View source for Oka coherence theorem

{{Short description|Theorem in complex analysis about the sheaf of holomorphic functions}}

In mathematics, the '''Oka coherence theorem''', proved by {{harvs|txt|authorlink=Kiyoshi Oka|first=Kiyoshi|last=Oka|year=1950}}, states that the [[sheaf (mathematics)|sheaf]] <math>\mathcal{O}_{\mathbb{C}^n}</math> of  [[holomorphic function]]s on <math>\mathbb{C}^n</math> (and subsequently the sheaf <math>\mathcal{O}_{X}</math> of holomorphic functions on a [[complex manifold]] <math>X</math>) is [[coherent sheaf|coherent]].<ref>{{harvtxt|Noguchi|2019}}</ref><ref>In {{harvtxt|Oka|1950}} paper it was called the [[idéal de domaines indéterminés]].</ref>

==See also==
* [[Cartan's theorems A and B]]
* [[Function of several complex variables|Several complex variables]]
* [[Algebraic geometry and analytic geometry#Main statement|GAGA]]
* [[Oka–Weil theorem]]
* [[Weierstrass preparation theorem]]

== Note ==
{{reflist}}
==References==
{{ref begin}}
*{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|publisher=Springer }}
*{{Citation | last1=Hörmander | first1=Lars | author1-link=Lars Hörmander | title=An introduction to complex analysis in several variables | publisher=North-Holland | location=Amsterdam | isbn=978-0-444-88446-6 |mr=0344507 | year=1990}}
*{{Citation | last1=Noguchi | first1=Junjiro | title=A Weak Coherence Theorem and Remarks to the Oka Theory |journal=Kodai Math. J.|volume=42|url=https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf | year=2019 | issue=3 | arxiv = 1704.07726|doi =10.2996/kmj/1572487232|pages=566–586| s2cid=119697608 }}
*{{Citation | last1=Oka | first1=Kiyoshi | title=Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques | url=http://www.numdam.org/item?id=BSMF_1950__78__1_0 |mr=0035831 | year=1950 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=78 | pages=1–27| doi=10.24033/bsmf.1408 | doi-access=free }}
*{{eom|id=c/c022990|title=Coherent analytic sheaf|first=A.L.|last= Onishchik}}
{{ref end}}
[[Category:Theorems in complex analysis]]
[[Category:Theorems in complex geometry]]


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