Editing Cartan's theorems A and B

{{short description|Coherent sheaf on a Stein manifold is spanned by sections & lacks higher cohomology}}
In [[mathematics]], '''Cartan's theorems A and B''' are two results [[mathematical proof|prove]]d by [[Henri Cartan]] around 1951, concerning a [[coherent sheaf]] {{mvar|F}} on a [[Stein manifold]] {{mvar|X}}. They are significant both as applied to [[Function of several complex variables|several complex variables]], and in the general development of [[sheaf cohomology]].

{{math theorem | name = Theorem A | math_statement = {{mvar|F}} is [[sheaf spanned by global sections|spanned by its global sections]].}}

Theorem B is stated  in cohomological terms (a formulation that Cartan ([[#CITEREFCartan1953|1953]], p.&nbsp;51) attributes to J.-P. Serre):
{{math theorem | name = Theorem B | math_statement = {{math|1=''H''{{i sup|''p''}}(''X'', ''F'') = 0}} for all {{math|''p'' > 0}}.}}

Analogous properties were established by [[Jean-Pierre Serre|Serre]] ([[#CITEREFSerre1957|1957]]) for coherent sheaves in [[algebraic geometry]], when {{mvar|X}} is an [[affine scheme]]. The analogue of Theorem B in this context is as follows {{harv|Hartshorne|1977|loc=Theorem III.3.7}}:
{{math theorem | name = Theorem B (Scheme theoretic analogue) | math_statement = Let {{mvar|X}} be an affine scheme, {{mvar|F}} a [[quasi-coherent sheaf]] of {{math|''O<sub>X</sub>''}}-modules for the [[Zariski topology]] on {{mvar|X}}. Then {{math|1=''H''{{i sup|''p''}}(''X'', ''F'') = 0}} for all {{math|''p'' > 0}}.}}

These theorems have many important applications.  For instance, they imply that a holomorphic function on a closed complex submanifold, {{mvar|Z}}, of a Stein manifold {{mvar|X}} can be extended to a holomorphic function on all of {{mvar|X}}. At a deeper level, these theorems were used by [[Jean-Pierre Serre]] to prove the [[GAGA]] theorem.

Theorem B is sharp in the sense that if {{math|1=''H''{{i sup|1}}(''X'', ''F'') = 0}} for all coherent sheaves {{mvar|F}} on a complex manifold {{mvar|X}} (resp. quasi-coherent sheaves {{mvar|F}} on a noetherian scheme {{mvar|X}}), then {{mvar|X}} is Stein (resp. affine); see  {{harv|Serre|1956}} (resp. {{harv|Serre|1957}} and {{harv|Hartshorne|1977|loc=Theorem III.3.7}}).

== See also ==
* [[Cousin problems]]

==References==
*{{citation |first=H. |last=Cartan |authorlink=Henri Cartan |title=Variétés analytiques complexes et cohomologie |journal=Colloque tenu à Bruxelles |year=1953 |pages=41–55|zbl=0053.05301}}.
* {{Citation | author1-link = Robert C. Gunning | last1=Gunning | first1=Robert C. | author2-link = Hugo Rossi | last2=Rossi | first2=Hugo | title=Analytic Functions of Several Complex Variables | publisher=[[Prentice Hall]] | year=1965|doi=10.1090/chel/368| isbn=9780821821657 }}.
*{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry  | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52 | url={{Google books|7z4mBQAAQBAJ|Algebraic Geometry|page=215|plainurl=yes}}|doi=10.1007/978-1-4757-3849-0}}.
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free }}
*{{citation |last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur la cohomologie des variétés algébriques |journal=Journal de Mathématiques Pures et Appliquées |volume=36 |year=1957 |pages=1–16|zbl=0078.34604}}
**{{cite book |isbn=978-3-642-39815-5|title=Oeuvres - Collected Papers I: 1949 - 1959|chapter= 35. Sur la cohomologie des variétés algébriques|last1=Serre|first1=Jean-Pierre|date=2 December 2013|pages=469–484|publisher=Springer |url={{Google books|eaUoAKOAbUsC|Oeuvres - Collected Papers I: 1949 - 1959|plainurl=yes}}}}

{{DEFAULTSORT:Cartan's Theorems A And B}}
[[Category:Several complex variables]]
[[Category:Topological methods of algebraic geometry]]
[[Category:Theorems in algebraic geometry]]