Proposition 7.2.7. Let \(X\) be a spectrum and let \(Y\) be a \(P\)-divisible spectrum. Then precomposition with the canonical map \(X \to X[P^{-1}]\) induces an isomorphism of mapping spectra \[ \hom (X[P^{-1}],Y) \iso \hom (X,Y). \] In particular, the inclusion \(\Sp [P^{-1}] \hookrightarrow \Sp \) admits a left adjoint given by \[ (-)[P^{-1}]\colon \Sp \to \Sp [P^{-1}]. \]
Proof. The mapping spectrum \(\hom (X[P^{-1}],Y)\) is the limit obtained by applying \(\hom (-,Y)\) to the defining diagram for \(X[P^{-1}]\). The transition maps in this limit are given by precomposition with the corresponding multiplication maps on \(X\). By bilinearity of composition in spectra, these maps agree with postcomposition by the corresponding multiplication maps on \(Y\).
All transition maps are induced by multiplication by products of primes in \(P\) on \(Y\), hence are isomorphisms since \(Y\) is \(P\)-divisible. The limit is therefore isomorphic to the first term \(\hom (X,Y)\).
Passing to underlying mapping animae and using Corollary 7.2.5 gives the claim. □
Generated from the authoritative LaTeX source.