Vol. 9, No. 1, 2015

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

Sho Saito

Vol. 9 (2015), No. 1, 1–11

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 0th 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 0th K-group of the exact category of Tate modules.

negative $K$-theory, delooping, Tate vector space
Mathematical Subject Classification 2010
Primary: 19D35
Secondary: 14C35
Received: 16 July 2013
Accepted: 10 December 2014
Published: 18 February 2015
Sho Saito
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602