# Cartan theorem A

Let ${\mathcal{O}}_{z}$ denote the ring of germs of
holomorphic functions^{} at $z$

###### Theorem (Cartan).

Suppose $\mathrm{F}$ is a coherent analytic sheaf on a Stein
manifold^{} $M$.
For every $z\mathrm{\in}M$, the the stalk ${\mathrm{F}}_{z}$
is generated as an ${\mathrm{O}}_{z}$ module by the germs at $z$
of the sections (http://planetmath.org/Sheaf) $\mathrm{\Gamma}\mathit{}\mathrm{(}M\mathrm{,}\mathrm{F}\mathrm{)}$.

Philosophically, this theorem says that there is good supply of of a coherent analytic sheaf on a Stein manifold.

## 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 | Cartan theorem A |
---|---|

Canonical name | CartanTheoremA |

Date of creation | 2013-03-22 17:39:10 |

Last modified on | 2013-03-22 17:39:10 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32Q28 |

Classification | msc 32C35 |

Synonym | Cartan’s theorem A |

Related topic | CartanTheoremB |