On Previdi's delooping conjecture for K-theory

TL;DR

This paper proves a modified version of Previdi's delooping conjecture for K-theory, relating the K-theory space of an exact category to that of Beilinson's category of Tate vector spaces, extending known results to negative K-groups.

Contribution

It establishes a delooping result for non-connective K-theory spectra, nearly confirming Previdi's conjecture and extending Drinfeld's theorem to all negative K-groups.

Findings

Negative K-groups are given by 0-th K-groups of iterated Beilinson categories.

The delooping holds for non-connective K-theory spectra.

Extends the understanding of negative K-theory for exact categories.

Abstract

We prove a modified version of Previdi's conjecture stating that the Waldhausen space (K-theory space) of an exact category is delooped by the Waldhausen space (K-theory space) of Beilinson's category of generalized Tate vector spaces. Our modified version states the delooping with non-connective K-theory spectra, almost including Previdi's original statement. As a consequence we obtain that the negative K-groups of an exact category are given by the 0-th K-groups of the idempotent-completed iterated Beilinson categories, extending a theorem of Drinfeld on the first negative K-group.