We prove the following theorem in several steps: let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mcf\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mcf'\) on \(X\) such that \(\mcf'|_U\cong\mcf\).
On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mcf\) is the union of subsheaves \(\mcf_\alpha\) if for every open set \(U\) the group \(\mcf(U)\) is the union of the subgroups \(\mcf_\alpha(U)\).
Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mcf\) coherent on \(U\). Then there exists a coherent sheaf \(\mcf'\) on \(X\) with \(\mcf'|_U\cong\mcf\).
With \(X,U,\mcf\) as in (b), suppose further we are given a quasi-coherent sheaf \(\mcg\) on \(X\) such that \(\mcf\subseteq\mcg|_U\). Show that we can find \(\mcf'\) a coherent subsheaf of \(\mcg\) with \(\mcf'|_U\cong\mcf\).
Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mcf\) a coherent sheaf on \(U\), and \(\mcg\) a quasi-coherent sheaf on \(X\) with \(\mcf\subseteq\mcg|_U\). Show that there is a coherent subsheaf \(\mcf'\subseteq\mcg\) on \(X\) with \(\mcf'|_U\cong\mcf\). For the inclusion \(i:U\hookrightarrow X\), taking \(\mcg=i_*\mcf\) proves the announced extension theorem.
As an extra corollary, show that on a noetherian scheme any quasi-coherent sheaf \(\mcf\) is the union of its coherent subsheaves.
Hint
For part (b), let \(i:U\to X\) be the inclusion; show that \(i_*\mcf\) is quasi-coherent, then use (a). For part (c), use the same method with \(\rho^{-1}(i_*\mcf)\), where \(\rho:\mcg\to i_*(\mcg|_U)\) is the restriction morphism. For part (d), cover \(X\) with open affines and extend over one of them at a time. For part (e), given a section \(s\) on \(U\), apply (d) to the subsheaf of \(\mcf|_U\) generated by \(s\).
Solution
For an open inclusion \(i:U\hookrightarrow X\) of noetherian schemes, direct image takes quasi-coherent sheaves to quasi-coherent sheaves [@Har10a, Proposition II.5.8(c)]. Indeed, \(i\) is quasi-compact because every open subset of a noetherian space is quasi-compact, and it is separated. Kernels, cokernels, and images of morphisms of quasi-coherent sheaves are quasi-coherent [@Har10a, Proposition II.5.7].
1. Part (a) holds with the stated sectionwise meaning of union.
Write \(X=\Spec A\) with \(A\) noetherian and \(\mcf\cong\widetilde M\) [@Har10a, Proposition II.5.4]. Every finitely generated submodule \(N\subseteq M\) gives a coherent subsheaf \(\widetilde N\subseteq\widetilde M\), since localization is exact.
Fix an open set \(W\subseteq X\) and a section \(s\in\widetilde M(W)\). Choose a finite cover \(W=\bigcup_{j=1}^aD(f_j)\) by distinguished opens; such a cover exists because \(W\) is quasi-compact. On each member write \[ s|_{D(f_j)}=m_j/f_j^{e_j},\qquad m_j\in M,\quad e_j\ge0. \] Let \(N\) be the submodule generated by the finitely many \(m_j\). These restricted sections lie in \(\widetilde N\) and agree on overlaps, because they agree in \(\widetilde M\) and \(\widetilde N\to\widetilde M\) is injective. They therefore glue to a section of \(\widetilde N(W)\) whose image is \(s\). For \(W=\varnothing\) its only section is already in the zero subsheaf. This proves the required equality of section groups on every open set. The family is directed: the sum of two finitely generated submodules is finitely generated and contains both.
2. If \(X\) is affine noetherian, \(\mathcal H\) is quasi-coherent on \(X\), and \(\mathcal H|_U\) is coherent, then some coherent subsheaf \(\mathcal H'\subseteq\mathcal H\) satisfies \(\mathcal H'|_U=\mathcal H|_U\).
Take a finite affine open cover \(U=\bigcup_{j=1}^aV_j\). On each \(V_j\), choose finitely many sections generating \(\mathcal H|_{V_j}\); coherence and the affine module description provide them. By step 1, each of these finitely many sections belongs to a coherent subsheaf of \(\mathcal H\) on \(X\). Take the sum \(\mathcal H'\) of those finitely many subsheaves. On the affine noetherian scheme \(X\), this is the associated sheaf of a sum of finitely many finite submodules, so it is coherent. Its restriction to each \(V_j\) contains a generating set of \(\mathcal H|_{V_j}\) and is a subsheaf of that sheaf, hence equals it. Thus \(\mathcal H'|_U=\mathcal H|_U\) as subsheaves. When \(U=\varnothing\), take \(\mathcal H'=0\).
