# Cartan theorem A

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

###### Theorem (Cartan).

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

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 CartanTheoremA 2013-03-22 17:39:10 2013-03-22 17:39:10 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 32Q28 msc 32C35 Cartan’s theorem A CartanTheoremB