Moore spectra are p-local

Proof. Let \(q\) be a prime number different from \(p\). We must show that the map \(q\colon X/p^n \to X/p^n\) is an isomorphism, which we may check on homotopy groups.

The long exact sequence on homotopy groups associated to the exact sequence \(X \xrightarrow {\cdot p^n} X \to X/p^n\) contains, for every \(m \in \Z \), a natural short exact sequence \[ 0 \to \pi _m(X)/p^n \to \pi _m(X/p^n) \to \pi _{m-1}(X)[p^n] \to 0. \] Here \(\pi _m(X)/p^n\) denotes the cokernel of multiplication by \(p^n\), and \(\pi _{m-1}(X)[p^n]\) denotes the kernel of multiplication by \(p^n\). By Exercise 7.2.2, homotopy groups carry multiplication by \(q\) on spectra to multiplication by \(q\) on abelian groups, so this natural short exact sequence is compatible with multiplication by \(q\).

Now multiplication by \(q\) is an isomorphism on every abelian group killed by \(p^n\): choose integers \(a,b\) with \(aq+bp^n=1\), and then multiplication by \(a\) is inverse to multiplication by \(q\). It follows that multiplication by \(q\) is an isomorphism on the two outer terms of the short exact sequence above. By the short five lemma, it is therefore also an isomorphism on \(\pi _m(X/p^n)\).

Since this holds for all \(m\), multiplication by \(q\) is an isomorphism of spectra. As this is true for every prime \(q\neq p\), the spectrum \(X/p^n\) is \(p\)-local. □