\(K\)-theoretic torsors for infinite dimensional vector bundles of locally compact type (Q1707274)

\textit{V. Drinfeld} [Prog. Math. 244, 263--304 (2006; Zbl 1108.14012)] observed that there were two notions of \(K\)-theory torsor, one might expect to associate to a topological module Tate \(R\)-module and these should be equivalent. These are \(B\mathcal K\) (the classifying space for \(\mathcal K\)-torsors) and \(\mathcal K_{\mathrm{Tate}}\) (for definition see \S2 of the paper). The author establishes this observation of Drinfeld. Precisely (S)He proves that there is an equivalence \(B\mathcal K\simeq\mathcal K_{\mathrm{Tate}}\) in the \(\infty\)-topos \(Shv_{Spaces}(N S_{Nis})\). The proof directly follows from an earlier result of the author about delooping \(\mathbb K\)-spectra and a result of Drinfeld which says negative \(K\)-groups vanish Nisnevich locally. The author also proved the following. Let \(\epsilon\) be an object of \(Vect_{Tate}(S)\) which gives rise to a \(\mathcal K\)-torsor \(\mathcal D_\epsilon\). Then the automorphism group \(Aut(\epsilon)\) acts on \(\mathcal D\). The proof uses a criterion from \textit{T. Nikolaus} et al. [J. Homotopy Relat. Struct. 10, No. 4, 749--801 (2015; Zbl 1349.18032)].