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 "O"_"X" 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.

:"A" = Γ("F","X")

then for any open set "U" of "X", ρ("A") spans "F"("U") as an "O"_"U"-module. Here

:ρ = ρ_"X","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 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.

In the theory of schemes, a related notion are ample line bundles.