mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Cartan's theorems A and B are two results proved by

Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...

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 refe ...

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 Stei ...

. 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-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...

, 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 when i ...

. Theorem B is stated in cohomological terms (a formulation that Cartan (

1953 Events January * January 6 – The Asian Socialist Conference opens in Rangoon, Burma. * January 12 – Estonian émigrés found a Estonian government-in-exile, government-in-exile in Oslo. * January 14 ** Marshal Josip Broz Tito i ...

, 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 y ...

) for coherent sheaves in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, 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 the ...

. 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 ).

References

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

See also

References