II.5.4

This problem has Hartshorne Height 1. The equivalence of the Ogus and Hartshorne definitions of (quasi)coherence.

HAPPY
The case of Quasicoherence For the one direction, fix $$F$$ that is Hartshorne qcoh. For the other direction, given a cover one which $$F$$ is globally presented, refine to cover of affines. The case of Coherence
 * There is affine cover $$\{U_i\}$$ s.t. $$F \cong \widetilde{M_i}$$ on the $$U_i$$. Pick a presentation for $$M_i$$ and show this presentation can be sheafified to give Ogus qcoh of $$F$$.
 * Use that direct sums of the structure sheaf are trivially Ogus qcoh and that qcoh of first term in the exact sequence $$0 \to \mathcal{O}^I \to \mathcal{O}^J \to F \to 0$$ means applying global sections preserves exactness (Thm2 here)
 * Argue on these affines $$\mbox{Spec }A$$ that $$ F \cong \widetilde{M}$$ where $$M = \mbox{coker}(A^I \to A^J)$$
 * Apply the same argument show Noetherian implies things are finitely generated.