Localizations and completions of stable $\infty$-categories

Abstract

We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathcal{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $\mathcal{M}$, we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded-below objects when $E$ satisfies some reasonable conditions.