Let \(X\) be a topological space, let \(\mathfrak{U} = \ts{U_i}\) be an open cover of \(X\), and suppose we are given for each \(i\) a sheaf \(\mcf_i\) on \(U_i\), and for each \(i, j\) an isomorphism \[ \phi_{ij}: \ro{\mcf_i}{U_i \intersect U_j} \to \ro{\mcf_j}{U_i \intersect U_j} \] such that
for each \(i\), \(\phi_{ii} = \id\), and
for each \(i, j, k\), \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on \(U_i \intersect U_j \intersect U_k\).
Show that there exists a unique sheaf \(\mcf\) on \(X\), together with isomorphisms \(\psi_i: \ro{\mcf}{U_i} \to \mcf_i\), such that for each \(i, j\) one has \(\psi_j = \phi_{ij} \circ \psi_i\) on \(U_i \intersect U_j\). We say that \(\mcf\) is obtained by glueing the sheaves \(\mcf_i\) via the isomorphisms \(\phi_{ij}\).
Solution
First note that the cocycle conditions imply \[ \phi_{ji}=\phi_{ij}^{-1}. \] Indeed, on \(U_i\cap U_j\) one has \[ \operatorname{id} =\phi_{ii} =\phi_{ji}\circ\phi_{ij}. \]
1. For every open set \(V\subseteq X\), define\[ \mathcal F(V) \]to be the set of tuples\[ (s_i)_i, \qquad s_i\in\mathcal F_i(V\cap U_i), \]such that for every pair \(i,j\),\[ \boxed{ \phi_{ij} \bigl(s_i|_{V\cap U_i\cap U_j}\bigr) = s_j|_{V\cap U_i\cap U_j}. } \]
This is the natural compatibility condition saying that the local sections \(s_i\) describe the same putative global section after the prescribed identifications of the sheaves on overlaps.
2. Componentwise restriction makes \(\mathcal F\) a presheaf on \(X\).
Let \(W\subseteq V\) and let \[ s=(s_i)_i\in\mathcal F(V). \] Define \[ s|_W = \bigl(s_i|_{W\cap U_i}\bigr)_i. \]
For every \(i,j\), restriction of the compatibility equation from 1 gives \[ \phi_{ij} \bigl(s_i|_{W\cap U_i\cap U_j}\bigr) = s_j|_{W\cap U_i\cap U_j}, \] because \(\phi_{ij}\) is a morphism of sheaves and hence commutes with restrictions. Thus the restricted tuple lies in \(\mathcal F(W)\).
Identity and composition laws hold componentwise, so these maps define a presheaf.
3. The presheaf \(\mathcal F\) satisfies the sheaf uniqueness axiom.
Let \[ V=\bigcup_\alpha V_\alpha \] and suppose \[ s=(s_i)_i, \qquad t=(t_i)_i \] in \(\mathcal F(V)\) have equal restrictions to every \(V_\alpha\).
Fix \(i\). Then \[ s_i|_{V_\alpha\cap U_i} = t_i|_{V_\alpha\cap U_i} \] for every \(\alpha\). The opens \(V_\alpha\cap U_i\) cover \(V\cap U_i\), and \(\mathcal F_i\) is a sheaf, so \[ s_i=t_i. \] This holds for every \(i\), hence \(s=t\).
4. The presheaf \(\mathcal F\) satisfies the sheaf gluing axiom.
Let \[ V=\bigcup_\alpha V_\alpha \] and suppose compatible sections \[ s^{(\alpha)} = (s_i^{(\alpha)})_i \in\mathcal F(V_\alpha) \] are given.
Fix \(i\). The sections \[ s_i^{(\alpha)} \in \mathcal F_i(V_\alpha\cap U_i) \] agree on overlaps because the tuples \(s^{(\alpha)}\) do. Since \(\mathcal F_i\) is a sheaf, they glue uniquely to \[ s_i\in\mathcal F_i(V\cap U_i). \]
1. Fix \(i,j\). On every open\[ V_\alpha\cap U_i\cap U_j, \]one has\[ \phi_{ij}(s_i) = \phi_{ij}(s_i^{(\alpha)}) = s_j^{(\alpha)} = s_j. \]These opens cover\[ V\cap U_i\cap U_j, \]so uniqueness in the sheaf \(\mathcal F_j\) gives\[ \phi_{ij}(s_i)=s_j \]on the whole overlap. Thus\[ s\in\mathcal F(V). \]
By construction it restricts to every \(s^{(\alpha)}\), and uniqueness follows from 3.
