Theorem 10.4.10 (Thom isomorphism). Let \(V\) be an \(R\)-oriented virtual vector bundle of rank \(d\) on an anima \(A\). Then there is an isomorphism of \(R\)-modules \[ \th (V)\otimes R \cong (\S [A]\otimes R)[d], \] and an isomorphism of spectra \[ \hom (\th (V),R)\cong \hom (\S [A][d],R). \]
Proof. Passing to colimits in the chosen trivialization \(\xi _{V,R}\cong R_A\) gives \[ \th (V)[-d]\otimes R \cong \colim _A\xi _{V,R} \cong \colim _A R_A \cong R[A], \] which proves the first isomorphism. The second follows by applying \(\hom _R(-,R)\) and using the free-forgetful adjunction between spectra and \(R\)-modules. □
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