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.

See also

In the theory of schemes, a related notion is ample line bundle.