Prime inversion of spectra

Definition 7.2.3. Consider a spectrum \(X\).

• For \(n > 0\), we define \(X[\frac {1}{n}]\) as the colimit \[ X[\frac {1}{n}] \; := \; \colim (X \xrightarrow {n} X \xrightarrow {n} X \xrightarrow {n} \dots ) \qin \Sp . \]
• If \(P\) consists of finitely many primes, we set \(N := \prod _{p \in P} p\) and set \[ X[P^{-1}] \; := \; X[\frac {1}{N}] \qin \Sp . \]
• If \(P\) consists of infinitely many primes, say \(p_1, p_2, p_3, \dots \), we define \(X[P^{-1}]\) as the colimit \[ X[P^{-1}] \; := \; \colim (X \xrightarrow {\cdot p_1} X \xrightarrow {\cdot p_1p_2} X \xrightarrow {\cdot p_1p_2p_3} \dots ) \qin \Sp . \]

When \(P\) consists of all primes, we refer to \(X[P^{-1}]\) as the rationalization of \(X\) and denote it by \(X_{\Q }\).

When \(P\) consists of all primes except a single prime \(p\), we refer to \(X[P^{-1}]\) as the \(p\)-localization of \(X\), and denote it by \(X_{(p)}\).