Characterizations of E-equivalences

Exercise 7.1.5. Show that for every spectrum \(Y\), the mapping spectrum \(\hom (E,Y)\) is \(E\)-local. Using this, show that for a morphism \(f\colon X \to X'\) of spectra the following conditions are equivalent:

(1)

The morphism \(f\colon X \to X'\) is an \(E\)-equivalence;

(2)

The induced map \(f^*\colon \hom (X',L) \to \hom (X,L)\) is an isomorphism for all \(E\)-local spectra \(L\);

(3)

The induced map \(f^*\colon \Hom _{\Sp }(X',L) \to \Hom _{\Sp }(X,L)\) is an isomorphism for all \(E\)-local spectra \(L\).

Hint: use that we have \(\pi _n \hom (X,Y) \cong \pi _0 \hom (X,Y[n]) \cong \pi _0 \Hom _{\Sp }(X,Y[n])\), and similarly for \(X'\).