For a scheme \(X\), write \(h_X = \Hom(-, X) \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) for its functor of points (PR-SCHFOP).
(Zariski sheaf)
A functor \(F \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) is a Zariski sheaf if for every scheme \(Y\) the presheaf \(U \mapsto F(U)\) on the open subsets \(U \subseteq Y\), with restriction maps \(F(U) \to F(U')\) induced by the inclusions \(U' \subseteq U\), is a sheaf of sets on \(Y\).
(Open subfunctor)
Let \(F, G \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) be functors. A natural transformation \(\iota \colon G \to F\) exhibits \(G\) as an open subfunctor of \(F\) if for every scheme \(X\) and every natural transformation \(h_X \to F\), the fibre product \(h_X \times_F G\) is represented by a scheme \(U\) and the induced morphism \(U \to X\) is an open immersion. A family of open subfunctors \(\{G_i \to F\}\) is an open cover of \(F\) if for every \(h_X \to F\) the open subschemes \(U_i \subseteq X\) representing \(h_X \times_F G_i\) cover \(X\).
(Representability by gluing)
Let \(F\) be a Zariski sheaf with an open cover by open subfunctors \(\{G_i \to F\}_{i \in I}\) such that each \(G_i\) is represented by a scheme \(X_i\). Then \(F\) is represented by a scheme \(X\) covered by open subschemes isomorphic to the \(X_i\).
The functor \(\PP^n\) of FE-SCHREPFUNCTORS is a Zariski sheaf because line bundles and their sections glue. For \(0 \leq i \leq n\) the subfunctor \(G_i\) of pairs \((\mathcal{L}, s_0, \ldots, s_n)\) in which \(s_i\) generates \(\mathcal{L}\) is open, its fibre product with \(h_X\) being represented by the open set \(X_{s_i} = \{x \in X : s_i(x) \neq 0\}\). Each \(G_i\) is represented by \(\AA^n_\ZZ\), via \((\mathcal{L}, s) \mapsto (s_j/s_i)_{j \neq i}\), and the \(G_i\) cover because generating sections have no common zero, so the theorem constructs \(\PP^n_\ZZ\) by gluing \(n+1\) copies of \(\AA^n_\ZZ\).
Dependencies
PR-SCHFOPThe functor of points, and what small test schemes detectuses
Backlinks
FE-SCHREPFUNCTORSThe functors of points of $\AA^n$, $\AA^n \setminus \{0\}$ and $\PP^n$related-to