Template:Short description In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf Template:Mvar on a Stein manifold Template:Mvar. 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): Template:Math theorem
Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when Template:Mvar is an affine scheme. The analogue of Theorem B in this context is as follows Template:Harv: Template:Math theorem
These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, Template:Mvar, of a Stein manifold Template:Mvar can be extended to a holomorphic function on all of Template:Mvar. 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 Template:Math for all coherent sheaves Template:Mvar on a complex manifold Template:Mvar (resp. quasi-coherent sheaves Template:Mvar on a noetherian scheme Template:Mvar), then Template:Mvar is Stein (resp. affine); see Template:Harv (resp. Template:Harv and Template:Harv).