Let \(X\) be a noetherian topological space, and let \(\ts{\mci_\alpha}_{\alpha \in A}\) be a direct system of injective sheaves of abelian groups on \(X\). Then \(\colim_\alpha \mci_\alpha\) is also injective.
Hint
First show that a sheaf \(\mci\) is injective if and only if for every open set \(U \subseteq X\), and for every subsheaf \(\mcr \subseteq \ZZ_U\), and for every map \(f: \mcr \to \mci\), there exists an extension of \(f\) to a map \(\ZZ_U \to \mci\). Secondly, show that any such sheaf \(\mcr\) is finitely generated, so any map \(\mcr \to \colim_\alpha \mci_\alpha\) factors through one of the \(\mci_\alpha\).
Solution
For an open inclusion \(j_U:U\hookrightarrow X\), put \(\ZZ_U=j_{U!}\underline{\ZZ}\), the extension by zero of the constant sheaf on \(U\). The extension-by-zero adjunction gives \[ \Hom_X(\ZZ_U,A)\cong\Gamma(U,A) \] for every abelian sheaf \(A\). Here a finite family of local generators means a surjection \(\bigoplus_{i=1}^t\ZZ_{U_i}\twoheadrightarrow A\); it need not consist of global sections on \(X\). The direct system in the statement is indexed by a directed set.
1. An abelian sheaf \(I\) is injective if and only if every map \(R\to I\) from a subsheaf \(R\subseteq\ZZ_U\) extends to \(\ZZ_U\), for every open \(U\).
Necessity follows from injectivity. For sufficiency, let \(A\subseteq B\) be a monomorphism of sheaves and let \(f:A\to I\) be given. Order pairs \((C,f_C)\), with \(A\subseteq C\subseteq B\) and \(f_C\) extending \(f\), by extension. A chain has an upper bound: take the subsheaf union and glue its compatible maps to \(I\). Zorn’s lemma gives a maximal pair.
If \(C\ne B\), some local section \(b\in B(U)\) has a germ not in \(C\). It defines \(t:\ZZ_U\to B\) by the displayed adjunction. Let \(R=t^{-1}(C)\) and extend the map \(f_C\circ t|_R\) to a map \(h:\ZZ_U\to I\) by the assumed property. The map \[ C\oplus\ZZ_U\longrightarrow I,\qquad(c,z)\longmapsto f_C(c)+h(z) \] annihilates the kernel of \((c,z)\mapsto c+t(z)\): that kernel consists, stalkwise, of \((-t(z),z)\) with \(z\in R\), on which the two contributions cancel. It therefore descends to \(C+\im t\subseteq B\) and extends \(f_C\). This subsheaf strictly contains \(C\) by the choice of \(b\), contradicting maximality. Hence \(C=B\), proving the required extension for every monomorphism and thus injectivity.
2. Every subsheaf \(R\subseteq\underline{\ZZ}_X\) is generated by finitely many local sections.
Let \(V=\{x:R_x\ne0\}\). This is open: a nonzero germ is represented locally by a nonzero constant integer, which remains nonzero on a smaller open neighborhood. If \(V=\varnothing\), then \(R=0\) and the assertion is immediate. Otherwise, quasi-compactness of the open \(V\) gives finitely many opens \(W_i\) covering \(V\) and nonzero constant integers \(a_i\in R(W_i)\). Let \(N=\prod_i\abs{a_i}>0\). On each \(W_i\), the constant \(N\) is an integer multiple of \(a_i\) and belongs to \(R\). These sections agree in the constant sheaf and glue, so \(N\in R(V)\).
For each integer \(0\le a<N\), put \[ V_a=\{x\in V:a\in R_x\subseteq\ZZ\}. \] This is open, because membership is witnessed by a local section equal to the constant \(a\) on a neighborhood. The same local sections glue to \(a\in R(V_a)\). The section \(N\) on \(V\), together with these finitely many sections \(a\) on \(V_a\), generates every stalk of \(R\). Indeed, for \(x\in V\), write any integer \(b\in R_x\) as \(b=qN+a\) with \(0\le a<N\). Then \(a\in R_x\), so \(x\in V_a\) and both summands are generated by the listed sections. Outside \(V\) all stalks are zero. The adjunction preceding step 1 gives the required finite generating morphisms from the corresponding \(\ZZ_V\) and \(\ZZ_{V_a}\).
