Determinant map for the prestack of Tate objects

Abstract

We construct a map from the prestack of Tate objects over a commutative ring $k$ to the stack of ${\mathbb{G}}_{\mathrm{m}}$-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by Schürg-Toën-Vezzosi with a relative $S_{\bullet }$-construction for Tate objects as studied by Braunling-Groechenig-Wolfson. Along the way we prove a result about the K-theory of vector bundles over a connective ${\mathbb{E}}_{\infty }$-ring spectrum which is possibly of independent interest.

arXiv commentSection 2 rewritten, some results now hold more generally

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