On Previdi’s delooping conjecture for K-theory

Saito, Sho

doi:10.2140/ant.2015.9.1

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

Sho Saito

Vol. 9 (2015), No. 1, 1–11

DOI: 10.2140/ant.2015.9.1

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.

Keywords

negative $K$-theory, delooping, Tate vector space

Mathematical Subject Classification 2010

Primary: 19D35

Secondary: 14C35

Milestones

Received: 16 July 2013

Accepted: 10 December 2014

Published: 18 February 2015

Authors

Sho Saito
Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya 464-8602 Japan

Vol. 9, No. 1, 2015