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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.

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