Generalize the glueing procedure of the text as follows. Let \(\ts{X_i}\) be a possibly infinite family of schemes. For each \(i \neq j\) suppose given an open subset \(U_{ij} \subseteq X_i\) with its induced scheme structure. Suppose also given for each \(i \neq j\) an isomorphism of schemes \(\phi_{ij}: U_{ij} \to U_{ji}\) such that
for each \(i, j\), \(\phi_{ji} = \inverseof{\phi_{ij}}\), and
for each \(i, j, k\), \(\phi_{ij}(U_{ij} \intersect U_{ik}) = U_{ji} \intersect U_{jk}\) and \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on \(U_{ij} \intersect U_{ik}\).
Then show that there is a scheme \(X\), together with morphisms \(\psi_i: X_i \to X\) for each \(i\), such that
\(\psi_i\) is an isomorphism of \(X_i\) onto an open subscheme of \(X\),
the \(\psi_i(X_i)\) cover \(X\),
\(\psi_i(U_{ij}) = \psi_i(X_i) \intersect \psi_j(X_j)\), and
\(\psi_i = \psi_j \circ \phi_{ij}\) on \(U_{ij}\).
We say that \(X\) is obtained by glueing the schemes \(X_i\) along the isomorphisms \(\phi_{ij}\). An interesting special case is when the family \(X_i\) is arbitrary but all the \(U_{ij}\) and \(\phi_{ij}\) are empty; then \(X\) is called the disjoint union of the \(X_i\) and is denoted \(\coprod X_i\).
Solution
For convenience set \[ U_{ii}=X_i, \qquad \phi_{ii}=\operatorname{id}_{X_i}. \]
1. On the disjoint union of underlying sets\[ S=\coprod_i|X_i| \]define a relation by declaring, for \(x\in X_i\) and \(y\in X_j\),\[ x\sim y \quad\Longleftrightarrow\quad x\in U_{ij} \text{ and } y=\phi_{ij}(x). \]Then \(\sim\) is an equivalence relation.
Reflexivity follows from \[ U_{ii}=X_i, \qquad \phi_{ii}=\operatorname{id}. \]
If \[ y=\phi_{ij}(x), \] then \[ x=\phi_{ji}(y) \] because \[ \phi_{ji}=\phi_{ij}^{-1}, \] so the relation is symmetric.
For transitivity, suppose \[ y=\phi_{ij}(x), \qquad z=\phi_{jk}(y), \] with \[ x\in U_{ij}, \qquad y\in U_{jk}. \] Since also \[ y\in U_{ji}, \] one has \[ y\in U_{ji}\cap U_{jk}. \] The overlap hypothesis says \[ U_{ji}\cap U_{jk} = \phi_{ij}(U_{ij}\cap U_{ik}). \] Since \(\phi_{ij}\) is injective and \(y=\phi_{ij}(x)\), it follows that \[ x\in U_{ik}. \] On this overlap the cocycle condition gives \[ z =\phi_{jk}(\phi_{ij}(x)) =\phi_{ik}(x). \] Thus \(x\sim z\).
2. Let\[ |X|=S/\!\sim \]and give it the quotient topology. Let\[ q:S\longrightarrow|X| \]be the quotient map, and put\[ \psi_i=q|_{X_i}:|X_i|\longrightarrow|X|. \]Each \(\psi_i\) is injective.
If \[ x,x'\in X_i \] have the same image in the quotient, then \[ x\sim x'. \] By the definition with \(i=j\), \[ x'=\phi_{ii}(x)=x. \] Thus \(\psi_i\) is injective.
3. The image\[ V_i:=\psi_i(X_i) \]is open in \(|X|\).
By definition of the quotient topology, it is enough to show that \[ q^{-1}(V_i) \] is open in the disjoint union \(S\).
Its intersection with the component \(X_i\) is all of \(X_i\). For \(j\ne i\), a point \[ y\in X_j \] maps into \(V_i\) exactly when it is equivalent to some point of \(X_i\), which by definition is exactly when \[ y\in U_{ji}. \] Hence \[ q^{-1}(V_i) = X_i \amalg \coprod_{j\ne i}U_{ji}. \] Every \(U_{ji}\) is open in \(X_j\), so this is open in the disjoint-union topology. Therefore \(V_i\) is open.
4. The map\[ \psi_i:X_i\longrightarrow V_i \]is a homeomorphism.
It is already a continuous bijection by 2 and the definition of the quotient topology. We show it is open.
Let \[ W\subseteq X_i \] be open. The inverse image in \(S\) of \[ \psi_i(W) \] has, in the component \(X_j\), the subset \[ \phi_{ij}(W\cap U_{ij}) \subseteq U_{ji} \] for \(j\ne i\), and has \(W\) in the component \(X_i\). Each of these sets is open because \[ \phi_{ij}:U_{ij}\xrightarrow{\sim}U_{ji} \] is a homeomorphism. Thus \[ q^{-1}(\psi_i(W)) \] is open in \(S\), so \(\psi_i(W)\) is open in \(|X|\).
Hence \(\psi_i\) is an open continuous bijection and therefore a homeomorphism onto \(V_i\).
