Oka coherence theorem

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

In mathematics, the Oka coherence theorem, proved by Kiyoshi Oka (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. (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 
• Noguchi, Junjiro (2019), "A Weak Coherence Theorem and Remarks to the Oka Theory", Kodai Math. J. 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, ISSN 0037-9484, http://www.numdam.org/item?id=BSMF_1950__78__1_0 
• 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