Show that a sheaf of modules F on a scheme X is quasi-coherent iff every point of X has a neighborhood U, such that is isomorphic to a cokernel of a morphism of free sheaves on U.
First note that locally free implies quasi-coherent. Basically if we have locally free, then we know that for some index set K. By Hartshorne II.5.2, and since is itself a sheaf, and the associated sheaf is uniquely isomorphic as a sheaf, .
Now suppose F is quasi-coherent. Let U be a neighborhood of a point, U = Spec A such that . (This is thm. II.5.4). Following Eisenbud, page 17, let be a generating set for the -module M (we can at least take the to be the elements of M). Let B index the kernel so that is exact.
As , the same sequence as above sheaf-ified is exact by II.5.2.a. Note that are free by above. As exact functors preserve cokernels, we have the result in one direction.
Conversely, if U = Spec A is the cokernel of a morphism of free sheaves on U, then it is a cokernel of locally free sheaves on U. We know that locally free implies quasi-coherent, thus it is a cokernel of quasi-coherent sheaves on U. Now using thm. II.5.7 the cokernel of quasi-coherents is quasi-coherent so is quasi-coherent.