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 F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.

Theorem A — 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):

Theorem B — H^p(X, F) = 0 for all p > 0.

Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when X is an affine scheme. The analogue of Theorem B in this context is as follows (Hartshorne 1977):

Theorem B (Scheme theoretic analogue) — Let X be an affine scheme, F a quasi-coherent sheaf of O_X-modules for the Zariski topology on X. Then H^p(X, F) = 0 for all p > 0.

These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, Z, of a Stein manifold X can be extended to a holomorphic function on all of 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 H¹(X, F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see (Serre 1956) (resp. (Serre 1957) and (Hartshorne 1977)).

See also

• Cousin problems

References

• Cartan, H. (1953), "Variétés analytiques complexes et cohomologie", Colloque tenu à Bruxelles: 41–55 .
• Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice Hall, doi:10.1090/chel/368, ISBN 9780821821657 .
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. https://books.google.com/books?id=7z4mBQAAQBAJ&pg=PA215. .
• Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 
• Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées 36: 1–16 
  • Serre, Jean-Pierre (2 December 2013). "35. Sur la cohomologie des variétés algébriques". Oeuvres - Collected Papers I: 1949 - 1959. pp. 469–484. ISBN 978-3-642-39815-5. https://books.google.com/books?id=eaUoAKOAbUsC.