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)).

References

Cartan, H. (1953), "Variétés analytiques complexes et cohomologie", Colloque tenu à Bruxelles: 41–55, Zbl 0053.05301.
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. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. Zbl 0367.14001..
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, MR 0082175
Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées, 36: 1–16, Zbl 0078.34604
- 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.

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