Sheaf spanned by global sections
Encyclopedia
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 group
s. 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(A) spans F(U) as an OU-module. Here
is the restriction map. In words, all sections of F are locally generated by the global sections.
An example of such a sheaf is that associated in algebraic geometry
to an R-module M, R being any commutative ring
, on the spectrum of a ring
Spec(R).
Another example: according to Cartan's theorem A, any coherent sheaf
on a Stein manifold
is spanned by global sections.
, a related notion is ample line bundle
.
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...
, a sheaf spanned by global sections is a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
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 group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. Then it is asserted that if A is the abelian group of global sections, i.e.
- A = Γ(F,X)
then for any open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U of X, ρ(A) spans F(U) as an OU-module. Here
- ρ = ρX,U
is the restriction map. In words, all sections of F are locally generated by the global sections.
An example of such a sheaf is that associated 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...
to an R-module M, R being any commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, on the spectrum of a ring
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
Spec(R).
Another example: according to Cartan's theorem A, any 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...
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 :...
is spanned by global sections.
See also
In the theory of schemesScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, a related notion is ample line bundle
Ample line bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...
.