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
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