Vol. 4, No. 3, 2019

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Vanishing theorems for the negative $K\mkern-2mu$-theory of stacks

Marc Hoyois and Amalendu Krishna

Vol. 4 (2019), No. 3, 439–472
Abstract

We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < dim(X), then Ki(X)[1n] = 0 if n is nilpotent on  X and Ki(X, n) = 0 if n is invertible on X. Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.

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Keywords
algebraic $K\mkern-2mu$-theory, negative $K\mkern-2mu$-theory, algebraic stacks
Mathematical Subject Classification 2010
Primary: 19D35
Secondary: 14D23
Milestones
Received: 3 May 2018
Accepted: 29 January 2019
Published: 17 December 2019
Authors
Marc Hoyois
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Amalendu Krishna
School of Mathematics
Tata Institute of Fundamental Research
Mumbai
India