Cartan's theorems A and B
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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
- Hp(X,F) = {0} for all p > 0.
The analogous properties also hold for coherent sheaves in algebraic geometry, when X is an affine scheme.