ be an analytic sheaf. For $z\in M$, denote by $\mathcal{F}_{z}$ the stalk of $\mathcal{F}$ at $z$. By $\mathcal{O}$ 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 $z$.

$\mathcal{F}$ 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 $f_{1},\ldots,f_{k}\in\Gamma(U,\mathcal{F})$ such that for each $w\in U$, $\mathcal{F}_{w}$ is a finitely generated module (as an $\mathcal{O}_{w}$-module).

Let $U$ be a neighbourhood in $M$ and Suppose that $f_{1},\ldots,f_{k}$ are sections in $\Gamma(U,\mathcal{F})$. Let $\mathcal{R}(f_{1},\ldots,f_{k})$ be the subsheaf of ${\mathcal{O}}^{k}$ over $U$ consisting of the germs

 $\{(g_{1},\ldots,g_{k})\in{\mathcal{O}}^{k}_{z}\mid\sum_{j=1}^{k}g_{j}(f_{j})_{% z}=0\}.$

$\mathcal{R}(f_{1},\ldots,f_{k})$ is called the sheaf of relations.

###### Definition.

$\mathcal{F}$ is called a coherent analytic sheaf if $\mathcal{F}$ is locally finitely generated and if for every open subset $U\subset M$, and $f_{1},\ldots,f_{k}\in\Gamma(U,\mathcal{F})$, the sheaf $\mathcal{R}(f_{1},\ldots,f_{k})$ is locally finitely generated.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title coherent analytic sheaf CoherentAnalyticSheaf 2013-03-22 17:39:05 2013-03-22 17:39:05 jirka (4157) jirka (4157) 4 jirka (4157) Definition msc 32C35 locally finitely generated sheaf sheaf of relations