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 proved by Henri Cartan around 1951, concerning a coherent sheaf {{mvar|F}} on a Stein manifold {{mvar|X}}. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
{{math theorem | name = Theorem A | math_statement = {{mvar|F}} is spanned by its global sections.}}
Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p. 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 Serre (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|OX}}-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
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