5. Hence \(\mathcal F\) is a sheaf on \(X\).
Steps <1>2–<1>4 give the presheaf structure and both sheaf axioms.
6. For every \(i\), projection to the \(i\)th component defines a morphism of sheaves on \(U_i\),\[ \psi_i: \mathcal F|_{U_i} \longrightarrow \mathcal F_i. \]
If \[ V\subseteq U_i \] and \[ s=(s_j)_j\in\mathcal F(V), \] define \[ (\psi_i)_V(s)=s_i. \] These projections commute with restriction because restrictions in \(\mathcal F\) are componentwise.
7. Each \(\psi_i\) is an isomorphism.
Fix \[ V\subseteq U_i. \] Given \[ t\in\mathcal F_i(V), \] define for every \(j\) \[ s_j = \phi_{ij} \bigl(t|_{V\cap U_j}\bigr) \in \mathcal F_j(V\cap U_j). \] Here on \(V\cap U_j\) the map \(\phi_{ij}\) is understood after restricting to \[ U_i\cap U_j; \] this is legitimate since \(V\subseteq U_i\).
For \(j,k\), the cocycle condition gives \[ \phi_{jk}(s_j) = \phi_{jk}\phi_{ij}(t) = \phi_{ik}(t) = s_k \] on \(V\cap U_j\cap U_k\). Thus \[ (s_j)_j\in\mathcal F(V). \]
Its \(i\)th component is \[ s_i =\phi_{ii}(t)=t, \] so this construction is a right inverse to \((\psi_i)_V\).
Conversely, if \(s=(s_j)_j\in\mathcal F(V)\), compatibility with the \(i\)th component says \[ s_j = \phi_{ij}(s_i) \] on \(V\cap U_j\). Thus the whole tuple is uniquely reconstructed from \(s_i\), and the same construction is a left inverse. Hence \((\psi_i)_V\) is bijective for every \(V\subseteq U_i\), naturally in \(V\), so \(\psi_i\) is an isomorphism of sheaves.
8. On every overlap \(U_i\cap U_j\), the local identifications satisfy\[ \boxed{ \psi_j = \phi_{ij}\circ\psi_i. } \]
For a compatible tuple \(s=(s_k)_k\), the defining condition of \(\mathcal F\) gives \[ s_j=\phi_{ij}(s_i) \] on the overlap. Since \(\psi_i\) and \(\psi_j\) are the corresponding component projections, this is exactly the displayed identity.
9. The pair\[ (\mathcal F,\{\psi_i\}) \]is unique up to a unique isomorphism compatible with all the \(\psi_i\).
Suppose \(\mathcal G\) is another sheaf on \(X\) with isomorphisms \[ \chi_i:\mathcal G|_{U_i}\xrightarrow{\sim}\mathcal F_i \] satisfying \[ \chi_j=\phi_{ij}\circ\chi_i \] on every overlap.
1. Hence\[ \Theta_V(t)\in\mathcal F(V). \]The construction commutes with restrictions, so it defines a sheaf morphism\[ \Theta:\mathcal G\longrightarrow\mathcal F. \]
On \(U_i\) one has \[ \psi_i\circ\Theta|_{U_i} =\chi_i. \] Since both \(\psi_i\) and \(\chi_i\) are isomorphisms, \(\Theta\) restricts to an isomorphism on every member of the cover. A morphism of sheaves which is locally an isomorphism on an open cover is an isomorphism globally.
If \[ \Theta':\mathcal G\to\mathcal F \] is another morphism compatible with all local identifications, then \[ \Theta'|_{U_i} =\psi_i^{-1}\chi_i =\Theta|_{U_i} \] for every \(i\). Since the \(U_i\) cover \(X\), the two sheaf morphisms are equal. Thus the compatible isomorphism is unique.
10. Therefore the sheaves \(\mathcal F_i\) glue along the cocycle \(\phi_{ij}\) to a sheaf \(\mathcal F\) on \(X\), uniquely up to the unique compatible isomorphism:\[ \boxed{ \mathcal F|_{U_i}\cong\mathcal F_i, \qquad \psi_j=\phi_{ij}\psi_i. } \]
Existence is <1>5–<1>8 and uniqueness is 9.
11. Q.E.D.
Step 10 is precisely the sheaf-gluing assertion of the exercise.