Let \(X\) be a noetherian scheme.
Show that the sheaf \(\mcg\) constructed in the proof of (3.6) is an injective object in the category \(\QCoh(X)\) of quasi-coherent sheaves on \(X\). Thus \(\QCoh(X)\) has enough injectives.
Show that any injective object of \(\QCoh(X)\) is flasque.
Conclude that one can compute cohomology as the derived functors of \(\Gamma(X,\wait)\), considered as a functor from \(\QCoh(X)\) to \(\Ab\).
Hint
For (b), the method of proof of (2.4) will not work, because \(\mco_U\) is not quasi-coherent on \(X\) in general. Instead, use (II, Ex. 5.15) to show that if \(\mci \in \QCoh(X)\) is injective, and if \(U \subseteq X\) is an open subset, then \(\ro{\mci}{U}\) is an injective object of \(\QCoh(U)\). Then cover \(X\) with open affines.
Solution
The category \(\QCoh(X)\) is a full abelian subcategory of \(\OO_X\)-modules, with kernels and cokernels computed as sheaves [@Har10a, Proposition II.5.7]. Choose a finite affine open cover \(X=\bigcup_{a=1}^r U_a\), where \(U_a=\Spec A_a\), and let \(j_a:U_a\hookrightarrow X\) be the inclusions. For a quasi-coherent \(F\), write \(F|_{U_a}=\widetilde{M_a}\) and choose an embedding \(M_a\hookrightarrow J_a\) into an injective \(A_a\)-module. The sheaf used in the affine-cover construction of [@Har10a, Corollary III.3.6] is \[ G=\bigoplus_{a=1}^r j_{a*}\widetilde{J_a}. \]
1. Each \(j_{a*}\widetilde{J_a}\) is a quasi-coherent sheaf injective in \(\QCoh(X)\).
An open immersion into a noetherian scheme is quasi-compact and separated. Its direct image preserves quasi-coherence [@Har10a, Proposition II.5.8(c)], as used in P-AGH2515EXTCOH. Thus \(j_{a*}\widetilde{J_a}\) is quasi-coherent. The affine equivalence \(\operatorname{Mod}(A_a)\simeq\QCoh(U_a)\) is exact, so the module injectivity of \(J_a\) makes \(\widetilde{J_a}\) injective in \(\QCoh(U_a)\).
For \(E\in\QCoh(X)\), restriction and direct image give the adjunction \[ \Hom_X(E,j_{a*}\widetilde{J_a}) \cong\Hom_{U_a}(E|_{U_a},\widetilde{J_a}). \] Given a monomorphism \(E\hookrightarrow E'\) in \(\QCoh(X)\), restriction is exact and gives a monomorphism on \(U_a\). A morphism from \(E|_{U_a}\) to \(\widetilde{J_a}\) therefore extends to \(E'|_{U_a}\) by injectivity on that affine open. Adjunction turns this extension into an extension from \(E'\) to \(j_{a*}\widetilde{J_a}\). This is injectivity in \(\QCoh(X)\).
2. The sheaf \(G\) is injective in \(\QCoh(X)\), and the construction gives a monomorphism \(F\hookrightarrow G\).
A finite direct sum is also a finite product. To extend a map into \(G\) across a monomorphism, extend each of its finitely many components using step 1 and assemble them. Hence \(G\) is injective.
The local embeddings \(F|_{U_a}\hookrightarrow\widetilde{J_a}\) induce maps \(F\to j_{a*}\widetilde{J_a}\) by adjunction. Combining them gives a morphism \(F\to G\). At any point \(x\in X\), choose \(a\) with \(x\in U_a\). The \(a\)th component on that stalk is the injective map \(F_x\to(\widetilde{J_a})_x\), because restriction of \(j_{a*}\) back to \(U_a\) is the identity. Thus the combined map is injective at every stalk and is a monomorphism. Every quasi-coherent sheaf therefore embeds into an injective object of \(\QCoh(X)\), proving (a).
3. The sheaf \(G\) is flasque, and every injective object of \(\QCoh(X)\) is flasque.
Since \(A_a\) is noetherian and \(J_a\) is an injective module, \(\widetilde{J_a}\) is flasque on \(U_a\) [@Har10a, Proposition III.3.4]. For open sets \(W\subseteq V\subseteq X\), the restriction map of its direct image is \[ \widetilde{J_a}(V\cap U_a)\longrightarrow\widetilde{J_a}(W\cap U_a), \] which is surjective by flasqueness on \(U_a\). Thus each summand of \(G\) is flasque, and so is the finite direct sum.
Now let \(I\) be injective in \(\QCoh(X)\). Step 2, applied to \(F=I\), gives a monomorphism \(e:I\hookrightarrow G\) with \(G\) flasque and quasi-coherent. Injectivity of \(I\) extends \(\id_I\) along \(e\) to a morphism \(p:G\to I\) satisfying \(pe=\id_I\). For any \(W\subseteq V\) and \(s\in I(W)\), extend \(e(s)\) to \(t\in G(V)\) by flasqueness, then apply \(p\). The section \(p(t)\) restricts to \(pe(s)=s\). Hence every restriction map on \(I\) is surjective, proving (b).
4. For every \(F\in\QCoh(X)\) and \(i\ge0\), the derived functor computed in \(\QCoh(X)\) agrees naturally with sheaf cohomology:\[ R^i\bigl(\Gamma(X,-)|_{\QCoh(X)}\bigr)(F)\cong H^i(X,F). \]
By step 2, choose an injective resolution \(F\to I^\bullet\) in \(\QCoh(X)\). The exact inclusion of \(\QCoh(X)\) into all module sheaves makes this an exact resolution of the underlying abelian sheaf. Step 3 makes its terms flasque. They are consequently acyclic for ordinary global sections, so this same resolution computes ordinary sheaf cohomology [@Har10a, Proposition III.2.5 and Remark III.2.5.1]. On the other hand, its global-section complex defines the derived functor in \(\QCoh(X)\). The two computations are the identical complex \(\Gamma(X,I^\bullet)\), giving the asserted isomorphism in all degrees. Comparison of resolutions gives naturality in \(F\).
5. Q.E.D.
Steps <1>1–<1>2 prove (a), step 3 proves (b), and step 4 proves (c).