Mannheimer Manuskripte 177 gk-mp-9403/3 SOME CONCEPTS OF ...

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 29then there exists a f ∈ F(U) withf |Ui = f i .Given two sheaves of rings F and G on X. By a sheaf homomorphismψ : F → Gwe understand an assignment of a ring homomorphism ψ U (for every open set U)ψ U : F(U) → G(U),which is compatible with the restriction homomorphismsUF(U)ψ U−→G(U)⋃ ⏐ ⏐↓⏐ ⏐↓V F(V )ψ V−→ G(V )More information you find in [Sch].Appendix B. The structure sheaf O R . In this appendix I like to show that thesheaf axioms for the structure sheaf O R on X = Spec(R) are fulfilled if we consider onlythe basis open sets X f = Spec(R) \ V (f) . Recall that the intersection of two basisbasis open sets X f ∩ X g = X fg is again a basis open set. The sheaf O R on the basisopen sets was defined to be O R (X f ) = R f and the restriction maps were the naturalmapsR f → (R f ) g = R fg , r ↦→ r 1 .Here I am following very closely the presentation in [EH].Lemma 1. The set {X f | f ∈ R} is a basis of the topology.Proof. We have to show that every open set U is a union of such X f . By definition,U = Spec(R) \ V (S) = Spec(R) \ ( ⋂V (f)) = ⋃(Spec(R) \ V (f)) = ⋃X f .f∈Sf∈Sf∈S□Obviously, only a set of generators {f i | i ∈ J} of the ideal generated by the set Sis needed. Hence, if R is a noetherian ring every open set can already be covered byfinitely manx X f .