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

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

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)).

References

Zbl 0053.05301
.

ISBN 9780821821657
.

Zbl 0367.14001
..

MR 0082175

Zbl 0078.34604
- Serre, Jean-Pierre (2 December 2013). "35. Sur la cohomologie des variétés algébriques". Oeuvres - Collected Papers I: 1949 - 1959. Springer. pp. 469–484.
  ISBN 978-3-642-39815-5
  .

See also

References