Oka coherence theorem

Short description: Theorem in complex analysis about the sheaf of holomorphic functions

In mathematics, specifically algebraic geometry, the Oka coherence theorem, proved by Kiyoshi Oka in 1950, states that the sheaf ${{𝒪}}_{{\mathbb{ℂ}}^n}$ of holomorphic functions on ${\mathbb{ℂ}}^n$ (and subsequently the sheaf ${{𝒪}}_X$ of holomorphic functions on a complex manifold $X$) is coherent.

See also

• Cartan's theorems A and B
• Several complex variables
• GAGA
• Oka–Weil theorem
• Weierstrass preparation theorem

References

• Grauert, H.; Remmert, R. (1984). Coherent Analytic Sheaves. A Series of Comprehensive Studies in Mathematics. 265. Berlin, Heidelberg: Springer. ISBN 978-3-642-69582-7. 

• Hörmander, Lars (1990). An Introduction to Complex Analysis in Several Variables (3rd ed.). Amsterdam: North-Holland. ISBN 978-0-444-88446-6. 

• Noguchi, Junjiro (2019). "A weak coherence theorem and remarks to the Oka theory". Kodai Mathematical Journal 42 (3): 566–586. doi:10.2996/kmj/1572487232. https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf. 

• Oka, Kiyoshi (1950). "Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques". Bulletin de la Société Mathématique de France 78: 1–27. doi:10.24033/bsmf.1408. 

• Hazewinkel, Michiel, ed. (2001), "Coherent analytic sheaf", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=c/c022990