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 states that F is spanned by its global sections. Theorem B states that

H^p(X,F) = 0 for all p > 0.

Analogous properties were established by Jean-Pierre 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):

• 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.

Similar results hold for the étale and flat sites after suitable modifications are made to the sheaf F.^{[citation needed]}

These theorems have many important applications. Naively, 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 (resp. quasicoherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see 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 .
• Chirka, E.M. (2001), "Cartan theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=c/c020570 .
• 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 (1955 1956), "Géométrie algébrique et géométrie analytique", Université de Grenoble. Annales de l'Institut Fourier 6: 1–42, ISSN 0373-0956, MR0082175, http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 .
• Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques", J. de Maths. Pures et Appl. 36: 1–16 .