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 ∞-category M admitting a multiplicative left-complete t-structure. If E is a homotopy commutative algebra in 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. Contents 1. Introduction 1 2. Preliminaries 3 3. Moore objects 9 4. Localization at some homotopy commutative algebras 15 5. Examples and applications 20 6. The E-based Adams-Novikov spectral sequence 23 7. Nilpotent Resolutions 25 Appendix A. Pro-Objects 35 Appendix B. Categorical Recollection 37 References 39