Sheaves of modules, and the operations on them (Tag D-MODOX)
An \(\OO_X\)-module is a sheaf \(\mcf\) on \(X\) with each \(\mcf(U)\) an \(\OO_X(U)\)-module, compatibly with restriction: \(\ro{(rm)}{V} = \ro{r}{V}\ro{m}{V}\). The tensor product \(\mcf \tensor_{\OO_X} \mcg\) is the sheafification of \(U \mapsto \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\). The sheaf hom \(\sheafhom_{\OO_X}(\mcf, \mcg)\) is \(U \mapsto \Hom_{\ro{\OO_X}{U}}(\ro{\mcf}{U}, \ro{\mcg}{U})\). A sheaf of ideals is a subsheaf \(\mci \subseteq \OO_X\) of \(\OO_X\)-modules.
The presheaf \(U \mapsto \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\) need not be a sheaf: on \(X=\PP^1_k\), its value on \(X\) for \(\mcf=\OO(1)\) and \(\mcg=\OO(-1)\) is \(H^0(\OO(1))\tensor_kH^0(\OO(-1))=0\), while \(\OO(1)\tensor\OO(-1)\cong\OO_X\) has the global section \(1\). Morphisms of sheaves defined on the members of an open cover and agreeing on overlaps glue uniquely, so \(U\mapsto\Hom_{\ro{\OO_X}{U}}(\ro{\mcf}{U}, \ro{\mcg}{U})\) is already a sheaf.
The functor \(\wait\tensor_{\OO_X}\mcg\) is right exact, and \(\sheafhom_{\OO_X}(\mcf,\wait)\) is left exact. For every \(x\in X\), \((\mcf \tensor \mcg)_x \cong \mcf_x \tensor_{\OO_{X,x}} \mcg_x\). If \(X\) is Noetherian and \(\mcf\) is coherent, then \(\sheafhom(\mcf,\mcg)_x \cong \Hom_{\OO_{X,x}}(\mcf_x, \mcg_x)\) [@Har10a, Proposition III.6.8].
\(\mods{\OO_X}\) has enough injectives [@Har10a, Proposition III.2.2], but it need not have enough projectives. On \(\PP_k^1\) over an infinite field, no projective object surjects onto \(\OO_X\), as proved in P-AGH362NOPROJECTIVES. In contrast, on a one-point ringed space with structure ring \(A\), a sheaf of modules is just an \(A\)-module; free-module surjections show that this category has enough projectives.
Pullback and pushforward
Pullback and pushforward of sheaves of modules (Tag D-MODPULL)
Let \(f: X \to Y\) be a continuous map and \(\mcg\) a sheaf on \(Y\). The inverse image sheaf \(\inverseof{f} \mcg\) is the sheafification of the presheaf \[ U \mapsto \colim_{V \supseteq f(U)} \mcg(V) , \] the colimit over open subsets \(V \subseteq Y\) containing \(f(U)\). If \(i: Z \injects X\) is the inclusion of a subspace, the restriction of a sheaf \(\mcf\) on \(X\) is \(\ro{\mcf}{Z} \da \inverseof{i} \mcf\).
\(\inverseof{f}\) is left adjoint to \(f_*\) on sheaves of abelian groups, and for \(x \in X\) the stalk is \((\inverseof{f} \mcg)_x = \mcg_{f(x)}\); in particular \((\ro{\mcf}{Z})_x = \mcf_x\) for \(x \in Z\). If \(\mcg\) is an \(\OO_Y\)-module, then \(\inverseof{f} \mcg\) is an \(\inverseof{f} \OO_Y\)-module, which need not be an \(\OO_X\)-module. [@Har10a, §II.1, Exercise II.1.18]
(Direct and inverse image)
For \(f: X \to Y\) of ringed spaces, \(f_*\mcf\) is the \(\OO_Y\)-module \(U \mapsto \mcf(\inverseof{f} U)\). For \(\mcg \in \mods{\OO_Y}\), the inverse image is \[ f^*\mcg \da \inverseof{f} \mcg \tensor_{\inverseof{f} \OO_Y} \OO_X , \] the sheaf-theoretic inverse image, base changed along \(\inverseof{f}\OO_Y \to \OO_X\).