3. Every subsheaf of a finite direct sum of sheaves \(\ZZ_{U_i}\) is generated by finitely many local sections.
2.For larger \(r\), project onto the last summand and let \(Q\subseteq\underline{\ZZ}_X\) be the image sheaf.Its finite local generators exist by step
2.Each such section on an open \(V\) lifts locally to \(R\), since \(R\to Q\) is a sheaf surjection.As \(V\) is quasi-compact, finitely many of these local lifts cover its domain.The kernel is a subsheaf of \(\underline{\ZZ}_X^{\oplus(r-1)}\) and has finitely many local generators by induction.Those kernel generators and the finitely many lifted generators together generate \(R\), as is checked at every stalk of the exact sequence \(0\to\ker\to R\to Q\to0\).
Finally, \(\ZZ_{U_i}\to\underline{\ZZ}_X\), induced by the constant section \(1\) on \(U_i\), is injective: its stalk is the identity on \(U_i\) and has zero source elsewhere. Thus a finite direct sum of the \(\ZZ_{U_i}\) embeds in \(\underline{\ZZ}_X^{\oplus r}\). Every subsheaf of that direct sum is covered by the result just proved. In particular, this applies both to \(R\subseteq\ZZ_U\) and to the relations of any finite generating presentation of \(R\).
4. For every open \(V\) and directed system of abelian sheaves \((A_\alpha)\), one has\[ \varinjlim_\alpha\Gamma(V,A_\alpha) \xrightarrow{\cong}\Gamma(V,\varinjlim_\alpha A_\alpha). \]
Every open subset of the noetherian space, including every finite intersection of opens, is quasi-compact. Hence every open covering has a finite subcover. The sectionwise direct-limit presheaf satisfies the gluing and uniqueness axioms for a finite cover: put finitely many representative sections at one common index, and then move to a further common index where all their finitely many overlap equalities hold. The sheaf axiom at that index glues the representatives. Uniqueness is proved in the same way, by moving to an index where the finitely many restrictions agree. The finite-cover axioms imply the full sheaf axiom, using a finite subcover and uniqueness on its intersections with the other members of a cover. Thus the sectionwise direct-limit presheaf is already a sheaf and is the sheaf colimit. This gives the stated formula [@Har10a, Exercise II.1.11].
5. If \(R\subseteq\ZZ_U\), every morphism \(R\to\varinjlim_\alpha A_\alpha\) factors through some \(A_\beta\).
By step 3 there is a surjection \[ P=\bigoplus_{i=1}^t\ZZ_{V_i}\twoheadrightarrow R. \] Its kernel \(K\) is a subsheaf of the finite direct sum \(P\), so another application of step 3 gives a surjection \(\bigoplus_{j=1}^s\ZZ_{W_j}\twoheadrightarrow K\). This supplies finitely many local generators and finitely many local relations.
Let \(f:R\to\varinjlim A_\alpha\) be given. The composite on each summand of \(P\) is a section over \(V_i\). By step 4 these finitely many sections are represented at one common index \(\alpha\), giving \(h:P\to A_\alpha\). The image under \(h\) of each chosen relation is a section on \(W_j\) whose image in the colimit is zero. Again by step 4, this section becomes zero at a sufficiently large index. Choose one index \(\beta\) at which all these finitely many relations vanish. The resulting map \(P\to A_\beta\) annihilates \(K\) and hence descends to \(R\to A_\beta\). Its composite with the colimit map is \(f\), because they agree after the surjection from \(P\). This proves factorization even when the transition maps are not monomorphisms.
6. Q.E.D.
Put \(I=\varinjlim_\alpha\mci_\alpha\) and take \(R\subseteq\ZZ_U\) with a map \(f:R\to I\). Step 5 factors \(f\) through one of the injective sheaves \(\mci_\beta\). Injectivity of that sheaf extends the factorization to \(\ZZ_U\to\mci_\beta\). Composing with \(\mci_\beta\to I\) gives an extension of \(f\) to \(\ZZ_U\). The criterion in step 1 therefore proves that \(I\) is injective.