Lemma 10.4.6 (Orientations of inverse virtual bundles). Let \(V\) be a virtual vector bundle on \(A\). An \(R\)-orientation of \(V\) canonically induces an \(R\)-orientation of \(-V\).
Proof. By Remark 10.2.17, including the normalization by virtual rank, there is an isomorphism \(\xi _{-V,R} \cong \xi _{V,R}^{-1}\). Inverting a trivialization of \(\xi _{V,R}\) gives the required trivialization. □
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