# Cartan's theorems A and B

In

mathematics ,**Cartan's theorems A and B**are two results proved byHenri Cartan around 1951, concerning acoherent sheaf "F" on aStein manifold "X". They are significant both as applied toseveral complex variables , and in the general development ofsheaf 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 anaffine scheme . The analogue of Theorem B in this context is as follows:**Theorem B**: Let "X" be an affine scheme, "F" aquasi-coherent sheaf of "O"_{"X"}-modules for theZariski 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".

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