Question

Are filtered colimits exact in all abelian categories?

In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase elements round diagrams), in particular A-Mod where A is a commutative ring. This implies that filtered colimits are exact in A-Mod.

I am aware of a vague principle that things that are true in A-Mod are true for all abelian categories, but I have never seen a precise statement of this principle so I am not sure if it applies in this case.

Answer 1

Here's a dumb counterexample. If C is an abelian category, so is C^op. In C^op, filtered colimits are filtered limits in C. And, of course, there are many examples of abelian categories (such as abelian groups) where filtered limits aren't exact.

Of course, your question is really: when is an abelian category C sufficiently close to Set, so that we can ratchet up the fact that filtered colimits are exact in Set to a proof for C.

Any category of sheaves of abelian groups on a space (or on a Grothendieck topos) will have exact filtered colimits, for instance.

Answer 2

A counterexample which is non-trivial is given in Chapter 6 of Neeman's book Triangulated Categories. The category in question is the full subcategory of additive functors Cat(S^{op}, Ab) where S satisfies some hypotheses (e.g. is an essentially small triangulated category) and we take those functors which are product preserving for small enough products (so as contravariant functors S-> Ab they send sufficiently small coproducts to products). This category is complete and cocomplete but has neither exact filtered limits or exact filtered colimits. More precisely it satisfies [AB4] and [AB4*] but neither [AB5] nor [AB5*].