\(f^*\) is left adjoint to \(f_*\): there is a natural isomorphism \[ \Hom_{\OO_X}(f^*\mcg, \mcf) \cong \Hom_{\OO_Y}(\mcg, f_*\mcf) . \] \(f^*\) preserves quasicoherence always, and coherence when \(f\) is a morphism of Noetherian schemes; \(f_*\) preserves quasicoherence for \(f\) quasicompact and separated.
For a morphism \(\Spec B\to\Spec A\) of affine schemes, \(f^*\tilde M\cong\widetilde{M\tensor_AB}\), so \(f^*\) is right exact on quasicoherent sheaves. The functor \(f_*\) is left exact, and its right derived functors are the higher direct images \(R^i f_*\). For \(f: \AA^1_k \to \Spec k\), \(f_*\OO_{\AA^1}\) corresponds to the \(k\)-module \(k[t]\), which is not finitely generated, so \(f_*\) does not preserve coherence. If \(Y\) is Noetherian, \(f\) is projective, and \(\mcf\) is coherent, then every \(R^i f_*\mcf\) is coherent [@Har10a, Theorem III.8.8]; the same holds for \(f\) proper.
On line bundles \(f^*\) is a group homomorphism \(\Pic(Y) \to \Pic(X)\). A very ample line bundle is obtained by pulling back \(\OO(1)\) along an immersion into projective space.
For a morphism of ringed spaces \(f\colon X\to Y\), the module pullback \(f^*\mcg=f^{-1}\mcg\otimes_{f^{-1}\OO_Y}\OO_X\) is left adjoint to \(f_*\). Hence \(f^*\) is right exact and \(f_*\) is left exact; the higher direct images \(R^if_*\) are the right derived functors of \(f_*\).
The inverse image \(f^{-1}\) of abelian sheaves is exact, but the tensor product in \(f^*\) need not preserve kernels. For the closed point \(f\colon\Spec\FF_p\to\Spec\ZZ\), the injection \(\ZZ\xrightarrow{p}\ZZ\) pulls back to the zero map \(\FF_p\to\FF_p\). If \(f\) is flat, \(f^*\) is exact.
On schemes, \(f^*\) preserves quasicoherent sheaves, sheaves of finite type, and coherent sheaves when \(X\) and \(Y\) are locally noetherian. The pushforward \(f_*\) preserves quasicoherent sheaves when \(f\) is quasicompact and quasiseparated, and preserves coherent sheaves when \(f\) is proper and \(Y\) is locally noetherian.
The four functors for abelian sheaves
For \(f \colon X \to Y\) continuous:
\(f_* \colon \Sh(X) \to \Sh(Y)\), \((f_*\mcf)(V) = \mcf(f^{-1}V)\), the direct image or pushforward, left exact;
\(f^{-1} \colon \Sh(Y) \to \Sh(X)\), the inverse image, left adjoint to \(f_*\) and exact;
\(f_!\), the direct image with proper supports, defined for sheaves of abelian groups by taking the sections of \(f_*\mcf\) whose support is proper over \(Y\). For an open immersion \(j\), \(j_!\) is extension by zero, which is exact and left adjoint to \(j^{-1}\); for a closed immersion \(i\), \(i_!=i_*\);
\(f^!\), the exceptional inverse image. For a closed immersion \(i\colon Z\to X\), \(i^!\mcf\) is the restriction to \(Z\) of the sheaf of sections of \(\mcf\) supported in \(Z\), right adjoint to \(i_*\) on abelian sheaves. For a general \(f\) it exists only on derived categories, as a right adjoint to \(Rf_!\) when \(X\) and \(Y\) are locally compact Hausdorff and \(f_!\) has finite cohomological dimension (Verdier duality).
\(f_!\) and \(f^!\) use the zero section of an abelian sheaf, so they do not transfer to sheaves of sets. Together with \(\otimes\) and internal \(\mathcal{H}om\) these are the six operations. For \(\OO_X\)-modules the pullback is written \(f^*\).
