# Co-filtered limits in algebraic categories.

If $$R$$ is a commutative ring with unity, we know that filtered colimits are exact. We also know that in an algebraic category, filtered colimits commute with finite limits. Are the following statements true? 1) Co-filtered limits are exact in $$R-Mod$$. 2) Co-filtered limits commute with finite colimits in algebraic categories.

## 1 Answer

No, claim 1) is false, and thus so is claim 2). Consider the inverse system of exact sequences of abelian groups whose rows are the sequences $$\mathbf{Z}\to \mathbb{Z}\to\mathbb{Z}/p^n$$. The columns are respectively the inverse systems (1) whose objects are all $$\mathbf{Z}$$ and whose morphisms are all $$p$$ (2) whose objects are all $$\mathbf{Z}$$ and whose morphisms are all $$1$$ and (3) whose objects are $$\mathbf{Z}/p^n$$ and whose morphisms are all $$1$$. The limits are respectively $$0$$,$$\mathbb{Z}$$, and the $$p$$-adics $$\mathbb{Z}$$, so the limiting sequence is very far from exact on the right.

The failure of claim (1) is measured by the (first) derived functor of limit, $$\mathrm{lim}^1$$. For countable inverse systems the higher derived functors of limit vanish, though for general shapes there can be arbitrarily many nontrivial derived functors.

An intuitive reason why your claims should be false is that categories in which filtered colimits commute with finite limits are very often locally finitely presentable; but a locally presentable category with locally presentable opposite must be a poset. This is a theorem of, I think, Freyd, which can be found in Adamek and Rosicky's book on locally presentable categories.