Sheaves of Modules and Closed Subschemes

Definition 1.1. If X is a topological space with a sheaf of ring OX , an OX-module is a sheaf F of abelian groups on X, together with the structure of an OX(U)-module on each F (U), which is compatible with restriction, in the sense that if V ⊆ U , for any s ∈ F (U) and f ∈ OX(U) we have ρUV (fs) = ρUV (f) · ρUV (s) in F (V ). A morphism between OX -modules F and G is a morphism φ of the underlying sheaves of abelian groups such that for all U ⊆ X, s ∈ F (U), and f ∈ OX(U), we have φ(fs) = fφ(s).