Localizations and completions of stable \(\infty\)-categories (Q6123769)

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