# II.5.4

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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.

• 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 $F$.

For the other direction, given a cover one which $F$ is globally presented, refine to cover of affines.

• 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 $M = \mbox{coker}(A^I \to A^J)$

The case of Coherence

• Apply the same argument show Noetherian implies things are finitely generated.