Hartshorne problem II.5.4 — January 22, 2012

Hartshorne problem II.5.4

Show that a sheaf of O_X modules F on a scheme X is quasi-coherent iff every point of X has a neighborhood U, such that F|_U 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 F|_{U_i} \cong (O_X)^K for some index set K. By Hartshorne II.5.2, \tilde{O_X^K} \cong (\tilde{O_X})^K and since O_X is itself a sheaf, and the associated sheaf is uniquely isomorphic as a sheaf, O_X \cong \tilde{O_X}.

  Now suppose F is quasi-coherent. Let U be a neighborhood of a point, U = Spec A such that F|_U \cong \tilde{M}. (This is thm. II.5.4). Following Eisenbud, page 17, let {m_\alpha}_{\alpha\in A} be a generating set for the O_{X|U}-module M (we can at least take the m_\alpha to be the elements of M). Let B index the kernel so that O_{X|U}^B \overset{\psi}{\to} O_{X|U}^A \overset{\gamma}{\to} M\to 0 is exact.

  As M\cong coker(\psi) = O_{X|U}^A /im(\psi) = O_{X|U}^A / ker(\gamma) = O_{X|U}^A / O_{X|U}^B, the same sequence as above sheaf-ified is exact by II.5.2.a. Note that \tilde{O_{X|U}^A} and \tilde{O_{X|U}^B} are free by above. As exact functors preserve cokernels, we have the result in one direction.

  Conversely, if F|_U, 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 F|_U is quasi-coherent.