On Previdi's delooping conjecture for $K$-theory

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 nonconnective $K$-theory spectra, extending and 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 that the first negative $K$-group of a ring is isomorphic to the $0$th $K$-group of the exact category of Tate modules.