Localization and completion

An abelian group can often be understood by separating its behavior at different primes. Its \(p\)-localization inverts all primes other than \(p\), and its \(p\)-completion records compatible information modulo the powers \(p^n\). For finitely generated groups, the arithmetic fracture square recovers the original group from its \(p\)-completion and its \(p\)-inversion. All three operations have analogues for spectra, and the aim of this chapter is to construct them and prove their basic properties.

The right level of generality is that of a Bousfield localization: given a spectrum \(E\), we invert the maps that are isomorphisms on \(E\)-homology, and thereby retain exactly those features of a spectrum that are ‘seen by’ \(E\). Rationalization, \(p\)-localization and \(p\)-completion all arise this way, for suitable choices of \(E\). Bousfield localizations are also the basic organizing device of chromatic homotopy theory, which we do not treat here.

Section 7.1 introduces the \(E\)-local and \(E\)-acyclic spectra and shows that the \(E\)-localization of a spectrum always exists. Section 7.2 treats inverting a set of primes; the homotopy groups localize algebraically, and rationally we obtain Serre’s theorem \(\S _{\Q } \simeq H\Q \) and the resulting splitting of rational spectra. Section 7.3 treats \(p\)-completion, which is less well behaved: it is not given by a naive algebraic operation on homotopy groups, and instead Theorem 7.3.10 expresses \(\pi _*(X^{\wedge }_p)\) through the Ext- and Hom-groups of Chapter 6. We close with the arithmetic fracture square, which reassembles a spectrum from its \(p\)-completions and its rationalization.

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