Corollary 4.4.27. The functors \(\pi _k\colon \Sp \to \Ab \) for \(k \in \Z \) are jointly conservative: a morphism of spectra \(f\colon X \to Y\) is an isomorphism if and only if the induced map \(\pi _k(f) \colon \pi _k(X) \to \pi _k(Y)\) is an isomorphism.
Proof. Since \(- \otimes -\) preserves cofibers in each variable, this follows by combining the previous lemma with the expression \(\Sigma ^{\infty }(Z) \simeq \cofib (\S \to \S [Z])\) from Remark 4.4.3 and the isomorphism \(X_+ \wedge Y_+ \cong (X \times Y)_+\) from Lemma 2.4.16. □
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