Let \(X\) be a topological space, let \(Z\) be a closed subset with inclusion \(i: Z \to X\), let \(U = X \sm Z\) be the complementary open subset, and let \(j: U \to X\) be its inclusion.
Let \(\mcf\) be a sheaf on \(Z\). Show that the stalk \((i_* \mcf)_P\) of the direct image sheaf on \(X\) is \(\mcf_P\) if \(P \in Z\) and \(0\) if \(P \notin Z\). Hence \(i_* \mcf\) is called the sheaf obtained by extending \(\mcf\) by zero outside \(Z\).
Now let \(\mcf\) be a sheaf on \(U\). Let \(j_!(\mcf)\) be the sheaf on \(X\) associated to the presheaf \(V \mapsto \mcf(V)\) if \(V \subseteq U\) and \(V \mapsto 0\) otherwise. Show that the stalk \((j_!(\mcf))_P\) equals \(\mcf_P\) if \(P \in U\) and \(0\) if \(P \notin U\), and show that \(j_! \mcf\) is the only sheaf on \(X\) with this property whose restriction to \(U\) is \(\mcf\). We call \(j_! \mcf\) the sheaf obtained by extending \(\mcf\) by zero outside \(U\).
Now let \(\mcf\) be a sheaf on \(X\). Show that there is an exact sequence of sheaves on \(X\), \[ 0 \to j_!\qty{\ro{\mcf}{U}} \to \mcf \to i_*\qty{\ro{\mcf}{Z}} \to 0. \]
Solution
1. If \(\mathcal F\) is a sheaf on the closed subspace \(Z\), then for every \(P\in Z\),\[ \boxed{(i_*\mathcal F)_P\cong\mathcal F_P.} \]
For an open set \(V\subseteq X\), \[ (i_*\mathcal F)(V)=\mathcal F(V\cap Z). \] Hence \[ (i_*\mathcal F)_P = \varinjlim_{P\in V}\mathcal F(V\cap Z). \]
The sets \(V\cap Z\), as \(V\) ranges over the open neighborhoods of \(P\) in \(X\), are exactly a cofinal family of open neighborhoods of \(P\) in the subspace \(Z\). Therefore this direct limit is the stalk \[ \mathcal F_P = \varinjlim_{P\in W\subseteq Z}\mathcal F(W). \]
2. If \(P\notin Z\), then\[ \boxed{(i_*\mathcal F)_P=0.} \]
Because \(Z\) is closed, its complement \[ U=X\setminus Z \] is an open neighborhood of \(P\). For every open neighborhood \[ P\in V\subseteq U, \] one has \[ V\cap Z=\varnothing, \] so \[ (i_*\mathcal F)(V)=\mathcal F(\varnothing)=0. \] These neighborhoods are cofinal among all neighborhoods of \(P\), hence the stalk is zero.
3. Thus \(i_*\mathcal F\) is extension by zero outside \(Z\):\[ \boxed{ (i_*\mathcal F)_P \cong \begin{cases} \mathcal F_P,&P\in Z,\\ 0,&P\notin Z. \end{cases} } \]
Combine 1 and 2.
4. Now let \(\mathcal F\) be a sheaf on the open subspace \(U\). Define the presheaf \(\mathcal P\) on \(X\) by\[ \mathcal P(V)= \begin{cases} \mathcal F(V),&V\subseteq U,\\ 0,&V\not\subseteq U, \end{cases} \]and let\[ j_!\mathcal F=\mathcal P^+ \]be its associated sheaf. If \(P\in U\), then\[ \boxed{(j_!\mathcal F)_P\cong\mathcal F_P.} \]
Sheafification does not change stalks, so \[ (j_!\mathcal F)_P\cong\mathcal P_P. \] Since \(U\) is open and contains \(P\), the neighborhoods \[ P\in V\subseteq U \] are cofinal among all neighborhoods of \(P\) in \(X\). On them \[ \mathcal P(V)=\mathcal F(V). \] Therefore \[ \mathcal P_P =\varinjlim_{P\in V\subseteq U}\mathcal F(V) =\mathcal F_P. \]
5. If \(P\notin U\), then\[ \boxed{(j_!\mathcal F)_P=0.} \]
Every neighborhood \(V\) of \(P\) contains \(P\), so no such \(V\) can be contained in \(U\). Hence \[ \mathcal P(V)=0 \] for every neighborhood of \(P\). Thus \[ \mathcal P_P=0, \] and sheafification preserves this stalk.
