Given a presheaf \(\mcf\) on \(X\), define a topological space \(\operatorname{Spe}(\mcf)\), called the espace étalé of \(\mcf\), as follows. As a set, \(\operatorname{Spe}(\mcf) = \Union_{P \in X} \mcf_P\). Define a projection map \(\pi: \operatorname{Spe}(\mcf) \to X\) by sending \(s \in \mcf_P\) to \(P\). For each open set \(U \subseteq X\) and each section \(s \in \mcf(U)\) we obtain a map \(\bar{s}: U \to \operatorname{Spe}(\mcf)\) sending \(P \mapsto s_P\), the germ of \(s\) at \(P\). This map satisfies \(\pi \circ \bar{s} = \id\), so it is a section of \(\pi\) over \(U\). Give \(\operatorname{Spe}(\mcf)\) the strongest topology such that all the maps \(\bar{s}\), for all \(U\) and all \(s \in \mcf(U)\), are continuous.
Show that the sheaf \(\mcf^{+}\) associated to \(\mcf\) can be described as follows: for any open \(U \subseteq X\), the group \(\mcf^{+}(U)\) is the set of continuous sections of \(\operatorname{Spe}(\mcf)\) over \(U\).
In particular, the original presheaf \(\mcf\) was a sheaf if and only if for each \(U\) the group \(\mcf(U)\) equals the set of all continuous sections of \(\operatorname{Spe}(\mcf)\) over \(U\).
This exercise connects Hartshorne’s definition of a sheaf with the espace étalé definition used elsewhere in the literature, for example in Godement.
Solution
Write \[ E=\operatorname{Spe}(\mcf)=\coprod_{P\in X}\mcf_P \] and let \[ \pi:E\longrightarrow X \] be the projection. For \(s\in\mcf(U)\) write \[ \bar s:U\longrightarrow E, \qquad P\longmapsto s_P. \]
1. A subset \(W\subseteq E\) is open in the specified topology if and only if\[ \bar t^{-1}(W) \]is open for every open \(V\subseteq X\) and every \(t\in\mcf(V)\).
This is exactly the definition of the strongest, or final, topology on \(E\) for which every map \[ \bar t:V\to E \] is continuous.
2. For every \(s\in\mcf(U)\), the image\[ \bar s(U)\subseteq E \]is open.
By 1, it is enough to show that for every \(t\in\mcf(V)\) the subset \[ \bar t^{-1}(\bar s(U)) \] is open in \(V\).
A point \(P\in V\) belongs to this inverse image exactly when \[ P\in U\cap V \qquad\text{and}\qquad t_P=s_P \] as germs in \(\mcf_P\).
If \(t_P=s_P\), then by definition of equality in the stalk there is an open neighborhood \[ P\in W_P\subseteq U\cap V \] such that \[ t|_{W_P}=s|_{W_P}. \] For every \(Q\in W_P\), therefore, \[ t_Q=s_Q, \] so \[ W_P\subseteq\bar t^{-1}(\bar s(U)). \] Thus every point of the inverse image has an open neighborhood contained in it, proving that the inverse image is open.
3. The projection\[ \pi:E\longrightarrow X \]is continuous, and for every \(s\in\mcf(U)\) the map\[ \bar s:U\longrightarrow\bar s(U) \]is a homeomorphism with inverse \(\pi|_{\bar s(U)}\).
Let \(U\subseteq X\) be open. For every \(t\in\mcf(V)\), \[ \bar t^{-1}(\pi^{-1}(U)) =V\cap U, \] which is open. By 1, \(\pi^{-1}(U)\) is open, so \(\pi\) is continuous.
2. Hence\[ \bar s:U\to\bar s(U) \]is a continuous bijection whose inverse is the continuous restriction of \(\pi\). It is therefore a homeomorphism.
4. Let \(\sigma:U\to E\) satisfy\[ \pi\circ\sigma=\operatorname{id}_U. \]If \(\sigma\) is continuous, then every point \(P\in U\) has a neighborhood \(W\subseteq U\) and a section \(s\in\mcf(V)\) on an open \(V\subseteq X\) such that\[ W\subseteq V \qquad\text{and}\qquad \sigma|_W=\bar s|_W. \]
Fix \(P\in U\). The point \[ \sigma(P)\in E_P=\mcf_P \] is a germ, so choose an open neighborhood \(V\) of \(P\) and a section \[ s\in\mcf(V) \] representing that germ. Thus \[ \sigma(P)=s_P\in\bar s(V). \]
By 2, \(\bar s(V)\) is an open neighborhood of \(\sigma(P)\) in \(E\). Since \(\sigma\) is continuous, \[ W:=\sigma^{-1}(\bar s(V)) \] is an open neighborhood of \(P\) in \(U\).
