Cartan's theorems A and B [SEP] named after
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 (a formulation that Cartan (1953, p.51) attributes to J.-P. Serre):
Analogous properties were established by Serre (1955) for coherent sheaves in algebraic geometry, when is an affine
of results.
Vanishing theorems in the affine case.
Complex analysis was revolutionized by Cartan's theorems A and B in 1953. These results say that if "E" is a coherent analytic sheaf on a Stein space "X", then "E" is spanned by its global sections, and "H"("X","E") = 0 for all "i"  0. (A complex space "X" is Stein if and only if it is isomorphic to a closed analytic subspace of C for some "n