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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 on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. :Theorem A. is spanned by its global sections. Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p.51) attributes to J.-P. Serre): :Theorem B. for all . Analogous properties were established by Serre (1955) for coherent sheaves in algebraic geometry, when is an affine scheme. The analogue of Theorem B in this context is as follows : :Theorem B (Scheme theoretic analogue). Let be an affine scheme, a quasi-coherent sheaf of -modules for the Zariski topology on . Then for all . These theorems have many important applications. Naively, they imply that a holomorphic function on a closed complex submanifold, , of a Stein manifold can be extended to a holomorphic function on all of . 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 for all coherent sheaves on a complex manifold (resp. quasi-coherent sheaves on a noetherian scheme ), then is Stein (resp. affine); see (resp. and ). * See also Cousin problems ==References== *. * . *.. *. * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cartan's theorems A and B」の詳細全文を読む スポンサード リンク
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