5. The opens \(V_i\) cover \(|X|\), and\[ \boxed{ V_i\cap V_j =\psi_i(U_{ij}) =\psi_j(U_{ji}). } \]Moreover,\[ \boxed{ \psi_i =\psi_j\circ\phi_{ij} \quad\text{on }U_{ij}. } \]
The images cover because every equivalence class has a representative in some component \(X_i\).
A point of \(V_i\) belongs to \(V_j\) exactly when its representative \[ x\in X_i \] is equivalent to some point of \(X_j\). By definition, this is exactly the condition \[ x\in U_{ij}, \] and the corresponding representative in \(X_j\) is \[ \phi_{ij}(x). \] This proves both the intersection formula and \[ \psi_i(x)=\psi_j(\phi_{ij}(x)). \]
6. Transport the structure sheaf \(\mathcal O_{X_i}\) along the homeomorphism\[ \psi_i:X_i\xrightarrow{\sim}V_i \]to obtain a sheaf \(\mathcal G_i\) of rings on \(V_i\).The scheme isomorphisms \(\phi_{ij}\) induce compatible sheaf isomorphisms\[ \mathcal G_i|_{V_i\cap V_j} \xrightarrow{\sim} \mathcal G_j|_{V_i\cap V_j}. \]
Define \[ \mathcal G_i=(\psi_i)_*\mathcal O_{X_i} \] on \(V_i\), using the homeomorphism \(\psi_i\).
By 5, the overlap \(V_i\cap V_j\) corresponds under \(\psi_i\) to \(U_{ij}\) and under \(\psi_j\) to \(U_{ji}\). The scheme isomorphism \[ \phi_{ij}:U_{ij}\xrightarrow{\sim}U_{ji} \] therefore transports the structure sheaf from the \(i\)-description of the overlap to the \(j\)-description.
The inverse condition \[ \phi_{ji}=\phi_{ij}^{-1} \] and the triple-overlap cocycle \[ \phi_{ik}=\phi_{jk}\circ\phi_{ij} \] imply the corresponding inverse and cocycle identities for these sheaf isomorphisms.
7. The local sheaves \(\mathcal G_i\) glue to a sheaf of rings\[ \mathcal O_X \]on \(|X|\), together with isomorphisms\[ \mathcal O_X|_{V_i} \xrightarrow{\sim} \mathcal G_i. \]
Apply the sheaf-gluing theorem of Hartshorne II.1.22 to the open cover \[ |X|=\bigcup_iV_i \] and the compatible sheaf isomorphisms from 6.
8. The locally ringed space\[ X=(|X|,\mathcal O_X) \]is a scheme, and each\[ \psi_i:X_i\longrightarrow X \]is an isomorphism of schemes onto the open subscheme \(V_i\).
By construction, \[ (V_i,\mathcal O_X|_{V_i}) \cong (X_i,\mathcal O_{X_i}) \] as locally ringed spaces. Thus every \(V_i\) is itself a scheme and \(\psi_i\) is an isomorphism onto it.
The opens \(V_i\) cover \(X\). Each \(X_i\) has an affine open cover, and transporting all those affine opens through the \(\psi_i\) gives an affine open cover of \(X\). Hence \(X\) is a scheme.
9. The maps \(\psi_i\) satisfy all four required properties:\[ \begin{aligned} &\psi_i:X_i\xrightarrow{\sim}V_i\subseteq X,\\ &X=\bigcup_iV_i,\\ &V_i\cap V_j=\psi_i(U_{ij}),\\ &\psi_i=\psi_j\circ\phi_{ij}\text{ on }U_{ij}. \end{aligned} \]
The first property is 8, and the remaining three are 5.
10. The glued scheme is unique up to a unique isomorphism compatible with the maps \(\psi_i\).
Suppose \[ X' \] with maps \[ \psi_i':X_i\to X' \] has the same four properties. On the open subset \[ V_i=\psi_i(X_i)\subseteq X, \] define \[ F_i = \psi_i'\circ\psi_i^{-1}: V_i\longrightarrow X'. \] On \[ V_i\cap V_j=\psi_i(U_{ij}), \] the compatibility conditions give \[ \psi_i =\psi_j\phi_{ij}, \qquad \psi_i' =\psi_j'\phi_{ij}, \] so \[ F_i=F_j \] on the overlap. Hence the morphisms \(F_i\) glue to a morphism \[ F:X\to X'. \]
Reversing the roles of \(X\) and \(X'\) gives \[ G:X'\to X. \] Both composites restrict to the identity on the open covers by the \(V_i\) and \(V_i'\), hence \[ GF=\operatorname{id}_X, \qquad FG=\operatorname{id}_{X'}. \] Thus \(F\) is an isomorphism. Any compatible isomorphism must agree with \(F_i\) on every \(V_i\), so it is unique.
11. If all \(U_{ij}\) for \(i\ne j\) are empty, then no distinct components are identified and\[ \boxed{ X=\coprod_iX_i } \]is their disjoint union as a scheme.
The equivalence relation of 1 becomes equality within each component and makes no cross-component identifications, so the underlying space is the topological disjoint union.
There are no nontrivial overlap isomorphisms of structure sheaves to impose. Thus \(\mathcal O_X\) restricts independently to \(\mathcal O_{X_i}\) on each open-and-closed component. This is precisely the scheme-theoretic disjoint union.
12. Q.E.D.
Steps <1>1–<1>9 construct the required scheme and maps, 10 gives the expected uniqueness, and 11 identifies the disjoint-union special case.