# 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.**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 (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):

**Theorem B (Scheme theoretic analogue).**Let X be an affine scheme, F a quasi-coherent sheaf of*O*-modules for the Zariski topology on X. Then_{X}*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 *H*^{ 1}(*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, Theorem III.3.7)).

- 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. - Hartshorne, Robin (1977),
*Algebraic Geometry*, Springer-Verlag, ISBN 0-387-90244-9. - Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques",
*Journal de Mathématiques Pures et Appliquées*,**36**: 1–16. - 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