Cartan's theorems A and B

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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.

The analogous properties also hold for coherent sheaves in algebraic geometry, when X is an affine scheme. The analogue of Theorem B in this context is as follows:

Theorem B: 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.

These theorems have many important applications. For example, they imply the following statement: Let X be a Stein manifold, let Z be a closed complex submanifold and let f be a holomorphic function on Z. Then there exists a holomorphic function F on X whose restriction to Z is precisely f.