In mathematics, Cartan's theorems A and B are two results

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d by Henri Cartan around 1951, concerning a

coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...

on a

Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a St ...

. They are significant both as applied to

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...

, and in the general development of

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...

. 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 (

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) for coherent sheaves in algebraic geometry, when is an

affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with t ...

. The analogue of Theorem B in this context is as follows : These theorems have many important applications. For instance, 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...

to prove the

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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 ).

References

*. * . *. * * ** {{DEFAULTSORT:Cartan's Theorems A And B Several complex variables Topological methods of algebraic geometry Theorems in algebraic geometry

See also

References