Writting objects in abelian category as limit of injective objects

Question

In "EVERY MODULE IS AN INVERSE LIMIT OF INJECTIVES" (https://arxiv.org/pdf/1104.3173.pdf) it is proven that every modules is an inverse limit of injective modules. A natural question is whether this can be generalized to arbitrary abelian categories. Maybe the most natural hope would be that if we consider abelian category with enough injectives $\mathcal{C}$ and choose an object $M$, we take the limit over all injective objects such we have an injection $M\hookrightarrow I$, i.e. we ask $$M=\text{lim}_{M\hookrightarrow I_\alpha} I_{\alpha}?$$ Of course, there might be other approaches, but this seems quite natural to me. Also, the case that interests me the most is the category of sheaves of abelian groups on a site.

Answer 1

This is false. To my mind it's a little easier to think about the dual question: in an abelian category with enough projectives, is every object a filtered colimit of projective objects?

In the category of modules over a ring $R$, which has enough projectives, we have that

1. every projective module is flat,
2. a filtered colimit of flat modules is flat, and
3. conversely, by Lazard's theorem every flat module is a filtered colimit of free modules.

Hence the filtered colimits of projective objects are precisely the flat modules, and so any ring $R$ with a non-flat module $M$ (precisely the rings which are not von Neumann regular) is a counterexample. Very explicitly and dualizing, $\text{Ab}^{op}$ is a counterexample to your original question.

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In general $\text{Ab}^{op}$ is a pretty nice counterexample to a bunch of conjectures you might have about how abelian categories behave; I've used it a couple times in other MO and math.SE answers. Note that by Pontryagin duality it's equivalent to the category of compact Hausdorff abelian groups, so we can think about it a bit more concretely than just as an opposite category.

In $\text{Ab}$ the projective objects are the free abelian groups and the flat modules are the torsion-free abelian groups. The Pontryagin dual statements are that in $\text{Ab}^{op}$ the injective objects are the products of copies of $S^1$ and the cofiltered limits of injective objects are the connected compact Hausdorff abelian groups. Hence any disconnected compact Hausdorff abelian group, e.g. any finite nontrivial abelian group, is a counterexample; these are Pontryagin dual to the abelian groups which have torsion.