Mt845: Algebraic Curves

1.1. Sheaves. Let X be a topological space. A sheaf F on X associates to each open set U an abelian group F (U), called the sections of F over U , along with a restriction map rV,U : F (V ) → F (U) for any open sets U ⊂ V (for a section σ ∈ F (V ), we often write σ|U to denote rV,U (σ)), satisfying the following conditions: (1) For any open sets U ⊂ V ⊂W , rV,U ◦ rW,V = rW,U ; (2) For a collection of open sets {Ui}i∈I and sections αi ∈ F (Ui), if αi|Ui∩Uj = αj |Ui∩Uj for any i, j ∈ I, then there exists a unique α ∈ F (∪iUi) such that α|Ui = αi for any i.