Why the Category of Coherent Sheaves Is Not Grothendieck – Algebraic Geometry

algebraic-geometrycategory-theoryschemessheaf-theory

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:

  1. The category of coherent sheaves on $\operatorname{Spec} A$, where $A$ is a noetherian ring, is equivalent to the category of finitely generated (= finitely presented) $A$-modules.
  2. The category of finitely presented $A$-modules is almost never cocomplete, simply because the direct sum of infinitely many finitely presented $A$-modules is almost never a finitely presented $A$-module!
  3. Grothendieck abelian categories are, by definition, cocomplete.

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...

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