Show that a sheaf of \(\OO_X\)-modules \(\mcf\) on a scheme \(X\) is quasi-coherent if and only if every point of \(X\) has a neighborhood \(U\) such that \(\mcf|_U\) is isomorphic to a cokernel of a morphism of free sheaves on \(U\).
If \(X\) is noetherian, then \(\mcf\) is coherent if and only if it is locally a cokernel of a morphism of free sheaves of finite rank. These properties were originally the definition of quasi-coherent and coherent sheaves.
Solution
For an open set \(U\subseteq X\), put \(\OO_U=\OO_X|_U\). For an index set \(I\), write \(A^{(I)}=\bigoplus_{i\in I}A\) and \(\OO_U^{(I)}=\bigoplus_{i\in I}\OO_U\). On \(U=\Spec A\), the associated-sheaf functor is exact and commutes with direct sums, so \[ \widetilde{A^{(I)}}\cong\OO_U^{(I)} \] [@Har10a, Propositions II.5.1 and II.5.2].
1. A quasi-coherent sheaf has the required local presentation by free sheaves.
For each \(x\in X\), choose an affine neighborhood \(U=\Spec A\) and an \(A\)-module \(M\) with \(\mcf|_U\cong\widetilde M\). Choose a free module \(A^{(J)}\) surjecting onto \(M\), and a free module \(A^{(I)}\) surjecting onto the kernel. This gives an exact sequence \[ A^{(I)}\xrightarrow{u}A^{(J)}\longrightarrow M\longrightarrow0. \] Applying the associated-sheaf functor gives \[ \OO_U^{(I)}\xrightarrow{\widetilde u}\OO_U^{(J)} \longrightarrow\mcf|_U\longrightarrow0. \] Thus \(\mcf|_U\) is the asserted cokernel.
2. A sheaf locally presented as a cokernel of free sheaves is quasi-coherent.
Fix \(x\in X\) and an open neighborhood on which the given presentation exists. Shrink it to an affine neighborhood \(U=\Spec A\) of \(x\). Restriction is exact and commutes with direct sums, as can be checked on stalks, so there is still an exact sequence \[ \OO_U^{(I)}\xrightarrow{v}\OO_U^{(J)}\longrightarrow\mcf|_U\longrightarrow0. \] By P-AGH253TILDEADJ, the morphism \(v\) corresponds to an \(A\)-linear map \[ u:A^{(I)}\longrightarrow\Gamma(U,\OO_U^{(J)})\cong A^{(J)}. \] Its associated-sheaf morphism is \(v\), since both have this effect on global sections and the correspondence is injective. Set \(M=\operatorname{coker}u\). Exactness of the associated-sheaf functor identifies \(\widetilde M\) with the cokernel of \(v\), hence with \(\mcf|_U\). These affine neighborhoods give the defining local descriptions of a quasi-coherent sheaf.
3. On a noetherian scheme, coherence is equivalent to the existence of such presentations with \(I\) and \(J\) finite.
Suppose first that \(\mcf\) is coherent. For each point choose an affine neighborhood \(U=\Spec A\) with \(\mcf|_U\cong\widetilde M\), where \(M\) is finitely generated and \(A\) is noetherian [@Har10a, Proposition II.5.4]. Choose a surjection \(A^n\to M\) with \(n\) finite. Its kernel is finitely generated because it is a submodule of the noetherian module \(A^n\). Choose finitely many generators of this kernel to obtain \[ A^m\longrightarrow A^n\longrightarrow M\longrightarrow0 \] with \(m,n\) finite. The construction of step 1 gives the required finite-rank sheaf presentation.
Conversely, shrink a given finite-rank presentation to an affine neighborhood \(U=\Spec A\). Step 2 identifies \(\mcf|_U\) with \(\widetilde M\) for the cokernel \(M\) of a map \(A^m\to A^n\) with \(m,n\) finite. In particular, \(M\) is finitely generated. Over the noetherian ring \(A\), every kernel of a map \(A^r\to M\) with \(r\) finite is finitely generated, being a submodule of \(A^r\). Thus \(M\) is a coherent module, and the affine local descriptions make \(\mcf\) coherent.
4. Q.E.D.
Steps 1 and 2 prove the characterization of quasi-coherence, and step 3 proves the noetherian finite-rank characterization.