>>10739060Thanks, but unfortunately the book hasn't introduced pullbacks or pushforwards, or in particular how they behave at the level of quasicoherent modules.
However, I thought of another plan of attack, please correct me if I'm wrong:
The condition of quasi-coherence of an -module is equivalent to the fact that for every point , there exists an open affine neighbourhood such that , that is, it is associated to some -module . As noted before, we need only care for the specified point , and suppose is such an affine neighbourhood. Naturally, since , the associated module will be . Checking at the level of basic open sets, if , a basic open set in , we have for elements . Therefore, by taking the limit, it is always the case that for primes that are not , .
Conversely, take any affine neighbourhood of and we have by definition that , and if is in the basic open set and otherwise. But note that implies that is a unit. These two pieces of data show that is the associated module, which gives us quasicoherence.
