# Determinant map for the prestack of Tate objects

## TL;DR

This paper constructs a determinant map for Tate objects over a ring, linking it to $\

## Contribution

It introduces a new determinant map for Tate objects by combining existing K-theory constructions with a relative $S_{ullet}$-construction.

## Key findings

- Established a determinant map from Tate objects to $\
- paper_type":"theoretical"}}# Answer: {
- contribution":"It introduces a new determinant map for Tate objects by combining existing K-theory constructions with a relative $S_{ullet}$-construction.",

## Abstract

We construct a map from the prestack of Tate objects over a commutative ring $k$ to the stack of $\mathbb{G}_{\rm m}$-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by Sch\"urg-To\"en-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.