Lemma 109.11.1. Let $A$ be a ring and let $I \subset A$ be an ideal. The category $\mathcal{C}$ of derived complete modules is abelian and the inclusion functor $F : \mathcal{C} \to \text{Mod}_ A$ is exact and commutes with arbitrary limits. If $I$ is finitely generated, then $\mathcal{C}$ has arbitrary direct sums and colimits, but $F$ does not commute with these in general. Finally, filtered colimits are not exact in $\mathcal{C}$ in general, hence $\mathcal{C}$ is not a Grothendieck abelian category.

Proof. See More on Algebra, Lemma 15.92.1 and discussion above. $\square$

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