Cartan's theorems A and B

In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf "F" on a Stein manifold "X". They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.

Theorem A states that "F" is spanned by its global sections.

Theorem B states that

:"Hp"("X","F") = {0} for all "p" > 0.

The analogous properties also hold for coherent sheaves in algebraic geometry, when "X" is an affine scheme. The analogue of Theorem B in this context is as follows:

Theorem B: Let "X" be an affine scheme, "F" a quasi-coherent sheaf of "O""X"-modules for the Zariski topology on "X". Then "Hp"("X", "F") = {0} for all "p" > 0.

Similar results hold for the étale and flat sites after suitable modifications are made to the sheaf "F".

These theorems have many important applications. For example,they imply the following statement: Let "X" be a Stein manifold, let "Z" be a closed complex submanifold and let "f" be a holomorphic function on "Z". Then there exists a holomorphic function "F" on "X" whose restriction to "Z" is precisely "f".


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cartan's theorem — In mathematics, there are two basic results in Lie group theory that go by the name Cartan s theorem. They are both named for Élie Cartan.:1. The theorem that for a Lie group G , any closed subgroup is a Lie subgroup.:2. A theorem on highest… …   Wikipedia

  • Cartan, Élie-Joseph — ▪ French mathematician born April 9, 1869, Dolomieu, Fr. died May 6, 1951, Paris       French mathematician who greatly developed the theory of Lie groups and contributed to the theory of subalgebras.       In 1894 Cartan became a lecturer at the …   Universalium

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Timeline of algebra and geometry — A timeline of algebra and geometryBefore 1000 BC* ca. 2000 BC Scotland, Carved Stone Balls exhibit a variety of symmetries including all of the symmetries of Platonic solids. * 1800 BC Moscow Mathematical Papyrus, findings volume of a frustum *… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Several complex variables — The theory of functions of several complex variables is the branch of mathematics dealing with functions : f ( z1, z2, ..., zn ) on the space C n of n tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct… …   Wikipedia

  • 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 P. Cousin in 1895. They are… …   Wikipedia

  • List of complex analysis topics — Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied …   Wikipedia

  • Nash functions — In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ Rn is an analytic function f: U → R satisfying a non trivial polynomial equation P(x,f(x)) = 0 for all x in U (A semialgebraic subset of Rn is a subset obtained from… …   Wikipedia

  • Stein manifold — In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein. Definition A complex manifold X of complex… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.