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 (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 III.3.7):

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 $H 1 (X, F) = 0$ 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