Ideal sheaves
The ideal sheaf of a closed subscheme (Tag D-MODIDEAL)
For a closed immersion \(i: Z \injects X\), the ideal sheaf is \[ \mci_Z \da \ker\qty{\OO_X \mapsvia{i^\sharp} i_*\OO_Z} . \]
\(Z \mapsto \mci_Z\) is a bijection between closed subschemes of \(X\) and quasicoherent sheaves of ideals on \(X\), and there is a short exact sequence \[ 0 \to \mci_Z \to \OO_X \to i_*\OO_Z \to 0 . \]
On \(X=\Spec A\), the bijection sends \(V(I)\) to \(\tilde I\) for ideals \(I\subseteq A\). A sheaf of ideals that is not quasicoherent is the ideal sheaf of no closed subscheme: for \(X=\AA^1_k\), \(U=X\sm\ts{0}\), and \(j\colon U\injects X\) the inclusion, the extension by zero \(j_!\OO_U\subseteq\OO_X\) is a sheaf of ideals with \(\Gamma(X,j_!\OO_U)=0\) and \(j_!\OO_U|_U=\OO_U\), so it is not quasicoherent.
For a hypersurface \(Z\subseteq\PP^n\) of degree \(d\), \(\mci_Z \cong \OO_{\PP^n}(-d)\), and for \(n\ge2\) the twisted sequence \(0\to\OO(m-d)\to\OO(m)\to\OO_Z(m)\to0\) gives \(h^0(\OO_Z(m))=h^0(\OO_{\PP^n}(m))-h^0(\OO_{\PP^n}(m-d))\), since \(H^1(\PP^n,\OO(m-d))=0\). The restriction \(i^*\mci_Z=\mci_Z/\mci_Z^2\) is the conormal sheaf of \(Z\) (D-MODCONORM).
For a closed subscheme \(i\colon Z\hookrightarrow X\) with ideal sheaf \(\mci_Z\), \(H^j(Z,\OO_Z)\cong H^j(X,i_*\OO_Z)\), and the long exact sequence of \(0\to\mci_Z\to\OO_X\to i_*\OO_Z\to0\) relates \(H^j(Z,\OO_Z)\) to \(H^j(X,\OO_X)\) and \(H^{j+1}(X,\mci_Z)\).
What separates the classes
Separating quasicoherent, coherent, and locally free (Tag FE-MODZOO)
On \(X = \Spec A\) with \(Z \subseteq X\) closed and \(j: U = X \setminus Z \injects X\) open, the sheaf \(j_!\OO_U\) extended by zero is an \(\OO_X\)-module that is not quasicoherent: its sections are not a localization at any point of \(Z\). The constant sheaf \(\ul{\ZZ}\) on a positive-dimensional \(X\) is another, for the same reason — restriction maps do not localize.
(Quasicoherent, not coherent)
\(\OO_{\AA^1}\) pushed to a point: \(f: \AA^1_k \to \Spec k\) has \(f_*\OO_{\AA^1} = \widetilde{k[t]}\), quasicoherent but not finitely generated over \(k\). On any integral \(X\), the constant sheaf of rational functions \(\mck_X\) with \(U \mapsto K(X)\) is quasicoherent and not coherent.
(Coherent, not locally free)
On \(X = \Spec k[t]\), the skyscraper \(\widetilde{k[t]/(t)}\) at the origin is coherent, with fibre dimension \(1\) at the origin and \(0\) elsewhere. Any torsion module gives such an example, and the non-constant fibre dimension is exactly the obstruction. On a singular curve the maximal ideal sheaf at the singular point is coherent, torsion-free, and still not locally free.
The three examples show that each implication in the hierarchy is strict. They separate the failure mechanisms: quasicoherence can fail through incompatibility with localization, coherence through failure of finite generation, and local freeness through varying fibre dimension. Over a reduced Noetherian scheme, a coherent sheaf is locally free exactly when \(x \mapsto \dim_{\kappa(x)} \mcf \tensor \kappa(x)\) is locally constant.