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Apr 15, 2019 at 17:03 comment added Tim Campion It does seem interesting to ask if there is a purely cohomological criterion for an $\mathcal O_X$-module to be quasicoherent... Since quasicoherence is a local condition, the criterion would presumably use local cohomology or something like that. The "surjectivity" condition seems to be an "almost flasque"-ness condition, and the "injectivity" condition seems to be a "controlled torsion" condition.... both of which sound "cohomological" in nature...
Apr 15, 2019 at 16:52 history edited Tim Campion CC BY-SA 4.0
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Apr 14, 2019 at 19:31 history answered Tim Campion CC BY-SA 4.0