For \(Q\in W\), the point \(\sigma(Q)\) belongs to \(\bar s(V)\). Because \(\sigma\) is a section of \(\pi\), \[ \pi(\sigma(Q))=Q. \] The unique point of \(\bar s(V)\) lying over \(Q\) is \(s_Q\). Therefore \[ \sigma(Q)=s_Q. \] In particular \(Q\in V\), so \(W\subseteq V\), and \[ \sigma|_W=\bar s|_W. \]
5. Conversely, any section \(\sigma:U\to E\) which is locally of the form \(\bar s\) for sections of the presheaf \(\mcf\) is continuous.
Suppose \(U\) has an open cover \(\{U_\alpha\}\) such that \[ \sigma|_{U_\alpha}=\bar s_\alpha|_{U_\alpha} \] for suitable presheaf sections \(s_\alpha\). Each \(\bar s_\alpha\) is continuous by the definition of the topology on \(E\). Hence every restriction \[ \sigma|_{U_\alpha} \] is continuous.
Continuity is local on the source: for any open \(W\subseteq E\), \[ \sigma^{-1}(W) =\bigcup_\alpha (\sigma|_{U_\alpha})^{-1}(W), \] which is open in \(U\). Thus \(\sigma\) is continuous.
6. Therefore the continuous sections of \(\pi\) over \(U\) are exactly the functions\[ \sigma:U\longrightarrow\coprod_{P\in U}\mcf_P \]such that\[ \sigma(P)\in\mcf_P \]and \(\sigma\) is locally represented by a section of the presheaf \(\mcf\).
The forward implication is 4, and the reverse implication is 5.
7. The assignment\[ U\longmapsto \{\text{continuous sections }U\to E\text{ of }\pi\} \]is a sheaf.
Uniqueness is pointwise: two sections agreeing on an open cover have the same value at every point.
For gluing, let continuous sections \[ \sigma_\alpha:U_\alpha\to E \] agree on overlaps. Define \[ \sigma(P)=\sigma_\alpha(P) \] for any \(\alpha\) with \(P\in U_\alpha\). Agreement on overlaps makes this well-defined, and clearly \[ \pi\circ\sigma=\operatorname{id}. \] The restriction of \(\sigma\) to each \(U_\alpha\) is the continuous map \(\sigma_\alpha\), so <1>5’s locality argument shows that \(\sigma\) is continuous.
8. The canonical map\[ \mcf(U)\longrightarrow \{\text{continuous sections of }E\text{ over }U\}, \qquad s\longmapsto\bar s, \]is the sheafification map, and the target is naturally \(\mcf^+(U)\).
By 6, the target consists exactly of locally representable choices of germs. This is the standard germ description of the associated sheaf.
For completeness, it has the sheafification universal property. Let \[ \phi:\mcf\longrightarrow\mcg \] be a presheaf morphism to a sheaf \(\mcg\). Given a continuous germ section \(\sigma\) over \(U\), choose an open cover \(U=\bigcup U_\alpha\) and sections \[ s_\alpha\in\mcf(V_\alpha) \] such that \[ \sigma|_{U_\alpha}=\bar s_\alpha|_{U_\alpha}. \]
On an overlap, the germs of \(s_\alpha\) and \(s_\beta\) agree at every point. Hence they agree locally near every point of the overlap. Applying \(\phi\), the sections \[ \phi(s_\alpha),\quad\phi(s_\beta) \] agree locally; since \(\mcg\) is a sheaf, they agree on the whole overlap. They therefore glue uniquely to a section of \(\mcg(U)\).
This construction gives a unique sheaf morphism from the continuous-section sheaf to \(\mcg\) extending \(\phi\). Thus the continuous-section sheaf satisfies the universal property of \(\mcf^+\) and is canonically isomorphic to it.
9. Hence\[ \boxed{ \mcf^+(U) = \{\sigma:U\to\operatorname{Spe}(\mcf) \mid \pi\circ\sigma=\operatorname{id}_U, \ \sigma\text{ continuous}\}. } \]
This is the identification established in 8.
10. In particular, \(\mcf\) is already a sheaf if and only if every continuous section of the espace étalé over every open \(U\) is of the form \(\bar s\) for a unique\[ s\in\mcf(U). \]
A presheaf is a sheaf exactly when its canonical map to its sheafification is an isomorphism. Under 9, this canonical map is \[ s\longmapsto\bar s. \] Therefore it is an isomorphism on every open set exactly under the stated condition.
11. Q.E.D.
Steps <1>4–<1>6 identify continuity with local representability by germs, <1>8–<1>9 identify those sections with the associated sheaf, and 10 gives the final characterization of when \(\mcf\) was already a sheaf.