3. Part (b) follows by applying step 2 to \(i_*\mcf\).
The sheaf \(\mathcal H=i_*\mcf\) is quasi-coherent by the direct-image result, and \(\mathcal H|_U\cong\mcf\) is coherent. Step 2 gives a coherent \(\mcf'\subseteq i_*\mcf\) whose restriction is \(\mcf\).
4. In part (c), the extension can be chosen as a subsheaf of the given \(\mcg\) and agrees with \(\mcf\) on \(U\) as a subsheaf of \(\mcg|_U\).
Let \(i:U\hookrightarrow X\) be the open inclusion and let \(\mathcal Q=(\mcg|_U)/\mcf\). The quotient \(\mathcal Q\) is quasi-coherent. Define \[ \mathcal H=\ker\bigl(\mcg\longrightarrow i_*\mathcal Q\bigr), \] where the map is restriction followed by the quotient map. This is a quasi-coherent subsheaf of \(\mcg\) because \(i_*\mathcal Q\) is quasi-coherent. Restriction to \(U\) is exact, so \[ \mathcal H|_U=\ker(\mcg|_U\to\mathcal Q)=\mcf \] as subsheaves of \(\mcg|_U\). Apply step 2 to \(\mathcal H\) and let \(\mcf'\) be the resulting coherent subsheaf. Then \(\mcf'\subseteq\mathcal H\subseteq\mcg\), with the stated equality on \(U\).
5. Part (d) holds, and therefore every coherent sheaf on an open subset extends to the noetherian scheme.
Choose a finite affine open cover \(X=V_1\cup\cdots\cup V_a\). Begin with \(W_0=U\) and the given coherent subsheaf \(\mcf_0=\mcf\subseteq\mcg|_{W_0}\). Suppose a coherent subsheaf \(\mcf_{j-1}\subseteq\mcg|_{W_{j-1}}\) has been constructed, where \(W_{j-1}=U\cup V_1\cup\cdots\cup V_{j-1}\). Apply step 4 on the affine scheme \(V_j\) to the coherent subsheaf \[ \mcf_{j-1}|_{W_{j-1}\cap V_j}\subseteq\mcg|_{W_{j-1}\cap V_j}. \] It gives a coherent subsheaf \(\mathcal H_j\subseteq\mcg|_{V_j}\) whose restriction equals that subsheaf on the intersection. Consequently \(\mcf_{j-1}\) and \(\mathcal H_j\) glue, through their equal inclusions in \(\mcg\), to a subsheaf \(\mcf_j\subseteq\mcg|_{W_j}\) on \(W_j=W_{j-1}\cup V_j\). It is coherent because it is coherent on each member of this open cover. It still restricts to the original \(\mcf\) on \(U\).
After the finitely many steps, \(W_a=X\) and \(\mcf_a\) is the required coherent subsheaf of \(\mcg\). For the extension assertion without a prescribed ambient sheaf, take \(\mcg=i_*\mcf\), which is quasi-coherent, and use its canonical identification with \(\mcf\) on \(U\).
6. The sectionwise union assertion in part (e) holds.
Let \(W\subseteq X\) be open and let \(s\in\mcf(W)\). The image \(\mathcal H\) of the morphism \[ \OO_W\longrightarrow\mcf|_W,\qquad a\longmapsto as, \] is quasi-coherent and locally generated by one section. On a noetherian affine open its module is therefore finite, so \(\mathcal H\) is coherent. Step 5, with ambient sheaf \(\mcf\), extends \(\mathcal H\) to a coherent subsheaf \(\mathcal H'\subseteq\mcf\) on \(X\) with \(\mathcal H'|_W=\mathcal H\). The section \(s\) is the image of \(1\in\OO_W(W)\) and thus belongs to \(\mathcal H'(W)\). Every section on every open set is consequently contained in a coherent subsheaf on \(X\). The reverse inclusion follows from these being subsheaves of \(\mcf\), proving the asserted union.
7. Q.E.D.
Step 1 proves part (a), step 3 proves part (b), step 4 proves part (c), step 5 proves part (d) and the announced extension theorem, and step 6 proves part (e).