Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's Tohoku paper). When $X$ is a scheme, the full subcategory $\mathbf{Qcoh}(\mathscr{O}_X)$ of quasi-coherent $\mathscr{O}_X$-modules is also grothendieck abelian (even if $X$ is not quasi-compact and quasi-separated, see here).
Now consider the full subcategory $\mathbf{Coh}(\mathscr{O}_X) \subset \mathbf{Mod}(\mathscr{O}_X)$ of coherent $\mathscr{O}_X$-modules. This is an exact abelian subcategory but is not, in general, grothendieck. My first question is: why not?
Intuitively speaking, why is $\mathbf{Coh}(\mathscr{O}_X)$ not grothendieck abelian? What goes wrong?
Secondly, can the situation be rescued?
Are there assumptions on $X$, e.g. noetherian, or proper over a noetherian base, etc., under which $\mathbf{Coh}(\mathscr{O}_X)$ is grothendieck abelian?
Any reference that addresses these or related questions is very welcome.
Best Answer
The category of coherent sheaves is almost never a Grothendieck abelian category, but for essentially trivial reasons:
Of course, I have to keep saying "almost" because there is in fact an example of a scheme for which the category of coherent sheaves is a Grothendieck abelian category: the empty scheme $\emptyset$. But this scheme has some rather bizarre properties; for instance, every sheaf is automatically quasi-coherent, coherent, and even free...