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.

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Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p. 51) attributes to J.-P. Serre): $$

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 : $$

These theorems have many important applications. For instance, 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 $H(X, F) = 0$ for all coherent sheaves $F$ on a complex manifold $X$ (resp. quasi-coherent sheaves $F$ on a noetherian scheme $X$), then $X$ is Stein (resp. affine); see  (resp.  and ).