Let \(\mcf, \mcg\) be sheaves of abelian groups on \(X\). For any open set \(U \subseteq X\), show that the set \(\Hom(\ro{\mcf}{U}, \ro{\mcg}{U})\) of morphisms of the restricted sheaves has a natural structure of abelian group. Show that the presheaf \[ U \mapsto \Hom(\ro{\mcf}{U}, \ro{\mcg}{U}) \] is a sheaf. It is called the sheaf of local morphisms of \(\mcf\) into \(\mcg\), or “sheaf hom” for short, and is denoted \(\sheafhom(\mcf, \mcg)\).
Solution
For an open set \(U\subseteq X\), write \[ \mathcal H(U) =\operatorname{Hom}(\mathcal F|_U,\mathcal G|_U). \]
1. The set \(\mathcal H(U)\) is naturally an abelian group.
Let \[ \phi,\psi:\mathcal F|_U\longrightarrow\mathcal G|_U \] be morphisms of sheaves of abelian groups. Define \[ (\phi+\psi)_V:\mathcal F(V)\longrightarrow\mathcal G(V) \] for every open \(V\subseteq U\) by \[ (\phi+\psi)_V(s)=\phi_V(s)+\psi_V(s). \]
Because the restriction maps of \(\mathcal G\) are homomorphisms and both \(\phi\) and \(\psi\) commute with restrictions, so does \(\phi+\psi\). Hence it is a morphism of sheaves.
The zero morphism is defined sectionwise by zero, and \[ (-\phi)_V(s)=-\phi_V(s). \] All abelian-group identities hold sectionwise in the groups \(\mathcal G(V)\). Thus \(\mathcal H(U)\) is an abelian group.
2. If \(V\subseteq U\), restriction of sheaf morphisms defines a homomorphism\[ \rho_{UV}:\mathcal H(U)\longrightarrow\mathcal H(V), \qquad \phi\longmapsto\phi|_V. \]These maps make \(\mathcal H\) a presheaf of abelian groups.
1. Also\[ \phi|_U=\phi, \qquad (\phi|_V)|_W=\phi|_W \]for \(W\subseteq V\subseteq U\). These are exactly the presheaf identities.
3. Let\[ U=\bigcup_iU_i \]and suppose morphisms\[ \phi_i:\mathcal F|_{U_i}\longrightarrow\mathcal G|_{U_i} \]agree on every overlap:\[ \phi_i|_{U_i\cap U_j} = \phi_j|_{U_i\cap U_j}. \]For every open \(V\subseteq U\) and every section \(s\in\mathcal F(V)\), the local sections\[ t_i := (\phi_i)_{V\cap U_i}(s|_{V\cap U_i}) \in \mathcal G(V\cap U_i) \]agree on overlaps.
On \[ V\cap U_i\cap U_j, \] the restrictions of \(\phi_i\) and \(\phi_j\) are the same morphism. Therefore \[ t_i|_{V\cap U_i\cap U_j} = (\phi_i)(s|_{V\cap U_i\cap U_j}) = (\phi_j)(s|_{V\cap U_i\cap U_j}) = t_j|_{V\cap U_i\cap U_j}. \]
4. The sections in 3 glue uniquely to a section\[ \phi_V(s)\in\mathcal G(V). \]This defines a homomorphism\[ \phi_V:\mathcal F(V)\longrightarrow\mathcal G(V). \]
The opens \[ V\cap U_i \] cover \(V\). By 3, the sections \(t_i\) are compatible, so the sheaf axiom for \(\mathcal G\) gives a unique section \[ t\in\mathcal G(V) \] restricting to every \(t_i\). Define \[ \phi_V(s)=t. \]
For \(s,s'\in\mathcal F(V)\), the restrictions of \[ \phi_V(s+s') \] and \[ \phi_V(s)+\phi_V(s') \] to every \(V\cap U_i\) are equal because \((\phi_i)_{V\cap U_i}\) is a homomorphism. Uniqueness in the sheaf axiom gives \[ \phi_V(s+s')=\phi_V(s)+\phi_V(s'). \] Similarly \(\phi_V(0)=0\), so \(\phi_V\) is a group homomorphism.
5. The homomorphisms \(\phi_V\) commute with restrictions, so they define a sheaf morphism\[ \phi:\mathcal F|_U\longrightarrow\mathcal G|_U. \]
Let \(W\subseteq V\subseteq U\) and \(s\in\mathcal F(V)\). Compare \[ \phi_V(s)|_W \qquad\text{and}\qquad \phi_W(s|_W). \] On each open \[ W\cap U_i, \] both restrict to \[ (\phi_i)_{W\cap U_i}(s|_{W\cap U_i}), \] because each local morphism \(\phi_i\) commutes with restrictions. Since the \(W\cap U_i\) cover \(W\), uniqueness for the sheaf \(\mathcal G\) gives \[ \phi_V(s)|_W=\phi_W(s|_W). \] Thus the family \((\phi_V)\) is a morphism of sheaves.
6. The morphism \(\phi\) restricts to \(\phi_i\) on every \(U_i\).
Let \(V\subseteq U_i\) and \(s\in\mathcal F(V)\). In the construction of \(\phi_V(s)\), one of the local sections is \[ (\phi_i)_V(s). \] The glued section restricts to this section on \(V\) itself, hence \[ \phi_V(s)=(\phi_i)_V(s). \] Therefore \[ \phi|_{U_i}=\phi_i. \]
7. The morphism \(\phi\) is the unique morphism on \(U\) restricting to all the \(\phi_i\).
If \[ \psi:\mathcal F|_U\to\mathcal G|_U \] has the same restrictions, then for every open \(V\subseteq U\) and \(s\in\mathcal F(V)\) the two sections \[ \phi_V(s),\quad\psi_V(s)\in\mathcal G(V) \] have equal restrictions on every \(V\cap U_i\). Since those opens cover \(V\) and \(\mathcal G\) is a sheaf, \[ \phi_V(s)=\psi_V(s). \] Thus \(\phi=\psi\).
8. Consequently\[ \boxed{ U\longmapsto \operatorname{Hom}(\mathcal F|_U,\mathcal G|_U) } \]is a sheaf of abelian groups, denoted\[ \mathcal Hom(\mathcal F,\mathcal G). \]
Step 2 gives the presheaf structure, while <1>3–<1>7 prove existence and uniqueness of gluing for arbitrary open covers.
9. Q.E.D.
Steps 1 and 8 are exactly the two assertions of the exercise.