Let \(\ts{\mcf_i}\) be a direct system of sheaves on a noetherian topological space \(X\). Show that the presheaf \(U \mapsto \varinjlim \mcf_i(U)\) is already a sheaf. In particular, \[ \Gamma\qty{X, \varinjlim \mcf_i} = \varinjlim \Gamma(X, \mcf_i). \]
Solution
Let \[ \mcp(U)=\varinjlim_i\mcf_i(U) \] be the sectionwise direct-limit presheaf. We assume, as usual for a direct system, that the index set is directed.
1. Every open subset of a noetherian topological space is quasicompact.
Let \(U\subseteq X\) be open and suppose \[ U=\bigcup_{\alpha\in A}U_\alpha \] is an open cover with no finite subcover.
Choose \(\alpha_1\). Since \(U_{\alpha_1}\) does not cover \(U\), choose \(\alpha_2\) such that \[ U_{\alpha_2}\not\subseteq U_{\alpha_1}. \] Inductively, because no finite union covers \(U\), choose \(\alpha_{n+1}\) so that \[ U_{\alpha_{n+1}} \not\subseteq U_{\alpha_1}\cup\cdots\cup U_{\alpha_n}. \] Then the open subsets \[ U_{\alpha_1} \subsetneq U_{\alpha_1}\cup U_{\alpha_2} \subsetneq\cdots \] form a strictly ascending chain. A noetherian topological space satisfies the ascending-chain condition on open subsets, contradiction. Thus a finite subcover exists.
2. The presheaf \(\mcp\) satisfies uniqueness for a finite open cover.
Let \[ U=U_1\cup\cdots\cup U_r \] and suppose \[ s,t\in\mcp(U) \] have equal restrictions to every \(U_a\).
Because the index system is directed, after passing to a common stage we may represent both \(s\) and \(t\) by sections \[ s_i,t_i\in\mcf_i(U) \] for one index \(i\).
For each \(a\), equality \[ s|_{U_a}=t|_{U_a} \] in the direct limit means that there is some index \(j_a\ge i\) at which the images of \[ s_i|_{U_a},\quad t_i|_{U_a} \] become equal. Since there are only finitely many \(a\), directedness supplies one index \(j\) dominating all \(j_a\).
Thus the images \[ s_j,t_j\in\mcf_j(U) \] have equal restrictions on every member of the finite cover. Since \(\mcf_j\) is a sheaf, \[ s_j=t_j. \] Hence \(s=t\) in \(\mcp(U)\).
3. The presheaf \(\mcp\) satisfies gluing for a finite open cover.
Let \[ U=U_1\cup\cdots\cup U_r \] and let \[ s_a\in\mcp(U_a) \] be sections whose restrictions agree on every overlap \(U_a\cap U_b\).
Because there are finitely many sections, directedness lets us choose one index \(i\) and representatives \[ s_{a,i}\in\mcf_i(U_a) \] for every \(a\).
For every pair \((a,b)\), compatibility in the direct limit means that at some later index \(j_{ab}\ge i\) the two restrictions \[ s_{a,i}|_{U_a\cap U_b}, \qquad s_{b,i}|_{U_a\cap U_b} \] become equal. There are only finitely many pairs, so choose one index \(j\) dominating all \(j_{ab}\).
The resulting sections \[ s_{a,j}\in\mcf_j(U_a) \] are genuinely compatible on every overlap. Since \(\mcf_j\) is a sheaf, there is a unique \[ s_j\in\mcf_j(U) \] restricting to all \(s_{a,j}\). Its image \[ s\in\mcp(U) \] restricts to the original sections \(s_a\).
4. The presheaf \(\mcp\) is a sheaf for arbitrary open covers.
Let \[ U=\bigcup_{\alpha\in A}U_\alpha \] be any open cover. By 1, choose a finite subcover \[ U=U_{\alpha_1}\cup\cdots\cup U_{\alpha_r}. \]
Uniqueness for the original cover follows immediately from uniqueness for this finite subcover, proved in 2.
Now suppose a compatible family \[ s_\alpha\in\mcp(U_\alpha) \] is given. By 3, the finite subfamily indexed by \(\alpha_1,\ldots,\alpha_r\) glues to a section \[ s\in\mcp(U). \] For any other \(\alpha\), both \[ s|_{U_\alpha} \quad\text{and}\quad s_\alpha \] restrict to the same section on each open \[ U_\alpha\cap U_{\alpha_a}, \qquad 1\le a\le r, \] because the original family was compatible. These finitely many intersections cover \(U_\alpha\), so 2 gives \[ s|_{U_\alpha}=s_\alpha. \] Thus \(s\) glues the entire family.
Hence \(\mcp\) satisfies both sheaf axioms for every cover.
5. Therefore no sheafification is needed:\[ \boxed{ \varinjlim_i\mcf_i =\left(U\longmapsto\varinjlim_i\mcf_i(U)\right). } \]
Hartshorne II.1.10 defines the direct limit sheaf as the sheafification of \(\mcp\). By 4, \(\mcp\) is already a sheaf, so its sheafification map is an isomorphism.
6. In particular,\[ \boxed{ \Gamma\!\left(X,\varinjlim_i\mcf_i\right) =\varinjlim_i\Gamma(X,\mcf_i). } \]
Evaluate the equality of sheaves in 5 on the open set \(X\): \[ \left(\varinjlim_i\mcf_i\right)(X) =\mcp(X) =\varinjlim_i\mcf_i(X). \] By definition, these are the two sides of the displayed identity.
7. Q.E.D.
Step 5 proves that the sectionwise direct limit is already a sheaf, and 6 gives the stated consequence for global sections.