They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when X is an affine scheme.
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.
Theorem B is sharp in the sense that if H1(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.