Let \(X\) be a topological space, let \(P\) be a point, and let \(A\) be an abelian group. Define a sheaf \(i_P(A)\) on \(X\) by \(i_P(A)(U) = A\) if \(P \in U\) and \(0\) otherwise.
Verify that the stalk of \(i_P(A)\) is \(A\) at every point \(Q \in \cl\qty{\ts{P}}\) and \(0\) elsewhere, where \(\cl\qty{\ts{P}}\) denotes the closure of the set consisting of the point \(P\). Hence the name “skyscraper sheaf”.
Show that this sheaf can also be described as \(i_*(A)\), where \(A\) denotes the constant sheaf \(A\) on the closed subspace \(\cl\qty{\ts{P}}\) and \(i: \cl\qty{\ts{P}} \to X\) is the inclusion.
Solution
Let \[ \mathcal S=i_P(A). \] Thus \[ \mathcal S(U)= \begin{cases} A,&P\in U,\\ 0,&P\notin U, \end{cases} \] with the evident restriction maps: identity when both opens contain \(P\), and the zero map whenever the smaller open does not contain \(P\).
1. The presheaf \(\mathcal S\) is a sheaf.
Let \[ U=\bigcup_\alpha U_\alpha \] be an open cover.
If \(P\notin U\), then \(P\notin U_\alpha\) for every \(\alpha\), so every group of sections involved is zero and the sheaf axiom is trivial.
Suppose \(P\in U\). Choose an index \(\alpha_0\) with \[ P\in U_{\alpha_0}. \] A compatible family of sections has a component \[ a_{\alpha_0}\in A. \] For every other \(\alpha\) containing \(P\), the overlap \[ U_\alpha\cap U_{\alpha_0} \] contains \(P\), so compatibility forces \[ a_\alpha=a_{\alpha_0}. \] For indices not containing \(P\), the only section is zero. Hence the family glues uniquely to the element \[ a_{\alpha_0}\in\mathcal S(U)=A. \] Thus \(\mathcal S\) is a sheaf.
2. If\[ Q\in\overline{\{P\}}, \]then every open neighborhood of \(Q\) contains \(P\).
The condition \[ Q\in\overline{\{P\}} \] means exactly that every open neighborhood of \(Q\) meets the set \(\{P\}\). Meeting that singleton means containing \(P\).
3. For every\[ Q\in\overline{\{P\}}, \]one has\[ \boxed{\mathcal S_Q\cong A.} \]
By 2, every neighborhood \(U\) of \(Q\) contains \(P\), so \[ \mathcal S(U)=A. \] Every restriction map between such neighborhoods is the identity on \(A\). Therefore the stalk, which is the direct limit over all neighborhoods of \(Q\), is \[ \mathcal S_Q =\varinjlim_{Q\in U}A \cong A. \]
4. If\[ Q\notin\overline{\{P\}}, \]then\[ \boxed{\mathcal S_Q=0.} \]
Choose an open neighborhood \[ Q\in V \] with \[ P\notin V. \] Then \[ \mathcal S(V)=0. \]
Any germ in \(\mathcal S_Q\) is represented by a section over some neighborhood \(U\) of \(Q\). Restrict that representative to \[ U\cap V, \] which is still a neighborhood of \(Q\) but does not contain \(P\). The restriction lands in \[ \mathcal S(U\cap V)=0, \] so the germ is zero. Hence the entire stalk is zero.
5. Thus the stalks are\[ \boxed{ (i_P(A))_Q \cong \begin{cases} A,&Q\in\overline{\{P\}},\\ 0,&Q\notin\overline{\{P\}}. \end{cases} } \]
Combine 3 and 4.
6. Put\[ Z=\overline{\{P\}} \]with its subspace topology. Then \(P\) belongs to every nonempty open subset of \(Z\).
Let \(W\subseteq Z\) be nonempty and open, and choose \(Q\in W\). Write \[ W=V\cap Z \] for some open \(V\subseteq X\) containing \(Q\). Since \[ Q\in Z=\overline{\{P\}}, \] step 2 says every neighborhood of \(Q\) in \(X\), in particular \(V\), contains \(P\). Since \(P\in Z\) as well, \[ P\in V\cap Z=W. \]
7. Let\[ i:Z\hookrightarrow X \]be the inclusion and let \(\underline A_Z\) be the constant sheaf with value \(A\) on \(Z\). Then\[ \boxed{i_*\underline A_Z\cong i_P(A).} \]
For an open \(U\subseteq X\), \[ (i_*\underline A_Z)(U) =\underline A_Z(U\cap Z). \]
If \(P\in U\), then \[ U\cap Z \] is a nonempty open subset of \(Z\). By 6 it contains the generic point \(P\), and \(Z\) is irreducible because it is the closure of one point. Hence the constant sheaf has \[ \underline A_Z(U\cap Z)=A. \]
If \(P\notin U\), then \[ U\cap Z=\varnothing. \] Indeed, if \(Q\in U\cap Z\), then \(Q\in\overline{\{P\}}\), so 2 would force the neighborhood \(U\) of \(Q\) to contain \(P\), contradiction. Hence \[ \underline A_Z(U\cap Z)=0. \]
Therefore \[ (i_*\underline A_Z)(U) = \begin{cases} A,&P\in U,\\ 0,&P\notin U, \end{cases} \] with exactly the same restriction maps as \(i_P(A)\). The two sheaves are canonically isomorphic.
8. Q.E.D.
Step 5 gives the stalk calculation, and 7 gives the pushforward description.