6. The restriction of \(j_!\mathcal F\) to \(U\) is naturally isomorphic to \(\mathcal F\).
On the open subspace \(U\), the presheaf \(\mathcal P\) restricts to the sheaf \(\mathcal F\) itself: for every open \[ V\subseteq U, \] one has \[ \mathcal P(V)=\mathcal F(V), \] with the same restriction maps. Therefore sheafification does nothing after restricting to \(U\), and \[ (j_!\mathcal F)|_U\cong\mathcal F. \]
7. The sheaf \(j_!\mathcal F\) is unique, up to unique isomorphism compatible with the given identification on \(U\), among sheaves \(\mathcal G\) on \(X\) such that\[ \mathcal G|_U\cong\mathcal F \qquad\text{and}\qquad \mathcal G_P=0\quad(P\notin U). \]
Fix an identification \[ \theta_U:\mathcal F\xrightarrow{\sim}\mathcal G|_U. \] Define a morphism of presheaves \[ \mathcal P\longrightarrow\mathcal G \] as follows. If \(V\subseteq U\), use \[ \mathcal P(V)=\mathcal F(V) \xrightarrow{\theta_U(V)} \mathcal G(V). \] If \(V\not\subseteq U\), use the unique zero map \[ 0=\mathcal P(V)\longrightarrow\mathcal G(V). \] These maps commute with restrictions. Indeed, the only mixed case is \[ W\subseteq V, \qquad W\subseteq U, \qquad V\not\subseteq U, \] and there the section from \(\mathcal P(V)=0\) restricts to zero in both presheaves.
By the universal property of sheafification this induces \[ \theta:j_!\mathcal F=\mathcal P^+\longrightarrow\mathcal G. \]
For \(P\in U\), the map on stalks is the isomorphism \[ \mathcal F_P\xrightarrow{\sim}\mathcal G_P \] induced by \(\theta_U\). For \(P\notin U\), both stalks are zero by 5 and the hypothesis on \(\mathcal G\). Hence \(\theta\) is an isomorphism on every stalk, and therefore an isomorphism of sheaves.
Any isomorphism compatible with the fixed identification on \(U\) must induce the same stalk maps at every point: those in \(U\) are prescribed, and those outside \(U\) are maps \(0\to0\). A morphism of sheaves is determined by its stalk maps, so this isomorphism is unique.
8. Let \(\mathcal F\) now be a sheaf on all of \(X\). There is a natural morphism\[ \alpha: j_!(\mathcal F|_U) \longrightarrow \mathcal F \]which is the identity on stalks over \(U\) and the zero map on stalks over \(Z\).
Let \(\mathcal P\) be the presheaf defining \[ j_!(\mathcal F|_U). \] If \(V\subseteq U\), define \[ \mathcal P(V)=\mathcal F(V) \longrightarrow\mathcal F(V) \] to be the identity. If \(V\not\subseteq U\), the source is zero, so use the zero map. These maps define a presheaf morphism \[ \mathcal P\to\mathcal F, \] which sheafifies to \(\alpha\).
The stalk description follows from <1>4–<1>5 and the construction.
9. There is a natural morphism\[ \beta: \mathcal F \longrightarrow i_*(\mathcal F|_Z) \]which is the identity on stalks over \(Z\) and has zero target on stalks over \(U\).
For the closed inclusion \[ i:Z\hookrightarrow X, \] the restriction sheaf on \(Z\) is \(i^{-1}\mathcal F\). The unit of the adjunction \[ i^{-1}\dashv i_* \] gives \[ \beta:\mathcal F\longrightarrow i_*i^{-1}\mathcal F =i_*(\mathcal F|_Z). \]
By <1>1–<1>2, the target stalk is \(\mathcal F_P\) at \(P\in Z\) and zero at \(P\in U\). At a point of \(Z\), the adjunction unit is induced by restricting a local section to smaller neighborhoods in the subspace \(Z\), so the induced map on the stalk is the identity under the natural identification \[ (i_*i^{-1}\mathcal F)_P\cong\mathcal F_P. \]
10. The sequence\[ \boxed{ 0 \longrightarrow j_!(\mathcal F|_U) \xrightarrow{\alpha} \mathcal F \xrightarrow{\beta} i_*(\mathcal F|_Z) \longrightarrow0 } \]is exact.
Exactness of sheaves of abelian groups may be checked on stalks.
If \(P\in U\), then <1>4–<1>5 and 3 identify the stalk sequence with \[ 0\longrightarrow \mathcal F_P \xrightarrow{\operatorname{id}} \mathcal F_P \longrightarrow0 \longrightarrow0, \] which is exact.
If \(P\in Z\), the stalk sequence is \[ 0\longrightarrow0 \longrightarrow \mathcal F_P \xrightarrow{\operatorname{id}} \mathcal F_P \longrightarrow0, \] which is also exact.
Thus the original sequence is exact at every stalk and hence exact as a sequence of sheaves.
11. Q.E.D.
Steps <1>1–<1>3 prove part (a), <1>4–<1>7 prove part (b), and <1>8–<1>10 prove part (c).