Cartan's theorems A and B
Encyclopedia
In mathematics
, Cartan's theorems A and B are two results prove
d 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
Analogous properties were established by Jean-Pierre Serre
(1957) for coherent sheaves in algebraic geometry
, when X is an affine scheme. The analogue of Theorem B in this context is as follows :
Similar results hold for the étale
and flat sites
after suitable modifications are made to the sheaf F.
These theorems have many important applications. Naively, they imply that a holomorphic function on a closed complex submanifold Z of a Stein manifold X can be extended to a holomorphic function on all of X. 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 H1(X,F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasicoherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Cartan's theorems A and B are two results prove
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
d by Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
around 1951, concerning a coherent sheaf
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
F on a Stein manifold
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 :...
X. They are significant both as applied to several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, and in the general development of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
.
Theorem A states that F is spanned by its global sections
Sheaf spanned by global sections
In mathematics, a sheaf spanned by global sections is a sheaf F on a locally ringed space X, with structure sheaf OX that is of a rather simple type. Assume F is a sheaf of abelian groups. Then it is asserted that if A is the abelian group of global sections, i.e.then for any open set U of X, ρ...
. Theorem B states that
- Hp(X,F) = 0 for all p > 0.
Analogous properties were established by Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
(1957) for coherent sheaves in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, when X is an affine scheme. The analogue of Theorem B in this context is as follows :
- Let X be an affine scheme, F a quasi-coherent sheaf of OX-modules for the Zariski topologyZariski topologyIn algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on X. Then Hp(X, F) = 0 for all p > 0.
Similar results hold for the étale
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
and flat sites
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
after suitable modifications are made to the sheaf F.
These theorems have many important applications. Naively, they imply that a holomorphic function on a closed complex submanifold Z of a Stein manifold X can be extended to a holomorphic function on all of X. At a deeper level, these theorems were used by Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
to prove the GAGA
Gaga
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theorem.
Theorem B is sharp in the sense that if H1(X,F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasicoherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see .