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

:"Hp"("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 "Hp"("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".

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