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.

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