In mathematics, a sheaf spanned by global sections is a sheaf (mathematics) 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 ).