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This problem has Hartshorne Height 1. The equivalence of the Ogus and Hartshorne definitions of (quasi)coherence.
The case of Quasicoherence For the one direction, fix that is Hartshorne qcoh.
- There is affine cover s.t. on the . Pick a presentation for and show this presentation can be sheafified to give Ogus qcoh of .
For the other direction, given a cover one which 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 means applying global sections preserves exactness (Thm2 here)
- Argue on these affines that where
The case of Coherence
- Apply the same argument show Noetherian implies things are finitely generated.