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
Analogous properties were established by Jean-Pierre 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 (Hartshorne 1977, Theorem III.3.7):
Similar results hold for the étale and flat sites after suitable modifications are made to the sheaf F.
These theorems have many important applications. Naively, 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 H1(X,F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasicoherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see Hartshorne (1977, Theorem III.3.7).