Let X be a scheme. Let F be a quasi-coherent O_X-module. Let G ⊂ F be an O_X-submodule, not necessarily quasi-coherent. Then there exists a quasi-coherent submodule G’ ⊂ G which is universal for maps of quasi-coherent modules into G. This is Lemma Tag 01QZ in the stacks project.
The condition that G is a submodule of a quasi-coherent module is necessary in order to construct G’ (I think; explicit counter examples welcome). This result has two funny looking applications
- Any morphism f : X —> Y of schemes has a scheme theoretic image (Lemma Tag 01R6), and
- i_* : QCoh(Z) —> QCoh(Y) has a right adjoint when i : Z —> X is a closed immersion of schemes (Lemma Tag 01R0).
At first sight it may seem that 1 is too strong. But I think it isn’t simply because there is no way you can deduce anything from the existence of a smallest closed subscheme of Y through which f factors. It is only when f is quasi-compact and quasi-separated that the scheme theoretic image commutes with restriction to open subschemes for example.
It may be that the existence in 2 of a right adjoint i^! of i_* : QCoh(Z) —> QCoh(Y) follows from general facts, but it is cute that one can explicitly write it down as in the proof of the lemma referenced above. Again, the construction of i^! is not local on X, except in case the sheaf of ideals defining the closed immersion i is of finite type (to see this use Lemma Tag 01PO).