Approximation of subcategories by abelian subcategories

Abstract

Let $\mathcal{C}$ be an abelian category and let $\Lambda : \mathcal{C}\rightarrow\mathcal{C}$ be an idempotent functor which is not right exact, so that the zeroth left derived functor $L_0\Lambda$ does not necessarily coincide with $\Lambda$. In this paper we show that, under mild conditions on $\Lambda$, $L_0\Lambda$ is also idempotent, and the category of $L_0\Lambda$-complete objects of $\mathcal{C}$ is the smallest exact subcategory of $\mathcal{C}$ containing the $\Lambda$-complete objects. In the main application, where $\Lambda$ is the $I$-adic completion functor on a category of modules, this gives us that the category of "$L$-complete modules," studied by Greenlees-May and Hovey-Strickland, is not an ad hoc construction but is in fact characterized by a universal property. Generalizations are also given to the case of relative derived functors, in the sense of relative homological algebra