Oka coherence theorem

In mathematics, specifically algebraic geometry, the Oka coherence theorem, proved by Kiyoshi Oka in 1950, states that the sheaf ${\mathcal {O}}_{\mathbb {C} ^{n}}$ of holomorphic functions on $\mathbb {C} ^{n}$ (and subsequently the sheaf ${\mathcal {O}}_{X}$ 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. Vol. 265. Berlin, Heidelberg: Springer. ISBN 978-3-642-69582-7.

• Hörmander, Lars (1990) [1966]. An Introduction to Complex Analysis in Several Variables (3rd ed.). Amsterdam: North-Holland. ISBN 978-0-444-88446-6. MR 0344507.

• Noguchi, Junjiro (2019). "A weak coherence theorem and remarks to the Oka theory" (PDF). Kodai Mathematical Journal. 42 (3): 566–586. doi:10.2996/kmj/1572487232. MR 4025758.

• 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. MR 0035831.

• Onishchik, A.L. (2001) [1994], "Coherent analytic sheaf", Encyclopedia of Mathematics, EMS Press