Cartan's theorems A and B

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. Fest): Theorem B — Hp(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 OX-modules for the Zariski topology on X. Then Hp(X, F) = 0 for all p > 0.

These theorems have many important applications. Zum Beispiel, 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 (bzw. quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (bzw. affine); sehen (Fest 1956) (bzw. (Fest 1957) und (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, Zbl 0053.05301. Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Lehrlingshalle, doi:10.1090/chel/368, ISBN 9780821821657. Hartshorne, Robin (1977). Algebraische Geometrie. Abschlusstexte in Mathematik. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. HERR 0463157. Zbl 0367.14001.. Fest, Jean Pierre (1956), "Géométrie algébrique et géométrie analytique", Annalen des Fourier-Instituts, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, HERR 0082175 Fest, Jean Pierre (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées, 36: 1–16, Zbl 0078.34604 Fest, Jean Pierre (2 Dezember 2013). "35. Sur la cohomologie des variétés algébriques". Oeuvres - Collected Papers I: 1949 - 1959. pp. 469–484. ISBN 978-3-642-39815-5. Kategorien: Several complex variablesTopological methods of algebraic geometryTheorems in algebraic geometry