Let \(Z\) be a closed subset of \(X\) and let \(\mcf\) be a sheaf on \(X\). Define \(\Gamma_Z(X, \mcf)\) to be the subgroup of \(\Gamma(X, \mcf)\) consisting of all sections whose support is contained in \(Z\).
Show that the presheaf \(V \mapsto \Gamma_{Z \intersect V}\qty{V, \ro{\mcf}{V}}\) is a sheaf. It is called the subsheaf of \(\mcf\) with supports in \(Z\) and is denoted \(\mathcal{H}_Z^0(\mcf)\).
Let \(U = X \sm Z\) and let \(j: U \to X\) be the inclusion. Show that there is an exact sequence of sheaves on \(X\) \[ 0 \to \mathcal{H}_Z^0(\mcf) \to \mcf \to j_*\qty{\ro{\mcf}{U}}. \] Furthermore, if \(\mcf\) is flasque, show that the map \(\mcf \to j_*\qty{\ro{\mcf}{U}}\) is surjective.
Solution
For every open \(V\subseteq X\), put \[ \mathcal H(V) = \Gamma_{Z\cap V}(V,\mathcal F|_V) = \{s\in\mathcal F(V):\operatorname{supp}s\subseteq Z\cap V\}. \]
1. Restriction of sections makes \(\mathcal H\) a subpresheaf of \(\mathcal F\).
Let \(W\subseteq V\) and let \[ s\in\mathcal H(V). \] For every \(P\in W\setminus Z\), the germ \(s_P\) is zero because \[ \operatorname{supp}s\subseteq Z\cap V. \] The germ of the restriction \(s|_W\) at \(P\) is the same germ \(s_P\), hence is zero. Therefore \[ \operatorname{supp}(s|_W)\subseteq Z\cap W, \] so \[ s|_W\in\mathcal H(W). \] Thus the restriction maps of \(\mathcal F\) restrict to maps for \(\mathcal H\).
2. The presheaf \(\mathcal H\) satisfies the sheaf uniqueness axiom.
Let \[ V=\bigcup_iV_i \] and let \[ s,t\in\mathcal H(V) \] have equal restrictions to every \(V_i\). They are also sections of the sheaf \(\mathcal F\), so the uniqueness axiom for \(\mathcal F\) gives \[ s=t. \]
3. The presheaf \(\mathcal H\) satisfies the sheaf gluing axiom.
Let \[ V=\bigcup_iV_i \] and let compatible sections \[ s_i\in\mathcal H(V_i) \] be given. Since \(\mathcal F\) is a sheaf, they glue uniquely to a section \[ s\in\mathcal F(V). \]
We must show \[ \operatorname{supp}s\subseteq Z\cap V. \] Let \[ P\in V\setminus Z. \] Choose \(i\) with \(P\in V_i\). Because \[ s_i\in\mathcal H(V_i), \] its germ at \(P\) is zero: \[ (s_i)_P=0. \] But \[ s|_{V_i}=s_i, \] so \[ s_P=(s_i)_P=0. \] Thus no point of \(V\setminus Z\) belongs to the support of \(s\), proving the required containment.
4. Hence \(\mathcal H\) is a sheaf, denoted\[ \boxed{\mathcal H_Z^0(\mathcal F).} \]
Step 1 gives the presheaf structure, and <1>2–<1>3 give the two sheaf axioms.
5. Let\[ U=X\setminus Z, \qquad j:U\hookrightarrow X. \]There is a natural restriction morphism\[ \rho:\mathcal F\longrightarrow j_*(\mathcal F|_U). \]On an open set \(V\subseteq X\), it is\[ \rho_V:\mathcal F(V) \longrightarrow \mathcal F(V\cap U), \qquad s\longmapsto s|_{V\cap U}. \]
Since \(U\) is open, \[ (j_*(\mathcal F|_U))(V) =(\mathcal F|_U)(V\cap U) =\mathcal F(V\cap U). \] The ordinary restriction maps of \(\mathcal F\) therefore define the displayed morphism, and compatibility with further restrictions is automatic.
6. For every open \(V\subseteq X\),\[ \boxed{\ker\rho_V=\mathcal H_Z^0(\mathcal F)(V).} \]
Suppose first that \[ s\in\ker\rho_V. \] Then \[ s|_{V\cap U}=0. \] Hence \[ s_P=0 \] for every \(P\in V\cap U=V\setminus Z\). Therefore \[ \operatorname{supp}s\subseteq Z\cap V, \] so \[ s\in\mathcal H_Z^0(\mathcal F)(V). \]
Conversely, suppose \[ s\in\mathcal H_Z^0(\mathcal F)(V). \] Then every germ of \[ s|_{V\cap U} \] is zero. A section of a sheaf whose germs are all zero is itself zero: each point has a neighborhood on which the section vanishes, and the sheaf uniqueness axiom glues these local zeroes. Thus \[ s|_{V\cap U}=0, \] so \(s\in\ker\rho_V\).
7. Therefore there is an exact sequence of sheaves\[ \boxed{ 0 \longrightarrow \mathcal H_Z^0(\mathcal F) \longrightarrow \mathcal F \xrightarrow{\rho} j_*(\mathcal F|_U). } \]
4. Step 6 identifies its image on every open set with the kernel of \(\rho\). Hence the sequence is exact at the first two nonzero terms; equivalently, it is exact as a sequence of sheaves.
8. If \(\mathcal F\) is flasque, then\[ \rho:\mathcal F\longrightarrow j_*(\mathcal F|_U) \]is surjective.
Let \(V\subseteq X\) be open. The restriction map occurring in 5 is \[ \mathcal F(V) \longrightarrow \mathcal F(V\cap U). \] Since \[ V\cap U\subseteq V \] and \(\mathcal F\) is flasque, this map is surjective.
Thus \(\rho\) is in fact surjective on sections over every open \(V\), and therefore certainly surjective as a morphism of sheaves.
9. In the flasque case the sequence of 7 extends to a short exact sequence\[ \boxed{ 0 \longrightarrow \mathcal H_Z^0(\mathcal F) \longrightarrow \mathcal F \longrightarrow j_*(\mathcal F|_U) \longrightarrow0. } \]
Combine exactness from 7 with the surjectivity in 8.
10. Q.E.D.
Steps <1>1–<1>4 prove part (a), <1>5–<1>7 prove the exact sequence in part (b), and <1>8–<1>9 prove its surjectivity assertion for flasque \(\mathcal F\).