230,030pages on
this wiki
Add New Page
Discuss this page0 Share

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 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.

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.