Oka coherence theorem

In mathematics, the Oka coherence theorem, proved by Kiyoshi Oka (1950), states that the sheaf ${\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n}}}$ of holomorphic functions on ${\displaystyle \mathbb {C} ^{n}}$ (and subsequently the sheaf ${\displaystyle {\mathcal {O}}_{X}}$ of holomorphic functions on a complex manifold ${\displaystyle X}$) is coherent.^[1][2]

See also

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

Note

1. Noguchi (2019)
2. In Oka (1950) paper it was called the idéal de domaines indéterminés.

References

• Grauert, H.; Remmert, R. (6 December 2012). Coherent Analytic Sheaves. Springer. ISBN 978-3-642-69582-7.
• Hörmander, Lars (1990), An introduction to complex analysis in several variables, 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 Math. J., 42 (3): 566–586, arXiv:1704.07726, doi:10.2996/kmj/1572487232, S2CID 119697608
• 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, ISSN 0037-9484, MR 0035831
• Onishchik, A.L. (2001) [1994], "Coherent analytic sheaf", Encyclopedia of Mathematics, EMS Press