In
mathematics, Cartan's theorems A and B are two results
proved 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 Ste ...
. 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
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, p. 51) attributes to J.-P. Serre):
Analogous properties were established by
Serre (
1957
1957 ( MCMLVII) was a common year starting on Tuesday of the Gregorian calendar, the 1957th year of the Common Era (CE) and ''Anno Domini'' (AD) designations, the 957th year of the 2nd millennium, the 57th year of the 20th century, and the 8th year ...
) for coherent sheaves in
algebraic geometry, when is an
affine scheme. 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 ina ...
to prove the
GAGA
Gaga ( he, גע גע literally 'touch touch') (also: ga-ga, gaga ball, or ga-ga ball) is a variant of dodgeball that is played in a gaga "pit". The game combines dodging, striking, running, and jumping, with the objective of being the last perso ...
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
In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They ...
References
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{{DEFAULTSORT:Cartan's Theorems A And B
Several complex variables
Topological methods of algebraic geometry
Theorems in algebraic geometry