# 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
${f}_{1},\mathrm{\dots},{f}_{k}\in \mathrm{\Gamma}(U,\mathcal{F})$ 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 ${f}_{1},\mathrm{\dots},{f}_{k}$ are sections in $\mathrm{\Gamma}(U,\mathcal{F})$. Let $\mathcal{R}({f}_{1},\mathrm{\dots},{f}_{k})$ be the subsheaf of ${\mathcal{O}}^{k}$ over $U$ consisting of the germs

$$\{({g}_{1},\mathrm{\dots},{g}_{k})\in {\mathcal{O}}_{z}^{k}\mid \sum _{j=1}^{k}{g}_{j}{({f}_{j})}_{z}=0\}.$$ |

$\mathcal{R}({f}_{1},\mathrm{\dots},{f}_{k})$ 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 ${f}_{1},\mathrm{\dots},{f}_{k}\in \mathrm{\Gamma}(U,\mathcal{F})$,
the
sheaf $\mathcal{R}({f}_{1},\mathrm{\dots},{f}_{k})$ 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 |
---|---|

Canonical name | CoherentAnalyticSheaf |

Date of creation | 2013-03-22 17:39:05 |

Last modified on | 2013-03-22 17:39:05 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32C35 |

Defines | locally finitely generated sheaf |

Defines | sheaf of relations |