(Gluing data)
Let \(\ts{X_i}\) be schemes, and for each pair \(i,j\) let \(U_{ij} \subseteq X_i\) be open with isomorphisms \(\varphi_{ij}: U_{ij} \to U_{ji}\) satisfying \(\varphi_{ii} = \id\), \(\varphi_{ji} = \inverseof{\varphi_{ij}}\), and the cocycle condition \(\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}\) on \(U_{ij} \intersect U_{ik}\). Then there is a scheme \(X\) with an open cover by copies of the \(X_i\) inducing the \(\varphi_{ij}\), unique up to isomorphism.
(Projective space by charts)
Take \(n+1\) copies of \(\AA^n\slice k = \Spec k[x_0/x_i, \dots, x_n/x_i]\) and glue \(U_{ij} = D(x_j/x_i)\) to \(U_{ji} = D(x_i/x_j)\) by inverting the ratio. The result is \(\PP^n\slice k\), and its global sections are \(k\), so it is not affine for \(n\geq1\).
The same charts with different transition maps produce different schemes: gluing two copies of \(\AA^1\) along \(\AA^1 \sm \ts{0}\) by the identity gives the line with a doubled origin, while gluing by \(t \mapsto \inverseof{t}\) gives \(\PP^1\). The line with a doubled origin is not separated: its two open embeddings of \(\AA^1\) agree on \(\AA^1\sm\ts{0}\) and differ at the origin. The scheme \(\PP^1\) is separated, and it is proper over the base field.
Dependencies
D-AN662Schemes, and morphisms of schemesuses
Backlinks
T-SCHPRJC$\Proj S$ is a scheme, with affine charts $D_+(f) \cong \Spec S_{(f)}$related-toFE-SCHLINEThe line with a doubled originuses