Cartan's theorems A and B
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 states that 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). It says that:
- Hp(X,F) = 0 for all p > 0.
Analogous properties were established by Serre (1955) 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 III.3.7):
- Let X be an affine scheme, F a quasi-coherent sheaf of OX-modules for the Zariski topology on X. Then Hp(X, F) = 0 for all p > 0.
These theorems have many important applications. Naively, 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. quasicoherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see Serre (1952) (resp. Serre (1957) and Hartshorne|1977|loc=Theorem III.3.7).
- See also Cousin problems
- Cartan, H. (1953), "Variétés analytiques complexes et cohomologie", Colloque tenu à Bruxelles: 41–55.
- Chirka, E.M. (2001), "Cartan theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
- Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice Hall.
- Hartshorne, Robin (1977), Algebraic Geometry, Springer-Verlag, ISBN 0-387-90244-9.Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Université de Grenoble. Annales de l'Institut Fourier 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175.
- Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées 36: 1–16.