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 on a Stein manifold. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
Theorem B is stated in cohomological terms :
Analogous properties were established by Serre for coherent sheaves in algebraic geometry, when 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,, of a Stein manifold can be extended to a holomorphic function on all of. 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 for all coherent sheaves on a complex manifold , then is Stein ; see .
